No title

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

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ACADEMIC PRESS, INC.

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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.

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

.

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

44

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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
H 0.754209 0.6180

Li

Be
Na

Mg

0.54793


B

C

F

Ne

1.2629

N
0

0.277

1.4611215

3.399


A1 0.441

Si

P

S

1.385

0.7465

2.077120

C1 3.617

Ar
Source: After Hotop and Lineberger (1985).

NEGATIVE IONS: STRUCTURE AND SPECTRA

3

study using infrared radiation. It gave 0.7539 f 0.0020 eV. Scherk (1979) has made a novel determination. He derived an expression for the lifetime of a spherical negative ion in a weak static electric field. On combining this with experimental data of Stinson et al. (1969), he deduced an EA of 0.75451 & 0.00041 eV, which is in excellent agreement with Pekeris’s result (given in Table I). The extremely high accuracy attained for 0 and S was by the tunable laser photodetachment electron spectrometry method. Hotop and Lineberger (1985) promise “It is now possible to determine EA(H) to a level (better than 0.01 cm- ’) that provides a challenge to theorists.” 1. Some Surprising Anions

Since the review by Hotop and Lineberger (1985) was published, there has been an important development in connection with the alkaline-earth atoms. Contemporaneous experimental research by Pegg et al. ( 1987) and theoretical research by Froese Fischer et al. (1987) has shown that the old dogma that none of these atoms possess a stable negative ion is false. The existence of the Ca- ion has been known for many years: it is produced, for example, in double electron-capture collisions between Ca’ ions and alkali-metal vapors (Alton et al., 1986). By ab initio calculations Bunge et al. (1982) predicted that the 4s 4p2 4P state is metastable. A search for the signal from electrons ejected in autodetachment failed although a search for a signal associated with the corresponding 2s 2p2 4P metastable state of Be- was successful (Kvale et al. 1985). When the research was being done it was thought that the lifeftime of (4s 4p2 “P) Ca- is in the subnanosecond range. Now in experiments such as that of Alton et al. (1986) mircoseconds elapsed between the Ca- ions being formed and their being detected; and Heinicke et al. (1974) has earlier put a lower limit of 10 ,us to the lifetime. It was therefore presumed that the ions involved were not metastable. Accurate measurement of the EA when the ion is fragile is difficult. Pegg et al. (1987) produced a Ca- beam by passing Ca’ ions in the energy range 60-80 keV through a Li charge transfer cell. They crossed this beam at right angles with a pulsed dye laser and examined the photoelectron energy spectrum. In the rest frame of the ion the photoelectron energy is E, = E, - E, - E,

where E , is the photon energy (1.9518 f 0.0008 eV), E , is the EA, and E, is the excitation energy of the residual atom. Setting E

= (m,/Mi) Ei

4

David R. Bates

it may be seen that in the laboratory frame the energy of a photoelectron ejected at an angle 8 with respect to the direction of the beam is EL = E

+ ~ ( E E , ) ~cos” 8 + E,.

(3)

Hence electrons ejected in forward and backward directions are separated in energy by an amount AE = ~(EE,.)~”.

(4)

Pegg et al. found a strong signal corresponding to the residual atoms being in the (4s 4p 3P0)state which strongly suggests Ca- ions in the 4s’ 4p configuration rather than the 4s’ 3d configuration that was commonly expected with some confidence. The expectation was misguided. Elementary considerations (Bates, 1947) show that the ordering in energy of the orbitals of a negative ion is not necessarily the same as the corresponding ordering for the isoelectronic neutral atom: thus because of the rapid falloff with distance of the attractive field, the extent of the penetration of the core by an outer orbital has a relatively more marked influence on the energy in the negative-ion case leading to an increased tendency for orbitals of low azimuthal quantum number to lie deeper than those of high azimuthal quantum number. Pegg et al. (1987) determined the EA by measuring AE of Eq. (4). This minimized errors associated with contact and surface potentials in the electron analyzer and avoided the need for precise knowledge of the analyzer constant. The final result of Pegg et al. is that the EA is 0.043 f 0.007 eV. Since 0.043 eV is only about 2 x the total energy of (4s’ ‘S) Ca or (4s’ 4p ’Po) Ca-, it is evident that the calculation of EA(Ca) is a formidable task (even taking into account that 99.6% of the (4s’ ‘S) Ca energy is the well-determined Hartree-Fock contribution). Nevertheless it has been established that (4s’ 4p ’Po) Ca- is indeed stable and the experimental EA(Ca) has been reproduced remarkably well. The first of the theoretical investigations (Froese Fischer et al. 1987) included multiconfiguration Hartree-Fock (MCHF) calculations (see Massey, 1979) on Ca and Ca- carried out in a systematic manner. In the initial step, configurations of the correct parity were constructed from the (4s, 4p, 3d, 4f) set of orbitals and variational calculations were done. Next the set was extended to the (5s, 5p, and 4d) orbitals and then to the (6s, 6p, 5d, and 5f) orbitals. Judgment was exercised on the configurations to be included. Finally relativistic shift corrections were made to the interaction matrix for the wave function expansion. Table I1 gives results. It also gives results of another MCHF investigation (Froese Fischer, 1989) in which somewhat

NEGATIVE IONS: STRUCTURE AND SPECTRA

5

TABLE I1

RESULTS OF MULTICONFIGURATION HARTREE-FOCK EA(Ca) CALCULATIONS

a

Orbitals

EA(Ca) in eV

4s, 4p, 3d, 4f 5s, 5p, 4d +6s, 6p, Sd, 5f last with relativistic shift 4s, 4p, 3d + 5s, 5p, 4d, 4f +6s, 6p, 5d, 5f, 5g last with relativistic shift

-0.224 0.012 0.056 0.045 -0.261 0.042 0.070 0.062

+

b

Source: (a) Froese Fischer et al. (1987). (b) Froese Fischer (1989).

different sets of orbitals were chosen and a somewhat different procedure was followed. Bauschlicher et al. (1989b) have also investigated Ca- using a large one-particle basis set and an extensive treatment of valence correlation (Table 111). The absolute errors in the .deduced values of EA(Ca) are small. Froese Fischer (1989) also calculated EA(Sr) and EA(Ba) by the MCHF method. Two other theoretical investigations of the alkaline earth atoms have been published. Vosko et al. (1989) applied density-functional theory (DFT; Kohn and Sham, 1965; Vosko and Wilk, 1983). While keeping the singleparticle picture of the simple Hartree-Fock (HF) approximation, this has the advantage of being able to include correlation. It gives the 4p orbital of Cato be bound whereas on the H F approximation, this orbital autoionizes. Kim and Greene (1989) used the R-matrix method and quantum defect theory. TABLE 111 ELECTRON AFFINITIES(eV) OF ALKALINEEARTHATOMS EA Atom Ca Sr Ba Ra

a

b

C

d

e

0.045

-

-

0.131 0.161 0.199 0.125

0.070 0.108 0.176 0.075

0.022

-

0.062 0.106 0.148

-

-

Source: (a) Froese Fischer et al. (1987); (b) Froese Fischer (1989); (c) Vosko et al. (1989); (d) Kim and Greene (1989); (e) Bauschlicher et al. (1989b).

6

David R. Bates

Their collision theory approach avoids the difficulty inherent in finding the EA by subtracting two large total energies (cf: Massey, 1976). Table I11 gives the theoretical results. The agreement between them is excellent, especially with regard to the rather peculiar variation down the Ca--Ra- column. This variation is quite different from the corresponding variation for the alkali atoms. (See Table IV.) Partly because scandium is adjacent to calcium in the Periodic Table, Froese Fischer et al. (1987) also calculated its EA though less accurately while Bauschlicher et al. (1989b) treated it to the same high approximation as they did calcium. An earlier theoretical study had been made by Jeung (1985) using a nonempirical Hartree-Fock pseudopotential for the core electrons and allowing for the configuration interaction of the four outer electrons. In harmony with a spectroscopic analysis by Feigerle et al. (1981), Jeung concluded that the ground state is (3d 4s2 4p) 'Do. He found the EA to be 0.16 eV. Contenting themselves with a (3d 4s 4p 4d 5s 5d) description (cf: Table 11), Frose Fischer et al. (1987) obtained an EA of 0.152 eV. Bauschlicher et al. (1989b) obtained 0.195eV. The measured value is 0.188eV (Hotop and Lineberger, 1985). A simpler MCHF calculation (Beck et al. 1987) failed to yield a stable negative ion and led to the validity of its detection being questioned. The closed outer shell of the inert gas atoms is unfavorable to a stable anion. In the set of experiments in which they detected metastable Ar(Section I.B.2), Bae et al. (1985) failed to detect Xe- but commented that this nu1 result was not conclusive evidence against the existence of the species. Haberland et al. (1989) used a different source. They allowed a mixture of

+

TABLE IV CALCULATED ELECTRON AFFINITIES (ev) OF ALKALIATOMSWITH, IN LAST ROW, CORRESPONDING EXPERIMENTAL VALUES (ROUNDED)

THE

EA Reference

Basis"

Li

Na

K

Rb

cs

Norcross (1974) Dulieu (1989) Ortiz (1988b) Guo and Whitehead (1989) Hotop and Lineberger (1985)

CT PI EPT LSD Exp

0.614 0.479 0.549 0.540 0.618

0.538 0.480 0.482 0.562 0.548

0.498 0.477 0.497 0.522 0.501

0.490

0.470

-

-

0.503 0.522 0.486

0.462 -

0.472

CT, collision theory; PI, Pluvinage method; EPT, electron propagator theory; LSD, generalized exchange local-spin-density functional theory; Exp, experimental values.

NEGATIVE IONS: STRUCTURE AND SPECTRA

7

5 1 0 % xenon in about equal amounts of argon and nitrogen to expand from a pulsed supersonic jet and crossed the gas pulse (pressure lo-’ Pa) by a 150-eV electron pulse. Secondary electrons were slowed by collisions, especially with nitrogen, and attached to form positive and negative clusters that traversed a skimmer and were detected by a time-of-flight mass spectrometer. Haberland et al. compared the signals of the Xe+ isotopes and those of an anion that they concluded was Xe- because of the good agreement for the isotopes 129, 130, 131, 132, and 136. However, they were unable to avoid impurities and the positive-ion spectrum had a signal at 128 and the negative-ion spectrum had additional intensity between 134 and 135. The strength of the electric field in the spectrometer showed the excess electron on Xe- to be bound by at least 1 meV. The time interval between formation and detection showed the lifetime to be at least 100 ps. The lifetime of the metastable species Ar- toward autodetachment is far shorter than this (Section I.B.2). Haberland et al. argued that it is therefore unlikely that the Xe- is metastable. They suggested that the extra electron might be in a 5d or 4f orbit. The polarizability of Xe is quite high (4.04 A3, see Miller and Bederson, 1977). 2. Rare-Earth Atoms

Little is known about the EAs of the rare earth atoms. Using horizontal analysis (see Massey, 1976) a few values were obtained by Zollweg (1969). In conflict with his results, Sen et al. (1980) argued that none of the rare earth atoms are likely to have stable negative ions. However, application of DFT by Cole and Perdew (1982) provided support for the work of Zollweg. Estimates for the lanthanides (La through Lu) have been made by Bratsch (1983) and for the actinides (Ac through Lr) by him and Lagowski (1984). By comparison with the d-block transition atoms, the EAs of Sm and Tm (whose anions have the electron configurations 4f7 6s’ and 4fI4 6s’) were both reckoned by Bratsch to be about 0.3 eV. Assuming that the energy associated with a given attachment process varies little along the series when the f-shell orbital population is conserved, the EAs for the other species were then obtained by considering data on the energy variation associated with changes in the f-shell orbital population. The estimated EAs for the lanthanides lie between -0.3 and 0.5 eV, while those for the actinides lie between -0.3 and 1.0 eV. The mean value in each set is 0.1 eV. Table V compares the four common results of Zollweg, of Cole and Perdew, and of Bratsch. Some accord is evident.

8

David R. Bates TABLE V ELECTRON AFFINITIES (eV) OF SOMERARE EARTH ATOMS EA Species Ce Pr Nd Gd

Atomic number

a

b

C

58 59 60

0.6 0.3

0.81 0.11 0.10 0.34

0.5 0.0 -0.3 0.5

64

0.1 0.2

Source: (a) Zollweg (1969); (b) Cole and Perdew (1982); (c) Bratsch (1983).

3. Calculations

The main calculations that have already been recorded are of direct relevance to this chapter in that they provide information on the EAs or the structures of negative ions, or both. Many interesting calculations have also been carried out on systems for which the EAs are accurately known from laboratory work and for which the structures are not in doubt. They will be reported here only briefly but merit a full article. Some of the problems have been discussed by Simons and Jordan (1987). In their classic investigation, Sasaki and Yoshimine (1974a, b) calculated the energies of B, C, 0, F and their anions using MCHF wave functions that are of the form

where QHF is the Hartree-Fock state function and QIare configuration state functions taken to be linear combinations of Slater determinates. Naturally, they ensured that Y is an eigenfunctiot of L2 and S 2 with the appropriate eigenvalues. An (8s, 7p, 6d, Sf, 4g, 3h, 2i) orbital set was used. The 0 1 were classified according to the electron excitation with respect to QHF: singledouble (SD) and triple-quadruple (TQ)-higher excitations were ignored. Massey (1979) has given their intermediate results at some length and only a few aspects of them need be mentioned here. The K-shell and KL intershell correlation contribution to EA is less than lo-’ eV, which is small enough to be neglected. In contrast the L-shell contribution is large. For example the

9

NEGATIVE IONS: STRUCTURE AND SPECTRA

calculated oxygen L-shell contribution is 1.67 eV; of this, 0.14 eV comes from TQ excitations. Sasaki and Yoshimine estimate that the true values of these contributions should be 1.82 and 0.27 eV, respectively. It is hence evident that high precision is difficult to attain. Table VI gives the EA calculated by Sasaki and Yoshimine and also those calculated by other scientists mentioned later in this section. The slow convergence of MCHF calculations, apparent from the work of Sasaki and Yoshimine, caused concern. Reasoning that the main defect in the Hartree-Fock description of the anion stems from its inability to represent the diffuse nature of the extra electron's charge distribution satisfactorily, Botch and Dunning (1982) examined the use of a compact multiconfiguration self-consistent field (MCSCF) function with a (4s, 4p, 3d) Gaussian basis set. They allowed for SD excitations (effectively including TQ excitations relative to the H F wave function). The mean error in their EAs for C, 0, and F (0.25 eV) is the same as that given by the large basis set calculations of Sasaki and Yoshimine. Feller and Davidson (1985) made a systematic study of the effects of variations in basis set. They recognized the importance of TABLE VI CALCULATED ELECTRON AFFINITIES (ev) OF SOME LIGHTATOMSWITH, IN LASTROW, CORRESPONDING EXPERIMENTAL VALUES (ROUNDED FROM TABLE I)

THE

EA Reference Sasaki and Yoshimine (1974b) Botch and Dunning (1982) Feller and Davidson (1985) Bauschlicher et al. (1986) and Taylor (1986) Feller and Davidson (1989) Jankowski and Polasik (1984) Raghavachari (1985) Ortiz (1987b) Adamowicz and Bartlett (1986a) Barnett et al. (1986) Yoshida et al. (1988) Ortiz (1988a) Guo and Whitehead (1989) Hotop and Lineberger (1985)

Basis"

B

C

0

F

V

0.15 -

1.11 1.08 1.22 -

3.12 3.18 3.16

V V V V MET P P P QMC QMC EPT LSD EXP

-

-

0.22 0.21 -

1.37 1.22

1.13 1.09 1.29 1.29 1.31 1.42 1.41

-

-

-

-

-

-

-

-

-

0.22 0.28

1.19 1.26

1.11 1.46

CI

-

3.48 3.47 3.37 3.37 3.45 -

4.72 3.13 3.40

-

3.53 -

3.62 3.75 3.52 3.62

V, variational; MET, many-electron theory; P, perturbation; QMC, quantum Monte Carlo (the error bars for F are kO.11 and for CI are f0.20); EPT, electron propagator theory; LSD, generalized exchange local-spin-density functional theory.

10

David R. Bates

attempting balance in the sense of overcoming the natural tendency to describe the neutral atom with much greater precision than its anion. Using balanced (8s, 5p, 4d, 2f, lg) contracted Gaussian basis sets and allowing for SD excitations, they calculated the EAs of C and 0 correct to 0.05 eV and 0.17 eV, respectively. Bauschlicher et al. (1986) obtained the same precision in the calculated EA(0) using a (6s, 5p, 3d, 2f) Slater-type orbital basis and correlating only the 2p electrons. They confirmed the presumption that relativistic effects are negligible (less than lo-’ eV). By a similar calculation Bauschlicher and Taylor (1986) got EA(F) to be 3.16 eV. In an extension of their earlier work on oxygen, Feller and Davidson (1989) employed a sequence of configuration wave functions designed so as to approach the complete basis set. About 95-96 % of the total O(3P) and O-(’P) correlation energies were recovered but the error in the calculated EA(0) remained as great as 0.15 eV. Estimates from multireference second-order perturbation theory suggest that the error might be reduced to 0.10 eV for their final basis set. Hardware and software limitations prevented this from being achieved variationally. Feller and Davidson note that their results accord with the expectation that EA(0) does not converge more rapidly than either E(0) or E(0-). As may be seen from Table VI, the EAs are underestimated by variational calculations. Recognizing the desirability of having a method of obtaining correlation energies that is computationally less demanding then MCHF or MCSCF, Sinanoglu (1961, 1976) developed his many-electron theory (MET). This exploits his finding that the correlation energy of a many-electron system can be represented to a good approximation as a sum of pair correlation energies corresponding to all possible electron pairs. Jankowski and Polasik (1984) applied MET to B, C, 0, and F with some success; thus, the mean of the moduli of the errors in their calculated EAs is only 0.07 eV (Table VI). The treatment of electron correlation in transition metal atoms is more involved. Nevertheless, Jankowski and Polasik (1985) calculated EA(Cu) to be 1.17 eV, which is in very good agreement with the experimental value of 1.23eV (Hotop and Lineberger, 1985). A different approach was adopted by Raghavachari (1985). Forming his wave function from a (5s, 4p) contraction of a (lOs, 6p) basis augmented with two sets of sp functions, four d functions, and two f functions, he applied many-body (Moller-Plesset) perturbation theory (Pople et al., 1976), which he carried to fourth order. His calculated EAs for B, C, 0, and F are marginally more accurate then those of Jankowski and Polasik (1984).

NEGATIVE IONS: STRUCTURE AND SPECTRA

11

Fourth-order many-order perturbation theory has also been used with success by Ortiz (1987b), who calculated EA(F) and EA(C), getting excellent agreement with the experimental values. Adamowicz and Bartlett (1 986a) have used many-body perturbation theory and a numerical orbital coupledcluster method (Adamowicz et al., 1985) to calculate EA(F) accurately (Table VI). Using nonrelativistic and relativistic pseudopotentials, Schwerdtfeger et al. (1986) have calculated EA(Br) and EA(1) to be 3.314 and 3.127 eV in good agreement with the measured nucleus 3.365 and 3.059 eV (Hotop and Lineberger, 1985). Migdalek and Baylis (1982) have treated all the halogens by a computationally efficient procedure in which the change in the total correlation energy associated with the added electron is modeled by the interaction of this electron in a polarization potential of the atom. The mean of the moduli of the errors in their EA values is as low as 0.07 eV. The quantum fixed-node Monte Carlo (QMC) method (see Reynolds et al., 1982; Barnett et al., 1985) is also powerful. It entails solving the Schrodinger equation for many interacting particles by the simulation of an appropriate diffusing random walk. By QMC Barnett et al. (1986) have obtained a value for EA(F) in good agreement with experiment. Combining QMC with a model potential, Yoshida et al. (1988) calculated EA(C1) quite accurately. However, in both cases (see Table VI) the Monte Carlo error bars are not inappreciable. Guo and Whitehead (1989) have applied generalized-exchange local-spindensity (LSD) functional theory (Manoli and Whitehead, 1986) with a correlation correction due to Vosko et al. (1980). Their results (Tables IV and V) are of only moderate accuracy. Electron propagator theory (EPT) is a method that generates corrections to Koopman’s theorem and provides a direct determination of the EA (Ohm and Born, 1981). It has been used by Ortiz (1988a) to calculate EA(F), for which the error in his result is as great as 1.3 eV, and EA(C), for which the error in his result is 0.13 eV (Table VI). Ortiz (1988b) has also applied EPT to the calculation of the EAs of the alkali atoms. His results do not have the remarkably high accuracy of the early collision theory results of Norcross (1974). They are rather better than those that Dulieu (1989) obtained by the Pluvinage method (Table IV). This method (Pluvinage, 1951) is based on an approximate solution of the Schrodinger equation that depends explicitly upon the inter-electronic distance. The sizes of the alkali anions have been calculated by Vosko and Wilk (1983) and by Christensen-Dalsgaard (1985); those of the anions of the 3d, 4d, and 5d metal atoms have been calculated by Sen and Politzer (1989).

12

David R. Bates

B. EXCITED STATES OF ATOMICANIONS

1. Resonance States

Much research has been done on auto detacting states. For instance, Pathak et al. (1988) have carried out a 15-state R-matrix calculation of the energies and widths associated with doubly excited states of H - converging to the n = 2,3, and 4 hydrogen thresholds. Table VII gives their n = 2 results and compares them with the results of correlated wavefunction calculations by Taylor and Burke (1967) and Seiler et al. (1971) and of measurements by Williams (1976) and Warner et al. (1986) from high-energy-resolutionspectra of electrons scattering elastically by atomic hydrogen. The agreement is excellent. Pathak et al. (1988) verified that (as had been demonstrated by Gailitis, 1980, for the n = 3 case) the distance eNof the Nth resonance below the threshold and rN, its width, satisfy the simple ratio law

where R, which depends on n and the total angular momentum L, may be obtained by diagonalizing a specified matrix (Gailitis and Damburgh, 1963). Pathak et al. (1989) have tabulated R for n I12, L I6. References to some other work on doubly excited states of H - may be found from Koyama et al. (1989). The description and classification of doubly excited states and the study of their correlation effects has aroused considerable interest (see Fano, 1983; Lin 1986) but lies rather outside the scope of this chapter. We shall therefore content ourselves with making brief mention of the role played by the potential ridge rl = -r2 where r, and r2 are the position vectors of the electrons. The importance of this ridge was first recognized by Wannier (1953) in his classic work on the threshold law for ionization by electron impact. Buckman et al. (1983) and Buckman and Newman (1987) carried out the key experimental research. They discovered doubly excited He- autodetaching states with very high (up to nine) principal quantum numbers by measurements on the yield of metastable helium atoms produced by electron impact excitation of helium. They are grouped. The lower energy resonances in each group are now known to have intrashell configurations, with n , = n2 = n (say), and are the simplest. Leaning on work by Fano (1980), Buckman et al. (1983) named them Wannier resonances. The sequence of

TABLE VII ENERGIES E (eV) AND WIDTHSr (mev) OF RESONANCESCONVERGING ON THE n = 2 HYDROGENIC THRESHOLD

ZSflLn

ISe -

3ss

'Po 3Po

IDC

i

s

Pathak et al.

Taylor and Burke

Burke et al.

Seiler et al.

Williams

Warner et al.

(1988)

(1967)

(1967)

(1971)

(1976)

(1986)

vl

E

r

E

r

E

r

E

r

E

r

E

r

9.557 10.177 10.147 10.176 9.741 10.126

52.0 2.6

9.560 10.178 10.150 10.177 9.740 10.125

47.0 2.2 0.02 0.04 5.9 8.8

9.575 10.178 10.151 10.179 9.768 10.160

54.0

9.574 10.178 10.151 10.181 9.768 10.160

54.0 2.3 0.02 0.02 8.0 7.7

9.549

63.0

9.557

45.0

* * 7.1 8.8

2.4 0.02

0.02 8.0 7.8

2; z

2

0 4

s

-

-

-

-

-

-

-

-

-

-

-

-

9.736 10.115

5.0 6.0

9.735

6.0

%U

-

-

v1

m

eel

m

* Widths narrower than 0.1 meV are not reported because of their being subject to computational errors.

2P P

14

David R. Bates TABLE VIII WANNIER TEIL FOR He-(lsnlz

's), H-(n12 's) AND Li-(lsznlz 's)

Energy (eV) below two-electron ionization threshold

Mean radius (au) (rl) = (rz) of pair

n

He-

H-

Li-

He-

H-

Li-

3 4 5 6 7 8 9

2.156(2.137) 1.180 (1.153) 0.745 (0.738) 0.511 (0.508) 0.375 (0.371) 0.287 (0.281) 0.227 (0.201)

1.885 1.088 0.706 0.493 0.366 0.282 0.224

2.285 1.213 0.755 0.517 0.376 0.287 0.227

14.2 26.5 42.6 62.9 87.0 114.7 144.1

16.3 28.7 44.9 66.5 90.9 120.1 152.0

13.3 25.8 41.8 62.0 86.9 114.9 144.0

Source: Nicolaides et a!. (1989). Note: The values in brackets (which are accurate to kO.010 eV) are the energies as measured by Buckman and Newman (1987).

such resonances of given symmetry is called a Wannier two-electron ionization ladder (TEIL). By considering the potential V = -Z/rl - Z / r , + l/rlz (7) in the vicinity of the ridge, Rau (1983) derived a simple expression for the TEIL energies that fits the data satisfactorily. Feagin and Macek (1984), Macek and Feagin (1985), and Lin and Watanabe (1987) have also treated the problem analytically; and Freitas et al. (1984) have done an eleven-state R-matrix computation. MCHF calculations on Wannier TEIL states for Heand also for H- and Li- have been carried out by Nicolaides and Komninos (1987), Komninos et al. (1987), and Nicolaides et al. (1989). Table VIII gives some of their results ('S for H- and Li-, ' S for He-) and in the case of Hecompares the energies with those measured by Buckman and Newman (1987). The agreement between theory and experiment is good. As the principal quantum number n is increased, the energies of the three systems (with respect to the appropriate ionization threshold) approach each other. The two-electron orbitals are strikingly large. 2. States Metastable or Stable toward Autodetachment

Many years ago, Drake (1970, 1973) showed that (2pz 3P) H- is bound and its lifetime toward decay by radiative detachment' into the 1s kp 3P0 Instead of radiative detachment, some writers use radiative autoionization. Both parts of the compound word autoionization are here inappropriate.

NEGATIVE IONS: STRUCTURE AND SPECTRA

15

continuum is 1.73 ns (see Massey, 1976). A search by Nicolaides and Beck (1978) did not reveal any other bound states of the ion. At variance with this, the existence of several bound states has been claimed. However, Bunge et al. (1982) have refuted the claim. They did so consequent on an important systematic search for excited states of the negative ions of the atoms H through Ca by performing nonrelativistic fixed-core valence-shell configuration interaction calculations. Their many predictions are consistent with all the results that will be reported here. Accurate configuration interaction calculations by Bunge and Bunge (1984), combined with results of a study of Chung (1984) on relativistic and mass polarization effects, give that (1s 2s 2p 4P0)He- lies 77.51 i-0.04 meV below (1s 2s %)He. It may be produced from fast He+ ions in a spinconserving charge transfer sequence such as

+ (IS)Ca + (%) He + ('S) Ca' (3S)He + (IS) Ca (4P0) He- + (2S) Ca',

('S) He'

+

(8)

(9)

and by a time-of-flight technique (see Massey, 1976), Blau et al. (1970) have measured the lifetimes z ( J ) toward autodetachment. They found z( 1/2,3/2) to be 11 & 0.5 p s and z(5/2) to be 345 & 90 ps. Chung (1984) has calculated the expectation values of the spin-orbit, spin-other-orbit and spin-spin operators. Early investigations (see Massey, 1976, 1979) indicated that (1s 2p2 4P and 'P) He- lie below (1s 2p 3P0) He. However, calculations by Bunge and Bunge (1978,1979) are against this being correct. These have been confirmed by photodetachment measurements on (1s 2s 2p 4P0) He- by Peterson et af. (1983) that showed a peak due to the 4P0+ 4P transition. The peak was located at 1.2344 eV in agreement with the calculations of Bunge and Bunge. Again Alton et al. (1983) made lifetime measurements on the He- ions formed when fast He+ ions pass through calcium vapor. They found no evidence for the ('P) He- ions expected from the spin-conserving charge transfer process ('S) He+ + (IS)Ca + ('P) He-

+ ('S) Ca2+.

(10)

It may now be taken as established that (1s 2p2 4P and 'P) He- are only core excited shape resonances. But there is a second bound excited state: spherically symmetric triply excited (2p3 4S0) He- (Beck and Nicolaides, 1978, Chung, 1979). Nicolaides et af. (1981) have done refined calculations that show that it is 0.317 eV below (2p' 3P) He (and 59.33 eV above (1s' 'S) He).

David R. Bates

16

They reckon that it decays by radiative detachment (2p3 4P0)He-

+ (1s

2p 3P0)He

+ e + hv

(11)

with a lifetime of 0.088 ns. In his collision theory work on the EAs of the alkali atoms, Norcross (1974) found that (3p2 3P) Na- is slightly (0.062 eV) below (3p 'Po) Na and pointed out that it is metastable toward autodetachment; (3s 3p 3P0)Na- is of course even lower in energy but is not metastable. Norcross obtained a similar result for the other alkali atoms with the exception of lithium. An explanation of the pattern has been provided by Bunge (1980a): the binding of the p-orbitals for the heavier alkali negative ions arises from the large correlation effects represented by ms-nd excitations with m 2 2, n 2 3, whereas in the case of lithium there are only ls-3d excitations that have a negligible effect. By a configuration interaction study Bunge (1980a, b) also discovered two excited states of Li- that are metastable toward autodetachment: (1s 2s 2p2 'P) Li- at 504.5 & 2.0 meV below (1s 2s 2p "Po) Li and (1s 2p3 'So) Li- at 291.0 & 2.0 meV below (1s 2p2 "P) Li. He calculated that the higher of these states decays mainly by a dipole transition to the lower: (1s 2p3 ' S o ) Li- + (1s 2s 2p2 'P) Li-

+ hv(A = 349.0 nm),

(12)

the lifetime toward this process being 2.9 ns while the lifetime toward radiative detachment, (1s 2p3 ' S o ) Li-

+ (1s 2s

2p "Po)Li

+ e + hv (A 2 406.4 nm),

(13)

is 56 ns (Nicolaides and Beck, 1978, Beck and Nicolaides, 1983). Seeking to check his prediction regarding the 1 349 nm line, Bunge (1980b) examined published beam-foil excited lithium spectra. (See review by Berry, 1975.) He noted that there is an unexplained line at 348.98 f 0.02 nm having an upper level that is not fed by cascading and has a measured lifetime of 2.9 ns. He suggested that this line is due to the ('So + 'P) Li- transition. His suggestion (the first regarding optical emission from an atomic anion) has been convincingly confirmed by the laboratory work of Mannervik et al. (1980) and of Brooks et al. (1980). Several crucial tests were applied: for instance, identification of the charge state of the emitter by use of an electric field; verification that the upper level has the required spherical symmetry by showing that the circular polarization fraction remains virtually zero when the line is excited by a tilted foil; and an improved determination of the lifetime of the upper level that gave it to be 2.3 ns in fairly good agreement with the calculated lifetime of 2.7 ns toward channels (12) and (13) together

NEGATIVE IONS: STRUCTURE AND SPECTRA

17

(Beck and Nicolaides, 1983). Fine and hyperfine structure calculations on the excited states of Li- have been done (Beck and Nicolaides, 1983; Cheng et al., 1984). Early moderate-sized configuration interaction calculations (Weiss, 1968) predicted that Be- has stable state but that (1s’ 2s 2p’ 4P) Be- is metastable with a binding energy relative to (1s’ 2s 2p 3P0) Be of 240 f 100 meV. Laboratory work by Bae and Peterson (1984) and by Kvale et al. (1985), in which a Be- beam was produced from a Be’ beam by sequential chargeexchange in caesium or lithium vapor confirmed the prediction. From the autodetachment decay rate as a function of time the former scientists showed the presence of more than one substrate with the lifetime of the longest greater than loops; and from the center of mass energy of autodetaching electrons, lithium showed the energy below (3P0) Be to be 195 f 90 meV. Configuration interaction computations by Beck and Nicolaides (1984) give this energy to be 217.17 f 57.1 meV while configuration interaction computations by Bunge (1986) give it to be 276.1 k 6.5 meV. By application of a state-specific theory of electron correlation and relativistic effects in the Breit-Pauli approximation, Aspromallis et al. (1986) have found that (consistent with the measurements of Bae and Peterson) the lifetimes of the J = 1/2, 3/2, and 5/2 levels are 0.080,2000, and 1.0 ps, respectively. They explain the wide dispersion of these values as being due to final-state correlation, cancellation, and nonorthonormality. The states (1s’ 2p3 4S0) Be- and (1s 2s 2p3 %’) Be- are also bound (Beck et al., 1981, Bunge et al., 1982, Bunge, 1986). The lower of these decays by an allowed dipole transition (4S0) Be-

+ (4P) Be-

+ hv(l = 365.3 nm)

(14)

(lifetime 1.49 ns) and also decays by radiative detachment. (4S) Be-

+ (3P) Be

+ e + hv

(15)

(lifetime 10 ns) so that its lifetime toward both channels together is 1.3 ns (Beck et al. 1981). The upper, (%O) Be-, decays by relativistic autodetachment into a six-channel continuum (Is’ 2s 2p ES 4P!&, Ed 4P&2,Ed 4D:/,, 1s’ 2p’ EP 4P&,, EP 4Dz/2,Ef 4D’&) toward which process its lifetime is 83 ns (Aspromallis et al., 1985). Gaardsted and Anderson (1989) have observed the predicted Be- optical transition in a Be beam-foil spectrum. The wave length given in Eq. (14) is that measured by them-the predicted value is 0.8 k 0.3 nm longer. They also determined the (4P) Be- lifetime to be 1.25 ns in excellent agreement with theory. Beck and Nicolaides (1984) have calculated the fine and hyperfine structure of the 4P and 4S0 states.

David R. Bates

18

Bound states of Mg- analogous to those of Be- are predicted (Bunge et al. 1982) but were not observed in a search (Bae et al. 1984a). Specifically they are 2p6 3s 3p' 4P,2p6 3p3 4S0, and 2p5 3s 3p3 6P. Beck (1984) calculated that they lie 0.360, 0.514, and 1.321 eV below the parent Mg states. He also 4P3/2and 4P5,2toward decay by calculated that the lifetimes of 4P1/2, relativistic autodetachment are as brief as 7.8, 9.8, and 1.6 ns, which hinders the detection of the species. The prediction of Bunge et al. (1982) that while Ar- has a long-lived metastable state, 3s2 3p5 4s 4p 4S, Ne- does not have such a state is supported by laboratory research. Bae et al. (1985) formed long-lived ( N 350 ns) Ar- from an Ar + beam by sequential charge transfer in Cs vapor but got a null, result from an Ne' beam. Ben-Itzhak et al. (1988) used a 600 keV neutral beam in the parent gas as their source: ArO

+ Aro

-+

Ar-

+ Arq+ + (q - 1)e.

