Advances in Atomic and Molecular Physics, Volume 16

Advances in

ATOMIC A N D MOLECULAR PHYSICS

VOLUME 16

CONTRIBUTORS TO T H I S VOLUME R. J. CELOTTA M. COHEN

R. DUREN E. N. FORTSON R. J. HUTCHEON M. H. KEY R. P. McEACHRAN

B. L. MOISEIWITSCH D. T. PIERCE S. SWAIN

L. WILETS

ADVANCES I N

ATOMIC AND MOLECULAR PHYSICS Edited by

Sir David R. Bates DEPARTMENT OF APPLIED MATHEMATICS A N D THEORETICAL PHYSICS THE QUEEN’S UNIVERSITY OF BELFAST BELFAST, NORTHERN IRELAND

Benjamin Bederson DEPARTMENT OF PHYSICS NEW YORK UNIVERSITY NEW YORK, NEW YORK

VOLUME 16

@

1980

ACADEMIC PRESS A Subsidiary of Harcourt Brace Jovanovich, Publishers

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London

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COPYRIGHT @ 1980, BY ACADEMIC PRESS,INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED I N ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM T HE PUBLISHER.

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ISBN 0- 12-003816-.1 PRINTED IN TH E UNITED STATES OF AMERICA

80818263

987654321

Contents

ix

1.IST OF CON.1RlBUTORS

Atomic Hartree-Fock Theory M . Cohen and R. P. McEachran 1. Introduction 11. The Hartree-Fock Method 111. Properties of Hartree-Fock Wave Functions

IV. V. VI. VII.

Properties of the Frozen Core Approximation The Extended Frozen Core Approximation Improved Frozen Core Approximations Conclusions Appendix: Relativistic Corrections to the Energy Levels References

2 4 12 16

23 34 49 50 52

Experiments and Model Calculations to Determine Interatomic Potentials R . Diiren 1. Introduction 55 11. Electronic Model Potentials and Interatomic Potentials 58 I l l . Experimental Sources 70 IV. Interatomic Potentials Determined with Model Potentials 91 V. Conclusions 96 References 97 Note Added in Proof 100

Sources of Polarized Electrons R. J . Celotta and D . T . Pierce 1. Introduction 11. Source Characteristics 111. Chemi-ionization of Optically Oriented Metastable Helium V

I02 I 04 107

CONTENTS

vi

IV. Photoionization of Polarized Atoms V. The Fano Effect Source VI. Field Emission from Ferromagnetic Europium Sulfide on Tungsten VII. Low-Energy Electron Diffraction VIII. Photoemission from GaAs IX. Summary References

112 1 I6 120 i27 134 152 154

Theory of Atomic Processes in Strong Resonant Electromagnetic Fields S . Swain I . Introduction 11. Master Equations 111. Resonance Fluorescence

IV. The Optical Autler-Townes Effect V. Conclusion References

159 165 171 190

196 196

Spectroscopy of Laser-Produced Plasmas M . H . Key and R , J . Hutcheon

I. Introduction 11. Ionization 111. Population Densities of Bound Levels IV. Intensity of Line Radiation V. Line Broadening VI. Continuum Emission VII. Radiative Transfer VIII. Structure and Spectroscopic Characteristics of Laser-Produced Plasmas IX. Spectroscopic Diagnostics of Laser-Produced Plasmas References Note Added in Proof

202 203 213 217 225 234 238 246 25 1 272 280

Relativistic Effects in Atomic Collisions Theory B . L. Moiseiwitsch I. Introduction 11. Excitation and Ionization 111. Electron Capture

References

28 1 282 307 3 I6

CONTENTS

vii

Parity Nonconservation in Atoms: Status of Theory and Experiment

E . N . Fortson and L . Wilets I. Introduction 11. The Neutral Current Interaction in Atoms 111. Observable Effects IV. Atomic Calculations V. Optical Rotation Experiments: Bismuth VI. Stark Interference Experiments: Cesium and Thallium VII. Atomic Hydrogen Experiments VIII. Conclusions References

INDEX C O N T E N T S O F PREVIOUS V O L U M E S

3 I9 32 1 324 328 338 357 367 3 70 37 1 375 387

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List of Contributors Numbers in parentheses indicate the pages on which authors’ contributions begin.