(16)

The projectiles emerging from the gas target were separated according to their charge states by a deflector. The mean lifetime of Ar- was determined by varying the deflector's position. A value of 260 +_ 25 ns was obtained. A similar experiment with neon showed that Ne- has a lifetime of less than 10 ns or that process (16) is orders of magnitude faster than Neo + Neo + Ne-

+ Neq+ + (q - 1)e.

(17)

By an experiment depending on sequential charge transfer between a 3 keV Ca' beam and Cs atoms in a cell Hanstorp et al. (1989) have confirmed the prediction that calcium has a metastable negative ion (4s 4p 4P)Ca- lying 0.550 eV below (4s 4p 3P0)Ca (Bunge et al., 1982). They measured its mean Ca- on the grounds lifetime to be 290 & 100 ps and attributed this to (4P5/2) that it autodetaches through the spin-spin interaction whereas (4P1/2, 3/2) Ca- autodetaches through the much stronger spin-orbit interaction. Hanstorp et al. made laser photodetachment measurements in the energy range 19,490-21,550 cm-'. Their results show a new channel opening near 20,800 cm-'. They noted that this energy would fit the ground-state photodetachment process

'

(4s' 4p 'Po)Ca-

+ hv

-+

(4s 3d 3D)Ca

+ ES, Ed

(18)

or the metastable-state photodetachment process (4s 4p2 4P)Ca-

+ hv -,(4s 5s %) Ca + EP

(19)

NEGATIVE IONS: STRUCTURE AND SPECTRA

19

and distinguished between the two by the Wigner law that the photodetachment cross section a(E) at energy E just above a threshold E , has the form a(E) = A(E - Eo)1+"2.

(20)

where A is a constant and 1 is the azimuthal quantum number of the ejected electron. From their results Hanstorp et al. concluded that they were observing process (19) and that E , is 20,755 k 40 cm-', which corresponds to an EA of 0.562 0.005 eV in good agreement with the calculations of Bunge et al. (1982). They further concluded that in curious contrast to the Ca- beam of Pegg et al. (1987) referred to in Section I.A.l, their Ca- beam contained less than 5 %, of ions in the ground state. They suggested that this might be because the relatively low energy of their Ca+ beam permitted certain energy-resonance charge transfer processes that favor the formation of (4P)Ca- to be dominant. Nicolaides and Komninos (1981) have proposed that negative ions having an upper state that decays by radiative detachment may be suitable for the construction of tunable high-frequency lasers (see also Beck, 1982). In some instances the energy of at least one of the excited states of the negative ion is below the energy of the ground state of the atom making the state stable toward autodetachment. (See Massey, 1976.) Thus, a tunable laser photodetachment study by Feldman (1977) showed that the excited (2p3 'DO)C- ion is stable by 0.033 eV; and a laser photodetachment electron spectroscopy study by Kasdan et al. (1975b) showed the excited (3p3 'Do) Si- and (3p3 'PO) Si- ions are stable by 0.523 and 0.029 eV, respectively. Configuration interaction calculations on ('DO) C- do not appear to have been done, but such calculations on (2Do)Si- and (2Po) Si- by Lewerenz et al. (1983) gave excitation energies of 0.92 and 1.39eV in quite good agreement with the measured values (0.86 and 1.36eV). Other anions for which ab initio calculations gave excited states are Cu- (Beck et al, 1987), Zn- (Beck, 1988), Mn- (Beck and Cai, 1988), Cr-, Fe-, Co-, Ni- (Cai and Beck, 1989), and Sc-, Y- (Bauschlicher et al., 1989b).

c. EFFECTOF ELECTRIC AND MAGNETIC FIELDS The dipole polarizability a of a negative ion is difficult to determine. Almost all the research has been theoretical. Electron correlation (cf: Sadlej, 1983) is important.

David R. Bates

20

A number of studies of H - have been made (see Glover and Weinhold, 1976) but only two will be mentioned. From accurately calculated photodetachment cross sections, Geltman (1962) found a to be 202.4 uO3.Glover and Weinhold (1976) have obtained a very precise value, 206.086 uO3by a variation-perturbation calculation. They also derived rigorous upper and lower bounds to a(w), the dynamic polarizability, at frequencies w, up to near the photodetachment threshold at 0.027751 au. Table IX gives some of their results. Moores and Norcross (1974) and Lamm et ul. (1978) obtained a for Li-, Na-, and K - (Table X) from photodetachment cross sections they calculated. The former used a configuration interaction wave function for the initial state and a close-coupling scattering wave function for the final states (and evaluated both the length and velocity formulae); the latter used an approximate asymptotic form for the initial wave function and a pseudo potential method to determine the final wave function. Table X gives the results. Pochan and Bishop (1984) have carried out all-electron configuration interaction calculations on Li- in the presence of a distant charge. From the energies they determined a to be 650 uO3. Maroulis and Bishop (1986) obtained 1097 uO3in an investigation on all the values of the Li- electric polarizability tensors up to the fourth rank using wave functions close to the Hartree-Fock limit. These two results straddle the photodetachment cross section results (Table X). They differ considerably in contrast to the similarily calculated polarizabilities of the parent atom (164 uO3 and 169.2 uo3) for which electron correlation is unimportant. Canuto et al. (1988) have investiTABLE IX BOUNDSTO DYNAMIC POLARIZABILITY a(o ) OF H- AT FREQUENCY 13 BOTHIN ATOMICUNITS Frequency (ox lo3) 0 2 4 6 8 10 12

Rigorous bounds

206.39 f 2.45 206.73 2.48 207.74 f 2.54 208.47 2.65 211.98 f 2.84 215.34 f 3.10 219.73 f 3.49

Frequency (w x lo3)

Rigorous bounds

14 16 18 20 22 24 26 27

225.34 f 4.07 232.51 f 4.96 241.81 &- 6.41 254.27 8.96 272.13 f 14.06 301.83 f 26.71 375.11 f 75.66 527.88 f 211.49

Source: After Glover and Weinhold (1976).

21

NEGATIVE IONS: STRUCTURE AND SPECTRA TABLE X STATIC DIPOLE POLARlZABILlTlES (a IN

a:)

OF

Li-, Na-,

AND

K- OBTAINED

FROM CALCULATED PHOTODETACHMENT CROSSSECTION CURVES

Moores and Norcross (1974)

Lamm et al. (1978)

Length {Velocity

Li -

Na-

K-

832 798 808

989 1058 1090

1805 1758 1780

gated the convergence of the calculated Li- polarizability with the size of the orbital basis set. Using polarization propagator methods based on a coupledcluster reference state, they get 547 aO3but reckon that the true value is 50 to 100 aO3greater. The case of F- has received much attention. The earlier work is referenced by Kucharski et al. (1984). Using many-body perturbation theory (MBPT), they themselves found o! to be 18-20 aO3.An MBPT calculation by Diereksen and Sadlej (1981) has given that a for C1- is 37.5 aO3 while a MCSCF calculation by ONeil et al. (1986) has given 34.6 aO3. Using an adiabatic approximation in hyperspherical coordinates, Park and Starace (1984) have investigated how H - is influenced by a strong uniform magnetic field.They found that the wave function contracts radially and experiences an angular distortion, its ss 'S character being partly replaced by an sd 'D character. They found also that the EA is reduced and obtained upper and lower bounds to its value. To a close approximation these may be expressed E f 0.150 eV in which E is the mean of the two bounds. At zero field (where the EA is 0.7542 eV), E is 0.853 eV, while at fields of 1, 2.35, 4.7, 9.0, and 11.0 x lO'G, E is 0.795, 0.713, 0.563, 0.408, and 0.123 eV, respectively.

D. DOUBLY CHARGED ANIONS The possible existence of long-lived ( 210 p s ) doubly charged negative atomic ions (DNAI) has long been of interest (Massey, 1976). An exotic species attracts curiosity; and from the practical viewpoint DNAIs would extend the use of medium-sized tandem accelerators for heavy-ion physics because of the overall increase in energy that could be achieved with them (Kutschera et al. 1984). Reviewing the subject, Kiser (1979) concluded the

22

David R. Bates

DNAIs exist. However, the apparent positive evidence for them cited was discounted by careful experiments by Spence et al. (1982b). These had ion sources that duplicated or closely duplicated those in the earlier experiments being checked but, unlike them, had mass resolution sufficient to identify any impurity ion from its mass defect and were designed so that artifact peaks due to collisional dissociation of molecular negative ions could not occur. In the early experiments the ratio [X2-]/[X -1 for various ions was claimed to be lo-' to for electron impact ionization sources and to be approximately for Penning sources. Spence et al. found around and 5 x l O - ' O to be the upper limits to these ratios. They concluded that the results of the early experiments were falsified by impurity ions, by artifact peaks, or by both. Discounting the apparent positive evidence for DNAIs does not of course in itself prove their nonexistence. A limitation on them has been set by experiments by Chang et al. (1987). The source consisted of a 6-keV caesium primary beam impinging on a solid sample (of, for example, TiH for H2-, of a few milligrams. This produced a current of up to 100 pA AI2O3for 02-) of singly charged negative ions and possibly also doubly charged negative ions. The negative ions were analyzed by a tandem-accelerator-based charge spectrometer. Ultrasensitive searches for doubly charged negative ions of H, 0, S, Se, Te, F, C1, Br, and I were made. None were found. The upper limits that Chang et al. (1987) set for [H2-]/[H-], [02-]/[0-], [S2-]/[S-], [Se2-]/[Se-], [Te2-]/[Te-], [F2-]/[F-], [C12-]/[Cl-], [BrZ-]/[Br-], and [12-]/[1-] were 1.0 x 10-l6, 1.1 x 10-l6, 7.1 x 5.8 x 10-14, 8.8 x 10-13,7.4 x 10-l6,8.O x 10-'5,2.0 x 10-13,and1.3 x 10-12,respectively. They point out, first, that because of the flight time of DNAIs from the ion source to the tandem terminal, the sensitivitiesjust given refer to DNAIs with lifetimes greater than about 10 ps; second that the electric field intensity in their tandem accelerator (3 keV/cm) would tend to destroy DNAIs with an eV, making them unobservable. Moreover, as EA of less than about 1 x they recognized, sputtering may not be a favorable DNAI source. Theorists have carried out some research on the DNAI problem. Bunge and Bunge (1978) reasoned that if a long-lived H Z - state exists, it must lie below the (2p2) 3P state of H- which Drake (1970) has shown itself lies 0.0095 eV below the 2p 2Postate of H; and they further reasoned that it must be a 4S0 state, this being the only species with symmetry and parity not present in the surrounding continuum. Configuration interaction calculations led them to conclude that the state is not bound. Cole and Perdew (1982) considered 0'- and Te2- on the grounds that the valence configuration of the neutral atom is p4 so that completion of p subshell might lead to a

NEGATIVE IONS: STRUCTURE AND SPECTRA

23

stable system. Bound states were not revealed by local spin density calculations they performed. Cole and Perdew treated 0 2 -further by considering a 10-electron system of nuclear charge Z which they varied. They found that the system is not stable unless Z is 8.73 or greater, which strongly suggests that 0'- is not close to stability. This is in accord with earlier theoretical work by Herrick and Stillinger (1979, who from analytic continuation of isoelectronic bound states predicted an 0'- resonance state at 5.38 eV above the 0-+ e continuum threshold. Lieb (1984) has proved that the maximum number of electrons N , that can be bound to a molecule consisting of NA atoms of total nuclear charge Z is given by N,

=22

+ NA - 1.

(21)

Applied to atomic hydrogen, this shows that H'- is not stable (in agreement with the calculations of Bunge and Bunge, 1982, and the experiment of Chang et al. 1987).

11. Diatomic Anions Many experimentally determined diatomic (and polyatomic) EAs are listed by Janousek and Brauman (1979) and by Drzaic et al. (1984). The calculation of molecular EAs has been reviewed by Baker et al. (1986) and by Nobes et al. (1987).

A. FEW-ELECTRON SYSTEMS As discussed by Massey (1976), the simplest molecular negative ion H; has a pair of states (Is a&' (2p a,) X2X: and (1s a,) (2p a")' B 'Xi that tend to H + H- in the separated nuclei limit. Both autodetach when the nuclei are X: and within around 7ao for close enough together: within 3a0 for X ' B 'Xi. The potential energy curve of neither state was known accurately (cf: Amaya-Tapia et al. 1986) until a few years ago when Senekowitsch et al. (1984) carried out a multireference configuration interaction calculation on X 'X: (in its stable region) finding that its potential is considerably more attractive than had been supposed (Table XI).

David R. Bates

24

TABLE XI TOTALENERGIES OF X 'Z; STATEOF H; AND XIZ: STATEOF H; AND X'Z; STATEOF H2

Negative of energy (a.u.)

R (ax.)

Hi

3.0 3.25 3.5 4.0 5.0 6.0 8.0 10.0 15.0 20.0

1.058682 1.054844 1.052385 1.048791 1.043107 1.038726 1.033340 1.030673 1.028386 1.027896 1.027736

00

H2

1.0573118 -

1.0318564 1.0163889 1.0037848 1.0008350 l.ooOo556 1.0000091

1.0000000

Source: After Senekowitsch et al. (1984). Senekowitsch et al. take the H, energies from Kolos and Wolniewicz (1975).

Harcourt (1987) has challenged the accepted view regarding the ground state of H; by proposing that there is a stable 'Xi state having a wave function constructed thus:

WG 1 = w

3 2Y

e ) + PCICI(H - H) + +(H H - )I

+ v[ll/(H2- H +) + $(H+ H2-)].

(22)

In this, IC/(H2,e) accommodates an electron in a diffuse 1s orbital centered midway between the protons. Calculations by Harcourt show that the EA is 0.068 eV and that re is 1.43 a, (as for H2). Aberth et al. (1975) have reported that HD- and D; , with lifetimes greater than 10 ps, were among the ions extracted from a duoplasmatron source. Their abundance was only lo-' to lo-' the D- abundance. Careful identification tests were made. On the other hand, Bae et al. (1984a) have advanced strong evidence for the nonexistence of metastable or stable H i from a series of experiments in which they passed a beam of positive ions through alkali vapor cells and searched for negative ions produced by charge transfer. In one experiment they found that the Heyield was about of a transmitted He+ beam; in another that the upper limit to any D; yield was only 4 x lo-' of the transmitted D: beam.

NEGATIVE IONS: STRUCTURE AND SPECTRA

25

There is a broad peak at 3-4eV in the total cross section for e-H, scattering. This is due to a ('C:)H; resonance state, which is important in dissociative attachment. (See Burke, 1979, Massey, 1979.) By a semi-empirical analysis of the experimental data, Bardsley and Wadehra (1979) derived the real part of the potential of this state and of a higher 'X; that contributes to vibrational excitation at around 10 eV. McCurdy and Mowrey (1982) treated the 'C: state by the complex self-consistent field method. They got excellent agreement with Bardsley and Wadehra. So too did Sabelli and Gislason (1984), who applied a novel computing procedure in which the system is initially embedded in a spherical positively charged cage that lowers the energy of the 'C: below the energy of the ground state of H, plus free electron. The resonant autodetaching states have been further discussed by De Rose et al. (1985), Amaya-Tapia et al. (1986), Domcke et al. (1986), and Komiha et al. (1987). A diatomics-in-molecules calculation by PaidarovB et al. (1982) indicates that the X lZ+state of HeH- is unbound. When investigating the negative ions gotten when an He' beam is passed through a Cs vapor cell, Bae et al. (1984b) made an interesting discovery: that long-lived metastable He; exists. Measurements on the autodetachment decay rates at several delay times after formation indicated the presence of more than one state and that the lifetime of the major component exceeds 100 ps. Bae et al. inferred that the He, is probably in one of the quartet states that may be formed when an electron is added to a He, molecule in the lowest C : ). This was confirmed by Michels (1984), who carried out a triplet state (a ' configuration interaction calculation that gave that ("n,) He; is bound relative to (a ' X: ) He, by 0.233 eV. Michels also estimated that the lifetime of the J = 5/2 substate toward autodetachment is about 350 ps and that of the J = 3/2, 1/2 substates is about 10 ps. In an investigation of autodetachment from He; produced by double charge transfer of energetic He; in a Li vapor cell, Kvale et d.(1986) observed two distinct channels. One was the channel studied by Bae et al. (1984b); the other they suggested as being ("&, u = 1) He; + (a , :X'

u = 0) He,

+ e.

(23)

They determined the EA to be 0.175 f 0.032 eV. More accurate calculations were then done by Michels (1986), who got an EA of 0.182 eV, checked that channel (23) is indeed open, and determined the spectroscopic constants. The ("n,)He; and (a ):X' He, potentials are closely similar (Pluta et al. 1988a). The "nuand "Xi symmetries of He; are not stable relative to (a ) :X' He, (Michels, 1984, Pluta et al., 1988a).

David R. Bates

26 B. HOMONUCLEAR ANIONS

Unlike Na;, which has been observed as a secondary ion in sputtering experiments by Leleyter and Joyes (1974), the lithium dimer anion does not seem to have been reported in the laboratory. Configuration interaction calculations on it have been done by a number of groups (Dixon et al. 1977; Partridge et al. 1983; Konowalow and Fish, 1984; Sunil and Jordan, 1984; Michels et al. 1985). The main results have been compared by Michels et al. They are in good accord so it will suffice to cite those of Michels et al. who find that for (X 'C:) Li;: re = 3.076A, o,= 231.3 cm-', o,x, = 2.36 cm-', Be = 0.508 cm-', a, = 0.0081 cm-', and Do = 0.818 eV. Since Do (Li,) is 1.04, eV (Huber and Herzberg, 1979) and EA(Li) is as in Table I it follows that EA (Li,) = 0.39 eV. The only other Li; state formed from Li('S) + Li-('S) is the A 'C; state. It has a complex potential at separations less than 3.45 A where a crossing with the X 'Xl state of Liz occurs. Considering the real part of the potential, Michels et al. calculated that re is 3.13 A, o, is 186 cm-', and nine vibrational levels are supported. From an analysis of the complex part, they deduced that autodetachment is slow, which favors the possibility of dissociative attachment e

+ Lif(X 'El)-+

Li;(A 'X;)

+ Li('S)

+ Li-('S).

(24)

They report higher states of Li; as being purely resonant. The anion of the carbon dimer occupies a special place among dimer X' - X 'Xl bands anions. Herzberg and Lagerqvist (1968) observed its B : both in absorption and inemission in flash discharges in methane-the first such anion transition to have been observed. There have been suggestions regarding astrophysical applications (Vardya and Krishna Swamy, 1980; Sarre, 1980; Wallerstein, 1982). Again C; is the only dimer anion having bound excited states that decay to the ground state by dipole-allowed radiative transitions that have been investigated both experimentally and theoretically. An important photodetachment spectroscopy study of C; has been carried out by Jones et al. (1980) in the photon energy range 1.75-2.5eV. They observed sharp resonances due to autodetachment at photon energies corresponding to transitions between high vibrational levels of the X 'Xi state and high vibrational levels of the B ' X: state. They studied the nine bands (5,6), (5,7), (5,8), (6,6), (6,7), (6,8), (6,9), (7,9), and (7,lO). From their results on these and a reanalysis of the data of Herzberg and Lagerqvist (1968), which provide information on the low vibrational levels, Jones et al,

NEGATIVE IONS: STRUCTURE AND SPECTRA

27

derived the spectroscopic constants of the two states. They also deduced that EA(C,) is between 3.374 and 3.408eV. Using a merged laser-ion beam spectrometer with sub-Doppler resolution, Mead et al. (1985) obtained spectra of the B 'Xi + X'Z; transitions and analyzed the strong perturbations of the B state by the hitherto unobserved A 'nu state to obtain the spectroscopic constants of the A state. Table XI1 lists the principal spectroscopic constants of the three states. Jones et al. (1980) additionally obtained their full potential energy curves and evaluated the B-X Franck-Condon factors. The usefulness of the Franck-Condon factors is however limited because of the strong variation of the electronic transition moment as calculated by Zeitz et al. (1979). Configuration interaction calculations by Zeitz et al. (1979) led them to predict that C; also has a state with its minimum (excitation energy about 3.5 eV) below the potential energy curve of (X 'Z8+)C2.They predicted some of the spectroscopic constants of the other states quite well (Table XIII). The more extensive highly correlated multireference calculations of Rosmus and Werner (1984) reproduce them accurately (Table XIII). Zeitz et al. (1979) got the EA to be 3.43 eV while by much more refined calculations Nichols and Simons (1987) got 3.1 eV which is in rather less good agreement with experiment. Leutwyler et al. (1982) have measured the (B 'Xu, u' = 0 and u' = 1) C; radiative lifetimes to be 77 f 8 and 73 & 7 ns. Rosmus and Werner (1984) have calculated them to be 76.5 and 75.8 ns. The dependence on u' is not marked (Rosmus and Werner). For u' 2 5, autodetachment occurs. The lifetimes toward this have been measured by Hefter et al. (1983). They TABLE XI1 EXPERIMENTALLY DETERMINED C; SPECTROSCOPIC CONSTANTS Constant T,

0

rc

0, 0,

Be ae

x 22,

xe

1.268 1781.2 11.7 1.746 0.0166

A

'nu

0.50 1.313 1656 10.8 1.630 0.0152

B 22Z: 2.28 1.223 1969.5 15.1 1.877 0.0189

Source: After Mead et al. (1985). Note: Except for T,, which is in eV, and re, which is in A, the constants are in wave numbers.

David R. Bates

28

TABLE XI11 CALCULATED C; SPECTROSCOPIC CONSTANTS A 'nu

x 'Z* Constant Te rs

0, we

a

b

0 1.278 1800

0 1.276 1780 12 1.726 0.016

-

x,

Be

-

%

-

B 'Z:

a

b

0.40

0.44 1.318 1646 11 1.619 0.016

1.32 1653 -

a

b

2.33 1.22 1968

2.35 1.231 1983 16 1.855 0.017

-

Source: a is from Zeitz et al. (1979); b is from Rosmus and Werner (1984). Note: Except for T,, which is in eV, and re, which is in A, the constants are in wave numbers.

decrease sharply as u' is increased and for u' 2 6 are much shorter than the radiative lifetimes. The theoretical study by Rosmus and Werner included the (A ,IT,,,u')C; and (X , :X' u')C, radiative lifetimes (Table XIV). There are strong theoretical arguments than N; is not stable (Massey, 1976). Evidence against the existence of long-lived N; was obtained by Bae et al. (1984a) from the series of experiments mentioned in Section 1I.A. Lauderdale et al. (1983) have applied the complex self-consistent field (CSCF) method to calculate the complex potential energy curve of the 'HE resonance state. Nestmann and Peyerimhoff (1985) have obtained the real parts of the potentials of the ,HSand ,Xu resonance states by a numerically very efficient procedure. They stabilized the resonance by increasing the nuclear charge from Z to Z(l + A), calculated the energy E(A) at four values of A, and then found E(0) by extrapolation. Photoelectric spectroscopy measurements by Travers et al. (1989) give that E A ( 0 , ) is 0.451 & 0.007 eV. Many bound states are predicted from configuration interaction calculations (Das et al., 1978, 1980, Michels, 1979). TABLE XIV OF (A CALCULATED RADIATIVELIFETIMES

'nu.d)C;

AND

(x 'El,

U')c;

~~

0

0,

1

3

2 ~~~

A lifetime @s) X lifetime (ms)

49.9 -

40.6

34.6

-

-

4

5

6

7

8

9

10

27.4 541

25.0 206

23.2 98

21.8 55

20.6 34

19.6 23

18.8 16

~

30.5 2000

Source: After Rosmus and Werner (1984).

29

NEGATIVE IONS: STRUCTURE AND SPECTRA

Results obtained by Michels (1979) are presented in Table XV. Experimental values of the spectroscopic constants of the X ,Zg state are available: thus, re = 1.347 k 0.005A, Be = 1.16 k 0.01 cm-' (Travers et al. 1989), we = 1089 cm-', wexe= 8.1 cm-' (Linder and Schmidt, 1971), and from D , ( 0 2 ) = 5.1 16 eV (Huber and Herzberg, 1979), EA(0) = 1.461 eV (Table I) and EA(0,) as above it follows that D , ( O ; ) = 4.106 k 0.007 eV. The agreement between these values and the corresponding values in the last row of Table XV is excellent. From measured threshold energies for charge transfer reactions of atomic halogen anions with F,, Chupka et al. (1971) found that EA(F,) is 3.08 k 0.10 eV (see Massey, 1976). This result has been satisfactorily reproduced by Nguyen and Ha (1987) in the course of a many-body perturbation theory calculation of the isotropic hyperfine splitting for the F nucleus in the dimer anion. Chen and Wentworth (1985) have utilized the EA together with experimental data on the vertical electron affinity, on dissociative electron detachment, and on photodetachment to calculate Morse potential energy curves for the bound states of F; (and the dimer anions of the other halogens). Four states arise from F('P0) F-('S): the 'X;, 'nu,'ng, and X 'Z: states. The first two of these are purely repulsive. Chen and Wentworth find that for the ,lI,: re = 3.02 A, o,= 54 cm-', D, = 0.02 eV; and for the X 'Xi state re = 1.92 A, o,= 462 cm-', and D, = 1.26 eV. The complex potential energy curve of the (,X:)F; resonance state has been

+

TABLE XV PREDICTED SPECTROSCOPIC CONSTANTS OF THE BOUNDSTATES OF 0; THAT Go TO O(") + O-(2Po) IN THE SEPARATED NUCLEILIMIT

T.

re

0,

%X,

4

a,

Do

States

(eV)

(A)

(cm-')

(cm-')

(cm-')

(cm-')

(eV)

41Tu 2C; 22, 211u '2:

3.85 3.77 3.02 2.93 2.88 2.81 2.61 2.60 2.05

2.48 1.99 2.18 1.92 2.00 1.98 2.13 2.12 1.88 1.34

345.6 484.4 451.5 452.1 514.4 524.7 504.1 503.7 604.8 1089.0

8.9 12.9 3.5 4.0 4.9 4.8 3.0 2.9 3.4 12.1

0.34 0.53 0.44 0.57 0.52 0.54 0.46 0.47 0.60 1.17

O.OOO4 0.0104 0.0045 0.0079 0.0063 0.0065 0.0037 0.0039 0.0061 0.017

0.29 0.36 1.11 1.20 1.25 1.32 1.52 1.53 2.07 4.09

2Au '2: 4As "2;

xzn,

0

Source: Michels (1979).

30

David R. Bates

calculated by Lauderdale et al. (1983) by the CSCF method. An ab initio study of the hyperfine coupling constants has been made (Karna et al., 1989). Azria et al. (1982, 1988) have discussed the low-energy Cl; and I; resonance states. Nimlos et al. (1987) have made photoelectron spectroscopy measurements that prove that EA(Si,) is 2.199 & 0.012 eV. As an aid to understanding their spectrum, they carried out MCSCF calculations. These give that the D 'nu state of Si, (cJ Huber and Herzberg, 1979) is only slightly above the X 3C; state and that the and states of Si; are nearly degenerate. Their experimental data requires that 211u is the ground state (with re = 2.187 A, and w e= 516 & 137cm-') and that the excitation energy of 'El (re = 2.127 A) is 0.117 & 0.016 eV. A photoelectron spectroscopy study of P; by Snodgrass et al. (1985) has given that EA(P,) = 0.589 & 0.025 eV and hence that Do (P;) = 4.88 f 0.06 eV. Values of some of the spectroscopic constants of P; were inferred from a peak due to a hot band and a Franck-Condon analysis. They are o,= 640 f 50 cm-', Be = 0.2776 & 0.0028 cm-', and re = 1.979 f 0.010 A. Laser photoelectron spectroscopy measurements have been made on some homonuclear metal dimer anions. They give the following values of electron affinities (correct to f0.008 eV) and vibrational frequencies (correct to f 15 cm-I): Fe,, 0.902 eV, 250 cm-'; Co,, 1.110 eV, 240 cm-' (Leopold and Lineberger, 1986); Re,, 1.571 eV, 320 cm-' (Leopold et al., 1986); Cu,, 0.842eV, 210cm-' (Leopold et al., 1987); Ag,, 1.028eV, 192cm-'; and Au,, 1.939eV, 149cm-' (K. Ervin and W. C. Lineberger cited by Bauschlicher et al., 1989a).Guidance on the electron configurations of the neutral and anion dimers is provided by this research. (See Leopold et al. 1988.) Bauschlicher et al. (1989a) have determined EA(Cu,), EA(Ag,), and EA(Au,) by ab initio calculations. Their values are too low by between 0.24 and 0.34 eV. From the experiment whose results led them to report the existence of Xe(Section LA), Haberland et aL(1989) also inferred that Xe, has an EA greater than 1 meV and the anion has a lifetime longer than 100 ps. C. MAINHETERONUCLEAR FAMILY For present purposes it is convenient to exclude hydrides from the main family. A general difference from homonuclear diatomic anions is not to be expected. Dipole-supported states are considered in Section 111.

31

NEGATIVE IONS: STRUCTURE AND SPECTRA

Using laser optogalvanic spectroscopy, Klein et al. (1983) determined that EA(CN) is 3.821 0.004 eV. Ortiz (1988a) has reproduced this value accurately by applying fourth-order electron-propagator theory. The only bound state that is stable toward autodetachment is the X 'X+ state, but ab initio calculations by Ha and Zumofen (1980) show there to be a bound a311and a bound AlII state of excitation energies 5.9 and 7.0 eV. Noting that in alkali halogenides crystals the former is manifest in luminescence spectra and the latter in absorption spectra, Ha and Zumofen calculated their spectroscopic contents and the 'X' - 'll transition array. Peterson and Woods (1987, 1989) have carried out many-body perturbation theory calculations on CN-, on another 14-electron anion BO- and on the 22-electron analogs AlO-, SiN-, CP-, and BS- Table XVI gives the derived EAs and spectroscopic constants. According to an indirect measurement EA(B0) is 3.1 eV (Srivastava et al., 1971). Included in Table XVI are the results of coupled cluster calculations on PO- by Adamowicz et al. (1988). Their values of EA, re, and o,agree satisfactorily with the values (1.092 f 0.010 eV, 1.54 f 0.01 A, and lo00 i-70 cm-') that Zittel and Lineberger (1976) obtained by fixedfrequency-laser photoelectron spectroscopy. They did not treat the a' A state (T,= 0.59 f 0.010 eV, o,= 1020 f 80 cm-') that Zittel and Lineberger discovered. Table XVI also gives the results of a large-scale-configuration calculation of CSi- by Anglada et al. (1983) that revealed a second bound state of excitation energy 0.4eV (much as might be expected for a system intermediate between C; and Si; ). Considerably less extensive configuration TABLE XVI OF SOMEHETERONUCLEAR ANIONS CALCULATED SPECTROSCOPIC CONSTANTS

Anion

State

CNBOAIOSiNCP BSPO CSi -

x IXt x l2+ x 1zt x 1zt x 12+ x '2+ x 32X 22+ A

EA (eV) 3.71 2.72 3.08 3.33 2.83 2.30 0.89 1.98 -

0,

(cm-I) 1.178 1.237 1.645 1.594 1.601 1.682 1.548 1.70 1.77

2080 1713 945 1121 1177 1008 1025 976 949

13.6 11.6 6.1 8.3 6.8 5.6 7.8 4.5 5.5

0.017 0.017 0.005 0.006 0.006 0.006 0.006 0.0 10 0.003

Source: Petersen and Woods (1989), Adamowin et al. (1988), Anglada (1983).

10.25 9.70 6.93 6.37 6.93 6.32 6.35 5.02 4.63 et

al.

32

David R. Bates

interaction calculations by Kame and Grein (1985) suggest the BN- likewise has two bound states: the X 'X+ state and a 211 state of excitation energy about 1.1 eV. An early photoelectron spectroscopy investigation (Siege1 et al. 1972) gave EA(N0) = 0.024 (+O.OlO, -0.005) eV. Measurements by Coe et al. (1987) are consistent with this and Travers et al. (1989) obtain 0.026 k 0.005 eV. Burnett et al. (1982) have studied NS- and CS- by laser photoelectron spectroscopy. As well as measuring that EA(CS) is 0.205 & 0.021 eV and EA(NS) is 1.194 k 0.011 eV, they carried out a Franck-Condon analysis of the relative intensities of the observed transitions and deduced that re (NS-) is 1.589 k 0.02 A and that re (CS-) is 1.627 k 0.02 A. In the case of NS- they observed photodetachment from several vibrational levels and hence found that o,is 880 f 70 cm-'. Since the alkali halides are closed-shell molecules, they would scarcely be expected to form anions except in that they are also highly polar molecules. Photoelectric spectroscopy measurements by Carlsten et al. (1976) gave EA(LiC1) to be 0.61 eV which is much less than the value of 6.5 eV for a dipole of the appropriate moment but was reproduced by the ab initio calculations of Jordan and Luken (1976). More generally, alkali halide anions in the X 2 X + state are not pure dipole-supported anions (Section 111), relatively short range forces being important. The anion may be described as Mo X- where M" is a neutral alkali atom strongly polarized by X-, a closed-shell halogen ion. Novick et al. (1979b) have observed resonances in the cross sections for photodetachment from NaCl-, NaBr-, and NaIabout 0.02 eV to the high-energy side of the neutral sodium 3s - 3p energy difference. They attributed the resonance to a transition to the autodetaching 'll state of the anion that goes to Na('P) + X- in the separated nuclei limit. eV independent of the The positions of the resonances are to within 1 x halide. As Novick et al. commented, this favors the correctness of the Mo X model. From the photoelectron spectra of 10 species of alkali halide anions, Miller et al. (1986) deduced the ground-state spectroscopic constants given in Table XVII. They pointed out that their measured EAs show no correlation with the dipole moments and reproduced them to within 0.1 eV by a simple electrostatic model that does not involve the dipole moment. Although the ground state is not a pure dipole-supported state, the first excited state (Section 111) corresponds to the second dipole-supported state. The position regarding the alkali hydrides (Section 1I.D) is similar in this regard. The earliest direct measurements on an alkaline earth halide, MgCl- was made by Miller and Lineberger (1988) using laser photoelectron spectro-

NEGATIVE IONS: STRUCTURE AND SPECTRA

33

TABLE XVII

MEASURED SPECTROSCOPIC CONSTANTS OF ALKALI HALIDE ANIONSIN x 2x+STATE

EA (eV)

re

Vo*l

Be

Do

Anion

(A)

(cm - ')

(cm-')

(eV)

LiclNaFNaCINaBrNaI KCIKBrKIRbCIcsc1-

0.593 0.520 0.720 0.788 0.865 0.582 0.642 0.728 0.543 0.455

2.123 2.005 2.497 2.660 2.915 2.815 2.985 3.250 2.940 3.060

500 430 265 215 175 210 165 140 180 160

0.640 0.403 0.195 0.134 0.102 0.115 0.072 0.053 0.079 0.065

1.75 2.12 1.34 1.19 0.99 1.42 1.18 1.oo 1.32 1.37

Source: Miller et al. (1986). *Miller et al. estimate that the o, values are within 30 cm- of the vol values.

scopy. They got that EA(MgC1) is 1.589 f 0.011 eV; and for the anion they got re = 2.37 f 0.01 A, we = 315 f 25 cm-', Be = 0.211 f 0.002 cm-', and Do = 1.27 f 0.10 eV. They also reported as preliminary results that EA(Mg1) is 1.90 f 0.04 eV; and that for the anion me = 190 f 45 cm-', Do = 0.79 f 0.26 eV. There have been a number of ab initio calculations on the anions of LiF and Be0 (another highly polar molecule), these being attractively simple systems. For EA(LiF) Yoshioka and Jordan (1980) and Hazi (1981) get 0.33 eV; for EA(Be0) Yoshioka and Jordan get 2.0 eV, while Adamowicz and Bartlett (1985) get 2.1 eV. Different methods were used. Laser photoelectron spectroscopy measurements by Engelking and Lineberger (1977) have given that EA(Fe0) is 1.492 f 0.020 eV and wefor FeOis 740 & 60 cm-'. Later unpublished work in the same laboratory, cited by Anderson et al. (1987), has narrowed EA(Fe0) to 1.4925 & 0.0050 eV. Anderson et al. found that the ground state of the anion is 4A712 with' Be = 0.4972(2) cm-' and that the 4A7/2 - 4A512 and 4A512- 4A312 separations are 226.123(7) and 465.82(1) cm-'. Facilitated by the already available accurate information on the optical spectrum of SeO Coe et al. (1986b) analyzed the photoelectron spectrum of Note that 0.4972(2)

= 0.4972 k 0.0002.