R. J. CELOTTA, United States Department of Commerce, National Bureau of Standards, Washington, D. C. 20234 (101) M. COHEN. * Institute for Advanced Studies, The Hebrew University, Jerusalem, Israel ( I ) R. DUREN, Max-Planck-Institut fur Stromungsforschung, D3400 Gottingen, West Germany (55) E. N. FORTSON. Department of Physics, University of Washington, Seattle, Washington 98195 (319) R. J. HUTCHEON,+ Physics Department, University of Leicester, Leicester, England (201) M. H. KEY, Science Research Council, Rutherford and Appleton Laboratories, Chilton, Didcot, Oxfordshire 0x1 1 OQX, England (201)

R. P. McEACHRAN,~Institute for Advanced Studies, The Hebrew University, Jerusalem, Israel ( I ) B. L. MOISEIWITSCH. Department of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Belfast BT7 INN, Northern Ireland (281). D. T. PIERCE, United States Department of Commerce, National Bureau of Standards, Washington, D. C. 20234 (101)

S. SWAIN, Department of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Belfast BT7 INN, Northern Ireland (159) L. WILETS. Department of Physics, University of Washington, Seattle, Washington 98195 (319) * Permanen[ address: Department of Physical Chemistry. The Hebrew University, Jerusalem. Israel. .pPresent address: Nuclear Power Company (Whetstone) Ltd.. Cambridge Road, Whetstone, Leicester. England. Permanent address: Department of Physics, York University, Toronto. Canada. IX

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li

.

ADVANCES IN ATOMIC A N D MOLECULAR PHYSICS VOL . 16

ATOMIC HARTREE-FOCK A4. COHEN* and R . P . M c E A C H R A N t Institute for Advanced Studies The Hebrew University Jerusalem, Israel

I . Introduction . . . . . . . . . . . . . . . . . . . . . . 2 I1. The Hartree-Fock Method . . . . . . . . . . . . . . . . . 4 A. Central Field Spin Orbitals . . . . . . . . . . . . . . . . 4 B . Derivation of the Hartree-Fock Energy . . . . . . . . . . . . 6 C . Derivation of the Radial Equations . . . . . . . . . . . . . 8 D . Integrals of the Radial Equations . . . . . . . . . . . . . . 10 E. Solution of the Radial Equations . . . . . . . . . . . . . . 11 I11. Properties of Hartree-Fock Wave Functions . . . . . . . . . . . 12 A . Ionization Potentials . . . . . . . . . . . . . . . . . . 12 B . Orthogonality of Excited State Functions . . . . . . . . . . . 13 C . Brillouin’s Theorem . . . . . . . . . . . . . . . . . . . 14 IV . Properties of the Frozen Core Approximation . . . . . . . . . . . 16 A . Ionization Potentials . . . . . . . . . . . . . . . . . . 16 B . Orthogonality of Frozen Core Functions . . . . . . . . . . . 17 C. Some Results of Frozen Core Calculations . . . . . . . . . . . 18 V . The Extended Frozen Core Approximation . . . . . . . . . . . . 23 A . Derivation of the Energy Expression . . . . . . . . . . . . . 24 B. The EFC Valence Radial Equation . . . . . . . . . . . . . 26 C . Orthogonality of the EFC Wave Functions . . . . . . . . . . . 28 D . Some Results of EFC Calculations . . . . . . . . . . . . . . 29 VI . Improved Frozen Core Approximations . . . . . . . . . . . . . 34 A. Multiconfiguration Frozen Cores . . . . . . . . . . . . . . 34 B . Valence Orbitals for Excited States . . . . . . . . . . . . . 37 C . The Ground State Problem . . . . . . . . . . . . . . . . 41 D . Polarized Frozen Core Approximations . . . . . . . . . . . . 43 E. More Elaborate Procedures . . . . . . . . . . . . . . . . 48 VII . Conclusions . . . . . . . . . . . . . . . . . . . . . . 49 Appendix: Relativistic Corrections to the Energy Levels . . . . . . . 50 References . . . . . . . . . . . . . . . . . . . . . . . 52

Permanent address: Department of Physical Chemistry. The Hebrew University. Jerusalem. Israel . Permnenf address: Department of Physics. York University. Toronto. Canada.

.

Copyright 0 1980 hy Academic Press Inc. All rights of reproduction in any form reserved. ISBN 0 -12 -003816-1

2

M . Cohen and R. P. McEachran

I. Introduction The Hartree-Fock (HF) approximation, in its many variations, forms the basis of the overwhelming majority of calculations of atomic, molecular, and crystal structures. The underlying physical assumptions and the main mathematical procedures were worked out in detail some fifty years ago, but it is only during the past twenty years that extensive variational calculations on excited as well as ground states of many-electron atoms have provided adequate data for comparison with experiment and to assess the strengths and weaknesses of the H F procedures. The rapid advances in digital computer technology during this latter period have resulted in the development of the numerical techniques and calculation procedures that have made tractable the solution of sets of equations much more complicated than the original H F integro-differential equations. At the same time, a number of fully automatic H F computer programs have become generally available and have been used to calculate numerous properties of atoms and ions. The recent rapid advances on the computational front have not been matched on the side of the theory, probably due to a widespread belief that explicit treatment of correlation effects will normally yield more accurate wave functions and calculated properties. Furthermore, there has been a natural tendency to prefer general expansion methods which may be applied equally to atoms, molecules, and crystals. This article deals exclusively with applications of atomic H F theory to calculations of atomic properties and in particular to oscillator strengths for electric dipole transitions. Much of the basic theory of the H F method has been described in detail in the monographs of Hartree (1957), Slater (1960), and more recently, Froese Fischer (1977), and we shall make extensive reference to these sources. Consequently, we have made no attempt to give a comprehensive description of the H F method. Rather, we take the opportunity to describe the development and application of a number of particularly simple variants of the H F approximation, namely the frozen core (FC) procedures. These F C approximations share with other H F approximations the ability to predict one-electron properties of atoms (including the rather sensitive oscillator strengths) with remarkable accuracy. The development of the various FC approximations parallels the earlier development of the H F approximations, and we will attempt to underline the similarities as well as the differences at each stage of the development. Conceptually, F C approximations emphasize the physical role of the valence electrons in determining many of the physical and chemical

ATOMIC HARTREE-FOCK THEORY

3

properties of atoms and molecules. For some purposes, the properties of a single valence electron are of crucial interest. For example, the ionization energy required to remove a valence electron from a neutral atom, leaving behind an ionized core, is calculated directly in FC procedures; the famous result of Koopmans (1933) emerges naturally from the FC theory. The formal mathematical properties of the elementary F C approximation, which applies strictly to the alkali atoms (McEachran et al., 1968), and some others with genuinely one-electron spectra (McEachran and Cohen, 1971) have been derived simply from those of the restricted Hartree-Fock (RHF) approximation. [We adopt the nomenclature of Froese Fischer (1977) in describing various H F approximations; we return to these names in the following sections.] For systems with complex spectra, and in particular for the effective description of valence shells containing two or more equivalent electrons, an extension of the theory is required, which closely parallels the extended Hartree-Fock (EHF) theory. The main novel feature of the extended frozen core (EFC) procedure lies in the fact that its wave functions contain nonorthogonal orbitals which therefore introduce overlap integrals as well as new one- and two-electron integrals into the energy expressions, leading ultimately to slightly more complicated equations for the radial functions. Even the derivation of the energy expression is no longer straightforward in the presence of nonorthogonal orbitals, but a general procedure has been developed (Jucys, 1967), and its application within the FC framework yields a description that retains much of the simplicity of the elementary FC model. In the last few years, there has been an increasing number of multiconfiguration Hartree-Fock (MCHF) calculations, whose main aim is to improve the H F treatment of electron correlation. Although the resulting wave functions often yield more accurate calculated properties than the corresponding H F wave functions, the simple orbital description is lost. By contrast, a relatively simple model in which the valence electron is attached to a multiconfiguration core is conceptually more attractive. We have felt it worthwhile to describe one such multiconfiguration frozen core (MCFC) approximation, even though its application presents serious difficulties in some ground state calculations. No such difficulties arise for excited states, however, and we present the results of some MCFC calculations for comparison with the corresponding FC and MCHF results. Efforts to improve the accuracy of H F calculations, either by including explicit correlation terms in the trial wave functions or through the use of several configurations, inevitably lead to a physically complicated model. A simpler procedure is to introduce a semiempirical polarization potential into the equation which describes the active valence orbital. Although such