34

David R. Bates

SeO-. They determined that EA(Se0) = 1.456 f 0.020 eV and that the (so constants for (X 'lI)SeOare Be = 0.4246 f 0.0050 cm-' re = 1.726 f 0.010 A), we = 730 f 25 cm-', wexe= 2 f 4 cm-' and Do = 3.84 f 0.09 eV. Rather similarly, Freidhoff et al. (1986a) determined that EA(Te0) = 1.697 f 0.022 eV and that for the anion Be = 0.334 f 0.010 cm-' (so re = 1.884 f 0.028 A), we = 690 f 80 cm-' and Do= 3.63 & 0.15 eV. They noted that as one goes down the (O', SO, SeO, TeO) group, the electron affinities of the monoxides increase in magnitude and approach closer to the electron affinities of their heavy atoms. From a weighted average of the corresponding homonuclear systems, Bauschlicher et al. (1989a) predict that EA(Cu Ag), EA(Cu Au), and EA(Ag Au) are 0.88, 1.30, and 1.44 eV, respectively. D. HYDRIDES Table XVIII gives measured and calculated EAs of hydrides. Iodine is the only halogen for which the EA of the hydride is positive. In order to avoid making the table cumbersome, the number of calculated values of an EA that are given is restricted to two. The very high precision of the measured EA(NH) and EA(0H) is noteworthy. The companion of the latter EA(0D) is 1.825486 0.000037 eV (Schulz et al., 1982). The photoelectron spectrometry researches of Kasdan et al. (1975a, b) show that CH- has a bound a 'A state and that Si H - has bound excited a 'A and b 'X' states all stable toward autodetachment. In the separated nuclei limit X 3X-goes to C- or Si- in the 4S0 state, a 'A to C - or Si- in the 'Do state, and b 'X' to Si- in the 'Po state. Refined ab initio calculations on CH- have been done by Rosmus and Meyer (1978), Sun and Freed (1982) and Manz et al. (1986); and on SiH- by Lewerenz et al. (1983). Table XIX gives a selection of the derived spectroscopic constants-the measured values for X 'X- of CH- and SiH- being corrected as suggested by Rosmus and Meyer (1978) by properly changing the Are sign assignment. It also gives laser photoelectron spectroscopy results for transition-metal hydrides (Stevens et al. 1983; Stevens Miller et al. 1987). Okumura et al. (1986) have studied CH- ions created in a methane discharge, mass selected and injected into a radio frequency octupole trap where they could be stored for a variable time ranging from 1 ms to 5 s. They probed the total excited-state population by observing the depletion of trapped CH- due to photodetachment at 1.16 eV, that is, 0.08 eV below the

TABLE XVIII HYDRIDE X H ELECTRON AFFINITIES(ev)

X=

Li

measured calculated

-

X=

measured calculated

X=

measured calculated

X=

0.28', 0.32' Na -

0.28', 0.37" K -

0.44'

calculated

Rb 0.42'

calculated

cs 0.36', 0.44'

X=

Be 0.70 f 0.10" 0.49', 0.808 Mg 1.05 f 0.06" 1.MU,0.83' Ca 0.93 f 0.05"

Cr measured 0.563 f 0.010"

B

C 1.238 f O.OOSb 1.19', 1.04'

-

0.07', 0.03'

A1

Si 1.277 f 0.W. 1.13h, 1.17"

-

0.03h

I >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.

',

+

+

+

NEGATIVE IONS: STRUCTURE AND SPECTRA

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,

NEGATIVE IONS: STRUCTURE AND SPECTRA

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

NEGATIVE IONS: STRUCTURE AND SPECTRA

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) =

'

NEGATIVE IONS: STRUCTURE AND SPECTRA

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. +

NEGATIVE IONS: STRUCTURE AND SPECTRA

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

NEGATIVE IONS: STRUCTURE AND SPECTRA

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-

+

+

NEGATIVE IONS: STRUCTURE AND SPECTRA

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

NEGATIVE IONS: STRUCTURE AND SPECTRA

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

+

'.

+

NEGATIVE IONS: STRUCTURE AND SPECTRA

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

NEGATIVE IONS: STRUCTURE AND SPECTRA

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.

NEGATIVE IONS: STRUCTURE AND SPECTRA

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

NEGATIVE IONS: STRUCTURE AND SPECTRA

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

NEGATIVE IONS: STRUCTURE AND SPECTRA

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

<6.3

>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-

NEGATIVE IONS: STRUCTURE AND SPECTRA

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.

NEGATIVE IONS: STRUCTURE AND SPECTRA

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.

REFERENCES

Aberth, W., Schnitzer, R., and Anbar, M. (1975). Phys. Rev. Lett. 34, 1600. Acharya, R. K., Kendall, R. A., and Simons, J. (1984). J. Am. Chem. SOC.106,3402. Adamowicz, L., and McCullough, E. A. (1984a). Chem. Phys. Lett. 107, 72. Adamowicz, L., and McCullough, E. A. (1984b). J. Phys. Chem. 88,2045. Adamowicz, L., and Bartlett, R. J. (1985). J. Chem. Phys. 83, 6268. Adamowicz, L., Bartlett, R. J., and McCullough, E. A. (1985). Phys. Reu. Lett. 54, 426.

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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

82

Joach im Kessler

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

ELECTRON-POLARIZATION PHENOMENA

"

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).

86

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

ELECTRON-POLARIZATION PHENOMENA

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).

ELECTRON-POLARIZATION PHENOMENA

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.

ELECTRON-POLARIZATION PHENOMENA

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

ELECTRON-POLARIZATION PHENOMENA

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

ELECTRON-POLARIZATION PHENOMENA

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

ELECTRON-POLARIZATION PHENOMENA

polarization effects for positrons are orders of magnitude smaller (<0.01%) than for electrons (Hasenburg, 1986;Hasenburg et al., 1987). Moreover, the polarization effects based on exchange, which will be discussed in the following section, do not appear in positron-atom scattering because exchange effects result from the indistinguishability of identical particles and cannot therefore occur in positron-electron interactions.

B. SPIN-DEPENDENT SCATTERING DUETO EXCHANGE INTERACTION In the preceding section we concentrated on scattering from spinless targets. If we drop this restriction, we may find polarization phenomena that are governed by the exchange interaction. An obvious example is the scattering of unpolarized electrons by polarized alkali atoms where polarization of the scattered beam results from exchange processes between the incident electrons and the spin-oriented atomic valence electrons: the scattered electrons become polarized while the target atoms become depolarized. In general one has the more complicated situation where an electron beam of polarization P, is scattered by a target of polarization Pa. In order to clearly illuminate the role of exchange, we shall assume in this section that the spin-orbit interaction is negligible, which is a good approximation for light alkali atoms. In terms of the scattering amplitudes f = Ifleiyland g = Jgleiy2 governing E q s . (3) and (2), the scattering cross section is given by

a = .,,(0)[1

- A(0)P,.Pa]

(13)

with

and

where f + g and f - g are the singlet and triplet scattering amplitudes, respectively. The polarization of the scattered electrons is given by

(1 p =

- $)Pa

+ (1

- %)Pe

-

1 - A P;P,

i fg*2a, -f*spe pa

.

(16)

Joachim Kessler

98

The derivation of these formulae is straightforward if the density-matrix formalism is applied (Kessler, 1985). For an unpolarized target, Pa = 0, one has from Eq. (16)

which expresses the partial depolarization of the polarized incident beam due to exchange processes with the unpolarized target electrons. If, on the other hand, an unpolarized electron beam is scattered by a polarized target, one has from Eq. (16)

If there is no exchange scattering, g = 0, one has from Eq. (14) cru = If[’ so that Pi = 0, which expresses the obvious fact that exchange is necessary for a polarization transfer from the target to the scattered electron beam to take place. In the other extreme, f = 0, the electrons are scattered by exchange only, which implies that the scattered beam has the original target polarization, P, = Pa, because all the scattered electrons were originally target electrons. The differences between formulae (6) and (16) describing the change of the electron-polarization vector from P, to Pe in two basic scattering processes are obvious. The orientation of the scattering plane described by A in Eq. (6) no longer plays a role in Eq. (16). It is replaced by the atomic polarization P,. If P, is either parallel or antiparallel to Pa, it is only the magnitude of the polarization that is changed by scattering. In all other cases one has a rotation of the electron polarization out of the initial plane spanned by P, and Pa by an angle u given by3 tan u =

-f*d lf12- 191’ + Ifi(fg*

912

In discussing the differences between Eqs. (6) and (16) we reemphasize that the physical origin of the amplitudes g is quite different in the two cases though it is customary to denote them by the same letter g. For a complete experiment yielding If[, lgl, and the relative phase y1 - y z , one has again to measure four observables. One of them is the (absolute) cross section 0, for unpolarized beams given by Eq. (14). Another observable The factor Pa was omitted by misprint in Kessler (1985), Eq. (4.42).

99

ELECTRON-POLARIZATION PHENOMENA

on which much attention has been focused is the scattering asymmetry. The authors in this field usually define it by

(where the subscripts denote antiparallel or parallel spin orientations of electrons and target atoms), which is in accordance with the negative sign in Eq. (13). According to Eq. (15) an asymmetry measurement yields information on the relative phase. The missing information on the complex scattering amplitudes may be obtained by observing the polarization of the scattered electrons, Eq. (16). Such a measurement may be performed with one of the colliding beams unpolarized which yields [l - (lglz/au)] if the electrons are polarized, Eq. (17), and [l - (lflz/ou)]if the target atoms are polarized, Eq. (18). Since c0s-l is not a single-valued function, the cos(y, - y z ) factor obtained from A is not sufficient to determine y1 - y z unambiguously. One also needs sin(?, - y z ) which may be found from the scattering of polarized electrons by atoms with polarization neither parallel nor antiparallel to P, and measuring the polarization component normal to P, and Pa.According to Eq. (16) such a measurement yields

Unlike the case of the spin-orbit processes discussed in Section A, here one does have other options for obtaining complete information on the collisions. Instead of performing all the measurements with the scattered electrons, one may also observe the polarization of the scattered atoms Pa. This has been done in a series of pioneering experiments in Bederson's group (Bederson, 1973). From the general expression (Kessler, 1985),

()::I 1--

p =

P,+

( ) : :-I 1-1

P a + ifs* -f*spe A P;P, 2%

pa 9

(16')

which follows from Eq. (16) by interchanging P, and Pa, it is seen that measurements of the electron polarization may be replaced by equivalent measurements of the atom polarization. We shall now consider the state of present knowledge about electron-atom collisions in which the exchange interaction is the only spin-dependent mechanism. The uncertainties of absolute cross sections have been discussed in Section A. For light atoms, for which the assumptions of the present

Joachim Kessler

100

section hold, the uncertainties are usually somewhat smaller than for heavy atoms, but the situation for sodium shows that there are still problems to overcome. Substantial disagreement between experimental and theoretical differential cross sections has been found for scattering angles larger than 40" (Mitroy et al., 1987a), necessitating further cross section studies. The spin-dependent observable on which by far most of the attention has been focused is the asymmetry A. Spin asymmetries have been measured for different types of processes in the scattering of polarized electrons from polarized atomic beams of hydrogen, lithium, sodium, and potassium. Experimental results on integrated spin asymmetries of electron-impact ionization are compiled in Fig. 10. They were obtained by two different groups by ionization of the polarized target atoms with polarized electrons. By counting the ions, the relative ionization cross sections for antiparallel and parallel spins of the colliding beams were determined regardless of the

0.

0.01

1

I

2

1

I

I

5

I

I

1 I

I llrrIIllrll

10

15

20

INCIDENT ENERGY, E / 1 FIG.10. Spin asymmetry of electron-impact ionization of various targets ( I energy). (From Baum et al., 1985, and Fletcher et al., 1985).

=

ionization

ELECTRON-POLARIZATION PHENOMENA

101

scattering angles and final energies of the electrons. That is why the terms in the asymmetry Eq. (15) must be integrated over these variables. The large asymmetries in the lower energy range of Fig. 10 mean that the interaction has a significant spin-dependence at such energies. All the ionization asymmetries observed are roughly energy-independent within the first eV above threshold. The asymmetry for sodium is nearly identical to that for lithium, while for potassium it is clearly different at energies from threshold up to four times the ionization energy. One may speculate that the different behavior of potassium is caused by the influence of the spin-orbit interaction which at the higher atomic number Z of this element is certainly of greater importance than for the low-Z alkalis. A theoretical answer to this problem cannot be given as of 1989, in particular since at the moment there are no theoretical approaches that describe correctly the asymmetries in the energy range of Fig. 10. With lithium, the spin asymmetry has not only been measured for direct ionization but also for ionization via electron-impact excitation of the , et al., 1988a). Beautoionizing metastable state 6Li(ls 2s ~ P ) ~ P , ,(Baum tween the excitation threshold at 57.3 eV and the second ionization threshold at 64.42 eV, an average asymmetry of A = -0.47 has been observed, which reveals a spin-dependence that is markedly different from that in direct impact ionization where, according to Fig. 10, only positive asymmetries were found. This difference results from the fact that the formation of a quartet state requires excitation of and exchange with a core electron, while in direct ionization only the outer electron is involved. The state of the experimental art of diflerential spin-asymmetry measurements is illustrated in Fig. 11, showing apparatus suitable for measuring such asymmetries in either elastic or inelastic electron scattering. The experiment of Fig. 11 uses hydrogen atoms of 50% polarization with a beam density of 2 x 10'0cm-3 as the target. The atomic polarization is generated by a six-pole magnet and measured by a Stern-Gerlach polarimeter. In order to minimize instrumental asymmetries, the direction of the hydrogen polarization can be reversed periodically through reversal of the solenoid guide fields along the beam line. Further details of the apparatus can be found in the caption of Fig. 11. Some of the results to be discussed in the following were obtained with a different technique of producing the target polarization: optical pumping with circularly polarized laser light. Sodium atom beams of 61 % polarization were obtained in this way by McClelland et al. (1987a). The polarization of the atoms was diagnosed by measuring the polarization of the resonance

d FIG.11. Schematic diagram of the measurement of differential spin asymmetries: (1) rf hydrogen source, (2) hexapole high-field state selector, (3) beam chopper, (4)upstream solenoid guide field, ( 5 ) direction of hydrogen polarization vector, (6) hemispherical electron monochromator, (7) channel multiplier ion detector, (8) hemispherical electron spectrometer, (9) downstream solenoid guide field, (10) hydrogen spin precessor, (1 1) Stern-Gerlach polarimeter, (12) quadrupole mass analyzer, (13) 787-nm laser beam, (14)GaAs crystal, (15) 90" electrostatic bender, (16) solenoidal electron spin precessor, (17)direction of electron polarization vector, (18) Lyman-a detector (for 1s-2Pmeasurements), (19) Faraday cup, and (20) electron beam exiting to Mott-scattering polarimeter. (Crowe et a/., 1989).

ELECTRON-POLARIZATION PHENOMENA

103

fluorescence that was generated with a second laser. Reversal of the atom polarization was easily achieved by reversal of the light polarization. Figure 12 displays differential spin asymmetries of polarized electrons scattered through 107.5' from lithium atoms that were polarized by a six-pole magnet. It shows that this observable can reach significant values. The asymmetry 1 obtained for elastic scattering (Fig. 12a) at electron energies corresponding to the 2P threshold implies according to Eq. (15) that here the triplet contribution is negligible (f- g = 0). Singlet scattering dominates up to 4 eV where triplet scattering starts to increase. Within a few eV above this energy, triplet scattering becomes dominant since according to Eq. (15) an asymmetry of - 1/3 means that the singlet contribution is negligible ( f + g = 0). In Fig. 12b the competition between singlet and triplet scattering is seen for the inelastic case (2P excitation) at the same angle. Triplet scattering dominates near 10eV, while it decreases again as the electron energy decreases. Comparison of theoretical results with the experimental data shows good agreement in the lower energy range with the five-state close-coupling calculation by Moores (1986). At scattering angles of 65" and 90", which have also been covered by the experiment, the agreement for inelastic scattering above 6eV is less satisfactory. As seen in Fig. 12, calculations using polarized-orbital methods are at variance with the experimental data and so is the two-potential localized exchange approach (Mathur, 1989). In the experiment discussed, measurements have also been made of the spin asymmetry in the intensity of the radiation that was emitted after excitation of the 2P state. They yield the asymmetry integrated over the scattering angles of the electrons and clearly reflect the dominance of singlet scattering near threshold. The example given in Fig. 12 is one of the few cases where one finds theoretical data that are reliable. Even for the simplest case, the hydrogen atom, the theoretical asymmetries depend very much on the model used, as shown by McDowell et al. (1984) with detailed calculations using different theoretical approaches. Measurements of elastic e-H scattering at 90" between 4.4 and 30.3 eV by Fletcher et d. (1985) illuminate this situation: While most of the theoretical predictions for the spin-averaged cross section agree with each other and with the experiments, there is disagreement among the same approximation methods in the predictions of the spin asymmetries. The sensitivity of spin asymmetries to the theoretical approach shows the great potential of polarization studies for confronting the theory with very stringent experimental tests. In a later calculation, van Wyngaarden and Walters (1986) were able to give a satisfactory theoretical description of the elastic

+

Joach im Kessler

104

Energy ( e V )

Energy ( e V

)

FIG.12. Spin asymmetry for (a) elastic and (b) inelastic (2P excitation) scattering of polarized electrons from polarized Li atoms. Experimental points (Baum et al., 1986, 1987, 1989b) and theoretical results: .... two-state close-coupling calculation (Burke and Taylor, five-state close-coupling calculation (Moores, 1986), - - - modified polarized1969), orbital calculation (Bhatia et al., 1978), x distorted-wave polarized-orbital model (Kennedy et al., 1977). ~

ELECTRON-POLARIZATION PHENOMENA

105

e-H asymmetries of Fletcher et al. (1985) using the multipseudostate closecoupling approximation, a method in which the entirety of target eigenstates is represented by pseudostates. But still the reliable prediction of hydrogen spin asymmetries is a difficult problem for present theories. When even for the simplest electron-atom scattering process we can think of, e-H scattering, our quantitative understanding is so rudimentary then we cannot expect the situation to be more favorable for heavier atoms. Of the heavier alkali atoms, differential spin asymmetries have been published only for sodium, but experiments for other targets are underway. Figure 13 shows the angular dependence of the spin asymmetry for elastic scattering of polarized electrons from polarized sodium atoms at 54.4 eV together with theoretical results obtained with a nonrelativistic two-state close-coupling approximation by Mitroy et al. (1987b). Recalling the previously mentioned discrepancies between theory and experiment that occur even for the spinaveraged hydrogen cross sections, it is not surprising that the asymmetries are described by theory only qualitatively. Besides, for sodium (2 = 11) the

A(%)

'1 6

c

-I

0 scat

(degrees)

FIG. 13. Spin asymmetry for elastic scattering of polarized 54.4-eV electrons from polarized Na atoms. Experimental points (McClelland et al., 1987a) and theoretical results of a two-state close-coupling calculation (Mitroy et al., 1987b).

106

Joachim Kessler

influence of the spin-orbit interaction is not completely negligible. McClelland et al. (1987a) have checked this by averaging the measured cross sections over the atomic spin orientation in order to obtain the asymmetries for scattering from an unpolarized target. By this procedure they find asymmetries up to 3 1% in a small angular range around 110" that cannot be caused by exchange scattering. This indicates that both exchange and spin-orbit interactions must be taken into account to predict the experimental results. In fact, situations where the spin effects are caused by only one of the basic spin-dependent interactions are the exception. They are presented at the beginning of this chapter because they are the least complicated. Spin effects that are caused by the combined action of spin-orbit and exchange interaction like the spin asymmetries in superelastic scattering will be treated later on. Beyond the asymmetries for spin-1/2 atoms discussed so far, we can report only one spin-asymmetry measurement with a spin-1 target. It has been performed by Baum et al. (1988b), who used a polarized metastable He(2%) atomic beam that was produced by means of a six-pole magnet and had a polarization of 90 %. Crossing the polarized helium beam with a polarized electron beam, they measured the integrated spin asymmetry of electronimpact ionization of helium. The first results are shown in Fig. 14 and are compared with the aforementioned results for lithium. The helium asymmetry is clearly smaller, in particular near the ionization threshold. Since, according to Fig. 10, the asymmetries of H, Li, and Na are very similar to each other one would not anticipate a major difference, if the incident electrons interacted only with the outer 2s electron of the metastable helium. It seems that the influence of the polarized 1s electron is appreciable as well. Theoretical calculations of the spin asymmetries in scattering from He(23S) have so far been restricted to excitation processes (Mathur et al., 1987) so that at the moment comparison between theory and experiment for spin-1 targets is not possible.* Since most of the attention has so far been focused on spin asymmetries, knowledge of the other observables that are necessary for a complete understanding of the scattering processes considered in this chapter is poor. Even such important quantities as the differential exchange and direct cross sections lg(0)I2 and lf(0)I2 have been measured only in some pioneering experiments dating back from the 1970s. In a series of experiments using the

* Note added in proof For latest work on the subject, seeBaum,etal. (1989~)and Bartschat (1989).

107

ELECTRON-POLARIZATION PHENOMENA

0.6

I

I

1

I

I

-

-

0.5 -

-

-

-

-

0.4

a

H

* 0.3aJ

L +

E E 0.2)r

v)

a

0.1 o.oo

-

He (23 S ) He(Z3S)

-

Li

1; II

5

I

I

I

I

I

10

15

20

25

Incident Energy ( e V 1 FIG.14. Spin asymmetry of electron-impact ionization of polarized metastable He(2%) (Baum et al., 1988b). The data points are compared with the results for lithium from Baum et al. (1985). The respective ionization thresholds are indicated.

method of polarized recoil atoms, Bederson and coworkers measured lg(0)(2 for potassium at energies between 0.5 and 1.2eV. (For a summary, see Bederson, 1973). By scattering unpolarized electrons from polarized atoms and observing the polarization P,' of the recoil atoms they determined Ig(0)lz/o,(O) according to Eq. (21). After a separate absolute measurement of o,(8) the differential exchange cross section Ig(0)lz could be derived with an overall error of 27 %. Relative measurements of If(0)12 by Kleinpoppen's group were made by scattering unpolarized electrons from polarized potassium atoms and observing the polarization Pe of the scattered electrons, which according to Eq. (18) yields If(0)2/a,(@ (Hils et al., 1972). The large uncertainties of these early experiments can certainly be reduced by applying modern techniques. Thanks to the advent of the GaAs polarized-electron

108

Joachim Kessler

source in particular, measurements of exchange and direct cross sections can now be made with much better accuracy. If from measurements of (according to Eq. (17)), A(B), and o,(O) one has determined If], 191, and cos(y, - y2), there remains still one measurement to be done in order to complete the experiment: the determination of sin(?, - y 2 ) by observation of the electron-polarization component normal to the plane spanned by Paand P,according to Eq. (21). Since this is a type of measurement similar to that described in Section A, it should be feasible with comparable uncertainties. Measurements with the final goal of a complete experiment are currently being prepared in several laboratories. We conclude this section by pointing out that the polarization phenomena discussed in Sections A and B give quite different pieces of information. The spin-orbit interaction of the scattered electrons is most important in the inner part of the atom. Consequently, the polarization effects caused by this force hardly depend on the outer electrons, though they are very sensitive to the exact shape of the atomic field made up of the nuclear field plus that of the charge-cloud distribution. On the other hand we have seen that in light alkali atoms polarization phenomena are primarily based on exchange and therefore are determined by the spin angular momentum of the valence electrons. In more complicated atoms it is the total angular momentum configuration that determines these phenomena. Though we have so far considered cases where the polarization effects are caused by only one of the spin-dependent interactions, one has in more complicated cases an interplay between exchange and spin-orbit interactions, so that in general it is not possible to assign the polarization effects to certain atomic regions. The next step of complication where the total angular momentum configuration plays the essential role will be discussed in the following section. c . POLARIZATION EFFECTSCAUSED BY THE INTERPLAY OF FINESTRUCTURE SPLITTING WITH EXCHANGE SCATTERING From the preceding section it may seem as though electrons cannot become polarized if they are scattered from an unpolarized target and their spin-orbit interaction in the atomic field is negligible. This is, however, not correct. It has been shown by Hanne (1976, 1983, 1984) that exchange scattering in conjunction with the fine-structure splitting of the target atoms gives rise to significant polarization effects, even if there is no spin-orbit interaction of the scattered electrons in the atomic field.

109

ELECTRON-POLARIZATION PHENOMENA

Figure 15 gives as a quantitative example the (theoretical) polarization of electrons that have excited the 23P,, fine-structure states of helium. The LS-coupled 3P states of helium can be excited from the singlet ground state 1' S o only by exchange processes. Although the spin-orbit interaction of the scattered electrons in the low-Z helium atom is negligible, the (initially unpolarized) electrons have significant spin polarization after they have excited certain fine-structure levels. For symmetry reasons the polarization vector is oriented normal to the scattering plane (Kessler, 1985). It can be seen from Fig. 15 that the polarization ratio P( J = 0) :P( J = 1) :P( J = 2) is 2 :1 : - 1. Since the corresponding ratio of the excitation cross sections of the fine-structure levels is 1 :3 :5, the average polarization of all the electrons that have excited the 3P, levels is zero. Accordingly, the polarization can only be observed in an experiment that separates electrons that have excited certain fine-structure levels. Figure 15 also contains the full information about the scattering asymmetry in helium because one has the relation P = -A (Hanne, 1983) when A is the asymmetry found by scattering of a totally polarized beam. Thus the ratio of the asymmetries associated with excitation of the J = 0, 1,2 fine-

0.81

P I-

-o.+ 4

-

-0 8-

-

I

I

I

I

I

I

I

I

I

I

I

I

I

t

1

I

I

1

110

Joachim Kessler

structure levels of helium is the same as given previously and the average asymmetry for the 3P,levels vanishes. We shall now explain the physical mechanism underlying the “finestructure effect” as the polarization phenomenon discussed in this section is called. For this purpose it is necessary to recall that the excitation process may result in an orientation of the orbital angular momentum of the excited atomic state. This has been studied in the past in a great number of electron-photon coincidence experiments and many theoretical papers. For a review we refer to Andersen et al. (1988). Assume that one has the situation depicted in Fig. 16. One observes electrons that excite a P level and are thereby scattered through a certain angle to the left while the atoms attain an orbital-angular-momentum orientation (L,) normal to the scattering plane. Assume further that the incident electrons are totally polarized normal to the scattering plane and that the detection system selects electrons that have excited a certain fine-structure state, say the 23P, level of helium. The atoms in this state have their spins and orbital angular momenta antiparallel to each other so that their average spin orientation is (S,) = - ( L J . Since the He states can be excited from the singlet ground state only by exchange, the excitation takes place by capturing an electron of suitable spin orientation and releasing the atomic electron with the opposite spin orientation. When the spin orientation of the incident electrons corresponds to that of the excited state, the excitation probability is higher than in the opposite case, because in helium there are no explicit spin-dependent forces by which

FIG. 16. Angular-momentumorientation in ’Po excitation.

I Is

ELECTRON-POLARIZATION PHENOMENA

111

“wrong” spin orientations might be flipped into the “right” direction. If, in the example of Fig. 16 where (S,) is oriented “down,” the incident electron beam is polarized “down,” the detector at the left will find more electrons that have excited 3P0than if the incident electrons are polarized “up.” In other words, the cross section at for spin-up electrons to excite the 3P0 finestructure level while being scattered to the left is less than the cross section u1 for spin-down electrons to undergo the same process and vanishes altogether in the case of total atomic orientation (S,) = -h. The asymmetry A = -Q t - Q1 Q t + Ql depends, of course, on the degree of the atomic orientation after the collision, clearly a function of scattering angle and energy. In cases where there is no atomic orientation, one has A = 0; for maximum atomic orientation ( L , ) = - (S,) = h, one has A = - 1; and in general A = - (L,)/h

=

(S,)/h

(22)

for excitation of He3Po. The spin-up-down asymmetry just considered is equivalent to a left-right asymmetry, because electrons scattered by the same angle to the left and to the right must have opposite spin orientations in order to have the same probability of exciting the same fine-structure level. This is easily seen by a mirror reflection as illustrated in Fig. 17. As a consequence, one finds different

OUT

00

i

I MIRROR FIG.17. Mirror reflection of impact excitation by polarized electrons.

112

Joachim Kessler

intensities at the same scattering angle to the right and to the left, when a finestructure level is excited by electrons of a well-defined spin orientation. The mechanism just described results in a polarization of the scattered electrons when the 3P, state is excited by unpolarized electrons. Assume for simplicity that at the scattering angle chosen in Fig. 16 one has the atomic orientation (L,) = h so that only the spin-down electrons can excite the He3P, state. Since the transition from the singlet ground state to the excited triplet state by exchange implies that a spin-down electron is captured while the atomic spin-up electron is released, the electrons arriving at the detector have the polarization P = + 1. Recalling that the asymmetry in this case is - 1, one has P = - A = 1 . In the general case where - h I(L,) Ih, one has by the same line of argumentation P = - A = (L,)/h.

(23)

While this intuitive explanation has the advantage of unveiling the polarization mechanism as a cooperative result of exchange scattering and spin-orbit interaction of the bound electrons, for a more rigorous presentation refer to Hanne (1983), who has also derived the corresponding relations for 3P, and 3P,. From the results (Csanak and Cartwright, 1987)

P=-A=

II

(L,)lh $(L,)lh

-$(L,}/h

for J = 0, for J = 1, and

(24)

for J = 2,

it is seen that for J = 1 and 2 the polarization effects are smaller than for J = 0, which is in accord with the intuitive model we have discussed. As described by the Clebsch-Gordan coefficients, an oriented L can combine with different orientations of the atomic spin S to arrive at a particular J # 0 (with different M , ) so that these states can be excited by electrons of both spin orientations, though with different probabilities. Equations (24) show that a measurement of either the polarization P after scattering of an initially unpolarized beam or the scattering asymmetry of a polarized beam yields directly the orientation (L,) of the excited atomic state, a quantity usually obtained by coincidence experiments. This is an example of how electron-photon coincidences and electron-polarization studies yield equivalent information in some special cases. (Another example has been studied by Hanne et al., 1981.) However, we shall see later that application of the two techniques to more complicated atoms generally yields

ELECTRON-POLARIZATION PHENOMENA

113

complementary information. Even in helium for excitation of L 2 3 levels the polarization P is not proportional to the atomic orientation (L,) any more. Instead, P is a linear combination of the atomic orientation and the atomic octopole moment (Csanak et al., 1988). A combination of electron-polarization experiments with electron-photon coincidence measurements makes possible the determination of these quantities. Figure 15 also illustrates the fine-structure effect with hydrogen atoms. This example applies to all atoms with one outer electron, where one has the ratio of the electron polarizations for the two fine-structure levels P(1/2):P(3/2) = 2 : - 1. The same relation holds for the scattering asymmetries. Since the cross section ratio eU(1/2):e,(3/2) is 1 :2,the net result is that the polarization effects vanish when averaged over the fine-structure states, as previously discussed. On the other hand, P = -A does not hold for one-electron atoms. Assume, for instance, that in Fig. 16 a ’P1/’state is excited by the electrons scattered to the left and that its preferential L orientation is “up.” Because J = 1/2,its preferential spin orientation is then “down,” so that the processes

+ A 1 -t’4*(2p1,2)1 + e t t e l + A 1 -tA*(’P1/2)1 + e l ,

If? If- Sl’

et

el

+A t

-tA*(2P1/2)J. +et,

(25)

191’

are the most relevant for its excitation. (The up and down arrows indicate the spin orientations of the electrons and atoms, while the last column in Eq. (25) gives the essential terms of the cross sections.) It is clearly seen that the scattering asymmetry of a polarized beam

is not so simply related to the polarization

P =N, N,

, - lfl’ + 191’ + N , lfl’ + I f -N

-

If-

91’

912

+ 191’

of an initially unpolarized beam after scattering. ( N t and N , are the numbers of spin-up and spin-down electrons after scattering.) In the limit of vanishing direct scattering amplitude f, one even has A = - 1, P = 0.Needless to say, this discussion of extreme cases only serves to elucidate the physical mechanism and has little to do with the situation met in practice.

Joach im Kessler

114

In heavy elements the fine structure in the energy spectrum of the scattered electrons is large enough to be experimentally resolvable, but in helium the fine structure is in the 100 peV range, while in sodium it is x 2 meV. Yet, it has been possible to directly observe the fine-structure effect in sodium where, unlike the case of heavy atoms, it is not superimposed on or even masked by other polarization mechanisms. Instead of exciting the fine-structure levels, the authors de-excited them by electron impact (superelastic scattering) after populating them by laser optical pumping. This method takes advantage of the fact that the time-reversed processes of excitation and de-excitation are equivalent: they are described by the same scattering matrix (Hertel et al., 1987), so that the polarization phenomenon discussed here can be observed in superelastic scattering as well. Evidence of the fine-structure effect was found by measuring the polarization of 20-eV electrons that were superelastically scattered through 30" from the 32P1,2state of sodium (Hanne et al., 1982). As is typical of the fine-structure effect, the electron polarization reversed sign when the laser was tuned to the 32P3,2 state. More comprehensive investigations were made by the NBS group (McClelland et al., 1985, 1986). The principle of their apparatus, which was used in slightly modified versions for different types of experiments, is shown in Fig. 18. A polarized electron beam (typical polarization P = 0.26 f 0.02)

Optical Pumping Laser N a Oven

I

GaAs Crystal

Pockels Cell Linear P olarizer Laser Diode FIG.18. Superelastic scattering of polarized electrons.