4

M . Cohen and R. P. McEachran

a polarized frozen core (PFC) procedure is not easy to justify a priori, its results for some simple systems are considerably more reliable than those of the elementary FC procedure. In the following sections, we describe the various FC procedures in some detail, illustrating their results by comparison both with the results of the corresponding H F procedures and with more accurate theoretical or experimental values whenever these are available.

11. The Hartree-Fock Method A. CENTRAL FIELDSPIN ORBITALS The spin-free nonrelativistic Hamiltonian operator that describes the motions of an N-electron atom or ion of nuclear charge Z may be written, in atomic units, as a sum of one- and two-electron operators:

The ith electron coordinate ri is measured from the nuclear position (which we assume fixed) and the interelectronic distances are rq = Iri - I.‘, The presence of the two-electron operators l / r v makes it impossible to obtain exact solutions to Schrodinger’s equation for the N-electron system. Essentially, all the H F methods replace these two-electron terms by approximate one-electron potentials giving an effective Hamiltonian which contains only one-electron operators: N

q2

H eff = I C Hieff= - i = = I

I

I

Z - Vleff vf + (rJ] I

‘i

(2)

The detailed form of V:fr(ri) differs for each variant of the H F method, and it will not be considered further here, except to note that, in general, I/ieff need not be a central field potential (depending only on the scalar distance ri).The exact eigenfunctions of Herrcan now be constructed from products of the single-electron eigenfunctions, ui(ri) of the H:“; they are expected to be approximate eigenfunctions of the physical Hamiltonian H , and should therefore display as many of the properties of the exact eigenfunctions as possible. Since H commutes with the spatial angular momentum operators L2 and L, and also (since it is spin-free) with the spin angular momentum operators S 2 and S,, the eigenfunctions of Herrare required to be eigen-

5

ATOMIC HARTREE-FOCK THEORY

functions of L', L,, s2, a nd S, as well. If H:" is spin-free, it is possible to attach a normalized spin function (conventionally written la) or I p ) ) to each eigenfunction uI(rI),so as to obtain spin orbitals Iu,a,) and IuIpl). Each spin orbital thus requires four coordinates in order to specify it completely (three space coordinates and one spin coordinate), .and those products of N such spin orbitals which are eigenfunctions of the angular momentum operators and are also antisymmetric under interchange of any pair of electrons (Pauli exclusion principle) are the required H F approximation eigenfunctions. The antisymmetry requirement is conveniently met by constructing Slater determinants of the form

I Ul(1)

UI(2)

... udN)

I

and it is understood that each spin orbital contains a specified ( a or p ) spin function. In general, linear combinations of such DN will be required to construct eigenfunctions of L2, L,, S 2 , and S,. For a highly ionized atom ( Z >> N ) , it is clear that the interelectronic terms in H will form only a relatively weak perturbation to the much stronger central field nuclear potential, suggesting that it should be a good approximation to choose yeffas a central field potential. If this can be achieved generally for neutral atoms as well as ions, then the spin orbitals may be written explicitly as products of spherical harmonics Ylm(O,+), spin functions %(a), and radial functions (whose forms are to be determined); thus in the traditional H F method, we find (cf. Hartree, 1957) un/m(r,',+;

0) =

('/r)~n/(r)'/~('~~)%(a)

(4)

We have used m rather than m, for the orbital angular momentum projection quantum number, and s instead of ms for the spin projection quantum number. The traditional form of the radial function, ( l/r)Pn/(r), leads later to some simplification of the radial equations. The dependence on quantum numbers of the radial functions may be more extensive (cf. Froese Fischer, 1977); we return to this point later. The advantage of having spin orbitals of central field type cannot be overestimated. The effect is to reduce the H F problem from N coupled integro-partial-differential equations (one for each orbital) in four variables to N coupled integro-differential equations in a single radial variable. A further substantial reduction results from treating shells of equivalent electrons (having the same values of n and I , but different values of m and s) by means of a single radial function (l/r)Pn/(r). This has the effect of

M. Cohen and R. P. McEachran

6

reducing the number of coupled radial H F equations to M , one for each shell of electrons usually designated by s, p, d, etc. for I = 0, 1,2, etc. The conventional description of the periodic table of the elements is a reflection of this shell structure approximation. As was pointed out by Delbriick (1930), choosing spin orbitals of central field type leads self-consistently to a central potential yeff(r,)whenever an atom consists entirely of closed shells, so that it may be described by a single HF determinant. In other cases, v.ef‘(r,) will not generally be of central field type, but the advantages of a central potential are so considerable that it has become standard procedure to perform a spherical averaging whenever the effective potential is not automatically spherically symmetric, as for open-shell systems, or whenever the MCHF procedure is employed. This averaging is implicit in the H F procedure, which constructs an energy expression out of central field type orbitals, expressing it entirely in terms of radial integrals over the unknown radial functions which are subsequently determined variationally. B. DERIVATION OF

THE

HARTREE-FOCK ENERGY

The orthogonality properties of the spherical harmonics Y/,,,(O, +) and of the spin functions &(a) ensure that two spin orbitals unrm and u,,.~.,,,.~, will be automatically orthonormal, provided that

It is a matter of great convenience to have mutually orthogonal radial functions, for it greatly simplifies the calculation of matrix elements of all types. Since the radial functions are to be determined variationally, the orthonormality conditions such as (5) must be included in the variation by means of Lagrange multipliers. In order to obtain the H F energy expression, it is necessary to calculate a stationary value of EN

where

QN

= <@N

IH I@N)/<@N I@N)

(6)

consists of a suitable linear combination of Slater determinants

D N .

The reduction of matrix elements over determinants containing orthonormal orbitals has been treated in detail by Hartree (1957) and Slater (1960), who has given a general prescription for the H F energy as a sum of two contributions, the “average energy of the configuration” E,,, and a specific multiplet correction term A E ( L S ) . For an N-electron atomic

7

ATOMIC HARTREE-FOCK THEORY

system consisting of A4 shells of electrons in the "configuration" M

{ ( r ~ , I ~ ) ~ ' ( .n.~ l(nMIM)q" ~ ) ~ * I L, S }; *

2I qj= N

(7)

i=

the average energy is given by (Froese Fischer, 1977) E,v

=

2 1,

Mi

c qj

i=

+ $ (qj - l )

z(njlj)

I

M

fk(lj9 lj)Fk(n;li?

I, + /I

i-1 Fo(njl;9n,$)

r=2

2

k=O

j=l

+

2

n,l;)

1

&(Ij,$)Gk(njlj?n,$)

k = I I, - $1

I

(8)

This expression for E,, is seen to contain both one-electron integrals, given by

I(n1) = I(nl,n l )

(9)

I(n1,n'l) =(owPnl(r)H'P,.,(r)dr= (nll H'In'l)

('0)

where

while

and two-electron Slater integrals, given by

F k ( n l , n ' l ' )= R k ( n l , n ' l ' ; n l , n ' l ' )

(12)

G k ( n l , n ' l ' ) = R k ( n l , n ' l ' ;n ' l ' , n l )

(13)

The F k and C k integrals are special cases of the general integral R ( a ,6 ; c , d ) =

s

0

w d r s mds P, ( r )P, ( r )Pb ( s )Pd ( s )U k( r , s ) 0

( 14)

with

U k ( r , s )= s k / r k + ' ( r

> s),

UA(r,s= ) r k / s k + ' ( s > r ) (15)

The numerical coefficients f k , for pairs of equivalent electrons, and gk, for pairs of nonequivalent electrons, result from integrations over angular and spin coordinates. They have been given by Slater (1960) and tabulated by Froese Fischer (1977).