Detector

ELECTRON-POLARIZATION PHENOMENA

115

of 2mm diameter intersects an atomic sodium beam of density 10'0atoms/cm3 and diameter 4mm produced by an effusive oven. The scattering region is illuminated with light from a frequency-stabilized,singlefrequency ring dye laser locked to the Na2SlI,(F = 2) + 3P,,,(F = 3) transition, creating a significant population of sodium atoms in the excited state. The rotatable electron detector (channel electron multiplier) incorporates a retarding-field analyzer to reject all elastically and -inelastically scattered electrons while allowing detection of the 2.1-eV more energetic superelastically scattered electrons. Figure 19 presents one of the central results of these experiments: a significant spin asymmetry of polarized electrons that de-excite unpolarized sodium atoms from the 3'P3/, fine-structure state and are thereby superelastically scattered. In accordance with what one anticipates from reflection symmetry (Fig. 17), the asymmetries for scattering to the left (positive angle) and to the right (negative angle) differ (only) in sign. For sodium the spinorbit interaction of the scattered electrons in the atomic field is almost negligible and cannot produce asymmetries of that size, especially since the scattering angles are relatively small (McClelland et al., 1987a). Accordingly,

h

L c Q,

0.2

E €x

0.1

.-CQ

-0.1

vl 4

m

0.0

-0.2 -40

- 20

e scat

0

20

40

(degrees)

FIG.19. Spin asymmetry of superelastic scattering of 10-eV electrons from unpolarized sodium atoms in the 32P,,,state (McClelland et al., 1985, 1986; Hertel et al., 1987).

116

Joachim Kessler

the asymmetry is an outcome of the “fine-structure effect,” the cooperative result of exchange scattering and fine-structure splitting. An advantage of studying these polarization phenomena by superelastic instead of inelastic scattering is that, by using different optical pumping schemes, the atoms can be prepared in well-defined initial states. In another version of the experiment the excited atoms were produced in polarized excited states by optical pumping with circularly polarized light while the scattering asymmetries of the polarized electrons were observed. This is a more complex situation because the asymmetries caused by spin-dependent interactions are superimposed on an asymmetry caused by the orientation of the atomic orbital angular momentum which also occurs when unpolarized electrons are scattered (Hermann et al., 1980; Hermann and Hertel, 1982; McClelland et al., 1987b). By a suitable combination of measurements with different preparations of the initial state (characterized by polarization and/ or alignment of the excited atoms and by polarization or non-polarization of the electrons) and by exploiting the equivalence of excitation and deexcitation, it was possible to evaluate essential parameters for inelastic e-Na scattering. The angular-momentum transfer ( L , ) for excitation of 32P3/2by singlet and triplet scattering and the ratio of triplet-to-singlet excitation cross sections could be determined at incident electron energies between 1.26 and 11.76 eV for a scattering angle of 30” and at 2.0 and 9.26 eV over the angular range 5 to 40”. In Fig. 19 the triangles were obtained by reprocessing the experimental asymmetries obtained with polarized excited atoms while the circular points were directly measured by the method described in the preceding paragraph. By combining the preparation of different initial states with a polarization measurement of the scattered electrons, one has an elegant method for a complete analysis of the process under study. Such a “perfect” experiment is being prepared (Kelley, 1989). An experiment that clearly demonstrates the difference in sign of the polarization effects associated with the and ’P3/2 fine-structure levels has been performed by Nickich et al. (1990). Polarization phenomena due to the fine-structure effect in its pure form are limited to special cases in which the following conditions are fulfilled: 1. The spin-orbit interaction of the scattered electron in the atomic field is negligible. 2. Internal spin-orbit coupling within the target atom is weak so that the LS-coupling scheme is valid.

ELECTRON-POLARIZATION PHENOMENA

117

With increasing atomic number the validity of these conditions breaks down so that the polarization effects are simultaneously generated by the mechanisms I have so far been discussing in their pure forms. A comparative calculation of the fine-structure effect in He, Mg, and Hg was made by Meneses et a/. (1987).

111. Combined Effects of Several Polarization Mechanisms A. THEORETICAL DESCRIPTION OF ELECTRON SCATTERING FROM UNPOLARIZED ATOMS HAVING ANGULARMOMENTUM The discussion of the preceding Section I1 illuminated the basic mechanisms responsible for the polarization phenomena in electron-atom collisions. It does not, however, give a description of the wealth of effects caused by the interplay of the spin-dependent interactions, namely 0 0 0

exchange interaction, spin-orbit interaction of the scattered electrons in the atomic field, internal spin-orbit coupling in the target.

It was only be selecting cases where certain limiting conditions are fulfilled that we were able to separate the effects of the diverse polarization mechanisms. We shall now deal with the general case in which all of the mechanisms have to be considered in order to understand the polarization phenomena. First we shall outline how the results of Section 1I.A have to be generalized if the target atoms, though still unpolarized, are no longer spinless and if inelastic scattering is included. Theoretical treatments of this situation were given by Huang (1987) and by Bartschat and Madison (1988), who applied the formalism of the reduced-density matrices (Blum, 1981; Blum and Kleinpoppen, 1983). We shall follow here the latter approach describing (elastic or inelastic) scattering of electrons from unpolarized targets when only the scattered electrons are observed and when the target atoms may have spin and orbital angular momentum in the initial and/or final state. An essential result of the generalized theory is that the four observables nu,S, IT: and U defined in Section 1I.A are no longer sufficient for a description of the scattered electrons, but that one needs eight observables as will be seen from Eq. (38). This is not surprising since one needs more scattering amplitudes in order to describe the more complex process illustrated in Fig. 20 leading from an initial state IJ,, M , ; po, m o ) to a final state

Joachim Kessler

118

INCIDENT TARGET ELECTRON INITIAL STATE

OUTGOING TARGET ELECTRON FINAL STATE

FIG.20. Diagram of scattering process.

IJ1,M , ; pi, ml). The projections of the initial and final electron spins on the quantization axis z parallel to the direction of the incident beam are denoted here by m, and m, while J,, J, and M,, M , are the initial and final total angular momenta and their z components. The scattering plane is defined to be the x-z plane. It no longer sufficesto have just two transition amplitudes, one with and one without spin flip of the scattered electron. Now, one must account for transitions between the various angular momentum orientations of the initial and final atomic states. It is therefore expedient to define a scattering amplitude f(M1m1; Mom,)

= (J1M1;P I % I TIJoM,;

Porn,)

(28)

where T is the transition operator. In the argument offwe have omitted those parameters that are well defined by either the geometry of the experiment (p,, pl) or the energy analysis (J1,.Io). We shall now discuss the eight observables by which scattering of spin-112 particles from an unpolarized target is completely described and express them in terms of the preceding scattering amplitudes. The differential scattering cross section of an electron beam of polarization P is given by

.(e, 4) = Gu[i + S,(B)P.BI

(29)

ELECTRON-POLARIZATION PHENOMENA

119

which formally looks like Eq. (11). The difference is that now the differential cross section for scattering of unpolarized incident particles is .Is@

=

c

kl If(M1m1; Momo)12 2k0(2J0 + 1) MlMornlrno

(30)

(where the electron wave vectors k are not incorporated in the definition of the scattering amplitude) and that the scattering asymmetry of an electron beam with polarization P is given by

The sums over the atomic quantum numbers M, and M, result from the assumption made throughout this section that the atoms are initially unpolarized and that the final orientation of the atoms is not observed. The additional sum over rn, and m1 in Eq. (30) is due to the fact that nurefers to scattering of initially unpolarized electrons while no spin analysis of the scattered electrons is made. In contrast, in Eq. (31) one has no sum over rn, because the incident electrons are polarized. If, on the other hand, the incident electrons are unpolarized and the scattered electrons are spinanalysed, one finds the polarization

normal to the scattering plane. In contrast to the discussion in Section II.A, the asymmetry Eq. (31) and the polarization Eq. (32) are no longer described by one and the same Sherman function S. That is why occasionally the two quantities Sp and S , are introduced. It is easily seen that the results discussed in the preceding chapter are special cases of the processes considered here. According to Eq. (30) one has for elastic scattering from a target with J , = J , = 0, after dropping the arguments M , = M , = 0, 1

1

all(@ = If(r;T)I

2

1 1 2 + If(- r; r)l

(33)

where use has been made of the relation f ( m , ; m,)

= (- l ) l - r n l - r n o . f ( - m 1 ;

following from the more general f ( ~ , ~M, , ;~ , ) = n,.n,.(-

-mo)

(34a)

~)JI-MI+(~/Z)-~I+JO-MO+(~/~)-~O

x f(-MI - m , ; - M o - m,)

(34b)

Joachim Kessler

120

which implies reflection invariance with regard to the scattering plane when IIo(ll,) is the parity of the initial (final) atomic state (Bartschat and Madison, 1988). With the notation f( - 1/2; 1/2) = g for the spin-flip amplitude and f(1/2; 1/2) = f for the non-spin-flip amplitude one has Eq. (5). Similarly it follows from (31) and (32) that for elastic scattering from a target with J, = J, = 0 one has 1 1-- 1 S A = S, = S = -Im(f(L*27 12 ) f *(2, 2) + f(- f;3)f*<3; - 3)) old

which is seen to be identical with the Sherman function of Eq. (7) when use is made of the preceding relations. Applying the formalism to pure exchange scattering from alkali atoms as discussed in Section 1I.B we find for the elastic cross section

Ten of the 16 terms following from Eq. (30), such as f(1/2 1/2; 1/2 - 1/2), violate the conservation of total spin angular momentum and can therefore be omitted according to the assumption of pure exchange scattering. Recalling the definitions of the exchange and direct amplitudes - g andfby Eqs. (2) and (3), one has f ( '

2

- r. r - r) =f 292 2 9

f ( '

2

- L. - 11 2, 22)

- -97

f(ff;3f>=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

ELECTRON-POLARIZATION PHENOMENA

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.

ELECTRON-POLARIZATION PHENOMENA

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).

126

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

ELECTRON-POLARIZATION PHENOMENA

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)

128

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

ELECTRON-POLARIZATION PHENOMENA

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).

ELECTRON-POLARIZATION PHENOMENA

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

ELECTRON-POLARIZATION PHENOMENA

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

ELECTRON-POLARIZATION PHENOMENA

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

ELECTRON-POLARIZATION PHENOMENA

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).

ELECTRON-POLARIZATION PHENOMENA

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

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...

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-

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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;

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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).

ELECTRON-POLARIZATION PHENOMENA

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

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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

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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.

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ELECTRON-POLARIZATION PHENOMENA

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

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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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Joachim Kessler

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

ELECTRON-POLARIZATION PHENOMENA

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

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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

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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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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

.

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

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I. E. McCarthy and E. Weigold

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*))

ELECTRON-ATOM SCATTERING

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.

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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)

ELECTRON-ATOM SCATTERING

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,)

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I. E. McCarthy and E. Weigold

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

ELECTRON-ATOM SCATTERING

(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.

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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.

ELECTRON-ATOM SCATTERING

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

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I. E. McCarthy and E. Weigold

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

ELECTRON-ATOM SCATTERING

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).

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I. E. McCarthy and E. Weigold

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

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I. E. McCarthy and E. Weigold

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

ELECTRON-ATOM SCATTERING

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).

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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

ELECTRON-ATOM SCATTERING

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.

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I. E. McCarthy and E. Weigold

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).

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ELECTRON-ATOM SCATTERING

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

I. E. McCarthy and E. Weigold

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

ELECTRON-ATOM SCATTERING

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

I. E. McCarthy and E. Weigold

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

ELECTRON-ATOM SCATTERING

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.

I. E. McCarthy and E. Weigold

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 <

*

N

N e *

Y

a

m 3 m7

' N N

0

7

4

0

rD

c

m -N

3

m

-

Q

3 0 n -.

- N

0

N (30

MetastabIe atom yield (arbitrary units) 0

0

P (30

0

P 0

Total cross section

*

b

0

c v

0 0

I. E. McCarthy and E. Weigold

192

Scattering angle (deg )

FIG. 6. Differential cross sections for the n = 1 and n = 2 singlet channels of helium at 30 eV. Filled circles: Trajmar (1973) (29.6 eV). Crosses: Cartwright et al. (1989). Error limits on these data are 15%. Open circles: Brunger et al. (1990). Full curves: 10-state CCO (Brunger et al., 1990). Long dashes: 10-state CC (McCarthy and Ratnavelu, 1989). Short dashes: UDWBA to 10-state CC (McCarthy and Ratnavelu, 1989). Chain curve: 19-state R-matrix (Fon et al. 1988) (29.6 eV).

-

N

10"

2 3s

E V

2 3 ~

z

-

c) .w

C

Lc

2

a 10-Lo

I

I

I

40

80

120

0

10

80

120

Scattering angle (deg ) FIG. 7. Differential cross sections for the n = 2 triplet excitations of helium at 30 eV. Other details are as for Fig. 6. Error limits on the data of Cartwright et al. (1989) are 22% for 23S and 16% for 23P.

ELECTRON-ATOM SCATTERING

193

quantitative descriptions of singlet excitations are obtained, but triplet cross sections are less successful. A 10-stateCCO calculation has been performed at 30 eV for the same cases (Brunger et al, 1990). The CI calculation of the 10 atomic states 1, 2, 3%; 2, 33S;2,3lP; 2, 33P;3lD was performed using all allowed excitations in a basis formed by the 1,2, 3,4s; 2, 3, 4p, and 3d Hartree-Fock orbitals, with higher excitations represented by S, p, and a pseudo-orbitals. Polarization potentials for the continuum were included for the following channel couplings: 11s-iis, 21s, 2 1 ~23s, , 2 3 ~21s-21s; ; 21p-21~;23s-23s; 2 3 ~ - 2 3 ~ . Since this calculation is a prototype for calculations of multi-electron atoms, two calculations of the same channels involving less computational labor are included in Figs. 6 and 7. They are the coupled-channels calculation (CC) without polarization potentials for the same 10 atomic states and the UDWBA to the 10-channel CC calculation. The 1's channel is included in the CC and CCO results. It is compared with experimental data due to Register et al. (1980). Singlet channels (Fig. 6) compare very well with the experiment for CCO, particularly in the case of the data of Brunger et al. The polarization potentials make the difference between the very good CCO description and the CC calculation, which is no better than the R-matrix result. The UDWBA is a surprisingly poor approximation to CC, indicating the invalidity of perturbative calculations in this case. The difference from the excellent approximation for the 2-channel sodium example of Fig. 1 is probably attributable to the large core potential of sodium, which dominates the diagonal matrix elements, rather than to the increased complexity of the helium case. The CCO description of the n = 2 triplet excitations (Fig. 7) is relatively poor. This is tentatively ascribed to the relatively poor exchange matrix elements of the dominant polarization potential, which play a minor role in singlet excitations but are decisive in the triplet case where spin flip is due to exchange. The main conclusion to be drawn from the 30-eV results is that a good description of the continuum is essential for helium, which has a large total ionization cross section: 0.076 & 0.003 mi (Montague et al., 1984) compared with the CCO estimate of 0.09 aa;. This is about twice the integrated cross section for the first dipole excitation. The direct terms of the half-onshell polarization potential, which dominate singlet channels, are good enough but an improved description of exchange is necessary for triplet channels.

194

I. E. McCarthy and E. Weigold

B. SODIUM There are reasons for believing that the 3s and 3p channels of sodium provide an ideal test case for theories of electron scattering from multielectron atoms. In a simple picture sodium is a one-electron atom and excitations are completely dominated by the first dipole excitation 3s-3p, whose integrated cross section is 78 % of the total reaction cross section at 54.4 eV, compared with 14% for ionization. One would expect to a good approximation that electron-sodium scattering is a two-channel problem for a one-electron target. Very detailed CCO calculations by McCarthy et al. (1985) and Mitroy et al. (1987) essentially confirm the simple picture within about 20%, but discrepancies of a factor of about 10 are observed between theory and early experiments, for example the detailed measurements of Buckman and Teubner (1979), at energies of 50 eV or more for the 3s and 3p channels in differential cross sections beyond about 20". Close agreement is obtained for integrated and total cross sections and for the later differential cross section measurements of Miller and Lorentz (1989). The 4-channel CCO calculation of McCarthy et al. (1985) at 54.4eV included 3s, 3p, 3d, and 4s channels in the Hartree-Fock (HF) approximation for the atomic states. This P-space includes all the lowest dipole couplings for 3s and 3p. The continuum polarization potential was included for 3s-3s, 3s-3p, and 3p-3p couplings. It made a differenceof not more than about 20 % to differential cross sections (see Fig. 8.) It was therefore omitted from the calculations with CI atomic states by Mitroy et al. (1987). The CI calculations for the 3s, 3p, 3d, and 4s states used the H F 3s, 3p, 3d, 4s, and 4p configurations as a reference set. The Slater determinants that could be formed by allowing all possible single and double excitations from the reference configurations into a S, p, and d pseudo-orbital space were included in the basis. The pseudo-orbitals were constructed by maximizing their overlap with the 2p and 3s orbitals. 1. Diflerential Cross Sections at 54.4 eV

No perceptible difference was found between differential cross sections calculated with HF and CI atomic states at 54.4 eV. This was also the case when core polarization was included by allowing excitation of a 2p electron. Although the CI basis was not large enough for convergence, the complete absence of effect argues strongly that the HF model is good enough for the

195

ELECTRON-ATOM SCATTERING

lo4

1 o3

lo2 10’ loo lo-’

10-2 t

10-3 -

+ * +4

I

I

I

Scattering angle ( d e g ) FIG.8. Differential cross sections for the 3s and 3p channels of sodium at 54.4 eV. Crosses (with the vertical line indicating the error limits): Miller and Lorentz (1989). Full circles: Buckman and Teubner (1979). Full curves: Qstate CC. Broken curves: four-state CCO.

atomic states. Maximum differences of about 20% were found between twochannel and four-channel CC calculations, confirming the rough validity of the one-electron, two-channel picture. It is important to note that the oneelectron picture involves the core-exchange potential as well as the coredirect potential. Figure 8 shows the comparison of the four-channel CCO and CC calculations with the experimental data of Buckman and Teubner (1979) and Miller and Lorentz (1989). The large discrepancies of the theory with the earlier data are not observed for the more recent data. Not shown in the figure are data of Marinkovii: et al. (1989) and Marinkovii: (1989) which lie between the other two data sets. The discrepancies between the three experimental results are serious and should be resolved. The earliest results, those of Buckman and Teubner, give the lowest cross sections at large angles and are in most serious conflict with the calculations. However, most of‘ the obvious sources of error, such as background contributions, would tend to lead to measured cross sections that are too large at backward angles. There appears at the moment to be no experimental reason for favoring one of the measurements above the other two.

196

I. E. McCarthy and E. Weigold

2. Integrated Cross Sections at 54.4 e V For comparison of calculations with experimental data we take the data of Srivastava and VugkoviC (1980) for the 3s channel and of Enemark and Gallagher (1972) for the 3p channel. The latter were put on an absolute scale by using semiempirical methods based on the known value of the optical oscillator strength to calculate the first Born cross section. The normalization energy was loo0 eV. The comparison is given in Table IV. 3. Quantities Related to Stokes Parameters

Information about the phase relationships of T-matrix elements for dipole excitations is contained in ratios, such as Stokes parameters, of bilinear combinations of T-matrix elements for different projection quantum numbers in the same transition. Examples are the parameters y, the alignment angle of the p orbital with respect to the incident beam direction, and L,, the angular momentum transferred to the target perpendicular to the scattering plane. They are defined in terms of the reduced Stokes parameters P I , P,, P , by y = arg(P,

+ ip,),

(57)

L, = - P 3 . (58) Figure 9 shows measurements of y in a 20eV superelastic scattering experiment on sodium, optically pumped to the 3p, rn = & 1 states (Scholten et al., 1990). Figure 10 shows L, for a 52.3-eV experiment on the same states (McLelland et al., 1989). In each case the experimental data are compared with four channel CC calculation by Mitroy, et al. (1987). The approximate validity of the relevant phase relationships between different calculated T-matrix elements is confirmed. TABLE IV INTEGRATED CROSS SECTIONS FOR THE SODIUM 3s AND 3p CHANNELS AT 54.4 eV

CC4-HF cc4

EXP

5.6 5.5 6.1 f 1.8

24.0 23.3 21.5 f 0.6

Notes: CC4: 4-channel CC calculation with CI target states. CC4-HF: 4-channel CC calculation with HF target states. Experimental data are described in the text.

197

ELECTRON-ATOM SCATTERING

Electron scattering angle ( d e g ) FIG.9. The alignment angle y for the 3p channel of sodium at 22.1 eV (Scholten et al., 1990) compared with the four-channel CC calculation (Mitroy et al. 1987).

1.0

I

I

I

I

3

-0.5 -

0

30

60 90 120 Scattering angle ( d e g 1

150

FIG.10. The parameter L , measured in a 52.3 eV superelastic experiment on sodium McClelland et al., 1989) compared with the four-channel CC calculation of Mitroy et al. (1987).

198

I. E. McCarthy and E. Weigold

VI. Conclusions After 60 years of quantum mechanics, it is interesting to review how well we can calculate scattering problems involving relatively few bodies whose interactions are completely known and whose boundary conditions have recently been considerably clarified. In view of the consistency of experiments on hydrogen, it is necessary to look for a common factor that may explain the still-significant discrepancies between the experiments and several quite distinct types of nonperturbative calculation that agree remarkably consistently among themselves. One such factor is that the correct boundary condition for three charged bodies is not satisfied by the contribution to scattering from ionized channels in any of the methods. For more complex targets, one does not perhaps require the same rigor in experimental-theoretical agreement as for hydrogen. Nevertheless, it is very encouraging to see the remarkable agreement of the CCO method with at least one set of differential cross section data for the singlet states of helium. This reaction is very sensitive to the treatment of the continuum, which can still be considerably improved using proven theoretical techniques. The case of sodium is very interesting because of the indications that the calculation of the 3s and 3p channels is, to a good approximation, an extremely simple problem, namely a two-channel problem with a oneelectron target. The agreement of absolute cross sections and phase relationships for the dipole excitation with certain experiments is excellent, but there are still very large disagreements among experiments. As a suggestion for future work it is important to remember that the theoretical and experimental understanding of absolute differential cross sections is far from complete and that these observables are necessary to establish our understanding in conjunction with more exotic spin- and projection-dependent ratios.

ACKNOWLEDGMENTS

We would like to thank the following people for prepublication results:I. Bray, M. J. Brunger, W. C. Fon, D. H. Madison, B. MarinkoviC, J. J. Mchlland, T. M. Miller, K. Ratnavelu, A. T. Stelbovics, P. J. 0. Teubner, and S. Trajmar. The work was supported by the Australian Research Council.

ELECTRON-ATOM SCATTERING

199

REFERENCES

Bass, A. (1988). PhD. thesis, Manchester. Brauner, M., Briggs. J. S., and Klar, H. (1989). J . Phys. B 22, 2265. Bray, I., Madison, D. H., and McCarthy, I. E. (1989a). Phys. Rev. A 40,2820. Bray, I., Madison, D. H., and McCarthy, I. E. (1989b). “XVI ICPEAC, New York,” Abstracts of contributed papers, p. 224. Bray, I., and McCarthy, I. E. (1989). Private communication. Bray, I., McCarthy, I. E., Mitroy, J., and Ratnavelu, K. (1989~).Phys. Rev. A 39,4998. Brunger, M. J., McCarthy, I. E., Ratnavelu, K., Teubner, P. J. O., Weigold, A. M., Zhou, Y., and Allen, L. J. (1990). J . Phys. B 23, 1325. Buckman, S. J., Hammond, P., Read, F. H., and King, G. C. (1983). J. Phys. B 16,4039. Buckman, S. J., and Teubner, P. J. 0. (1979). J. Phys. B 12, 1741. Burke, P. G., Noble, C. J., and Scott, M.P. (1987). Proc. Roy. Soc. Lond. A 410, 289. Burke, P. G., and Seaton, M. J. (1971). Meth. in Comp. Phys. 10, 1. Byron, F. W., Jr., Joachain, C. J., and Potvliege, R. M. (1982). J . Phys. B 15, 3916. Byron, F. W., Jr., Joachain, C. J., and Potvliege, R. M. (1985). J . Phys. B 18, 1637. Callaway, J., and Oza, D. H. (1985). Phys. Reo. A 32, 2628. Callaway, J., Unnikrishnan, K., and Oza, D. H. (1987). Phys. Reo. A 36, 2576. Cartwright, D. C., Csanak, G., Trajmar, S. and Register, D. F. (1989). Private communication. Enemark, E. A,, and Gallagher, A. (1972). Phys. Rev. A 7, 1573. Feshbach, H. (1962). Ann. Phys. (N.Y.) 19,287. Fite, W. L., and Brackmann, R. T. (1958). Phys. Rev. 112, 1141. Fon, W. C., Berrington, K. A., Burke, P. G., and Kingston, A. E. (1990). To be published. Fon, W. C., Berrington, K. A., and Kingston, A. E. (1988). J. Phys. B 21, 2961. Frost, L., and Weigold, E. (1980). Phys. Reo. Lett. 45, 247. Gell-Mann, M., and Goldberger, M. L. (1953). Phys. Rev. 91, 398. Glauber, R. J. (1959). In (W. F. Brittin, ed) pp. 315-414, Interscience, New York. de Heer, F. J., McDowell, M. R.C., and Wagenaar, R. W. (1977). J. Phys. B 10, 1945. Hood, S. T., Weigold, E., and Dixon, A. J. (1979). J . Phys. B 12, 631. Kauppila, W. E., Ott, W. R., and Fite, W. L. (1970). Phys. Rev. A 1, 1099. Kingston, A. E., and Walter, H. R.J. (1980). J . Phys. B 13, 4633. Lloyd, C. R., Teubner, P. J. O., Weigold, E., and Lewis, B. R. (1974). Phys. Rev. A 10, 175. Long, R. L., Cox, D. M., and Smith, S. J. (1968). J . Res. NBS A 72, 521. Lower, J., McCarthy, I. E., and Weigold, E. (1987). J. Phys. B 20,4571. Madison, D. H. (1979). J. Phys. B 12, 3399. Madison, D. H. (1989). “XVI ICPEAC, New York” invited papers. Madison, D. H. Winters, K. H., and Downing, S. L. (1989). J . Phys. B 22, 1651. MarinkoviE, B., (1989). Private communication. MarinkoviC, B., Pejkv, V., FilipoviC, D., Cadei, I., and VuSkoviC, L. (1989). In. “XVI ICPEAC, New York,” Abstracts of contributed papers, p. 133. McCarthy, I. E., Mitroy, J., and Stelbovics, A. T. (1985). J. Phys. B 18, 2509. McCarthy, I. E., Ratnavelu, K., and Weigold, A. M. (1988). J . Phys. B 21, 3999. McCarthy, I. E., Saha, B. C., and Stelbovics, A. T. (1981). Aust. J. Phys. 34, 135. McCarthy, I. E., and Stelbovics, A. T. (1980). Phys. Rev. A 22, 502. McCarthy, I. E., and Stelbovics, A. T. (1983a). Phys. Rev. A 28, 1322. McCarthy, I. E., and Stelbovics, A. T. (1983b). Phys. Reo. A 28, 2693. McCarthy, I. E., and Zhang, X. (1989). Aust. J. Phys. (to be published). McClelland, J. J., Kelley, M. H., and Celotta, R. J. (1989). Phys. Rev. A 40,2321.

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Miller, T. M., and Lorentz, S. R. (1989). “XVI ICPEAC, New York.” Abstracts of contributed papers, p. 198. Mitroy, J., McCarthy, I. E., and Stelbovics, A. T. (1987). J. Phys. B 20,4827. Montague, R. G., Harrison, M. F. A., and Smith, A. C. H. (1984). J . Phys. B 17,3295. Ratnavelu, K. (1989). Ph.D. thesis, Flinders University, Bedford Park, Australia. Register, D. F., Trajmar, S. and Srivastava, S. K. (1980). Phys. Rev. A 21, 1134. Scholten, R. E., Shen, G. F., and Teubner, P. J. 0. (1!390). J. Mod. Opt. To be published. Scott, M. P., Scholz, T. T., Walters, H. R. J., and Burke, P. G. (1989). J. Phys. E 22, 3055. Slim, A. H. and Stelbovics, A. T. (1987). J. Phys. B 20, L211. Srivastava, S. K., and VuSkoviC, L. (1980). J. Phys. B 13, 2633. Stelbovics, A. T. (1990). Phys. Rev. A 41, 2536. Stelbovics, A. T., and Bransden, B. H. (1989). J . Phys. B 22, L451. Trajmar, S. (1973). Phys. Rev. A 81, 191. Truhlar, D. G., Trajmar, S., Williams, W., Ormonde, S., and Torres, B. (1973). Phys. Rev. A 8, 2475. van Wingerden, B., Weigold, E., de Heer, F. J., and Nygaard, K. J. (1977). J. Phys, E 10, 1345. van Wyngaarden, W. L. and Walters, H. R. J. (1986). J. Phys. B 19, 929. Wallace, S. J. (1973). Ann. Phys (New York) 78, 190. Weigold, E., Frost, L., and Nygaard, K. J. (1980) Phys. Rev. A 21, 1950. Wigner, E. P., and Eisenbud, L. (1947). Phys. Rev. 72, 29. Williams, J. F. (1975). J. Phys. B 8, 2191. Williams, J. F. (1981). J. Phys. B 14, 1197. Williams, J. F., and Willis, B. A. (1975). J. Phys. B 8, 1641.

II II

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

ELECTRON-A TOM IONIZATION 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 . . . . . . . . . . . . . . . . . . . 20 1 11. Theory of Ionization . . . . . . . . . . . . . . . . 203 A. T-Matrix Approximations . . . . . . . . . . . . . . . . . 203 B. Wave-Function Approximations . . . . . . . . . . . . . . . 206 C. The Ion-Target Overlap. . . . . . . . . . . . . . . . . . 208 D. Differential Cross Sections . . . . . . . . . . . . . . . . . 210 111. Total-Ionization Cross Sections: Asymmetries with Spin-Polarized 21 1 Atoms and Electrons . . . . . . . . . . . . . . . . . . . . IV. Double Differential Cross Sections . . . . . . . . . . . . . . . 213 V. Triple Differential Cross Sections . . . . . . . . . . . . . . . . 214 A. Absolute-Scale Determination in TDCS Measurements . . . . . . . 215 B. Threshold Behavior . . . . . . . . . . . . . . . . . . . 218 C. Autoionization . . . . . . . . . . . . . . . . . . . . . 22 1 D. Coplanar Asymmetric Kinematics . . . . . . . . . . . . . . 225 E. Electron Momentum Spectroscopy of Atoms . . . . . . . . . . 228 239 VI. Conclusions . . . . . . . . . . . . . . . . . . . . . . . 24 1 Acknowledgments . . . . . . . . . . . . . . . . . . . . . 24 1 References . . . . . . . . . . . . . . . . . . . . . . . .

I. Introduction The ionization of atoms by electron impact is one of the most interesting processes in the field of atomic collisions. As well as being a process of great importance in diverse areas such as plasma physics, astrophysics, upper atmospheric physics, and radiation chemistry and biology, it has led to the development of a new spectroscopy, namely electron momentum spectroscopy, which has yielded a much deeper understanding of the structure of atoms and molecules. Electron impact ionization also provides challenging and enriching theoretical problems. The break-up channel exhibits all the difficulties of many-body scattering theory coupled with the special problem of the infinite range of the Coulomb interaction. Total electron impact ionization cross sections have been measured for many target atoms and molecules. Most of these measurements have been made by collecting the ions produced in the collisions in a mass spectrometer 20 1 Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-003827-7

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or by means of condenser plates (e.g., de Heer and Inokuti, 1985). Although these measurements are of great importance for many applications, they give relatively little information on the dynamics of the ionization process itself or on the structure of the target. In recent years the emphasis has been on obtaining differential cross sections, the so-called single, double, and triple differential cross sections, in order to determine the dynamics that most sensitively influence these cross section structures. In this chapter we discuss what can be learned from some examples of these cross sections, particularly the differential cross sections with the emphasis being on the triple differential cross section. The double differential cross section is obtained by measuring the intensity distribution of one electron in the final state as a function of energy and angle. This is equivalent to integrating the full differential cross section over the solid angle of the other electron, as well as its energy when more than one final ion state is excited. Although this leads to a consequent loss of information about the ionization process, such cross sections can still give valuable information on the ionization mechansism (e.g., Opal et al., 1972) and on the structure of the target (e.g., Bonham and Wellenstein, 1977). The most detailed information on the structure of the target and on the dynamics of the collision process is obtained by means of the triple differential or (e, 2e) cross section, in which the kinematics are completely determined. The energy E , of the incident electron is known, and the energies E, and E B and directions of motion (a,, $ A ) and (O,, 4,) relative to the incident direction are determined for the two emitted electrons. The electron separation energy to a final state f is defined by cf = E , - E, where the total energy E = E, + E,. Measurements of triple differential cross sections can be conveniently divided into two broad regimes, depending on the magnitude of the momentum transfer K = k, - k,, where by convention the “scattered” electron A is assumed to be the one of higher energy, i.e., E, 2 E,. The first regime is that of high-momentum transfer, say K 2 6 a.u. where the ionization mechanism can be approximated accurately by a simple model such as the impulse approximation (Hood et al., 1973, 1974). These high-momentum-transfer experiments are dominated by close encounters between the incident and struck electron and they yield structure information on the target and ion in the form of electron-momentum probability distributions to well-defined different ion states (McCarthy and Weigold, 1976, 1988). These experiments are sometimes referred to as binary (e, 2e) experiments or electron-momentum spectroscopy (EMS). The most important class of experiments in this category are those using symmetric kinematics, i.e., OA = OB and E, = E,, at

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high-incident and outgoing energies (McCarthy and Weigold, 1976, 1988). Very high energy asymmetric (e, 2e) collisions, where the momentum transfer is large and the kinetic energies of the free electrons are large as well, also belong to this category. If the scattered and ejected electron energies and angles are chosen so that the collision belongs to the Bethe ridge of the Bethe surface (Inokuti, 1971), then simple impulsive models can be applied to describe the reaction mechanisms and the data can be used to give structure information (Daoud et al., 1985; Lahmam-Bennani et al., 1986; Avaldi et al., 1987a). The other broad category of (e, 2e) collisions consists of those having lowmomentum transfer. Such collisions dominate the total ionization cross section at any given incident energy. At low energies the ion cannot be regarded as a spectator and the measurements are used to study the ionization mechanisms by using simple targets such as helium (Ehrhardt et al., 1969, 1972) or atomic hydrogen (Weigold et al., 1979). This class of events provides an extremely sensitive test of the details of reaction models, the simple first-order models generally being quite inadequate. In the limit as K + 0 at high energies, i.e., 8, x 0, k, k,, these experiments give structure information through measurements of electron-separation-energy spectra (Brion, 1975; Hamnett et al., 1976). These experiments are often called dipole (e, 2e) experiments since they simulate photo-electron spectroscopy, with the absorbed photon having energy “hv” = Eo - E A .

-

11. Theory of Ionization For electron-atom scattering it is possible to develop a formalism from first principles that is incomplete only in its treatment of the effect of ionization. This is not true for ionization. Theoretical approximations are developed, and their range of validity assessed, by an iterative process involving close collaboration between theory and experiment. An important theoretical development has been the establishment of the boundary condition for three charged particles by Brauner et al. (1989). A. T-MATRIX APPROXIMATIONS A useful starting point for the intuitive development of approximation methods for ionization of multielectron atoms is the T-matrix T/o(~A, kt19 ko) = (k,kBfI TlOko),

(1)

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where 0 denotes the ground state of the target, f is the observed final state of the ion, and ko, k,, k, are the momenta of the external electrons in obvious notation, k, being assigned to the faster of the two. The exact T operator represents the effect of the interaction of all electrons and the nucleus, but it is obvious that the major interaction, at least for large momentum transfer K, where

K = ko

- k,,

(2)

is a binary collision between the incident electron and one target electron. The binary encounter approximation represents T by the appropriate operator t for a two-electron collision in the target:

We now assume that the electrons are distinguishable and antisymmetrize the ion-two-electron final state, with the ion represented by a linear combination of independent-particle configurations. The ionization amplitude is now an antisymmetric sum of terms in which the incident electron encounters one each of the electrons in the final state. If in a particular term this is not one of the externally observed electrons, then the amplitude contains factors that are the overlap of the single-particle wave function of an outgoing electron with a bound single-particle wave function for the same electron in the initial target state. This is essentially the momentum-space orbital of the initial-state electron, which is extremely small for outgoing energies greater than about 100 eV. For such energies we may drop these exchange-collision terms. In the surviving terms one of the externally observed electrons is the projectile and the t-operator commutes with the ion-wave function since it operates only in the space of the projectile and the electron that is removed from the target. The plane-wave impulse approximation (PWIA) drops the exchangecollision terms and represents the two-electron collision operator t by the t-matrix for the collision of two free electrons:

Here we have explicitly represented the antisymmetrization operator for the observed electrons for total electron spin S. The space-exchange operator is Pr. The PWIA amplitude may be rearranged as the product of two factors:

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205

where

and t, denotes the antisymmetric t-operator for two electrons with spin S at a relative collision energy that corresponds to k , i.e., it is half on-shell. p is the ion recoil momentum. To include the distortion of the incident and two outgoing electron wave functions from plane waves by the potential due to the remainder of the system, we replace the plane waves J k ) by distorted waves IX(*)(k)), which are elastic scattering functions with appropriate boundary conditions for electrons in the appropriate potential. For simplicity we use the static potential of the target for IX(+)(ko)) and of the ion for Ix(-)(kA)) and IX'-'(kB)). The distorted-wave impulse approximation (DWIA) is

q % k A kB, ko) = (k' I t, Ik Xx'- '(k,)x' -)(kB)f IOx' + YkO)). 3

(7)

Note that in this approximation the target and ion structure appear only in the form of the ion-target overlap (flo), which is a one-electron function. Its momentum representation is

(PflO)

= (PI+>.

(8)

For a hydrogen-atom target I) is the 1s orbital. We may regard the reaction as a probe for the ion-target overlap function I++. The electronic structure of the target and ion may be represented by linear combinations of independent-particle configurations. This is the configuration-interaction (CI) representation. The ionization amplitude is then a linear combination of amplitudes for knockout of an electron in an orbital I++i. For example, the orbital amplitude in the DWIA is

T!s'(kA kB, ko) = (k I t , I kxx(-)(kA)x(-)(kB)I+d(+)(kO)). 9

(9)

In many cases one orbital is the overwhelming contributor to the linear combination. In general we can understand the reaction in the energy range for the binary encounter approximation as a linear combination of threebody amplitudes, the three bodies being two electrons and an inert core. For hydrogen this is exact. The problem has been divided into a structure problem to determine the coefficients in the combination of the 7';')and a

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three-body reaction problem to determine the TiS).We have seen the intuitive arguments leading to the DWIA for TIS).

B. WAVE-FUNCTION APPROXIMATIONS An alternative way to consider a three-body amplitude is to shift the emphasis from the three-body T-matrix to the three-body wave function Y&-)(kA,k,) for the final state. For the orbital we have (Gell-Mann and Goldberger, 1953) the orbital amplitude

where V is the projectile-core potential and u is the two-electron potential. Brauner Briggs, and Klar (1989) (referred to as BBK) showed that the asymptotic form of the wave function Y(-) for three charged particles is the product of the asymptotic forms (phase factors) for each of the three Coulomb two-body subsystems. BBK give references to earlier work on the boundary condition. They approximated Yc-)by the product of two electron-core distorted waves and the relative Coulomb factor 4(kA - k,) for the two-electron subsystem. This approximation has the correct boundary condition. They calculated the coplanar asymmetric ionization of hydrogen. The BBK approximation to the orbital amplitude Tis) is

zIS'(kA, k,, ko) = (X'-)(kA)X(-)(kB)4(kA - k,)lu

+ VI$,kO)?

(11)

where (rl$(q)) = r ( l + iv)exp(av/2),F,[-iv; v = l/q.

1; -i(qr

+ qer)]

(12)

To make the BBK approximation for the ionization amplitudes in the polarization potential for scattering (see Chapter 3) is a very promising direction for including the three-body boundary condition in the scattering calculation, which is the only serious omission at this stage. However, it presents serious numerical difficulties and has not been implemented as of 1989. It is possible to calculate the polarization potential if the relative Coulomb factor is omitted. It is therefore important to try this approximation for ionization experiments. This tests the ionization amplitudes on the three-body energy shell. In the case of scattering, it has been shown (Bray et a1.,1989) that low-order

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approximations converge rapidly in the distorted-wave representation in which a local, central potential U is added to the channel Hamiltonian and subtracted from the interaction potential. The entrance channel wave function in (10) becomes a distorted wave lf+)(ko)), which is an eigenstate of the potential U. A convenient definition of U is the ground-state average potential

u = (Olu

+ VlO),

(13)

which cancels as much as possible of the noncentral potential u + K The orbital amplitude in the distorted-wave representation is

Tjs’(k,, k,, ko) = <‘J’ii’(k,,

k,)Iv

+

-

ul$iX(+)(ko))-

(14)

The distorted-waue Born approximation (DWBA) is obtained from (14) by setting the two-electron Coulomb factor &(kA- k,) equal to unity in (11). The orbital is orthogonal to the final-state distorted waves for a true three-body system or approximately orthogonal for each orbital amplitude in a CI expansion, so the DWBA for the orbital amplitude is

TiS’(k,4

3

kB,

k0) = (x(-)(kA)f-)(kB) I I *ix(+)(kO))9

(15)

where us is the product of the antisymmetrization operator and the twoelectron potential. (See Eq. (4).) It is interesting to note that the two-electron potential u in (15) is the firstorder term in the Born expansion of the two-electron t-matrix t. At first sight it would seem to be more accurate to replace u by t, thereby including the two-electron interaction to full convergence. This cannot be done since the corresponding orbital amplitude involves fully off-shell matrix elements of t, which are not uniformly convergent to the half shell (McCarthy and Roberts, 1987). In any case a calculation with a nonlocal interaction operator is numerically too difficult. One could regard (9) as a factorization approximation to the form of (15) with u replaced by t. The numerical calculation for the local operator u is feasible and the factorization approximation can be tested in this case. It turns out to be very good on the bound-electron Bethe ridge (Madison et al. 1989). This is a set of kinematic conditions in which the momentum transfer K is equal to the momentum k, of the slower observed electron. The Bethe ridge condition is approximately satisfied by noncoplanar symmetric kinematics. Essentially it means that the atomic core behaves like a spectator in the collision.

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C. THEION-TARGET OVERLAP The overlap of the configuration-interaction expansions of the ion and target-wave functions has been discussed in detail by McCarthy and Weigold (1988). Here we give enough detail to define the ideas and terminology used in the present chapter. It is necessary first to choose an orthonormal basis set of independentparticle configurations for each wave function. We choose the Hartree-Fock orbitals of the target. A configuration is an antisymmetric product of singleparticle wave functions representing a particular distribution of particles and holes over a truncated set of orbitals. Each configuration has the total angular momentum and parity of the corresponding state. Experience has shown that the target Hartree-Fock orbitals are an excellent basis in the sense that (e, 2e) momentum distributions are often described by a single configuration rather than a linear combination. To answer the question of whether ion orbitals may give a better basis, it is sufficient to consider the hydrogen atom! Each state is represented by a linear combination of configurations in the suitably truncated basis. The binary-encounter approximation selects overlaps of configurations such that the ion configuration results from the removal of one electron from a target configuration, leaving the others undisturbed. Many observed ion excitations are quite well described by the overlap of a particular ion configuration with the lowest (Hartree-Fock) target configuration. If we represent the target only by its Hartree-Fock configuration, we have the target Hartree-Fock approximation (THFA). Experience with diagonalizations has shown that a better approximation is to consider such an ion state as resulting from the removal of an electron from a linear combination of target configurations, the coefficients of the relevant configurations in the ion eigenstate being almost the same as the coefficients of the corresponding configurations in the target eigenstate. This is the weak-coupling approximation. The corresponding weak-coupling expansion represents the ion eigenstate f as a linear combination of configurations consisting of a hole in orbital $ j coupled to a target eigenstate a.

The target-ion overlap in the binary encounter approximation is now

ELECTRON-ATOM IONIZATION

209

All the orbitals $ j in (17) must have the total angular momentum and parity off: They therefore differ only in their radial shape. The linear combination (17) could be represented in terms of a single normalized function @i, which we call the characteristic orbital. In many cases of interest, the experimentally determined characteristic orbital is a target Hartree-Fock orbital. In our formalism, we use the characteristic orbital, defined by

(sflo)

= 6&ll@i).

(18)

The experimental cross section is given by squaring the ionization amplitude. We are therefore interested in the spectroscopic factor S @ :

Using the orthonormality of the target eigenstates a we find the spectroscopic sum rule,

c s@

= 1,

I

and the definition of the orbital energy ci,

The analysis of an ionization experiment involves identifying a manifold of ion states for which S{{) is not zero. This manifold is characterized by the orbital *i. The validity of the binary-encounter and characteristic-orbital description of the reaction is verified by checking that there is a manifold i whose momentum-distribution shapes are all given by a single function + i . This means that S{$) is independent of momentum and given by the ratio of the experimental cross section for f to the summed cross sections for the manifold i. This is often the case. We then have an experimental definition of an orbital, whose energy is given by (21) and normalization by (20). If the definition is independent of experimental conditions, such as incident energy, we can define the orbital and its corresponding spectroscopic factors purely experimentally. It is an important test of a theory of the ionization reaction that it results in the same normalization (20) for several ion-state manifolds with different total angular momentum and parity. Thus far we have discussed ion states that could be said to observe finalstate correlations. They are ones in which the characteristic orbital is occupied in the target Hartree-Fock configuration. Another class of ion

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states depends on initial-state correlations. These are ones whose characteristic orbital is not occupied in the target Hartree-Fock configuration so that (qf 10) depends on orbitals that are occupied only in higher configurations. An example is the 2p state of the helium ion. For such states (qflO) is very sensitive to the coefficients of the higher target configurations and it is unlikely that the momentum distributions of two of them are so similar that a characteristic orbital description makes sense. In other words their spectroscopic factors, defined by the ratios of cross sections, are momentumdependent. Nevertheless, it is likely that the: overlap is dominated by a single ion configuration for which the momentum distribution resembles that for the basis orbital from which the electron is removed. This enables them to be identified. Analysis of such ion states requires detailed CI wave functions for both target and ion. D. DIFFERENTIAL CROSS SECTIONS The differential cross section for experimentally defined energies and momenta is given by an average and sum over inital- and final-state degeneracies, respectively.

where J o is the total angular momentum of the target ground state. For the DWIA we have for a singlet target state

Here N i is the occupation number of the orbital ll/i and the electron-electron collision factor is 1

1 1 1 + kAI4 Iko - kgI4 Iko - kAI2Iko - kBlz

The DWBA for a singlet target state is

ELECTRON-ATOM IONIZATION

21 1

For some purposes it is useful to have the double differential cross section (DDCS), defined by

At present we have labelled the faster outgoing electron by A. By convention we have the DDCS for the primary electron. It we choose A to be the slower electron, we have the DDCS for the secondary electron.

111. Total-Ionization Cross Sections: Asymmetries with Spin-Polarized Atoms and Electrons The status of total-ionization cross sections with unpolarized atoms and electrons has been reviewed by de Heer and Inokuti (1985). The major development since that time has been the removal of the spin degeneracy in such measurements by the use of polarized electrons and target atoms. Thus the spin asymmetry of the total-ionization cross section has been measured for hydrogen (Gay et al., 1982; Fletcher et al., 1985), the light alkali atoms (Baum et al., 1985), and more recently for cesium (Baum et al., 1989a,) and metastable He Q3Se)(Baum et al., 1989b). For light atoms the observed spin effects result from electron exchange in the collision, and they are quite pronounced from threshold to about ten times threshold. They provide a sensitive test for ionization calculations near threshold, where generally quite drastic assumptions have to be made about the nature of the interaction. The spin asymmetry A, determined in their measurements is obtained by observing the ion count rates, N T and N , , ,for respectively antiparallel and parallel spin configurations of the electron and atom beams of polarizations P , and Pa, respectively.

,

The asymmetry is related to the triplet (a,) and singlet (a,) ionization cross sections by A, = (a, - aJ/(a, + 3aJ, where the total ionization cross section is given by )(a, + 30,). The asymmetry is therefore a measure of the singlet and triplet contributions, A, = 1 corresponding to pure singlet ionization and A, = - 3 to pure triplet. Figure 1 shows the results obtained for the asymmetry by Fletcher et al. (1985) for hydrogen and by Baum et al. (1985) for the light alkalis, Li, Na, and

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I. E. McCarthy and E. Weigold

K, plotted as a function of the incident energy E, in units of the corresponding ground-state ionization energies 6 , . For €3, Li, and Na the asymmetries as a function of Eo/co are very similar, having a broad maximum of A, 0.45 at threshold and slowly decreasing to zero with increasing energy. The threshold value corresponds to roughly 80% singlet and only 20% triplet ionization. At threshold the only triplet contributions can come from the odd partial waves (3P0,3F0. . .), whereas the singlet contributions come from the even partial waves (lSe, IDe, . . .) (Stauffer, 1982; Green and Rau, 1983; Rau, 1984); therefore the singlet contribution should dominate. For potassium the measured asymmetries are much lower than for the lighter atoms. The reason for this behavior is not clear. For metastable He (z3Se)the variation of asymmetry with energy (Baum et al., 1989b) is very similar to that observed for Li (Baum et al., 1985), although it is slightly lower in magnitude. The differences can, however, be explained by systematic error in the determination of the atomic polarization Pa. The calculation of the spin asymmetry is not a trivial problem. Bray et al. (1990) obtain the asymmetry shown by the solid curve for atomic hydrogen from the total reaction cross section in an elastic e-H scattering calculation with the optical potential for ionization calculated in the DWBA. This is the weak-coupling approximation. It is equivalent to the DWBA (15) with X'+'(k,) calculated in a detailed nonlocal, spin-dependent potential. Bartschat (1989) used a simple DWBA to calculate the spin asymmetry A, as a function of the incident energy for Li and He Q3Se).The results for Li are shown in Fig. 1 as the dashed curve. For Eo/co > 4 (i.e., E > 20 eV) the agreement between theory and experiment is quite good, but near threshold the discrepancy is very large, much larger than the discrepancy for the total cross sections. In contrast to the weak-coupling calculation that of Bartschat significantly underestimates the asymmetry at low energies. The Wannier theory (Wannier, 1953), as extended to include spin (Green and Rau, 1983), predicts a spin-dependence that does not vary with energy near threshold. Briefly it predicts that the most probable two-electron escape occurs with the two electrons emerging on opposite sides of the ion core, and that the total energy E = E , + E , in the final state is uniformly distributed between the emitted electrons. Temkin (1982) proposed that close to threshold, the important configuration is that for which the two electrons have very different energies, with the slower electron seeing the charge of the nearby ion core, while the faster electron experiences the attractive longrange field due to the ion and slow electron. This Coulomb-dipole theory predicts an undulation of the spin asymmetry near threshold. This was not

-

213

ELECTRON-ATOM IONIZATION

I I

+

.

I

1

I 2

Li

3 I\

I

I

I 5

I

I

I

I

I

I I

10

IIIIIL 15

20

Incident energy , E, / E~ FIG.1. The ionization asymmetry A, plotted as a function of the incident energy E , in units of ground-state ionization energy to. The data for H are from Fletcher et al. (1985), and for the alkalis Li, Na, and K from Baum et al. (1985). The solid curve is the weak-coupling calculation for H by Bray et al. (1990). The dashed curve is a DWBA calculation for Li by Bartschat (1989).

observed in the high-precision measurements of Kelley et al. (1983) on sodium up to 2.0 eV above threshold. Their results, as well as the less precise results of Fletcher et al. (1985) for H and Baum et al. (1985, 1989a, b) for the alkalis and metastable He, are fully consistent with the Wannier theory (Wannier, 1953; Green and Rau, 1983; Rau, 1984).

IV. Double Differential Cross Sections Double differential cross sections (DDCS) for primary and secondary electrons are an important test of the understanding of ionization, particularly

214

I. E. McCarthy and E. Weigold

of the contribution of ionization to the imaginary part of the optical potential, which requires an integration of products of two on-shell ionization amplitudes over the whole range of the kinematic variables (McCarthy and Stelbovics, 1980). Figures 2 and 3 show the primary- and secondary-electron DDCS for helium at a range of incident energies and final-state energy sharings. The largest cross sections are very well represented by the DWBA, suggesting the validity of the DWBA for the integral over the whole kinematic range. Agreement of the DWBA (McCarthy and Zhang, 1989) with the smaller cross sections of Muller-Fiedler et al. (1986) and Opal et al. (1972) is at best qualitative, but much better agreement is achieved at 500 eV with the data of Avaldi et al. (1987b).

V. Triple Differential Cross Sections Ionizing collisions between electrons and neutral targets in which the kinematics of all the electrons are completely determined have provided in

FIG. 2. Primary-electron DDCS for helium at incident energies shown. Experimental data are due to Miiller-Fiedler et al. (1986) (open circles) and Avaldi et al. (1987b) (full circles). Full curves are DWBA (McCarthy and Zhang 1989). Cases illustrated are E, = 100eV: E , = 73.4 eV(a), 71.4 eV(b), 55.4 eV(c); E , = 300 eV: E, = 235.4 eV (cross section multiplied by 100) (a), 271.4 eV(b); E , = 500 eV: E, = 471.4 eV(a), 435.4 eV(b).

215

ELECTRON-ATOM IONIZATION

r-7

10’

E, = 500 eV

-

; 10’

10’

10’

100

L

“E z”

f

.

a 10’ n b 0

I

0

LO

80

120

Scattering angle (deg)

FIG.3. Secondary-electronDDCS for helium at the incident energies shown. Experimental data and curves are as for Fig. 2 with Opal et al. (1972) (crosses). The value of E , for an experimental point is indicated by a vertical line joining it to be corresponding curve. Cases illustrated are E , = 100 eV: E , = 4 eV(a), 10 eV(b), 20 eV(c); Eo = 300 eV: E, = 4 eV(a), 20 eV(b), 40 eV(c), 100 eV(d;) E , = 500 eV: E, = 4 eV(a), 20 eV(b), 40 eV(c), 102 eV(d), 205 eV(e).

recent years an increased understanding of the ionization mechanism and of the momentum distribution of target electrons. Significant advances have been made in the study of threshold effects, of autoionizing-resonances and Auger emission, of ionization at low to intermediate energies, and of structure applications. We now consider some of the important advances in these categories.

A. ABSOLUTE-SCALE DETERMINATION IN TDCS MEASUREMENTS

Generally when comparing the predictions of theoretical models with triple differential cross section (TDCS) measurements, the shapes of the triple differential cross sections are compared, since either the data are not absolute, or the absolute scale may have large errors. Such comparisons have provided extremely stringent tests of theoretical models. In some cases such as those concerned mainly with structure determination, e.g., electron momentum spectroscopy (Section V, E), relative normalizations are adequate and absolute data, although desirable, are not necessary. However, to thoroughly test electron-impact ionization theories it is necessary to have some absolute measurements. The accurate determination of the absolute scale in (e, 2e) experiments is a difficult problem. The direct method, which relies on no theoretical models to

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I. E. McCarthy and E. Weigold

estimate various parameters, is to use the relationship.

where I , and I , are the coincident and incident electron count rates respectively, n the target density, ARA and ARB the acceptance solid angles of the analyzers, AE, the effective coincidence energy resolution, t' the effective interaction length, and cA and cB the overall transmission and detection efficiencies of the analyzers. Direct measurement of some of these quantities is difficult in crossed-beam techniques, largely due to the uncertainty in determining the target gas density and the effective interaction length l. The latter depends on the viewing angles of both outgoing electron analyzers and their overlap, as well as the target gas density distribution and the incident electron beam profile. Beaty et af. (1977) and Stefani et al. (1978) were the first to report absolute (e, 2e) measurements, but the errors in the absolute scale were very large. Van Wingerden et af. (1979, 1981) used a static gas cell to determine coplanar symmetric (e, 2e) cross sections at 8, = 8, = 45" on He and H, to an accuracy of 20 % over a range of energies. However, the use of static gas cells severely limits the range of angles over which the two analyzers can be moved, and (e, 2e) measurements therefore generally employ crossed beam techniques. Various ways of normalizing the cross sections based on physical arguments have therefore been introduced. The first method is based on Bethe sum rule arguments and was introduced by Lahmam-Bennani et al. (1983b). In their method the (e, 2e) cross section is first integrated over all possible directions, including out-of-plane contributions. It is assumed that the plane-wave Born approximation is valid and that there is cylindrical symmetry about the momentum-transfer direction K for both the binary peak (in the direction K) and the recoil peak (aligned along - K). This assumption limits its applicability to high-energy asymmetric collisions. This integrated cross section is then equated to the double differential cross section. Since double differential cross sections do not discriminate against ion-excited states belonging to the same orbital under study, the further assumption is made that these contributions are small. Contributions from other orbitals for atoms other than He must also be subtracted. This can be done either by measuring the (e, 2e) cross section for these transitions separately (e.g. Daoud et al., 1985), or by estimating their contribution from other data (e.g., Compton scattering). The double differential cross sections are measured in the same apparatus used for the TDCS

ELECTRON-ATOM IONIZATION

217

measurements and are normalized using the Bethe sum rule (LahmamBennani et al., 1980). This then normalizes the triple differential cross section. The reliability of this method depends on the validity of the first Born approximation. The second normalization technique is based on extrapolations to the optical limit. One can define a triple differential generalized oscillator strength (TDGOS) in terms of the triple differential (e, 2e) cross section

f"'(K, k,; E) =

(h,)

E K 2 d S ) ( k , ,k,, E).

In the limit of zero momentum transfer (K + 0), the TDGOS can be simple related to the optical dipole oscillator strength (Lassettre et al., 1969). Further, for the two directions k , 11 K and k, )I - K, the TDGOS tends to the same value, which is proportional to the dipole oscillator strength. Jung et al. (1985) used this methed to normalize their data. At each 8, (i.e., K), 8, was adjusted to be in the K or - K direction, respectively. The measured cross section in these directions are converted to TDGOS by using (29) and plotted as a function of K, which is varied by varying OA. In practice the optical limit K = 0 cannot be reached, since the minimum value of K is Kmin= k , - k , when 8, is zero. Jung et al. used a polynomial fit to extrapolate their measurements to K = 0 where the recoil and binary cross sections parallel and antiparallel to K respectively should be equal. At this point the TDGOS is proportional to the dopile optical oscillator strength, which is obtained from photoionization transition probabilities. The third method, introduced by Stefani and coworkers (e.g., Avaldi et al., 1987a), also uses the optical limit but is basically a combination of the preceding two methods. The TDCS are first integrated, as in the method of Lahmam-Bennani et al., but the resulting DDCS measured as a function of K are transformed to double differential generalized oscillator strengths (DDGOS) and plotted as a function of K . Extrapolation to K = 0 gives the optical oscillator strength, which if known is then used to normalize the data. It should be emphasized that the preceding three normalization techniques do not give truly absolute cross sections, since they depend to a varying extent on the validity of the Born approximation. There is, however, no reason as of 1989 to believe that they are unreliable at high energies. When high-quality data are used, all three methods lead to absolute values of the TDCS in very good agreement with each other with an overall accuracy of about 8 % (Lahmam-Bennani et al., 1987). At low energies the only reliable procedure is to measure absolute cross sections using (28).

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I. E. McCarthy and E. Weigold

B. THRESHOLD BEHAVIOR The threshold behavior of ionization cross sections has been the subject of many investigations. Wigner (1948) showed that threshold laws depend largely on phase space considerations, arising from a feature of the escape process, namely the lack of kinetic energy for complete escape. They therefore do not generally require a detailed knowledge of the reaction dynamics and hence can only provide a limited amount of information on the dynamics of the collision. Wannier (1953) extended Wigner’s theory to electron impact ionization, where there are three particles in the final state. He divided configuration space into three regions. Out to a radius of a few atomic units we have the inner reaction zone, which requires a full quantum mechanical description. Next there is the “Coulomb zone” where the potential energy dominates over the kinetic energy and the behavior is classical. Beyond a critical radius, which varies as E-’ where E is the total energy of the emitted electrons, we get to the outer zone (also called the asymptotic or free zone) where the kinetic energy dominates and the two electrons are essentially free. Wannier argued in analogy with Wigner’s analysis that the details of the dynamics in the reaction zone are not important, and instead made the assumption that the distribution in phase space of the two electrons at the boundary with the Coulomb zone is essentially uniform (i.e. quasiergodic). Then, using purely classical arguments, he was able to obtain for the total ionization cross section the threshold law IS cc En, (30) where n = 1.127 for a singly charged ion. Furthermore the differential cross section has a maximum at e A B = n, where e A B is the angle between the two emitted electrons, and the angular spread varies as (Vinkalns and Gailitis, 1967). There have been many investigations into threshold behavior in electron impact ionization, and the earlier work is summarized in an excellent review by Read (1989, which gives details of both the theoretical and experimental work. Total ionization cross section measurements are in excellent agreement with the Wannier threshold law. Differential measurements at threshold are very difficult because of the low energy of the emitted electrons and the high energy resolution required in the incident electron beam. Nevertheless, because they provide a more stringent test of theory, several measurements have been made. CvejanoviE and Read (1974) used a coincidence time of flight technique to show that the energy-partitioning probability was indeed uniform to within approximately 5 % at E .- 0.37 eV and 0.60 eV for electron-

ELECTRON-ATOM IONIZATION

-

219

helium ionization. They were also able to verify that the angular correlation has a maximum at 6,, = with a width A 6 A B Ell4, and by measuring the yield of low-energy electrons they obtained accurate agreement with the Wannier theoretical value of 1.127 for the exponent n. Several (e, 2e) coincidence experiments have been performed near threshold in the coplanar geometry. Fournier-Lagarde et al. (1984) performed such measurements for ionization of helium down to E = 1 eV, and Selles et al. (1987) made measurements in the energy range 0.5 eV I E I 2 eV. The triple differential cross sections measured by Selles et al. are in reasonable agreement with the semiclassical calculation of Crothers (1986) at some scattering angles, but severe discrepancies were found at many scattering angles. Jones et al. (1989) measured (e, 2e) angular correlations for helium in the perpendicular plane (6, = 8, = 90") for values of 4AB= 4, - 4, in the range 120-240" for E = 1, 2, 4, and 6 eV. They were able to fit their experimental data with a parameterized cross section based on the Wannier model using s, p, and d waves. Their most important result was that the width of the angular correlation was given by A4,B = 70" f 3" at E = 1 eV, in good agreement with the predicted value of 69" by Crothers (1986) and Feagin (1984) and 67" by Altick (1985), but significantly smaller than the value of 85" predicted by Rau (1976). Ehrhardt et al. (1989) made a very interesting comparison of (e, 2e) coplanar angular correlations for helium and hydrogen at the same total energy E = 4 eV and with E A = E , = 2 eV and 6 , = 8, f 150" with 6, varying from about 30 to 140". Their results are shown in Fig. 4. There is a striking difference between the angular correlations for the two atoms. Those for helium have distinct maxima, whereas those for hydrogen have minima at the corresponding angles of 6,. These results show the importance of the inner interaction zone, since the scattering complexes of hydrogen and helium only differ in this reaction zone. They are therefore at variance with the classical Wannier model, which ignores the details in this zone. These measurements show that short-range interactions do give contributions to the angular correlations of the two electrons, and these can only be included within the framework of a theory that solves the scattering equation in the whole space, including the inner interaction zone. Ehrhardt et al. also find, in agreement with the conclusions of Jones et al., that it is necessary to include all partial waves up to d-waves to accurately describe the angular correlations, and that with the addition of f-waves there is some further improvement in the fit to the results for helium.

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I. E. McCarthy and E. Weigold

1

h

In

c

.-

C 3

d

L

b

v

C

.-0

He V

I

I

-E,=E,=ZeV 0. 0

0

0

0

0

€

0

I0 0

60 120 Scattering a n g l e , OA ( d e g )

180

FIG.4. The coplanar (e, 2e) cross section for H and He at E = 4 eV and E , = E , = 2 eV plotted as a function of 0 , with fixed difference angle OAB = 150". Filled circles: 0 , = 0 , - 150", crosses: 0, = OA + 150". The data are from Ehrhardt et al. (1989).

ELECTRON-ATOM IONIZATION

22 1

C . AUTOIONIZATION

The measurement of (e, 2e) cross sections in the region of an autoionizing resonance gives information on the resonance and direct cross sections as well as their interference as a function of the momenta k, of the ejected (or emitted) electron for known values of the momentum transfer K. In this instance the electron whose energy is different to that resulting from the decay of the resonance is referred to as the “scattered” electron, even though its energy may actually be lower than that of the ejected electron. TDCS measurements in the autoionizing region can therefore provide very sensitive information on details of the excitation process of the resonance as well as on interference in the decay channel with direct ionization. Thus in the absence of a direct contribution and interference between excitation amplitudes to the various magnetic substates of the resonance, the angular correlations between the scattered and emitted electrons should have the form [P,(cos @)I2in the scattering plane for an electron emitted with angular momentum L, where 8’ is the angle of emission relative to the symmetry axis K (Balashov et ai., 1972). Deviations from such a distribution are due to interference between excitation amplitudes of the different magnetic substates of the resonance as well as to interference with the direct channel. The simplest example of autoionization is provided by helium, and this was the target used in the first coincidence measurements of electron-impact autoionization cross sections, which were carried out by Weigold et ai. (1975) for the (2p2)’D and (2s 2p)’P resonances. More recently Pochat, et ai. (1982) and Moorhead and Crow (1985) reported some angular correlation measurements for helium. An extensive series of high-resolution measurements has been reported by Lower and Weigold (1989) at incident energies of 100,200, and 400 eV for scattering angles of 3,13, and 16”over a wide range of ejectedelectron angles in the region encompassing the (2s2)’S, (2p2)’D, and (2s 2p)’P resonances. For separable resonances (i.e., if resonances of the same LS do not overlap, and if there is no appreciable interaction between the two final state electrons), Tweed (1976) showed that the triple differential cross section can quite generally be written in the parameterized form originally due to Shore (19671,

222

I. E. McCarthy and E. Weigold

where

and I?, and E , are, respectively, the energies of the rth autoionizing resonance and the energy of the ion plus continuum-emitted electron (relative to the energy of the residual ion) with total angular momentum and spin quantum numbers denoted by p = {r; L, M ,S}. The full width at half maximum of the resonance is given by T,. The momentum-dependent parameters a, and b, have dimensions of cross sections and are related to the resonance contributions and the form of the resonance profiles, whereas f is simply the cross section for direct ionization. The parameters a, and b, depend upon matrix elements that contain both the direct and resonance amplitudes, the resonance amplitude being essentially given by the probability of transfer of momentum to the atomic target multiplied by the probability for subsequent decay into the continuum. The sum is over terms arising from the different products of incident and scattered-electron partial waves and includes the appropriate phase factors and angular dependent spherical harmonics (Tweed, 1976). Figure 5 shows some typical coincidence ejected-electron energy spectra obtained by Lower and Weigold (1989) at an incident energy of 100 eV and scattering angle 8, = 13". The change in the energy spectra in only a small change in ejected-electron angle (A6, = 18")is remarkable. In each case there are a series of resonance profiles superimposed on a direct contribution f: The data are fitted using Shore's parameterization (Eq. 31) and the known widths of the resonances, the fitted resonance profiles (the dashed lines in Fig. 5 ) being convoluted by the instrumental response function, the final fitted function being indicated by the solid line. These fits are used to extract the direct cross section and the cross section parameters a, and b,, where a, is a measure of the resonance asymmetry due to interference with the background and b, is a measure of the contribution of the resonance to the cross section. It should be noted that a, and b, can be either positive or negative depending on the nature of the interference. FIG.5. The 100 eV TDCS for electron impact ionization of helium at a scattered-electron angle of 8" = 13" (4" = 0) plotted as a function of the ejected-electron energy E , for two ejected-electron angles 8, = -24" and -42" (4, = 180"). The fits to the separate resonances using Shore's parameterization (eq. (31)) are shown by the dashed curves, the solid curve giving the overall fit (from Lower and Weigold, 1989).

500

I

I

-

He ( e , 2 e ) He’

ffl 4-

c 2

0

U

33

a,

U

C

3.4

a,

-u .-

35

36

1 (2p’)’D

R

u

.-C

0

u

600

LOO

200

(2s21’s (2s2p

1

I

0 I 33

I I I 3.4 35 36 Ejected electron energy (eV)

I. E. McCarthy and E. Weigold

224

The results at 100eV for the three resonances (2sz)'S, (2pZ)'D, and (2s 2p)'P are shown in Fig. 6. The ejected-electron angles for 4 = 0" and 4 = 180" are indicated by positive and negative fls angles, respectively. The data have been arbitrarily normalized by fixing the magnitude of the maximum in the direct cross section under the ' S resonance to unity. The direct cross section f shows the normal binary and recoil peak behavior observed in low energy asymmetric (e, 2e) collisions. (See section V.D.) The resonance parameters have even more rapid variations in magnitude as a function of the ejected-electron angles than does the direct cross section. Although the peak structures in a, and b, are correlated with the direction of the momentum transfer OK, as is that for the direct cross section f; the correlation is not simple. The influence of the direct amplitude in the resonance parameters is dominant. This can be seen immediately from the (2s')'S data. In the absence of an interfering direct component, the excitation and decay of this state would involve only 1' = 0 transitions, and therefore the angular correlation should be isotropic (a, = 0, b, = constant). A considerable amount of theoretical effort is needed to explain the data. First-order theories, such as the plane wave Born approximation, are inadequate since

2

1

0

s-o.Ll

I

,

+120 +60

T, 7 , 0

-60

,

1

-1

-120

'120

+60

0

-60

-120

Ejected electron angle, Be ( d e g )

FIG.6. The TDCS parameters for 100 eV electron-impact ionization of He at a scattered electron angle of 13" (4" = 0) plotted as a function of the ejected-electron angle $ B (-ve8, for $B = 180")for the (2s2)'S, (2p2)'D, and (2s2p)'P autoionizing resonances. The data have been normalized by fixing the maximum of the direct cross section f under the ' S resonance equal to unity.