8

M . Cohen and R. P. McEachran

The multiplet correction term, A E( LS), consists entirely of combinations of Slater integrals F k ( n l ,n'l') and G k ( n l ,n'l'), and the numerical coefficients have been listed for many configurations by Slater (1960). In summary, the energy expression for a particular multiplet of the configuration (7), assuming that all orbitals are orthonormal, may be written quite generally EN = E,,

+AE(LS)

(16)

and contains only the three types of radial integrals given in Eqs. (9), (12), and (13). C. DERIVATION OF THE RADIAL EQUATIONS The radial functions, which we have denoted both as In/) and as ( l / r ) P , , , ( r ) ,are now to be determined by means of the variational principle. Since the energy expression EN in (16) has been derived on the assumption that all the orbitals are orthonormal, it is necessary to restrict the variation to achieve this. We therefore introduce Lagrange multipliers A, and vary the composite quantity WN= EN +

2 AV(nili1 nj<) '*J

and note that each A,, automatically multiplies a vanishing integral unless I, = 4. Thus, there will be one diagonal multiplier A,, for each distinct radial orbital, and off-diagonal multipliers Ay between pairs of radial orbitals having the same I-value only. In carrying through the variation of the Slater integrals, it is convenient to introduce the "potential" function Y"(b,d)

=spd.U k ( r , s ) P b ( s ) P d ( s ) 0

Note that our definition of Y k ( b ,d ) differs slightly from the Y " ( b ,d ; r ) of Hartree (1957). In terms of this function, we have quite generally R " ( a , b ; c , d )= ~ m d r P o ( r ) P c ( r ) Y ' ( b , d=) ( a I Y " ( 6 , d )I c )

(19)

0

or, alternatively, R " ( a , b ; c , d )= I " d r P b ( r ) P d ( r ) Y * ( a , c ) = ( b l Y k ( a , c ) I d ) (20) 0

with similar results for F k ( a ,b ) and G k ( a ,6).

ATOMIC HARTREE-FOCK THEORY

9

Each term in (17) must now be varied individually to obtain a stationary expression with respect to arbitrary variations in each radial function Id). Since a given Inf) orbital occurs once in ( n f I n ' f ) , twice in ( n f I n f ) , I ( n f ) , F k ( n f , n ' f ' ) ,and G k ( n f , n ' f ' ) ,and four times in F k ( n f , n f )we have quite generally

S Z ( n f ) = 2(S I H'I n f )

(23)

correct to first order, where we have written 16) as the arbitrary variation in the orbital Inf). For a specific example, we consider members of the boron isoelectronic sequence in the configuration ls'2s'nf 'L. This is a case of a single valence electron outside closed shells and, as we shall see later, is particularly amenable to the FC treatment. The energy expression reduces to E,, and is given explicitly by

E(ls22s2nf;'L)

=2f(ls)

+ 2 f ( 2 s ) + f ( n f ) + Fo(Is, 1s)

+ F0(2s, 2s) + 4{ FO( Is, 2s) - 4 GO( Is, 2s)) + 2( F0(2s,n/)

-

[2(2f+ l)]-lG'(2s,n/)}

(27)

In addition to three diagonal Lagrange multipliers (one for each normalized orbital) there will be off-diagonal multipliers to ensure that (1s 12s) = 0 and also, in the case I n f ) = Ins), to ensure that (1s I n s ) = (2s I n s ) = 0. Carrying through the variations with respect to Ils), 12s), and I n / ) in

M . Cohen and R. P. McEuchrun

10

turn, we obtain the HF equations:

{ HO + YO( Is, 1s) + 2 Y0(2s, 2s) + YO(n/, n l ) - Z l S } P l S =

{ Y o ( l s , 2 s ) + E l s , 2 s } P 2 s + [ 2 ( 2 ~I)]-'{ + ~ ' ( h ~ ~ ) + ~ I S . , /(28) }~,/