ELECTRON-ATOM IONIZATION

225

they would give the K direction as a symmetry axis, which is clearly in violation of the measurements. D. COPLANAR ASYMMETRIC KINEMATICS In this kinematic arrangement ko, k,, and k, are coplanar, E , is much greater than E,, 6 , is fixed (usually at an angle of 20" or less), and 6s is varied. Differential cross sections are roughly proportional to K -4, so they are larger in more asymmetric situations. Thus this arrangement observes the kinematic region that contributes most to the effect of ionization on electron scattering. This effect is described in chapter 3. There is a special case of coplanar asymmetric kinematics that has a different motivation. This is the kinematics of the bound-electron Bethe ridge, in which the magnitude of the momentum transfer is equal to the magnitude of k,. K = kB.

(34)

Here the factorization approximation for the DWBA is valid (Madison et al., 1989), which suggests that the DWIA is valid. This enables the reaction to be used as a simple probe for the ion-target overlap, particularly as the electroncollision factor in the DWIA is almost constant. The relevance of the boundelectron Bethe ridge was suggested and established by Lahmam-Bennani et al. (1988). Coplanar asymmetric kinematics has an advantage for the spectroscopic application in its large cross section. Examples of its use are given by Lahmam-Bennani et al. (1983a, b, 1986), Daoud et al. (1985), and Avaldi et al. (1987a). The spectroscopic application is similar to that of noncoplanar symmetric kinematics (which is also near the bound-electron Bethe ridge) and is covered in section V.E. The use of coplanar asymmetric kinematics to simulate photoelectron spectroscopy has been summarized by Brion (1975). Here we concentrate on the general understanding of the ionization reaction at intermediate energy in the most fundamental case, hydrogen, with a view to the use of theoretical descriptions of ionization in other reactions such as scattering. Three types of description have been applied to scattering. They are the coupled pseudostate method (CPS) (Curran and Walters, 1987), the unitarized eikonal-Born series (Byron et al., 1985), and the DWBA (McCarthy and Zhang, 1990). These methods suffer from the disadvantage that they assume two-body Coulomb boundary conditions in the final state

I. E. McCarthy and E. Weigold

226

and ignore the asymptotic electron-electron interaction. CPS represents the initial-state three-body wave function by the result of a coupled-channels calculation with the ionization space represented by discrete pseudostates. The three-body ansatz (Eq. (11)) of Brauner et al. (1989) represents the finalstate wave function as a product of three Coulomb waves. It has the correct boundary condition. It has not yet been applied to scattering. Figure 7 gives an example of three calculations compared with the 250 eV experimental data of Ehrhardt et al. (1985, 1986), Klar et al. (1987), and Lohmann et al. (1984). All the experimental differential cross sections were obtained by relative measurements. Those of Ehrhardt et al. were put on an absolute scale as described in section V.A. The data of Lohmann et al. were normalized by comparison with the second Born calculations of Byron et al. (1983). Brauner et al. (1989) have shown that the first Born approximation to their three-body calculation is not valid at 250 eV and have suggested that their calculation should be used to put the experimental data on an absolute scale. The general conclusions to be drawn from Fig. 7 and further examples given by McCarthy and Zhang (1989b), as well as from Brauner et al. (1989),

I

I

-1 80

-120

I

- 60

I

0 8, (deg) FIG.7.

I

I

60

120

1 180

227

ELECTRON-ATOM IONIZATION

I -180

I

-120

1

- 60

0

I

I

60

120

1 180

FIG.7 (continued). The differential cross section for the ionization of hydrogen. Experimental data are due to Ehrhardt et al. (1985, 1986) and Klar et al. (1987) (closed circles) and to Lohmann et al. (1984) (open squares).The curves are DWBA full (McCarthy and Zhang, 1990); CPS (long dashes) (Curran and Walters, 1987); and three-body approximation (short dashes) (Brauner et al., 1989).

228

I. E. McCarthy and E. Weigold

are that the three-body calculation gives an excellent description of the shapes of the 8, distributions, but in many cases underestimates the magnitudes in comparison with the present normalization. The other two methods are comparable. They give a good description of the large cross sections in the binary peak, with DWBA being slightly larger in general.

E. ELECTRON MOMENTUM SPECTROSCOPY OF ATOMS We now turn to the discussion of high-energy (e, 2e) reactions for which the momentum transfer K is large compared to the magnitude p of the ionrecoil momentum. Under these conditions one has impulse-type collisions, or binary encounter collisions, in which the ion is largely a spectator and from which the orbital electron density of the target atom can be obtained (McCarthy and Weigold, 1976,1988). The kinematical arrangement normally used for these experiments is the symmetric one (E, = E,, 8, = OE), since this maximizes the momentum transfer for any given E. The noncoplanar symmetric geometry (Weigold et al., 1973), in which 8 = 8, = 8, is kept fixed at approximately 45" and the out-of-plane azimuthal angle 4 = 4A- 4E- n is varied to vary the ion-recoil momentum p, has the advantage that for constant E the momentum transfer K also remains constant for a transition to a given final state as 4 (i.e., p) is varied. As a result, the noncoplanar symmetric cross section as a function of p is directly proportional to the structure factor in the factorized impulse approximation. The simplest atom is of course hydrogen, and since its structure is known exactly, it provides a test case for EMS. The overlap function for hydrogen is simply the square of the momentum space wave function, I$l,(P)lZ = 8n-2(1

+ P2)-4,

(35)

and in the PWIA for noncoplanar symmetric kinematics with K s p , the differential cross sections should be simply proportional to this. In their noncoplanar symmetric (e, 2e) measurements at 400, 800, and 1200 eV, Lohmann and Weigold (198 1) obtained excellent agreement between their measured momentum distribution and the predicted one (35) at all three energies, thus confirming the validity of the PWIA at high enough energies and orbital mapping in the case where we know the orbital exactly.

I . Noble Gases There have been many experiments on the noble gases He, Ne, Ar, Kr, and Xe. In Ar, Kr, and Xe there are significant correlation effects in the valence s

229

ELECTRON-ATOM IONIZATION

orbital that lead to large splittings in the strengths and the excitation of many final ion states belonging to the valence s manifolds. The spectroscopic strengths for these valence s transitions are independent of the incident energy and the ion recoil, i.e., struck electron momentum, in the range 0-1.5 a.u. (See McCarthy and Weigold 1976, 1988; Cook et al. 1986). The case of He provides a sensitive test of correlations in the initial state, particularly for the transitions to n # 1 final ion states, for which the target Hartree-Fock approximation predicts that the overlap function should either vanish or be small. The momentum profiles for these states are in good agreement with those given by full overlap calculations using accurate correlated He wave functions in both shape and magnitude, and they are in serious disagreement with those given by the THFA (Cook et al., 1984). We shall examine the case of argon in some detail, since it contains all the complications of initial-state and final-state correlations in a case where many of the final states below the double-ionization continuum can be experimentally resolved. In addition, there are two occupied valence states, 3p and 3s, in the target Hartree-Fock configuration of the ground state, so that it is possible to test the consistency of the spectroscopic factor determination using two methods of normalization: the spectroscopic sum rule and comparison of the structure factor with the manifold structure factor for a different manifold The 3p manifold is dominated by the transition to the ground state of the ion. McCarthy et al. (1989) measured the spectroscopic factor for this transition to be 0.95 0.02. Its momentum profile shape at 1500 eV is described within experimental error by the DWIA and up to p 1.7 a.u. by the PWIA using the THFA, i.e., the H F 3p wave function. (See Fig. 8.) In the high-momentum region, above p 1.7 a.u., the PWIA significantly underestimates the cross section. McCarthy et al. normalized their measured cross sections at $J = 10" to the 3p-' DWIA cross section for the ground-state transition obtained by multiplying the calculated 3p- ' manifold cross section by the ion ground-state spectroscopic factor of 0.95. In order for the PWIA cross section to agree with the DWIA value at this point, it has to be multiplied by 0.83, the PWIA cross section being larger than the DWIA one since it neglects refraction of the electron waves. The 3s manifold, on the other hand has a number of states with significant spectroscopic strength, the separation energy spectra showing a lot of structure extending above the double-ionization threshold at t = 43.6 eV (Weigold et al., 1973; McCarthy and Weigold 1976, 1988; McCarthy et al., 1989). The shapes of the different momentum profiles for the strongly excited

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-

I. E. McCarthy and E. Weigold

230

\

I I

I 1

I

I

I

I

1

I

I

I

', I

I

states at large separation energy, and also that of the continuum above the double-ionization threshold, are independent of energy and have the characteristic shape of 3s momentum distributions up to a little over 1 a.u. of momentum. Figure 8 shows that the DWIA-THFA describes the shape and normalization of the whole 'Se manifold structure factor for all momenta (or 4). The cross section for the transition at 29.3 eV with dominant configuration 3s 3p6('Se) is also well described by the DWIA-THFA using a spectroscopic factor of 0.55. The momentum profiles for other 'Se final states as well as the satellite contributions in the range 42-55 eV are shown in Fig. 9. They are accurately described by the calculated 3s profile, except that for the 36.52 eV state with dominant configurations 3s2 3p4('S) 4s, which shows a sharp peak below 0.3 a.u. of momentum. The extra strength at low momentum can be explained by 4s ionization with a spectroscopic factor of about 0.0015 due to a small 3s' 3p4('S) 4s' component in the argon ground state.

m

I

~

-

I

( b ) E = 36.52 eV

352 3p'('S)4s

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500eV

0

I

I

I

(c)

~

I

2

1

1

1

E =38.6eV

1

1

( d ) f =41.21eV 35' 3p4('D14d

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+ ~

2.0

I

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I

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2.6

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'

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3s' 3p4('D)nd

'La I

1.2

2

1

0

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-

0.8

-

x 0.08

0.8

-

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-

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T, 2

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Momentum ( a u . )

FIG.9. Momentum profiles for the 2Se manifold of argon and various satellites compared with the calculated 3s momentum profiles multiplied by their respective spectroscopic factors (McCarthy et al., 1989). The dominant configurations for the corresponding ion states and the respective separation energies are as shown. The dominant 'Se transition to the 3s 3p6 ion state at t = 29.3 eV is shown in Fig. 8. For the 3s23p44stransition a 4s HF momentum profile scaled by the factor 0.0015 is also shown.

232

I. E. McCarthy and E. Weigold

The spectroscopicfactors for the 'S' manifold are found to be independent of energy in the range 500-1500 eV. They are also independent of momentum in the range 0-2 a.u. except that for the 3s' 3p4 4s state discussed previously, where initial state correlations are important, Table I compares the spectroscopicfactors for the 2Se manifold obtained in a number of many-body calculations with those measured by McCarthy et al. (1989), which are consistent with earlier, less accurate EMS measurements. (See, for example, McCarthy and Weigold, 1976, 1988.) The final state configuration interaction (FSCI) calculations include only final-state correlation, whereas the overlap and the Green's function (GF) calculations include initial-state correlations and relaxation as well as final-state correlations. None of the calculations adequately describe the data. Since initial-state correlations play a very small role in the 'S' manifold, the energy of the 3s orbital may be obtained using the weighted mean (21) of final states belonging to the 'S" manifold. The result is cSs = 35.2 f 0.2 eV, which compares very well with the H F value of 34.76eV (McCarthy and Weigold, 1988). Although the dominant features in the argon-separation energy spectra above the 2P" 3s' 3p5 ion ground state at 15.76 eV and 'S" 3s 3p6 ion excited state at 29.3 eV are due to transitions to other states belonging to the 'S" manifold, McCarthy et al. (1989) were able to identify transitions belonging to both the 'D' and 'Po manifolds. The 'D' transitions can only occur if there are d-wave correlations in the Ar ground state. The most prominent 'D" transition is to the 3s' 3p4 ('D) 4s state at 34.20 eV and its momentum profile, shown in Fig 10a, is very interesting. It has a narrow peak at p 0.25 a.u., which is to be expected from the diffuse (in coordinate space) spectroscopic 3d and 4d orbitals. The Hartree-Fock 3d and 4d momentum distributions are shown in Fig. 10a multiplied by the normalization factors indicated, which must be less than or equal to the square of the corresponding CI expansion coefficient for the (nd)2 d-wave configurations in the argon ground state. The low momentum region is best described by the 4d wave function, If the intensity of the low-momentum peak is attributed to, say, a ground state 3s' 3p4('D) 4s 4d configuration, it would require only a 0.4 % admixture of this configuration to account for its intensity. Dyall (1980) carried out a CI calculation of the argon ground state including double excitations of the form (no' with n = 3 and 4 for 1 = 2. He found that the major interaction in the argon ground state involved the d-wave configurations. The total d-wave strength given by the Dyall calculation is a factor of 3 higher than that observed for the 34.20 eV transition by

-

TABLE I SPECTROSCOPIC FACTORS FOR THE *Se MANIFOLD COMPARED WITH CALCULATED VALUES

Dominant ion state configuration

3s3p6 3p44s 3p43d 3p44d 3p45d 3046d i r + ++ e

Experiment McCarthy et al., (1989) EMS 4eV) 29.24 36.50 38.58 41.21

42'65 43.40

s,

}

Mitroy et al. (1984) Overlap FSCI

s,

s,

0.55(1) 0.02(1)* 0.16(1) 0.08(1)

0.649 0.13 0.161 0.083

0.600

0.08(1)

0.081

0.095

0.12(1)

0.013

0.08

0.006 0.142 0.075

Hibbert and Hansen (1987) FSCI

s,

0.618 0.006 0.112 0.057 0.02 1 0.009 0.18

Amusia and Kheifets (1985) GF

Sf

0.55

0.20 0.1 1 0.04

Von Niessen (1987) GF

s,

0.605 0.008 0.135 0.005 0.025 0.177

Note: The experimental values are independent of momentum in the range 0-2 a.u. except for the 3p44s transition, indicated by an asterisk, where S, decreases from 0.03 k 0.01 at p < 0.2 to 0.01 k 0.01 at p > 0.5 a.u. The error in the last figure is given in parenthesis.

h,

w w

I. E. McCarthy and E. Weigold

234

( b ) ~=35.63eV

6 6

.-

3s2 3p4 ('P)4p *PO

1500eV 1OOOeV 500eV

x

L

-

z2

v)

0

1

11

c o

.0

Y -1 0

2

1

0

1

2

FIG.10. The momentum profiles to final states at 34.2, 35.63, 37.15, and 39.57-eV separation energy compared with several calculated distributions (McCarthy et al., 1989). The factors following the 3p calculated distributions are 3p spectroscopic factors. The CI(I = 2) distribution is the total *Dgmanifold distribution obtained by Mitroy et al. (1984). The CI(l= 1) distributions are 0.67 of the total zP"satellite distribution obtained by Mitroy et al. The 3d, 4d, and 4p distributions shown are the spectroscopic Ar 3d, 4d, and 4p momentum distributions with their respective spectroscopic strengths.

-

McCarthy et al. (1989). Dyall's calculation does not predict any detectable cross section above p 0.5 a.u., in disagreement with the measurements. Mitroy et al. (1984) carried out a full calculation of the overlap function between the correlated Ar ground state and correlated final states. The 'De manifold momentum distribution calculated by Mitroy et al. is shown in Fig. 10a multiplied by a factor of 2. Clearly this calculation underestimates the intensity of the small momentum peak and overestimates the contribution at

ELECTRON-ATOM IONIZATION

235

high momentum. Mitroy et al. found that the dominant contribution of the correlation energy came from “correlating” pseudo natural orbitals (denoted by a bar) rather than the spectroscopic (Hartree-Fock) orbitals. These pseudo orbitals are localized in the same region of space as the spectroscopic 3s and 3p orbitals and therefore give rise to momentum contributions at much higher momenta than those given by the diffuse spectroscopic 3d and 4d orbitals. The measurements of McCarthy et al. show that the spectroscopic 4d orbital is more important than given by the calculation of Mitroy et al., who overestimate the large momentum components due to the 3d orbital. The transition at L = 39.5 eV also has a very similar momentum distribution to that for the 3s’ 3p4(’D) 4s 2Destate (Fig. 10d). This could be due to excitation of the 3s2 ~ P ~ (4d~ ‘D‘ P ) ion state at 39.64 eV. Some of the cross section is also probably due to the excitation of the 3s’ 3p4(’S) 4p ’Po ion state at 39.57 eV. Although the low momentum region cannot be explained by a 3p ionization process, there could be a 4p contribution from the 4s’ 3p4(’S) 4pz component in the Ar ground state. McCarthy et al. (1989) find two definite ’Potransitions (Figs. 10b and c) in addition to the dominant ground-state transition. Both transitions have the 3p momentum distribution. Shown in Fig. 10b are both the 3p DWIA-THFA momentum distribution multiplied by the spectroscopic factor of 0.01, and 0.67 of the total ’Po satellite intensity (marked CI(l = 1) calculated by Mitroy et al. (1984) in their full overlap calculation. There is a small difference in shape between the HF and CI calculations due to the effect of initial-state correlations. Both describe the data adequately. The spectroscopic factors are 0.01 and 0.03 for the 35.63 and 37.15 eV transitions, respectively. The observed spectroscopic factors in the ’Pomanifold are in very good agreement with a number of many-body calculations. We have discussed the case of argon in some detail since it demonstrates the richness of information available from EMS measurements. This includes momentum profiles for orbitals occupied in the H F ground state as well as those for orbitals that are not occupied in the HF ground state but that play an important role in correlation effects. It allows these initial-state correlations to be identified and measured quantitatively. This is also true for finalstate correlations, which are especially important in the inner valence region. Quantitative measurements can be made of the spectroscopic factors for transitions belonging to the different symmetry manifolds. These spectroscopic factors are energy-independent, as they must be if they are purely a function of the structure of the target and ion. When initial-state correlations are negligible, the spectroscopic factors are also momentum-independent.

I. E. McCarthy and E. Weigold

236

They can then be used to derive an unambiguous orbital energy, which is in good agreement with the H F orbital energy. The importance of relativistic effects in the outer valence orbitals of atoms was first demonstrated by Cook et al. (1984b). Their 1200 eV noncoplanar symmetric EMS measurements showed that the 5p,/, and 5p1/2 one-electron momentum distributions in xenon differed significantly from each other and that they could not be described by nonrelativistic wave functions, but that they were in excellent agreement with those given by relativistic Dirac-Fock wave functions. Figure 11 shows the 5p,,, :5p1/2 branching ratios obtained at 1OOOeV by Cook et al. (1986) compared with the Hartree-Fock and the Dirac-Fock predictions. The DWIA-DF calculation accurately reproduces the data from d, = 0" ( p 0.1 a.u.) to = 30" (p 2.2 a x ) . The 5p,,, wave function has significantly more low-momentum components than the 5p,,, wave function. The two HF wave functions are of course indistinguishable, so that the H F branching ratio is just the ratio of statistical weights and independent of p . An interesting aspect of the EMS results on Ar, Kr, and Xe (McCarthy and Weigold, 1988) is that they show the importance of collective quadrupole coupling in the core due to correlation effects. This is not only the case for the

-

-

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b

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n L n I

I 0

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1 40

q5 ( d e g ) FIG. 11. The ratio of the 5p,,, to 5p,,, cross sections for the 1000-eV noncoplanar symmetric (e. 2e) experiment on xenon plotted as a function of the out-of-planeazmuthal angle (Cook et al. 1986). The full curve is the DWIA using Dirac-Fock orbitals. The chain line is the non-relativistic Hartree-Fock ratio.

ELECTRON-ATOM IONIZATION

231

final-state correlations in the 'Se manifold where ns2 np4('D) core coupling is totally dominant, but also in the collective many-body effects in the ground state of argon, where again 3s2 3p4('D) nl n'l' configurations are dominant. It therefore seems essential that the many-body calculation of atomic structure should take care in treating the quadrupole core excitations accurately. 2. Other Atoms For atomic lead Frost et al. (1986) showed not only that it is necessary to use relativistic structure calculations, but that correlations must also be included in both the initial and final states. In lead the valence electrons are, in terms of the Hartree-Fock determinant, two in the 6p1/2 orbital and two in the 6s,/, orbital. For such a heavy atom, relativistic effects are important in the structure, so it is reasonable to base the discussion on j j coupling with relativistic multiconfigurational Dirac-Fock optimal level (MCDF-OL) orbitals. They found that the 6p-' transition is split with transitions to both the 6p,,, and 6~31,ion states. Whereas LS coupling would give a branching ratio of 0.5, pure j j coupling would imply a value of zero, i.e., no 6~31,excitation. They calculated the momentum-dependent branching ratio for a twoconfiguration (a I6p:/2) + b I6pgl2)) target ground-state wavefunction using both MCDF-OL and MCDF-EAL (extended-average-level) orbitals. When correlations were included in the final states as well as the initial state, complete agreement was obtained with the measurements by the optimal level calculation. The momentum distribution for the 6p manifold was very well described by the relativistic orbitals and much less well by the nonrelativistic HartreeFock orbital. Ion states at 14.6, 18.4 and 20.3 eV were identified as belonging to the 6s manifold. The relativistic orbital again gives a much better description of the measured 6s momentum profiles than the nonrelativistic orbital. Excellent agreement was obtained between the calculated and measured spectroscopic factors when correlations were included in both the initial and final states. Group I1 atoms can exhibit strong initial state CI in the ground state, and it is possible to observe transitions corresponding to orbitals that are not occupied in the Hartree-Fock ground state. Grisogono et al. (1988) and Pascual et al. (1988) measured the strength and momentum profiles of the (n + l)p2 CI component in the Zn and Mg ground states, respectively. In these cases the measured momentum distributions were governed by the momentum density of the unoccupied orbital corresponding to the transition.

238

I. E. McCarthy and E. Weigold

The magnitude of the cross section leading to the ion state (whose dominant configuration generally consists of a hole in the corresponding atomic “excited”-state configuration) is a direct measure of the strength of this contribution to the many-body ground-state wavefunction. Their results were in good agreement with their CI calculations. These results again showed that EMS provides a sensitive quantitative measure of many-body efforts in atomic wavefunctions, as well as determining the adequacy of the HF picture. 3. Excited Target States and Oriented Targets

By using lasers to excite atoms to well-defined states it is possible to measure electronic momentum distributions for targets in excited states. Further, by using polarized laser light it is possible to excite specific magnetic substates. This offers the possibility of measuring the momentum distributions from atoms in aligned and oriented states. Weigold et al. (1989) obtained the first results for an excited oriented target state, namely Na(3p). All previous measurements had been for spherically averaged ground-state targets. A ring dye laser provided 0’ laser incident at right angles on a well-collimated sodium beam, the light being in the plane of the EMS spectrometer. Approximately 12% of the sodium atoms were in the excited 3’P3/2 ( F = 3, m F = 3) state, the rest being in the ground state. Orientation of the excited state was achieved, since 100% of the excited atoms were in the II = 1, m, = 1) state, corresponding to 50% in the 3p, and 3p, states. This is shown schematically in Fig. 12. Coincidence separation energy spectra of the 32S1/2ground state, and 32P3/2excited state, at separation energies of 5.1 and 3.0eV, respectively, were simultaneously collected as a function of the out-of-plane azimuthal angle 4, using a multiparameter EMS spectrometer. Figure 13 shows the measured momentum distribution for the excited 3p(m, = 1) state compared with the calculated distribution using a Hartree-Fock wavefunction and allowing for the finite experimental momentum resolution. Agreement is excellent. The experiment measures the momentum distribution of the 3py orbital along the py axis (see Fig. 12) rather than the usual spherically averaged distribution. The study of laser-excited target opens exciting new avenues. It will be possible to study oriented and aligned targets and to examine how the remainder of the electron cloud adjusts to one of the electrons being in an excited state. At high laser intensities it should be possible to do measure-

239

ELECTRON-ATOM IONIZATION

px eo

-

Laser beam

ti Atomic

beam

React ion plane FIG.12. The oriented 3p distribution of excited Na in the measurements of Weigold et al. (1989). The population of atoms in the 3p, state is zero. The measured momentum distribution is averaged on a plane perpendicular to the scattering plane. The direction of the polarized laser beam, the atomic Na beam, the incident electron (q,)and , emitted electron directions (e”, ee) are also shown.

ments on “dressed” targets (See, for example, Joachain et al., 1988, and Chen, 1989.)

VI. Conclusions Electron-impact ionization of atoms shows a rich diversity of phenomena that permit both the detailed investigation of fundamental collision dynamics as well as the quantitative investigation of the structure of the target atoms and the resulting ions. The threshold behavior is in general well described by the Wannier theory and its extensions. However, recent data on H and He by Ehrhardt et al. (1989) shows that even near threshold it is important to accurately treat the

I. E. McCarthy and E. Weigold

240

-w 0.10

I

TI

I

1

1

1 I

.--w c

3

h

2

E '800 eV EA=E=, 400 eV

0.08

L

.-w

4 L

d

Y

0

.-w

O.O€

u

a,

w w

ul

2

0.04

0

.-0 + c

a,

L

a 0.02

Y-

.-

n

I

0

I

0.2

0.4

0.6

Momentum (a.u.) FIG.13. The momentum-densitydistribution of the 3p(m, = 1) orbital of sodium compared with the corresponding Hartree-Fock density (Weigold et al., 1989). The calculated distribution incorporates the known finite experimental momentum resolution.

ELECTRON-ATOM IONIZATION

24 1

collision region. The variation of the spin asymmetry of the total ionization cross section for polarized electrons on polarized atoms shows similar behavior for H, Li and Na when plotted as a function of the energy in units of the threshold energy. The weak-coupling approximation agrees well with the hydrogen data at all energies, showing that the calculations obtain correct ratios for singlet-to-triplet ionization. More experimental and theoretical work is required in the near-threshold region. At intermediate and high energies the cross section is dominated by lowmomentum transfer collisions with asymmetric kinematics in the outgoing electrons. In this kinematical region the DWBA and Eikonal-Born series calculations have proved to be the most satisfactory. The establishment of the boundary condition for the three charged bodies by Brauner, et al. (1989) has been a major advance. On the experimental side more accurate absolute data are required, particularly at low energies. For high-energy and momentum-transfer collisions, the DWIA gives a complete description of the shapes and relative magnitudes of the cross sections for the transitions belonging to different symmetry manifolds, such as the 3p and 3s valence manifolds of argon. This is particularly so for symmetric kinematics. The PWIA also gives an accurate description of the shapes (momentum profiles) out to well above 1 a.u. of ion-recoil momentum. This is the region of electron-momentum spectroscopy, which has been successfully applied to quantitatively probe the details of the many-body target and ion-wave functions.

ACKNOWLEDGMENTS

We are grateful to the Australian Research Council for supporting our work.

REFERENCES

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Bartschat, K. (1989). Phys. Rep. 180, 1. Baum, G., Granitza, B., Hesse, S., Leuer, B., and Raith, W. (1989a). In “XVIth Int. Conf. Phys. Electronic Atomic Coll New York 1989, Abstr. Contr. Papers” p. 154. Baum, G., Moede, M., Raith, W., and Schroder, W. (1985). J. Phys. B 8, 531. Baum, G., Raith, W., Steidl, H., and Taborski, J. (1989b). I n “XVIth Int. Conf. Phys. Electronic Atomic Coll.”, New York 1989 Abstr. Contr. Papers, p. 159. Beaty, E. C., Hesselbacher, K. H., Hong, S. P., and Moore, J. H. (1977). J. Phys. B 10, 611. Bonham, R. A., and Wellenstein, H. F. (1977). In “Campton Scattering’’ (B. Williams, ed.), pp. 234-272, McGraw-Hill, New York. Brauner, M., Briggs, J. S., and Klar, H. (1989). J. Phys. B 22,2265. Bray, I., Madison, D. H., and McCarthy, I. E. (1990). Private communication. Bray, I., McCarthy, I. E., Mitroy, J., and Ratnavelu, K. (1989). Phys. Rev. A 39,4998. Brion, C. E. (1975). Radiut. Res. 64, 37. Byron, F. W., Jr., Joachain, C. J., and Piraux, B. (1983). Phys. L e r r . *A, 627. Byron, F. W., Jr., Joachain, C. J., and Piraux, B. (1985). J. Phys. B 18, 3203. Chen, X. J. (1989). Phys. Rev. A 40, 1796. Cook, J. P. D., McCarthy, I. E., Mitroy, J., and Weigold, E. (1986). Phys. Res. A 33, 211. Cook, J. P. D., McCarthy, I. E.,Stelbovics, A. T., and Weigold, E. (1984a). J. Phys. B 17,2339. Cook, J. P. D., Mitroy, J., and Weigold, E. (1984b). Phys. Rev. Lett. 52, 1116. Crothers, D. S . F. (1986). J . Phys. B 19,463. Curran, E. P., and Walters, H. R. J. (1987). J. Phys. B 20, 337. Cvejanovii., S., and Read, F. H. (1974). J. Phys. B 7, 1841. Daoud, A., Lahmam-Bennani, A., Duguet, A., Dal Cappello, C., and Tavard, C. (1985). J. Phys. B 18, 141. Heer, F. J., and Inokuti, M. (1985). In “Electron Impact Ionization’’ (T. D. Mark and G. H. Dunn, eds.), pp. 232-276, Springer Verlag, Wien, New York. Dyall, K. G. (1980). Ph.D. thesis, Monash University, Melbourne, Australia. Ehrhardt, H., Jung, K., Knoth, G., and Schlemmer, P. (1986). Z. Phys. D 1, 3. Ehrhardt, H., Knoth, G., Schlemmer, P., and Jung, K. (1985). Phys. L e r r . llOA, 92. Ehrhardt, H., Rosel, T., Schlemmer, P., Agricola, R., and Jung, K. (1989). In “XVIth Int. Conf. Phys. Electronic Atomic Coll.”, New York 1989, Abstr. Contr. Papers, p. 232. Ehrhardt, H., Schultz, M., Tekaat, T., and Willmann, K. (1969). Phys. Rev. Leu. 22,89. Ehrhardt, H., Hesselbacher, K. H., Jung, K., Schulz, M., and Willmann, K., (1972). J. Phys. B 5, 2107-2116. Feagin, J. (1984). J. Phys. B 17, 2433. Fletcher, G. D., Alguard, M. J., Gay, T. J., Hughes, V. W., Wainwright, P. F., Lubell, M. S., and Raith, W. (1985). Phys. Rev. A31, 2854. Fournier-Lagarde, P., Mazeaq J., and Huetz, A. (1984). J. Phys. B 17, L591. Frost, L., Mitroy, J., and Weigold, E. (1986). J. Phys. B. 19, 4063. Fuss, I., McCarthy, I. E., Noble, C. J., and Weigold, E. (1978). Phys. Rev. A 17, 604. Gay, T. J., Feltcher, G. D., Alguard, M. J., Hughes, V. W., Wainwright, P. F., and Lubell, M. S. (1982). Phys. Rev. A 26, 3664. Cell-Mann, M., and Goldberger, M. L. (1953). Phys. Rev. 91, 398. Green, C. H., and Rau, A. R. P. (1983). Phys. Rev. Lerr. 48, 533. Grisogono, A. M., Mitroy, J., Pascual, R., Stefani, G., and Weigold, E. (1988). J. Phys. B. 21,895. Hamnett, A., Stoll, W., Branton, G., Brion, C. E., and van der Wiel, M. J. (1976). J. Phys. B 9, 945. Hibbert, A,, and Hansen, J. E. (1987). J. Phys. B. 20, L245. Hood, S. T., McCarthy I. E., Teubner, P. J. O., and Weigold, E. (1973). Phys. Rev. A 8, 2494. Hood, S. T., McCarthy I. E., Teubner, P. J. O., and Weigold, E. (1974). Phys. Rev. A 9, 260. Inokuti, M. (1971). Rev. Mod. Phys. 43,297.

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Klar. H., Roy, A. C., Schlemmer, P., Jung, K., and Ehrhardt, H. (1987). J. Phys. B 20,821. Lahmam-Bennani, A., Avaldi, L., Fainelli, E., and Stefani, G. (1988). J. Phys. B 21, 2145. Lahmam-Bennani, A., Cherid, M., and Duguet, A. (1987). J . Phys. B 20,2531. Lahmam-Bennani, A., Duguet, A., and Dal Cappello, C. (1986). J. Elec. Spectrosc. 40, 141. Lahmam-Bennani, A,, Duguet, A., Wellenstein, H. F., and Roualt, M. (1980). J. Chem. Phys. 72, 6398.

Lahmam-Bennani, A., Wellenstein, H. F., Duguet, A., and Rouault, M. (1983a). J. Phys. B 16, 121.

Lahmam-Bennani, A., Wellenstein, H. F., Dal Cappello, C., Rouault, M., and Duguet, A. (1983b). J. Phys. B 16, 2219. Lassettre, E. N., Skerbele, A., and Dillon, M. A. (1969). J . Chem. Phys. 50, 1829. Lohmann, B., McCarthy, I. E., Stelbovics, A. T., and Weigold, E. (1984). Phys. Rev. A 30,758. Lohmann, B., and Weigold, E. (1981). Phys. Lett. A 86, 139. Lower, J., and Weigold, E. (1989) In “XVIth Int. Conf. Phys. Electronic Atomic Coll., New York, 1989,” Abstr. Contr. Papers, p. 167. Madison, D. H., McCarthy, I. E., and Zhang, X. (1989). J. Phys. B 22, 2041. McCarthy, I. E., Pascual, R., Storer, P., and Weigold, E. (1989). Phys. Rev. A 40, McCarthy, I. E., and Roberts, M. J. (1987). J. Phys. B 20, L231. McCarthy, I. E., and Stelbovics, A. T. (1980). Phys Rev. A 22, 502. McCarthy, I. E., and Weigold, E. (1976). Phys. Rep. 27C, 275. McCarthy, I. E., and Weigold, E. (1988). Rep. Prog. Phys. 51, 299. McCarthy, I. E., and Zhang, X. (1989). J. Phys. B 22, 2189. McCarthy, I. E., and Zhang, X. (1990). Aust. J. Phys. 43. To be published. Mitroy, J. D., Amos, K. A., and Morrison, I. (1984). J. Phys. B 17, 1659. Moorhead, P. D. K., and Crow, A. (1985) In “XIVth Int. Conf. Phys. Electronic Atomic Coll.,” Berlin, 1985 Abstr. Contr. Papers, p. 160. Miiller-Fiedler, R., Jung, K., and Ehrhardt, H. (1986). J. Phys. B 19, 1211. Opal, C. B., Beaty, E. C., and Peterson, W. K. (1972). At. Data 4, 209. Pascual, R., Mitroy, J., Frost, L., and Weigold, E. (1988). J. Phys. B 21,4239. Pochat, A,, Tweed, R. J., Doritch, M., and Peresse, J. (1982). J. Phys. B 15, 2269. Rau, A. R. P. (1976). 3. Phys. B 9, L283. Rau, A. R. P. (1984). Phys. Rep. 110, 369. Read, F. H. (1985). In “Electron Impact Ionization” (T. D. Mark and G. H. Dunn, eds.), pp. 42-88, Springer-Verlag, Wien, New York. Selles, P., Huetz, A,, and Mazeau, J. (1987). J. Phys. B 20, 5195. Shore, B. W. (1967). J. Opt. Soc. Am. 57, 881. Stauffer, A. P. (1982). Phys. Leu. 91, 114. Stefani, G., Camilloni, R., and Giardini-Guidoni, A. (1978). Phys. Lett. A 64, 364. Temkin, A. (1982). Phys. Rev. Lerr. 49, 365. Tweed, R. J. (1976). J. Phys. B 9, 1725. van Wingerden, B., Kimman, J. T., van Tilburg, M., and de Heer, F. J. (1981). J . Phys. B 14,2475. van Wingerden, B., Kimman, J. T., van Tilburg, M., Weigold, E., Joachain, C. J., Piraux, B., and de Heer, F. J. (1979). J. Phys. B 12, L627.

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Vinkalns, I., and Gailitis, M. (1967). I n “Vth Int. Conf. Phys. Electronic Atomic. Coll,” (Nauka, Leningrad) Abstr. Contr. Papers, p. 648. von Niessen, W. (1987). Private communication. Wannier, G. H. (1953). Phys. Rev. 90,817. Weigold, E., Hood, S. T., and Teubner, P. J. 0. (1973). Phys. Reo. Lett. 30,475. Weigold, E., Noble, C. J., Hood, S. T., and Fuss, I. (1979). J . Phys. B 12, 291. Weigold, E., Ugbabe, A., and Teubner P. J. 0. (1975). Phys. Reu. Lett. 35,209. Weigold, E., Zhang, D., and Zheng, Y. (1989). I n “XVIth Int. Conf. Phys. Electronic Atomic Coll, New York, 1989,” Abstr. Contr. Papers, p. 152. Wigner, E. P. (1948). Phys. Rev. 73, 1002.

ADVANCES IN ATOMIC, MOLECULAR, A N D OPTICAL PHYSICS, VOL. 27

ROLE OF AUTOIONIZING STATES IN MULTIPHOTON IONIZATION OF COMPLEX ATOMS V. I. LENGYEL Uzhgorod University Uzhgorod, USSR

M . I. H A Y S A K Uzhgorod Branch of Institute for Nuclear Research Academy of Science, Ukraine Uzhgorod, USSR

I. Introduction . . . . . . . . . . . . . . . . . . . 11. Quasienergy Method . . . . . . . . . . . . . . . . 111. AIS Contribution . . . . . . . . . . . . . . . . . IV. Application of the Method to Calculation of the Two-Photon Ionization of Ca. . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

. . . .

245 246 250

. . . . . . . .

255 262

I. Introduction The advent of powerful lasers in a wide frequency range from infrared to ultraviolet gave a new impulse to investigations of the interaction of radiation with matter. Detection of double-photon excitation of atoms of cesium in the United States (Hall and Robinson, 1961), seven-photon ionization of Xe in Moscow (Voronov and Delone, 1965), two-electron many-photon ionization in Uzhgorod (Aleksahin et al., 1979), and the discovery of above-threshold ionization in Paris (Agostini et al., 1979) brought about the development of a new, extensive, thrilling field of research-multiphotonics. During a considerable period of time the main object of investigation here was atoms with one valence electron (for a review of such investigations, see, e.g., Rapoport et al., 1978, Delone and Krainov, 1984), though certain attention was paid to more complicated atoms as well, such as alkali-earth elements, i.e., systems with two-valence electrons. From the point of view of theoretical interpretation, there is an enormous difference between these objects. If in the first case one can easily use a oneparticle approach, then in the second case, electron correlations play an essential role. Atoms with two electrons in their outer shell have a richer 245 English translation copyright 0 1991 by Academic Press, Inc. All rights or reproduction in any form reserved. lSBN0-12-003827-7

246

V. I. Lmgyel and M. I. Haysak

spectrum of energy levels. In addition to the usual singlet and triplet terms, which correspond to a single excited state, terms are possible here in which both outer electrons are excited. Some of these states lie above the first ionization potential and this is why they are autoionizing. It is obvious that these autoionizing states (AIS) would reveal themselves in multiphoton experiments. Geller and Popov (1981) seem to be the first to have pointed out the theoretical possibility of AIS manifestation, while the first experimental evidence of it was obtained by Chin, et al. (1981). On the other hand, from the earlier works of Balashov and coworkers (1970) concerning the singlephoton ionization of He atoms, initiated by Fano (1961), it became evident that effects of the influence of AIS must be found in the multiphoton case as well. In 1982-1984, several works appeared (see, e.g., Andryushin et al., 1982; Haysak et al., 1984a) in which the analysis of the AIS effects was given, concrete calculations were carried out, and definite predictions were made. Some aspects of the problem were discussed at the same time in Lambropou10s and Zoller (1981). At first the arguments concerning the significance of AIS were met with some skepticism. The situation is not cleared up even as we enter the 1990s. On the other hand, it may be because for theoretical interpretation of this phenomenon, it is necessary to carry out rather cumbersome calculationscross sections for ionization depend not only on wave functions of the system in the discrete spectrum, but also on those in the continuum spectrum. To take into account the correlations, one has to go beyond the Hartree-Fock approach. This is the reason that as of 1989 very few calculations of this type had been done. The experimental situation is no better either; if in one group of experiments the influence of AIS on cross section is confirmed (Agostini et al., 1989), in another group such influence is not revealed at all (e.g., Bondar' et al., 1986). This is why we find it worthwhile to give a short review of the current state of this field of research and to examine the existing contradictions.

11. Quasienergy Method The behavior of atoms in a time-dependent external field is of course described by the time-dependent Schrodinger equation

a w = H(t)Y(t).

ih-

at

MULTIPHOTON IONIZATION OF COMPLEX ATOMS

247

The Hamiltonian H ( t ) may be presented in the well-known form H ( t ) = HA

+ w(t),

where

P2

e F = -A0eik”(&.p), 2mc

HA=-+

2m F is the time-independent operator and w is the angular frequency of the external field, p is momentum, and E is polarization vector. Evidently, the generalization to a many particle system does not present any difficulty. It is convenient to express the laser radiation in terms of intensity I (or density of flux, i.e., energy per unit of area per unit of time): A, =

Jy.

(3)

Further, we shall need the matrix element of the F operator on states of atomic Hamiltonian HA HAvA~) = &nVn(r)* (4) It is necessary to solve Eq. (1) with the Hamiltonian (2). Further, we shall follow the work of Haysak et al. (1984a). The initial condition for (1) is @(to) = 4i, with to being the time of switching on the interaction. Usually two different ways of switching on the interaction are considered, namely, instantaneous and adiabatic switching. We shall analyze only the first possibility, though the second one is interesting also. 4i is the wave function of the atom before switching on the interaction. The 4i is the eigenfunction of the operator HA,which corresponds to the eigenvalue c i . Here and throughout the chapter we use atomic units ( h = M e = e = 1). Following the arguments concerning the use of the Floquet theorem (Faisal, 1987) we can write down the partial solutions of (1) in the form (5)

+(t)= e-iE’4E(t),

where 4E(t+ T) = 4E(t),and T = (2n/o) is the periodic solution of the “stationary” equation

(A?,+ W ) 4 E= E 4 E ,

So = HA - i-

a

at’

(6)

which forms a complete set of functions with the real spectrum of quasienergies E. Functions 4Eare often called quasienergetical. The probability amplitude A(t, t o ) for finding the system in the 4, state at time t after switching the interaction on is defined by the equation A(t, t o ) = (+,I+@

+ to)).

(7)

248

V. I. Lengyel and M . I. Haysak

To determine the amplitude we shall expand $(t quasienergetical functions

+ to) in a complete set of

~rl/(t+ to>> = J d ~ e - i ~ t 1 4+, to)>(4E(tO)l4i> ~t

(8)

in which the coefficients are chosen to satisfy initial conditions. Taking into consideration (2), the amplitude (7) assumes the form (9) and is a periodic function of t with period equal to 17: That is why it is necessary to average it over this period, i.e.,

t IoT

If one formally integrates (10) over the parameter z = t the 6 - function a[w(z

(10)

dtoA(t, to).

J(t) = -

1 1 “ - t - to)] = -

+ to with the help of

eiko(r-t-to)

2nkz-m

(1 1)

then as a result one obtains an expression for the averaged amplitude, which contains only explicit time-dependence, namely

k=-m

J

where (bEk = dE(t)eikot, and l

(41 1 4 2 )

=

T

T 0

~ ~ 0 ~ 4 I 4 ~z ( ~t o )O> ) l

is a scalar product in the space of periodic functions. As is seen from (12), to obtain the amplitude A ( t ) it is necessary to find the solution 4Eof the equation (6). This solution can be expressed in terms of the Following the discussions of GoldGreen’s function of the operator So. berger and Watson (1964), with evident changes, we obtain (12) in a form

‘S

A(t)= - d E 271i

e-iEtR,i(E) ( E - E,)(E - E~ - Rii)’

(13)

where R,, = (4,l R I &-)) is the matrix element of the shift operator. The function 4iE from (13) obeys the “stationary” equation (6), which can be

MULTIPHOTON IONIZATION OF COMPLEX ATOMS

249

rewritten in the form:

I+E>

=

I4i> + GERWI 4 E > ,

(14)

where G: is the reduced Green function, which has the form GE" = (1 - I4i><4iIYE,

(15)

and 9, is the Green function of the operator X o ,i.e.,

where

(n = 1,2,..., k = 0, f1, f 2,...)

In, k)) = (pneikwt

is a complete set of quasienergy states (QES), which are the eigenfunctions of the operator X o ,corresponding to eigenvalues of the EIpk' = E, + ko,and cp, with E, are eigenfunctions and eigenvalues of the H A operator, i.e., quantities that satisfy (4). In (16) the summation also means integration over the continuum spectrum of the operator H A . The formal solution of (14)can be written in the form

I 4 d = (1 - GERW)- I4i>.

(17)

In this case, usually the initial state of the system is taken to be c$i. However, we shall use a more general approach and take in a form that corresponds to all possible states of the system. If this solution is found, one can write down the quasienergies approximately.

6,

'

= Ei

+ <4iIRI4i>,

(18)

where R is a shift operator with the form R=W

1 1 - G!W

=

W

+ WGiW +

Here one can clearly see the advantage of this approach: Eq. (19) can be used for calculation of photon processes of different multiplicity . the first term corresponds to one-photon processes, the second one corresponds to twophoton processes, etc. The integral in (13) has poles at E = E / and E

= ci

+ Ai - $ri,

Ai = Re R i i ,

Ti = 21m Rii.

Restricting ourselves to an approximation similar to that of Goldberger and Watson (1964), we can carry out the integration by the method of residues

V. I. Lengyel and M. I. Haysak

250

and obtain

A(t)=

exp(iEJt){1 - exp[i(ef - ci - Ai - E~ - Ai + i r i

+ iri)t]}

(20)

In the zero width approximation the expression (20) with t#+ = vieior, o >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-~ <4EI+E'>

= 6(E - E').

The transition amplitude from the initial (ground) state cple-2i"' to a final (singly ionized) atom, which corresponds to the absorption of two photons (Je3 in accordance with (22) is given by the expression ~ ( t =)

1

s.-i(E+k")~<(Je31~ne-ik"'>

n. k

A(t) =

s

O3

dE

-m

exp(-iEt) n2 Z 2 ( E )

+

1

+ Z(E)G(E - e3)

+ 0-11 + Y*l-21 + Yl-12 + I Y 1 2 r 2 2 ) - 1 . The term containing describes the direct two-photon transition from ground state into the continuum while the term Res2 describes the decay of AIS. The necessary integration in (31) is carried out by the method of residues. The poles of the integrand are defined by the condition Z ( E ) = i-in, while E1.2 = 2- '(E\O) - inrl + &Lo) - id-,,) f 2-l{[&\O)- &Lo) + i7t(r22 - rl1l2 (32) 4(A12 - inr12)(A21- inl-21)}1'2.

+

,

254

V. I. Lengyel and M . I. Haysak

Restricting ourselves to low intensity of the external field, which we consider to be linearly polarized, we obtain

In obtaining (33) we neglected the shifts due to the interaction of the discrete spectrum with the continuum spectrum. From the explicit form of these quantities one can easily deduce that theoretical expression for photon processes of large multiplicity would be rather complicated. As follows from (19) that for three-photon processes, double sums would be present in equations of the type (33), while for four-photon ones, triple sums would be present, etc. It is not surprising that for compound atoms such calculations have not been carried out as of 1989. The transition amplitude assumes the form (2.20) (Haysak et al., 1984a), where, however, the matrix element of direct transition of Resl is approximately substituted by the compound matrix element

The full expression for the transition amplitude has a rather complicated form. But the main feature of our result is already reflected in (34). Namely, the first term corresponds to the direct nonresonance two-photon transition through a virtual level. Sometimes for description of probability of this transition the “ionizating-width” term is used Ti = rll (Delone et al., 1984). The quantity r, = A12 is called the field width. Finally, the quantity l ( $ e 3 1 r ; ~Iqpzl2 = ra is called the autoionizing width because it characterizes the decay of AIS due to electron-electron interactions. Since the AIS decay due to absorption of additional photons is also possible, the additional term appears in (34), which we shall denote as r;= ((~e41zI(p2)12c9’2,i.e. we introduce the notation

rZ2 = r, = r: + r;.

MULTIPHOTON IONIZATION OF COMPLEX ATOMS

We then have

r12=

255

,/m Jrbr,. N

Should we analyze only transitions between ground state and AIS, we obtain a classical two-level scheme and expression (32) has the familiar form EL2 =

& (1O )

+ &LO) + 1

-&O)

2

-2

- E$o))*

+ 4r;.

This quantity defines the well-known Rabi frequency. The second term corresponds to a resonance transition through AIS. Remember that e3 is the energy of two photons in a chosen scale, EL') according to (27) is the quasienergy of AIS. If we neglect the upward AIS decay (ri= 0), then the result obtained coincides with the Fano result for two discrete-one continuum spectra (Balashov et al., 1970; Fano, 1961). Indeed, since Rest =

JT;= &,

Re32 =

Jrb,

bringing Eq. (34) to a common denominator, we obtain

where q=--

I-,

A12

xJr,rb = n J r , , r , is the profile index of Fano. In reality, the quasienergy would be defined by expression (32), in which different widths would be present and the Fano parameter assumes the more complicated form A12

=n

Jr,,Jrn'

(36)

As a result, for large d the interference character of cross section will change.

IV. Application of the Method to Calculation of the Two-Photon Ionization of Ca For the reader's convenience, we shall write down the normalized expressions for cross sections. It is well known that one can express the transition

256

V. I. Lengyel and M. I. Haysak

probability with the help of the Fermi golden rule in terms of many-photon matrix elements and intensity. In a complete analogy with (9,we obtain (see, e.g., Delone and Krainov, 1984; and Agostini and Petite, 1988)

where Resl is a multiphoton matrix element, presented in (33). Usually the probability is written in the form

(38) F = Z/o with F the flux and I the intensity. First, the “cross section” defined in such a way usually does not depend on intensity; second, it coincides with the real cross section in the one-photon case. In our case, the cross section will be expressed in cm4 sec; a typical value is 1.3 x cm4 sec. Taking into account that in the CI system the permittivity c0 = 1/4q we obtain for cross section (Agostini and Petite, 1988; Haysak et al., 1984a) W N =O(N)FN,

o ( ~=) 2n(87cao)’IX$)Ip,6(Ef - Ei

- 2ho),

(39)

where a is the fine structure constant, and is a composite matrix element of the second order, calculated according to (34) and divided by 8’. The case of linear polarization has been considered. In (39) it is necessary to carry out the averaging over initial and sum over final states, taking into account the 6function. This leads us to the differential cross section. To obtain the total cross section it is necessary to carry out an integration over an angle. This total cross section will be compared with experimental results later. It is clear from (34) that the matrix element has a resonance character in the region of energies e3 - EL’) N 0, which is the result of interference of direct ionization with the channel of AIS formation as well. These AIS decay because of interelectron interactions. These resonances differ from those formed by way of vanishing denominators of the type E, - c1 - k o in the expression (3.3). The last case is realized when intermediate levels fulfill the condition E, - N ko, k = 1,2,3, . .. . This case is called k-photon resonance. Resonances appearing because of autoionizing states we call autoionizing resonances. Let us consider first the cross section for ionization of the He atom from the excited metastable level 2 3S. In existing experiments on two-photon ionization, the electric field has a value of about 10 6V/cm. Besides, the process of two-photon ionization of singlet and triplet metastable states of He atom by the doubled frequency of a ruby laser has no resonance character. When calculating the composite matrix element in (39), only 2P to 8P states were taken into account in the sum over intermediate states. We

MULTIPHOTON IONIZATION OF COMPLEX ATOMS

257

neglect transitions between states of continuum spectra as well as other transitions, because the main contribution in the matrix element .Xyj is made by discrete states, particularly by 6 'P state. The results of calculations of cross sections are presented in Table I. While carrying out numerical calculations, the experimental values of the energies of discrete levels 2 '*'S and 2 1*3P to 8 l s 3 Pwere used. The experimental data concerning ionization by the doubled-frequency ruby laser with intensity I N 2 x 10"J/cm2 sec are taken from Lompre et al. (1980). These data are also presented in Table I. It is seen from the results that it is necessary to carry out more precise experiments in order to make a decisive conclusion about effectiveness of theoretical calculations. However, as is seen from Table I, the results obtained are within the limits of experimental errors. Numerical calculations of the just-mentioned cross sections have been carried out in Lompre et al. (1980). The method of a model Hamiltonian was used in these calculations. The results obtained in a configurational-interacLion method (Haysak et al., 1984a) are similar to theoretical results obtained by using the method of a model Hamiltonian. Further, we shall deal with two-photon ionization of Ca atom for such frequencies o that k - photon resonances are absent and only autoionizing resonances are realized. For obtaining photoionization cross section from (33), it is necessary to choose a definite basis of intermediate states of the Ca atom-both of excited type In) and of AIS type q z . In the dipole approximation in the case of two-photon ionization from the ground state only 'S, 'P, 'D states contribute. The numerical calculations for Ca were carried out using the CI method with a frozen core approximation (Haysak et al., 1985). One-particle wave functions are calculated taking into account the motion of electrons in a field of the model polarizing core potential

TABLE I CROSS SECTION OF TWO-PHOTON IONIZATION OF AN He ATOMBY A

DOUBLED FREQUENCY RUBYLASER(IN CM4 SEC)

States 2 1s 2 3s

Configurationalinteraction method o(2)=

n(2) =

4.86 10-49 5.1 10-50

Method of model Hamiltonian

Experiment

o(z)= 2.18 10-49 u(z)= 3.34 10-50

(2.7-2) 10-49 (15-14) 10-50

V. I. Lengyel and M. I. Haysak

258

Here a. is a static polarizability of doubly ionized Ca ion, while p is an adjustable parameter defined from experimental values of energies of lower excited states of Ca'. It turns out that for stabilization of about 10 lower states of a given AIS series, it is necessary to mix 40-60 configurations. For calculations of energies and widths of 'S, 'P, and 'D terms, the experimental value for the static polarizability of a. = 3.45~: was used. The parameters p in (40) were normalized to experimental values of ionization potentials from states 3p 64s, 4p, 3d, and 4f of Ca'. From these conditions it was found that po = 1.692, p1 = 1.63, p z = 1.909, and p3 = 2.3. Basis functions were chosen in the form h,t (core, nl, n'l'), where nl = 4s, 5s, 4p, 5p, 3d, 4d, 4f; n' = 4-12; I' = 0-3 and in a form of Hartree-Fock continuum wave functions with the configuration 4s 81. Parameters of some AIS are presented in Table I1 (see Haysak et al., 1985), where energies E, and widths raare presented in electron volts. The values of AIS parameters thereby obtained (which are calculated in the CI approximation) are in good agreement with the experimental values. In the quoted works a vast number of calculations of AIS parameters in a wide interval of energies from 6.6 to 8.4 eV, converging to a threshold n = 4 of Ca' ion with use of different number of basis functions, have been carried out. Differences in basis sets change the value of energy by about 0.03 eV. At the same time, values of r: change significantly, as seen in Table 11. To obtain more precise results it seems to be necessary to take into account the continuum spectrum and to broaden the initial set of basis functions. Moreover, the question of precise consideration of interelectron interaction within the CI method needs further elucidation. The method of hyperspherical coordinates seems to be very promising in this respect. Many interesting results have been obtained in this direction. (See, e.g., Haysak et al., 1988a, and literature cited there.) The results of TABLE I1 ENERGIES AND States 3d5d 3d6d 3d7s 3d8~ 3d5d 3d6d

'S -

ID -

WIDTHS (yo) OF

AIS OF C A

E eV*

lo3 eV*

lo3 eV**

7.102 7.311 7.197 7.381 7.15 7.36

9.03 4.59 54.1 30.7 0.126 0.011

11.1 13.8 -

0.94 0.17

lo3 eV*** 35 39 110

52.7 0.01 0.14

Sources: as in manuscript corrected *** Haysak et al., 1984b.

259

MULTIPHOTON IONIZATION OF COMPLEX ATOMS

calculations of a two-photon ionization cross section by linearly polarized light are given in Fig. 2. In this figure the results of an experimental determination of the cross section in (Alimov et al., 1988) are also given. Let us note that while experimental points are fixed with an accuracy eV, the position of AIS as given in CI approximation is fixed with the accuracy of only about 0.03 eV. That is why it would be very interesting to obtain experimental results in a broader energy region in order to obtain the identity of AIS. As is seen from Table 11, the widths in different CI approximations differ sometimes by even one order of magnitude. At the same time, the absolute cross section values depend significantly on these parameters. So, if the widths of resonances in the CI method are given with an accuracy of a factor of 2, the total cross section is obtained with an accuracy of a factor of 4. From this point of view the obtained agreement of theory and experiment seems to be quite satisfactory. We would like to stress that only differential cross sections of two-photon ionization are sensitive to different orbital momenta and that is why experimental determination of differential cross sections are of great interest. The obtained calculations show that cross section values in the region of autoionizing resonances are one to three orders higher than cross sections without AIS. While obtaining these cross sections of two-photon ionization, we neglected forced transitions from AIS contained in the analyzed frequency region w.

-

u1 ZOOZto

Z1*

%a

222

%a

230

134

238

Z42

FIG.2. The two-photon ionization cross section. The solid curve shows theoretical calculaa.u. Points with errors bars are experimental data. (Source:Alimov et al., tions at E 1988).

-

V. I. Lengyel and M . I. Haysak

260

But these results are correct only for intensities of the external fields smaller than lo5 V/cm, for which autoionizing rh widths are much larger than ionizing ones (rf I (p2~Z~i,be3)~2&'2). With increasing intensity of the external field, when ionizing width becomes equal to autoionizing widths, it is necessary to take into account the possibility of decay of the final state. Absorption of an additional photon leads to an excited final state. That is why in the expression for rN , in (33) such states are accounted for. It is interesting that after Andryushin et al. (1982), Agostini et al. (1989) has also taken into account the necessity of such a process in Sr. In Fig. 1, the level scheme of the Ca atom in the relevant energy region is shown. In this figure, the AISs, which can be formed by two-photon ionization in a weak field, are shown. For realization of this scheme one has to know the continuum-spectrum wave function up to 11.1 eV. In this case it is necessary to solve a multichannel Hartree-Fock problem for continuum spectra with the configurations (4SE111, 3dE2/2,4pE,l,. 5PE414). The calculations were carried out with the use of wave functions of a continuum spectrum of the core with polarizability (Haysak et al., 1988b). The obtained results show that increasing the field leads to drastic changes of the structure of cross section ionization. The dependence of calculated cross section on intensity is shown in Fig. 3. N

r

163

' 1

261

MULTIPHOTON IONIZATION OF COMPLEX ATOMS

1

471

-

1

1

I

-

V. I. Lengyel and M . I. Haysak

262

TABLE I11 IONIZINGWIDTHSr; eV x (lo&)-* at E = 10.5 eV at E = 10.9 eV

3d5d ‘S

3d6d ‘S

3d7s ID

3d6d ID

8.2 5

17 11

3.4 2.6

7.5 0.5

In Table 111, E is the triple frequency of the laser radiation. The values of l-1 presented in this table are calculated according to (33). If one compares the obtained values of F : with the autoionizing width Yo, the values of which are also shown in Table 11, one can see that for 8 a.u., these widths become equal to each other, while at stronger fields they increase even further, suppressing the resonance structure as it is seen from Fig. 3. In such a way the results (Chin et al., 1981; Agostini et al., 1989) for Sr could be understood, though direct experimental observation of this phenomena in Ca would be more useful. It is seen that qualitatively the probability of Ca ionization is much the same as the probability of Sr ionization. The AIS displayed themselves in two-photon ionization of Mg as well (Alimov et al., 1987). Proceeding from the obtained results one can understand the negative result of Bondar’ et al. (1986) on multiphoton ionization of Ba: it seems that intensities of the fields did not fit into that very narrow interval at which AISs display themselves. So, these results do not confirm the “confluence of coherence” Rzazewski and Eberly (1981). Pazdzewski and Voitkiv (1985) and Fedorov and Kazakov (1989) have come to the same conclusion. In the future, analogous calculations for other alkaliearth atoms could be carried out. N

REFERENCES

Agostini, P., and Petite, G. (1988). Contemp. Phys. 29, 58. Agostini, P., Fabre, F., Mainfray, G., Petite, G., and Rahman, N. (1979). Phys. Rev. Lert. 42,1127. Agostini, P.,Lambropoulos, P., and Tang, X. (1989). “Multiphoton Spectroscopy of Double Excited, Bound and Autoionizing States of Sr.” Saclay. Aleksahin, I. S., Delone, N. B., Zapesochny, I. P., Suran, V. V. (1979). JETP 49,447. Alimov, D. T., Bel’kovsky, A. I., Il’kov, F. A. er al. (1987). Preprint P-6-248. Inst. Nucl. Phys., Tashkent (in Russian). Alimov, D. T., Bel’kovsky, A. N., Haysak, M. I., et al. (1988). Ukr. Phys. J. 33, 658 (in Russian). Andryushin, A. I., Kazakov, A. E., and Fedorov, M. V. (1982). JETP 55,53. See also J. Phys. B: Ar. and Mol. Phys. 15 (1982), 2851.

MULTIPHOTON IONIZATION O F COMPLEX ATOMS

263

Balashov, V. V., Grishanova, S. I., Kruglova, I. M., and Senashenko, V. S . (1970). Optics and Spectroscopy 28,466. Bondar’, I. I., Dudich, M. I., Suran, V. V. (1986). In “Energy Levels and Transition Probabilities in Atoms and Ions,” p. 127. Spectroscopy Council, Moscow (in Russian). Chin, S. L., Feldman, D., Krautwald, J., and Welde, K. H. (1981). J. Phys. B: Atom. and Mol. Phys. 14,2353. Delone, N. B., and Krainov, V. P. (1984). “Atom in Strong Light Field,” p. 224. Energoizdat, Moscow (in Russian). Faisal, F. H. (1987). “Multiphoton Processes. Plenum, New York. Fano, U. (1961). Phys. Rev. 124, 866. Fedorov, M. V., and Kazakov, A. E. (1989). Prog. in Quant. Elec. 13, N1. Geller, Ju. N., and Popov, A. K. (1981). “Laser Induced Radiation.” Nauka, Novosibirsk (in Russian). Goldberger, M., and Watson, K. (1964). Collision Theory. Wiley, New York. Hall, J., and Robinson, E. J. (1961). Phys. Rev. Lett. 77, 229. Haysak, M. I., Zatsarinny, 0. I., Lengyel, V. I., Petrina, D. M., and Shuba, I. M. (1984a). “Nonlinear Processes in Two-Electron Atoms,” p. 115. Spectroscopy Council, Moscow (in Russian). Haysak, M. I., and Zatsarinny, 0.I., Libak, 0. A., Petrina, D. M., and Shuba, I. M. (1984b). In “Elementary Processes in Atomic Collisions,” p. 29. ChSU, Cheboksary (in Russian). Haysak, M. I., Zatsarinny, 0.I., Libak, 0.A., and Petrina, D. V. (1985). I n “AIS Spectroscopy of Atoms and Ions, p. 195. Spectroscopy Council, Moscow (in Russian). Haysak, M. I., Lengyel, V. I., Poyda, V. Yu.(1988a). Ukr. Phys. J. 33, 664 (in Russian). Haysak, M. I., Zatsarinny, 0. I., Lengyel, V. I., and Petrina, D. M. (1988b). “Theses of reports at XI11 International Conference on Nonlinear Optics,” part I, p. 346. Minsk (in Russian). Kazakov, A. E., Makarov, V. P., and Fedorov, M. V. (1976). JETP 43,20. Lambropoulos, P. and Zoller, P. (1981). Phys. Rev. A 24, 379. Lengyel, V. I., and Haysak, M. I. (1988). In “Multiphoton Processes” (S. J. Smith and P. L. Knight, eds.), p. 304. Cambridge University Press. Lompre, L. A., Mainfray, G., Mathieu et al. (1980). J. Phys. B: At. and Mol. Phys. 13, 799. Pazdzewski, V. A., and Voitkiv, A. B. (1985). J. Phys. B: At. and Mol. Phys. 18,455. Rapoport, L. P., Zon, B. A., and Manakov, N. L. (1978). Theory of many-photon processes in atoms. “Atomizdat,” Moscow (in Russian). Rzazewski, K., and Eberly, J. (1981). Phys. Rev. Lett. 47,40. Voronov, G. S., and Delone, N. B. (1965). JEPTLett. 1, 66.

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II

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

MULTIPHOTON IONIZATION OF ATOMIC HYDROGEN USING PERTURBATION THEORY E. KARULE Insliiute of Physics Laivian SSR Academy of Sciences Riga, Saiaspiis USSR

I. Introduction . . . . . . . . . . . . . . . . . . . . . 11. Multiphoton Ionization of Atomic Hydrogen within the Framework of Perturbation Theory . . . . . . . . . . . . . . . . . . 111. Sturmian Expansions . . . . . . . . . . . . . . . . . . A. Sturmian Transition Matrix Elements . . . . . . . . . . . B. Radial Integrals. . . . . . . . . . . . . . . . . . . C. Recurrence Relations for the Sturmian Transition Matrix Elements. IV. Analytical Continuation of the Transition Matrix Elements. . . . . A. Technique of Double Sum Resummation . . . . . . . . . . B. Analytical Continuation Remote from the Threshold. . . . . . C. Analytical Continuation near the Threshold . . . . . . . . . V. Theoretical Estimates and Experimental Data for Atomic Hydrogen . A. Ionization by Linearly Polarized Light. . . . . . . . . . . B. Ionization by Circularly Polarized Light . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

265

. . . . . . . .

261 215 215 216 . . 280 . . 281 . . 281 . . 281 . . 289 . . 291 . . 291 . . 295 . . 291

I. Introduction One-photon ionization was first observed a hundred years ago by Herz. It laid the foundation for the quantum treatment of light. Multiphoton ionization (MPI) of atoms by the minimum energetically necessary number of photons was first observed by Voronov and Delone (1965, 1966) only a few years after the invention of the laser. As atoms possess discrete negative energy spectra, there are nonresonant and resonance-enhanced multiphoton processes. In the case of nonresonant multiphoton ionization, no intermediate states have the same energy as atomic states in discrete spectrum. When one or more intermediate states coincide with atomic states, we have resonance-enhanced multiphoton ionization process. 265 English translation copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-003827-7

E. Karule

266

In strong fields atoms may absorb not only the minimum energetically necessary number of photons but also one or more excess photons. One-step multiphoton ionization process in which extra photons are absorbed is called above-threshold ionization (ATI). The energy of the extra absorbed photons is converted entirely into the kinetic energy of the emitted electron. Therefore photoelectron energy spectra must be studied, which became possible only with the development of methods of photoelectron spectroscopy with very high energy resolution. In the case of AT1 by monochromatic light the energy spectrum of photoelectrons consists of equally spaced peaks. The first experiments on above-threshold ionization were carried out by Agostini et al. (1979, 1981). Energy conservation mandates that the photoelectron leaves the atom with energy E = Kw - E,

= (N

+ S)w - E,,

s = 0, 1 , 2 ...

(1)

where w is the energy of a photon, E, is the ionization potential of an atom, N, is the minimum energetically necessary number of photons for the ionization, and S is the number of excess photons. Atomic units are used. Most MPI experiments are performed with rare gases. Until now there have been few MPI and AT1 experiments for atomic hydrogen (LuVan et al., 1973; Muller et al., 1986; Feldman et al., 1987; Wolf et al., 1988), though as the simplest of atoms atomic hydrogen is of interest in its own right. The wave functions of atomic hydrogen are well known. Therefore, rather good approximations may be obtained. Besides, the results of calculations of cross sections for atomic hydrogen provide an estimate of the lowest limits for cross sections of other atoms (Lambropoulos, 1985; Lambropoulos and Tang, 1987). At moderate intensities, nonresonant multiphoton ionization may be considered within the framework of perturbation theory (Section 11). In Section 111, the Sturmian transition matrix elements for atomic hydrogen are presented. In the case of ionization with excess photons, they diverge. In Section IV the method of analytical continuation of the Sturmian transition matrix elements in the case of ionization with an excess photon is first described in detail. In Section V, results of calculation using dipole approximation are compared with semiclassical estimations and experimental data. The applicability of perturbation theory is investigated. Perturbation theory proves to be valid up to intensity of light Z = 1013Wcm-’ in the case of ionization from ground state and N I8.

MULTIPHOTON IONIZATION OF ATOMIC HYDROGEN

261

11. Multiphoton Ionization of Atomic Hydrogen within the Framework of Perturbation Theory Starting with the pioneering work of Goppert-Mayer (193 1) on twophoton absorption, up until now perturbation theory is the most widely used in multiphoton ionization calculations for atomic hydrogen. At moderate light intensities, direct multiphoton ionization as well as above-threshold ionization may be satisfactorily treated by the methods of perturbation theory. There are three methods that are used to investigate multiphoton processes: the method of solving the set of inhomogeneous differential equations, the Green’s functions method, and the variational method. Zernik (1964) and Zernik and Klopfenstein (1965) started with twophoton ionization of atomic hydrogen in the 2s metastable state. To calculate transition matrix elements, they solved the first-order inhomogeneous differential equation. This technique was first proposed by Dalgarno and Lewis (1955) and later reformulated by Schwartz and Tieman (1959). The first to investigate multiphoton ionization of atomic hydrogen, rare gases, and alkali atoms when large number of photons participate were Bebb and Gold (1966) and Bebb (1967). They used a Green’s function method. The Green’s function energy eigenfunction expansion

was employed. In the case of atomic hydrogen R,, and RE, are the radial parts of the discrete and continuum-state wavefunctions of atomic hydrogen. Bebb and Gold (1966) summed the series (2) in transition matrix elements directly for 2 s N 5 12, but they did not take into account the continuum and only approximately took into account the contribution from those intermediate states that are far from any of the atomic states. Calculations were performed for linearly polarized light. Results of their calculations differ from the results of exact calculations carried out later, especially at minima between resonances where all intermediate states contribute significantly. Moreover, as the number of photons increases, the accumulation of errors also increases. Therefore, the discrepancy between exact perturbation calculation results and those obtained using approximate methods, as in Bebb and Gold (1966), is greater for large N.

E. Karule

268

Gontier and Trahin (1968, 1971, 1973) have generalized the method of solving the inhomogeneous differential equations to perturbation orders higher than two. In the case of K-photon ionization, the system of K-1 inhomogeneous differential equations must be solved. For two- and threephoton ionization of atomic hydrogen in the ground state this method was used also by Chan and Tang (1969) and Chang and Poe (1976,1977). Solution of the set of inhomogeneous differential equations takes a lot of time even on the big computers. Therefore, besides this method, Green’s function methods were developed. Green’s function methods enables one to obtain expressions for the transition matrix elements in closed form. The Coulomb Green’s function, besides the energy eigenfunction expansion in the form (2), has other representations. One of them is the Coulomb Green’s function integral representation (Hostler, 1964; Schwinger, 1964 in momentum space) GL(I, r’;Q) = - 2(rr’)- ’/’

jOm exp[ (I + r‘)cosh(t)/p](coth -

~/2)’~

x I z L + 1[2(r~’)’/’sinh(t)/p]dt

(3)

where p = (-2Q)-”’, and I z L + is a Bessel function of imaginary argument. The Green’s function integral representation (3) was applied by Rapoport et al. (1969) for two-photon ionization calculations for atomic hydrogen in the 2s state and by Klarsfeld (1969a, b) for two-photon ionization in the ground state. Later this representation was used by Arnous et al. (1973), who calculated two-photon ionization of atomic hydrogen in the Is, 2s, and 2p states. The integral representation of Coulomb Green’s function leads to a closed expression for transition matrix elements in terms of Appell’s hypergeometric functions in two variables. The hypergeometric series converges if intermediate states are in the discrete spectrum. In expression (3) radial variables are not separated. Therefore, it is not convenient for the consideration of multiphoton processes of order higher than two. Coulomb Green’s eigenfunction expansion (2) can be integrated over energies in the continuum, which yields (Mapleton, 1961) GL(I, r’; Q) = -

[w ++ r(2L

1 -PIP]

rp<)rp>)

2)rr‘ Mp.Lt 1/2 -

Wp,Ltl/’

__

(4)

where M and W are Whittaker’s functions as defined by Erdelyi et al. (1953). In expression (3), the radial variables separate but there are practically two different expressions for the Green’s function. Which one to use depends on which of the radial variables is greater or less than the other. Expressions for

MULTIPHOTON IONIZATION OF ATOMIC HYDROGEN

269

transition matrix elements may be obtained in closed form also in this case but they are complicated. Zon et al. (1971) used Green’s function in form (4) for three-photon ionization rate calculations of atomic hydrogen in 1s and 2s states in the cases of linearly and circularly polarized light. Laplanche et al. (1976) applied Eq. (4)to two- and three-photon ionization of the ground state. For multiphoton processes of order higher than three Green’s function representation (4)is inconvenient. A more appropriate method for investigating multiphoton processes in which a large number of photons participate is the Green’s function Sturmian expansion (Hostler, 1970)

where x (2r/p),F( -n

Sn,(2r/n)

+ L + 1,2L + 2; 2r/p)

= nRn,(2r/n).

S,, are the Sturmian functions (Rotenberg, 1970) which are the charge eigenfunctions. For fixed and negative energies R there are bound states only at the discrete and positive values of the charge 2 = n/p. It follows that the summation in expression (5) is only over a discrete spectrum. It is easy to get Green’s function integral representation (3) from the Sturmian expansion ( 5 ) if we express the denominator in an integral form and do summation over rn explicitly. Since the radial variables in expression (5) separate, the Sturmian expansion is more convenient than others to get transition matrix elements for high-order processes in closed form (Karule, 1971). In revised form these expressions for transition-matrix elements were used to calculate multiphoton ionization of atomic hydrogen in 1s state for N up to 16 (Karule, 1975) in the case of linearly polarized light. A method equivalent to the Green’s function Sturmian expansion was first used by Podolsky (1928), who calculated two-photon bound-bound transitions. For lower-order ionization processes of atomic hydrogen in ground and excited states the Sturmian expansion was used by Khristenko and Vetchinkin (1976), Klarsfeld and Maquet (1974), Maquet (1977), and Justum and Maquet (1977). The Green’s function method takes less time on computers but the advantage of the method of solving inhomogeneous differential equations or

E. Karule

270

the so-called implicit summation technique is that it can be likewise easily applied for heavier atoms (Chang and Poe, 1976, 1977; Crance and Aymar, 1979; Aymar and Crance, 1979). A third method employed to investigate high-order multiphoton processes within the framework of perturbation theory is the variational method used by Gao and Starace (1988, 1989). The agreement between results of authors who have used different methods is rather good (Table I). The discrepancies are mainly due to the different Rydberg constants used by authors. To calculate the wavelength at fixed photon energy we have equation

1 = lO8(2oR)-'(A)

(6)

where R is the Rydberg constant. Two different Rydberg constants are used in MPI calculations for atomic hydrogen: R, = 109737.3 (for infinite mass) and R, = 109677.58 (experimental value for atomic hydrogen). At fixed photon energy this causes some shift of the wavelength, which must be taken into account when comparison of results is made. Discrepancies in results due to the Rydberg constants are considerable only near resonances where transition rates change rapidly (Table I). To compare the results of different authors it is useful as well to know subsequent relations. Within the framework of perturbation theory the transition probability per unit time is expressed as WK= 6,FK = GKIK (sec-')

(7)

where I is the intensity of light in W cm-2 and F is the photon flux per cm2 per second. I

=Fho

(W cm-').

ho is the energy of a photon in joules (1 eV = 1.60219 lo-'' J). Therefore the generalized cross section may be expressed as

6, = d , ( h ~ ) ~(cmZKsecK-')

(8)

where cross section 8, is in units W-K cmZKsec-'. In this chapter we use the transition rate, which is defined by Q(,)/IK-' = 6K ( h ~ ) ' = - ~dKhw (W'-K crn2,).

(9)

The transition probability during the pulse cannot exceed unity. If the

TABLE I RATE Qjz)/I (W-'cm4) TRANSITION

(A)

Chan and Tang

Klarsfeld

(1969)

(1970)

FOR

TWO-PHOTON IONIZATION OF GROUND-STATE ATOMIC HYDROGEN BY LINEARLY POLARIZED LIGHT

Gontier and Trahin

Karule

Karule

(1975)

(1988)

(1971)

R,

R"

4.9 1 7.09 4.00 6.42

6.79 5.51 4.05 5.80 6.42 1.28 8.45 9.14 1.02

Laplanche et al.

Khristenko and Vetchinkin

Chang and Poe

GaoBo and Starace

(1976)

(1976)

(1977)

(1988)

4.92 7.08 4.04 6.42

6.77 5.52 4.05 5.80

4.91 7.08 4.00 6.42

4.96 4.15 4.02 6.44

Power of ten

~~~~~~~~~

975 1020 1100 1200 1200.65 1300 1400 1600 1700

5.15 6.75 4.01 6.30

-

-

-

5.52 4.05 5.80

4.05 5.80

-

-

-

-

1.28 8.45 9.15 1.03

1.28 8.45 9.14 1.02

1.28 8.45 9.14 1.02

1.27 8.45 9.15 1.03

-

-

-

-

1.28 8.47 9.18 1.03

1.28 8.45 9.14 1.02

1.28 8.45 9.18 1.02

1.28 8.45 9.15 1.03

(-33) (-33) (-33) (-32) ( - 32) (-32) (-33) (-33) (-32)

272

E. Karule

transition probability during the pulse is unity, then the atom is ionized and the saturation intensity I s is reached

where z K is the duration of the pulse. Saturation intensity increases with the shortening of pulse duration. Most of the papers discussed previously deal with multiphoton ionization when the minimum energetically necessary number of photons participate and all intermediate states have negative energies. In the case when ionization with excess photons occurs, there are intermediate states in the continuum. But the Green's function Sturmian expansion (5) and integral representation (3) are valid only for negative energies. Methods of implicit summation also have formally divergent integrals if there are intermediate states in the continuum. Therefore investigations of the preceding threshold multiphoton processes are more complicated than those with minimum energetically necessary number of photons. Zernik and Klopfenstein (1965) were the first to theoretically investigate above-threshold ionization. They calculated two-photon AT1 of atomic hydrogen in the 1s state, solving inhomogeneous differential equations. Klarsfeld (1970) used Green's function integral representation (Eq. 3) for this purpose. Transition-matrix elements were expressed by Appell's hypergeometric series F , , which was analytically continued above the photoelectric threshold into a series of the same type. However, in both cases calculations were not performed in the whole region of frequencies where ionization with an excess photon is possible. We proposed method of analytical continuation for the Sturmian transition matrix elements, which allows one to calculate two-photon ionization rates for atomic hydrogen in 1s and excited states (no I 9) in the whole region above the threshold as well as on the threshold (Karule, 1977, 1978, 1984, 1985). The first experimental works on AT1 (Agostini et al., 1979, 1981; Kruit et al., 1981) stimulated an interest in ionization with extra photons. Several different methods were proposed. Klarsfeld and Maquet (1979a) in the case of two-photon AT1 considered the Sturmian matrix elements as a series of one variable and analytically continued it above threshold as the series of another variable. But the convergence of these series was slow. Therefore Klarsfeld and Maquet (1979b, 1980) proposed to use for AT1 a method of Pad6 approximants. They carried out investigations for N + 1 = 2,3. But near threshold Pade approximants converge slowly. Costescu and Florescu (1978) used integral representation of Appell's function to calculate two-photon ATI.

MULTIPHOTON IONIZATION OF ATOMIC HYDROGEN

213

The method of solving the set of inhomogeneous differential equations was adjusted to the case of AT1 by Gontier et al. (1980), Gontier and Trahin (1980), Aymar and Crance (1980a), and Muller et al. (1986). Each of these authors overcame difficulties concerned with formally divergent integrals in a different way. The highest-order AT1 processes (N + S = 12, N = 6, S = 6) for atomic hydrogen in the ground state applying this technique were calculated by Gontier and Trahin (1980). Manakov et al. (1984) obtained analytical continuation for the Green's function Sturmian expansion as a double series of Laguerre polynomials with an arbitrary parameter. The parameter may be used to improve the convergence. By this method Fainshtein et al. (1984) calculated two-photon ionization of atomic hydrogen in highly excited states for A/ng < 60.2 nm. But there are no results for the region near the threshold 60.2 nm < A/ng I9 1.1 nm. For multiphoton processes of higher order than two, analytical continuation of the Sturmian expansion for transition matrix elements in the case of one excess photon (K = N S, S = 1, N 5 8) are given in my paper (Karule, 1988). Two analytical continuations have been obtained. For linearly polarized light calculations of MPI and AT1 cross sections and their ratio for the ionization of atomic hydrogen in the ground state are carried out in the whole region of frequencies where ionization with one extra photon may occur (Karule, 1988a). The original technique to overcome the divergence of Sturmian transition matrix elements in the case of AT1 was proposed by Shakeshaft (1985). He decomposed the continuum wave function into outgoing and ingoing waves

+

R E M =f k l W

+fm

(11)

wheref,, is the outgoing wave

(lk" + )

= (27~k)-'/~(-2k)'+' exp

iqI

~

(ki + 1 + 1,21+ 2; -2ikr

x ri exp(ikr)Y

-

$ (a, b; z) is the irregular Kummer's function (Gradshteyn and Ryzhik, 1965). Therefore, the transition matrix element for two-photon ionization of atomic hydrogen may be written in the form T(n,l,, L, El) = T,

+ Tb

(12)

274

E. Karule

where

& =
= (fkIIGLIRnolo)?

If for the Coulomb Green's function the Sturmian expansion in the form of Eq. ( 5 ) is used, then the series for matrix elements T. converge but for & diverge. To get convergent series for Tb,the other Sturmian expansion for the Coulomb Green's function must be used

+ 2ziRnL(r)RnL(r') where SnL(-2r/p) = -2p-'[(2L

+ 1)!]-'[(n

- L)2L+1]1/2exp(r/p)

x ( - 2 r / ~ ) ~ F ( - n + L + 1 , 2 L + 2 ; -2r/p)

+ 1)!]-1(2r)LF((i//d) + L + 1, 2L + 2; 2ik'r) L Ck = 2[1 - e x p ( - 2 ~ / k ' ) ] ~ /fl ~ [ ( ~ k '+) ~1]1/2y k' = (2R)'l2.

RnL(r) = ck[(2L

s= 1

For R > 0, SnL(- 2r/p) = S,*,(2r/p) but RRL(r)is the continuum wavefunction normalized on the energy scale. For R < 0 we have &(r)

= ck[(2L

+ 1)]-1(2r)L~(-p + L + 1 , 2 +~ 2; 2r/p)

where c k

= 2[1 - exp(2izp)l-

112 L - L - 1 / 2

ip

[(p - L)2L+1]1/2,

p = (-2sz)'/2.

The equality of expressions ( 5 ) and (13) for the radial Green's function follows from the equality of the contour integrals &7

where

f(Z)dz

=

fp

f(Z)dZ

(14)

+ it, r)S,,(R + k, r')/(Z - l)], R < 0 SzL(R,r) = Czk[(2L + 1)!]-1(2r)L~(~p + L + 1 , 2 +~ 2; 2r/p) f(Z) = [S,,(R

czk = 2 ~ 1 e x p ( 2 i ~ ~ p ) ] - ' / ~ i ~ p - ~ - ~ /-~ L [ ()~~p ~ + ~ ] ~ / ~ / z - ~ / ~ .

Contours in Eq. (14) may be closed only at the negative energies R. Integrating the contour integral at the left side of Eq. (14) in the lower part of

MULTIPHOTON IONIZATION OF ATOMIC HYDROGEN

275

the Z plane we obtain Eq. (5) for the Green's function. The integration of the contour integral on the right side of the equality in the upper part of the complex Z plane including pole Z = 1 leads to expression (13) for the Green's function. If R is negative, both Sturmian expansions for Green's function may be used to obtain convergent expressions for the transition matrix elements. At positive energies the R series in expression (13) are the complex conjugate to series in expression (5). Therefore, in the case of two-photon abovethreshold ionization we have

The Series in the transition matrix elements (15) converge slower than the series in the analytically continued Sturmian matrix elements (Karule, 1978). The convergence is worse if there are more excess photons. To improve the convergence, different techniques were introduced by Shakeshaft (1986a, b). Variational methods (Gao and Starace 1988, 1989) also proved to be successful in above-threshold calculations. Results of calculations using lowest-order perturbation theory are in a good agreement with results of experiments performed on the multiphoton above-threshold ionization of atomic hydrogen in the ground state (Muller et al., 1986; Feldman et al., 1987; Wolf et al., 1988). This proves the validity of perturbation theory for MPI and AT1 of atomic hydrogen in the ground state up to intensities 10l3 W cm-' in the case of linearly polarized light.

111. Sturmian Expansions A. STURMIAN TRANSITION MATRIXELEMENTS In investigating multiphoton processes theoretically the main problem is to calculate the radial part of the transition matrix elements. In the dipole approximation the radial part of transition matrix elements for K-photon ionization of atomic hydrogen using the Green's functions formalism may be written in the form

276

E. Karule

where G L is the radial part of the Coulomb Green's function, E j is the energy, and Lj is the orbital momentum of the set of intermediate states. To evaluate transition matrix elements for multiphoton processes, I proposed (Karule, 1971) to use the Coulomb Green's function Sturmian expansion (Eq. (5)) (Hostler, 1970). Then transition matrix elements (Eq. (16)) may be rewritten in the form (Karule, 1971, 1975)

The Sturmian expansions for transition matrix elements are convergent if all intermediate states have negative energies, as in the case of multiphoton ionization with the minimum energetically necessary number of photons.

B. RADIALINTEGRALS

First, we have to evaluate the radial integrals. In the Sturmian matrix elements (Eq. (17)), the integrals over radial variables separate and each of them is dependent only on four quantum numbers, namely m,n, L, and 1. Let us write each radial integral in a form

+ 1)!(21+ I)!]-' eip[ -(!P + 4

f(m,L,p ; n, 1, 4) = -2L+'+2[(2L x

JOm

!)]++l+3

- m + L + 1,2L+2,-

According to the selection rules for orbital angular momenta L = 1 & 1. In my papers, I derived and used different formulas for the radial integrals. Formula I . First, let us assume that L = I + 1 and then, using twice recurrence relation (Erdelyi et al., 1953), zF(a, c, z ) = (c - l)F(a, c - 1, z ) - F(a - 1, c - 1, z )

(19)

MULTIPHOTON IONIZATION OF ATOMIC HYDROGEN

277

to the confluent hypergeometric function with the parameter c = 2 L + 2 in Eq. ( 1 8 ) and integrating over r (Landau and Lifshitz, 1965), we obtain f ( m , L, p ; n, I, q ) = ( - 1)L-m221+2pL-1[pq/(q -P)]''+~

x [21+ l ) ! ] - l z - -

2

s=o

x 'Fl(l - m

+ s, 1 - n + 1,21+ 2; 1 - z'),

L=l+l

+

where z = (q p)/(q - p). If L = 1 - 1 m, L, p and n, I, q must be interchanged. Expression (20) was used in papers (Karule 1971, 1975). Formula 2. A more convenient expression may be obtained if in the case L = 1 - 1, instead of Eq. (15), another recurrence relation is used cF(a, c, z ) = aF(a

+ 1, c + 1 , z ) - ( a - c)F(a, c + 1, z).

(21)

Then we have f ( m , L, p ; n, 1, q ) = f ( n , 1, q ; m, L, p ) = (- 1)L-"221+2pL-'

x [pq/q - p)]"+4[(21 + l ) ! ] - l z - n s=o

x [ ( L- m

+ l),(s - m - L - 2)2-s

x,F,(-m+l+s,

(22)

-n+l+1,21+2;1-2').

Expression (22) is more convenient as in both cases L = 1 & 1 the same hypergeometric series must be evaluated (Karule, 1988). Formula 3. If one of the confluent hypergeometric functions in Eq. ( 1 8 ) is written in a form of a hypergeometric series, we obtain after the integration over the radial variable an expression in the form of Appell's hypergeometric function of two variables

+ + x F 2 ( L + 1 + 4, - m + L + 1, -n + 1 + 1,2L + 2,

f ( m , L, p ; n, 1,q) = -2L+1+2(2L 2)1-L+2[(21 ~ ) ! ] - ' [ p q / ( + p q ) ~ ~ + ' + ~ 21 + 2 ; 1

+ z-1,1

- z-1 1

(23)

E. Karule

278

where F , is Appell's function of two variables (Erdelyi et al., 1953). Formula 4. Integral (18) is a special case of integrals J:P(a, a') =

Jorn

exp( - h ~ ) z ' - ' + ~ F ( ac,, kz)F(a', c - p , k'z)dz

(24)

where the values of the parameters are supposed such that the integral converges absolutely; s and p are positive integers, Re h > 0. In the method proposed by Gordon (1929), the general formula for such integrals may be obtained but it is so complex that it cannot be used conveniently. Using a different method of derivation we can obtain the general formula for the integral (24) in a simple form. Let us use for one of the confluent hypergeometric functions the integral representation (Erdelyi et al., 1953)

Then we can rewrite the integral (24) in the form

where I:P(a', t ) =

Jorn

exp[-(h - kt)z]z'-'+'F(a', c - p , k'z)dz.

(27)

The integral (27) contains only one hypergeometric function and is easy to evaluate (Landau and Lifshitz, 1965)

The integral over t is the integral representation for Appell's series F , of two variables (Erdelyi et al., 1953) -a'-m

( h - k')

(29)

279

MULTIPHOTON IONIZATION OF ATOMIC HYDROGEN

Using Eqs. (26), (28), and (29) we obtain the formula for the integral (24) in the form of Appell's functions which are transformed into series of the same type to obtain the general formula in the form

- p),(k')" (c - p),(k' - h)"rn!

1

(d)"(-S

a, - s - rn, a'

+ rn, c ; -

As the second parameter of the F, function is a negative integer, we have the integral in terms of Gauss' functions.

There are no restrictions in Eq. (30) for a, c and c - a due to the gamma functions in Eq. (25), (26), and (29) as these functions appear in intermediate steps of derivation but do not appear in the final expression (30). In a case s = p = 0 and h = (k + k')/2 we have J,Oo(a,a') = 2T(c)(k

+ k')'+O'-'(k'- k)-"(k - 4kk' (k' - k)'

Equation (32) coincides with Eq. (f. 13) given by Landau and Lifshitz (1965) for this case. For integral (18), which is a special case of integral (24), we have

+

f ( m , L, p ; n, 1, q ) = -2L+1+2(2L 2)1-L+zC(21 + 1)!]-'Cpq/(q (- l)-m+L+IZ-m-n-2

-P

)]~+'+~

+ 1 + 1),(1- L - 2),(1 - z)s s=o (21 + 2),s! x F,(-rn + L + l , L - 1 - 2 - s, - n + 1 + 1 + s, 2L + 2; 1 + z, 1 - z2). (-n

Parameters rn, L, p and n, 1, q in Eq. (33) may be interchanged.

280

E. Karule

C. RECURRENCE RELATIONSFOR THE STURMIAN TRANSITION MATRIX ELEMENTS If we use expression (16) to calculate transition matrix elements, it is necessary to evaluate two matrix elements in the case of two-photon ionization of atomic hydrogen by linearly polarized light. With increasing number of photons, the number of transition matrix elements that have to be calculated rapidly increases. For K = 4, one must evaluate 6; for K = 6 one must calculate 20, but for K = 8, one must calculate 70 different matrix elements. The use of the Sturmian expansion (17) for transition matrix elements allows one to perform a summation over all angular variables step by step. Hence, for K = N (N is the minimum energetically necessary number of photons for ionization) we obtain the total transition matrix element in the form

m

where B(L,, L,- 1) = 1/{4 - l/[max(L,, L,- 1)]2}1/2 and 2C/(2E)L'2is the normalization factor of the Coulomb wavefunction. X j may be calculated recursively down to X, - :

x Xj- l(mj-

By the definition we have

1,

Lj-

MULTIPHOTON IONIZATION OF ATOMIC HYDROGEN

28 1

In the case of ionization by the energetically strictly necessary number of photons, all intermediate states have negative energies and the transition matrix elements (34) converge. For the radial integrals, formulas (20), (22), (23), and (33) may be used. If there are intermediate states in the continuum as in the case of above-threshold ionization, the Sturmian expansion for transition matrix elements diverges.

IV. Analytical Continuation of the Transition Matrix Elements A. TECHNIQUE OF DOUBLE SUM RESUMMATION The Sturmian transition matrix elements (34) contain series in terms of polynomials. Straightforward analytical continuation of such series is impossible. Therefore, in the case of ionization by excess photons, when there are intermediate states with positive energies, resummation must be carried out. If only one additional photon is absorbed (K = N + S , S = l), expression (34) for transition matrix elements may be rewritten in the form

where

Y is the divergent part of the AT1 matrix elements because all intermediate states mN are in the continuum and the sum over these states diverges.

E. Karule

282

Our method of the evaluation of the above-threshold transition matrix elements (36) is based on two techniques: the summation over intermediate states with negative energies as described in Karule (1975) and the analytical continuation of the divergent sum (37) over intermediate states in the continuum (Karule, 1978, 1984, 1985, 1988a). To continue Y analytically it is necessary to rewrite it in a form of hypergeometric series of two variables. To carry out the resummation in Eq. (37) one of expressions for f must be written in the integral form f(p, I, p ; m

+ L + 1, L, q ) = -2'+L+2[(21+ l ) ! ( m + I ) , ~ + J - ~

-p

+ 1 + 1,2L +

(38)

where q = qN,L = LN, p = q N + l , l = L N + I . Another expression for f is convenient to write in terms of Appell's functions F, (23). As mN- is an integer, f is a polynomial

LN

- 1-L

+S

+2

X n=O

(L-

LN-1

-s

- 2),(2L (2L

+ 2 + m),(l + 2;')"

+ 2),n!

d(L, q ; 1, p ) = p + 1 + 2 ( - ~ 1 ) ' - L + 2 ( + 2 12 ) L - 1 + 2 [ & ~ + ' + 4 ~

+

Let us divide (mN 2LN which yields

+ + + +

( 2 L 2 m), ( m L 1 - q)

1 + l)!

+ 2), in Eq. (39)by the denominator of Y in Eq. (37)

MULTIPHOTON IONIZATION OF ATOMIC HYDROGEN

283

Therefore, from (37), (38), (39), and (40)we have

A(q, L; p , 1; n) contains the quotient of Eq. (40):

x F( - p

+ 1 + 1,2z + 2;2r/p)M(k, r)dr.

(42)

M(k, I) contains a summation over intermediate states in the continuum in the form of series in terms of Laguerre polynomials. Summation over intermediate states may be performed using summation formulas for Laguerre polynomials (Erdelyi et d., 1953)

x F( - k , 2L

+ 2; [-2r/q(l + Zl)].

Rewriting the confluent hypergeometric function as a sum in Eq. (43), inserting Eq. (43) into Eq. (42), and integrating over the radial variable, we

284

E. Karule

2LN+l

+ 2; 1 - z3)

wherez1 = (qN-1 + q N ) / ( q N - l - qNbndz3 = (%+1 + q N - l ) / ( q N + l - q N - 1 ) . As a polynomial Eq. (44) is valid both below and above the N - photon ionization threshold. Between N - and (N - 1)-photon ionization thresholds, z1 and z3 are pure imaginary. At the N-photon ionization threshold, z1 = - 1 . To obtain T, we have to evaluate B, which contains the reminder of Eq. (40)

where

MULTIPHOTON IONIZATION OF ATOMIC HYDROGEN

285

To perform the summation over intermediate states in the continuum m = mN,L = LN in Eq. (46), the denominator is written in integral form and then the summation formula for the Laguerre polynomials is applied (Erdelyi et al., 1953). Then we replace N ( r ) in Eq. (45) by Eq. (46) and integrate over the radial variable. The resulting Gauss’s hypergeometric function is transformed to obtain the Gauss’s function with argument 1 - X instead of the variable X

=

[

(2LN+1

+ l)!(-qN+l + L N + l + ~ ) L N - L N + I + Z

(q;+l

,>,,+LN+1+4

+

-4s + I - L N + I

-1

(47)

):-

QN+ I - L N - 3

x (1

where z 2 = ( q N +

1

+ qN)/(qN+ 1 - qN).

Gauss’s function in Eq. (47) is a polynomial. To perform the integration over t, we also have to express in the form of a polynomial another function:

k-I+L-2

k)

( 1 - :Y/(1-

where L = LN, q = q N , 1 = LN+i, p = q N + l in Eqs. (48), (49), and (SO). From (47) and (48) we have

L-i+2 [( I - -s=o

L (p

+ + l),

2),(p 1 L - 2),z”,!

1

(49)

286

E. Karule

where J(q, p , L; s, k ) is the integral form of Appell's function F , in two variables

J=

Iol

tL-q(

&)

-p-L-s-k

1-

dt

(1 -

+ 1 -q)-'Fl(L + 1 - 4, p x L + 2 - q ; l / z , z , , z2/z1).

= (L

2 ) + + +

p-L-3+s+k

L

s

k

-

1, - p

+ L + 3 - s - k;

where

L-1-n-2,q+L+ly2L+2;1+-

(50)

MULTIPHOTON IONIZATION OF ATOMIC HYDROGEN

287

The convergence of Eq. (51) for T, and hence convergence of the transition matrix elements (36) is dependent on the convergence of the method, which is used to evaluate Appell's functions in Eq. (51).

B. ANALYTICALCONTINUATION REMOTEFROM THE THRESHOLD Appell's functions may be expressed as hypergeometric series of two variables

The hypergeometric series in Eq. (52) converges if both arguments are of modulus less than unity. l/z,z, is of modulus less than unity below as well as above the N-photon ionization threshold, but lzz/z,l < 1 holds only below the threshold. Therefore if the series expansion for F , is used in Eq. (51), T, converges below the N-photon ionization threshold but diverges above it. To get convergent expressions for T, above threshold, we expand Appell's function F , in terms of Gauss's hypergeometric functions

The double series in Eq. (53) may be analytically continued in the form of a series in terms of other Gauss's hypergeometric functions. Above the N photon ionization threshold 1z21 > 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.

MULTIPHOTON IONIZATION OF ATOMIC HYDROGEN

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

MULTIPHOTON IONIZATION OF ATOMIC HYDROGEN

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

MULTIPHOTON IONIZATION OF ATOMIC HYDROGEN

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

MULTIPHOTON IONIZATION OF ATOMIC HYDROGEN

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.

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Gontier, Y., Poirier, M., and Trahin, M. (1980). J. Phys. B: Atom. Molec. Phys. 13, 1381. Gontier, Y., and Trahin, M. (1968) Phys. Rev. 172, 83. Gontier, Y., and Trahin, M. (1971) Phys. Rev. A4, 1896. Gontier, Y., and Trahin, M. (1973) Phys. Rev. A7,2069. Gontier, Y., and Trahin, M. (1980) J. Phys. B: Atom Molec. Phys. 13,4383. Goppert-Mayer, M. (1931). Ann. Phys. (Leipzig) 9, 273. Gordon, W. (1929). Annalen der Physik 2, 1031. Gradshteyn, I. S., and Ryzhik, I. W. (1965). “Tables of Integrals, Series and Products”, Academic Press, New York. Hostler, L. (1964). J. Math. Phys. 5, 591. Hostler, L. C. (1970). J. Math. Phys. 11, 2966. Justum, Y., and Maquet, A. (1977). J. Phys. B: Atom. Molec. Phys. 10, L287. Karule, E. (1971). J. Phys. B: Atom. Molec. Phys. 4, L67. Karule, E. (1975). In “Atomic Processes” (R. K. Peterkop, ed.), pp. 5-24, Zinatne, Riga (in Russian). Karule, E. (1977). In “Multiphon Processes,” Proc. 1st Intern. Conf., Rochester, 1977 (J. H. Eberly and P. Lambropoulos, eds.), pp. 159-169, J. Wiley & Sons, New York. Karule, E.(1978). J. Phys. B: Atom. Molec. Phys. 11,441. Karule, E. (1984). In “Nonlinear Processes in Two-Electron Atoms” (N. B. Delone, ed.), pp. 209-235, USSR Academy of Sciences, Moscow (in Russian). Karule, E. (1985). J. Phys. B: Atom. Molec. Phys. 18,2207. Karule, E. (1988a). J. Phys. B: At. Mol. Opt. Phys. 21, 1997. Karule, E. (1988b). In “Abstracts of Contributed Papers of XI Intern. Conf. on Atomic Physics” (C. Fabre and D. Delande, eds.), Paris. Khristenko, S. V., and Vetchinkin, S . I. (1976). Opt. Spektrosk. 40,417. Klarsfeld, S . (1969a). Lett. Nuovo Cimento 1, 682. Klarsfeld, S . (1969b). Letr. Nuovo Cimento 2, 548. Klarsfeld, S . (1970). Left. Nuovo Cimenro 3, 395. Klarsfeld, S., and Maquet, A. (1974). J. Phys. B: Atom. Molec. Phys. 7 , L228. Klarsfeld, S., and Maquet, A. (1979a). Phys. Lett. 73A, 100. Klarsfeld, S., and Maquet, A. (1979b). J. Phys. B: Atom. Molec. Phys. 12, L553. Klarsfeld, S., and Maquet, A. (1980). Phys. Lett. 78A, 40. Kruit, P., Kimman, J. and Van der Wiel, M. J. (1981). J. Phys. B: Atom. Molec. Phys. 14, L597. Lambropoulos, P. (1972a). Phys. Rev. Leu. 28, 585. Lambropoulos, P. (1972b). Phys. Rev. Lett. 29,453. Lambropoulos, P. (1985). Phys. Rev. Lett. 55, 2141. Lambropoulos, P., and Tang, X . (1987). J. Opt. Soc. Am. B4, 821. Landau, L. D., and Lifshitz, E. M. (1965). “Quantum Mechanics”, 2nd ed. Pergamon, New York. Laplanche, G., Durrieu, A., Flank, Y., Jaquen, M., and Rachman, A. (1976). J. Phys. B: Atom. Molec. Phys. 9, 1263. LuVan, M., Mainfray, G., Manus, C., and Tugov, I. (1973). Phys. Rev. A7,91. Manakov, N. L., Marmo, S. I., and Fainshtein, A. G. (1984). Teor. and Math. Physics 59, 49. Mapleton, R. A. (1961). J. Math. Phys. 2,478. Maquet, A. (1977). Phys. Rev. A15, 1088. Muller, H. G., van Linden van de Heuvell, H. B., and van der Wiel, M. J. (1986). Phys. Rev. A34, 236. Podolsky, B. (1928). Proc. Natl. Acad. Sci. US 14, 253. Rapoport, L., Zon, B., and Manakov, N. (1969). Zh. Eksp. Teor. Fir. 56,400; Sou. Phys. - JETP 28,480. Rotenberg, M. (1970). Ado. Atom. Molec. Phys. 6,233. Schwartz, C., and Tieman, J. J. (1959). Ann. Phys. ( N . Y.) 6, 178.

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Schwinger, J. (1964). J. M u d . Phys. 5, 1606. Shakeshaft, R. (1985). J. Phys. E: Atom. Molec. Phys. 18, L611. Shakeshaft, R. (1986a). Phys. Rev. A34, 244. Shakeshaft, R. (1986b). Phys. Rev. AM,5119. Shakeshaft, R. (1987). J. Opt. SOC.Am. B4,705. Voronov, G. S. and Delone, N. B. (1965). Zh. Eksp. Teor.Fiz. Left. 1, 42. Voronov, G. S. and Delone, N. B. (1966). Zh. Eksp. Teor.Fiz.Lett. 50, 78. Wolf, B.,Rottke, H., Feldman, D., and Welge, K. H. (1988). 2. Phys. DlO, 35. Zernik, W. (1964). Phys. Reo. 135, 51. Zemik, W. (1968). Phys. Rev. 176,420. Zemik, W., and Klopfenstein, R. W. (1965). J. Math. Phys. 6, 262. Zon,B. A., Manakov, N. L., and Rapoport, L. P. (1971). Zh. Eksp. Teor.Fiz. 61,968; (1972) Sou. Phys. - JETP 34, 515.

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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

CONTENTS OF PREVIOUS VOLUMES

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

CONTENTS OF PREVIOUS VOLUMES

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

CONTENTS OF PREVIOUS VOLUMES

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.

CONTENTS OF PREVIOUS VOLUMES

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

CONTENTS OF PREVIOUS VOLUMES

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

CONTENTS OF PREVIOUS VOLUMES

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