{ H O + ~ Y O ( I SI S, ) + y0(2s,2s)+ Y o ( n / , n f ) - c 2 s } ~ 2 s =

{ Yo(ls,2s) + ~ l s , 2 s } ~ +l s [2(21+

1)]-'{ ~ ' ( 2 s , n / )+ c 2 s , n l } ~ n l (29)

{H'+2YO(ls, ls)+2Y0(2s,2s)- €,,}P,/ = (21

+ I)-'{[

Y'(ls,n/)

+ c , s , n l ] ~+[ l s ~ ' ( 2 s , n / )+ E ~ , . , ~ ] P ~ ,(30) }

We have suppressed the r dependence of the radial functions writing P,,/ for Pn/(r),etc., and we have rewritten all the Lagrange multipliers A,, in terms of c,,/,,!/; note that cia.,/ and E ~ occur ~ only , ~ if ~In/) = Ins).

D. INTEGRALSOF

THE

RADIALEQUATIONS

Multiplying Eq. (28) by PI, and integrating, we obtain (making use of the orthonormality of the orbitals) cis= I ( l s )

+ F0(ls, IS) + 2[ F0(ls,2s) - +G0(ls,2s)]

+ { ~ O ( l s , n / ) [2(2/+ l ) ] - ' ~ ' ( l s , n l ) } -

(31)

and similarly, using Eq. (29) E~~ =

I(2s)

+ F0(2s, 2s) + 2[ Fo( IS, 2s) -

! j

+ { F0(2s, n / ) - [ 2(2/ + l ) ] -'G'(2s,

Go( IS, 2 ~ ) ] nl)}

(32)

while from Eq. (30), we obtain en/=

I ( n / ) + 2 { F o ( l s , n f ) - [ 2 ( 2 f + l)]-'G'(ls,nf)} + 2 { ~ ~ ( 2 s , n-l[)2 ( 2 / + 1 ) ] - ' ~ ' ( 2 s , n / ) }

(33)

We see, therefore, that the diagonal Lagrange multipliers are composed of combinations of the same one- and two-electron integrals as is the total energy; they are conventionally described as "orbital energies" and may be related to ionization energies under certain circumstances. We return to this point later. Integral expressions for the off-diagonal Lagrange multipliers may be

ATOMIC HARTREE-FOCK THEORY

11

obtained in a similar way. For example, multiplying Eq. (28) by P,, and integrating, we obtain E , ~ , , ~ = I ( ~ S , ~ S ) +1s,2s)+ R ~ ( ~ Ro(ls,2s;2s,2s) S,~S;

+R0(ls,nl;2s,n/)-[2(2/+

1)]-1R'(ls,2s;nl,n/)

(34)

and an identical result is obtained from Eq. (29). However, if we d o not assume that (1s 12s) = 0 but multiply Eq. (28) by P 2 s ,and Eq. (29) by P I , , subtract, and integrate, we obtain (€2, -

%)(ls 12s) = 0

(35)

Since c l s# eZs. Eq. (35) implies that independent of the value of and it is conventional to choose (1s 12s) = 0 for any value of 'Is.2s = 0. and E ~ cannot ~ , be ~ set ~ equal to zero (when On the other hand, In/) = Ins)). For on multiplying Eq. (28) by Pns,and Eq. (30) by Pis, subtracting, and integrating, we obtain =

-

RO(ls,ns; n s , n s )

(36)

and similarly from Eqs. (29) and (30), c

~= ~ R O(~S, . ~ n s ;~ns, ns)

(37)

That E , ~ may , ~ ~ be set equal to zero is a direct consequence of the fact that each of the Is2 and 2s' shells is filled, whereas the ns shell is only half-filled. These results are completely general (Hartree, 1957). E. SOLUTION OF

THE

RADIAL EQUATIONS

The manipulations of the preceding section require the converged solutions of the coupled nonlinear integro-differential H F equations (28)-(30). These solutions must be obtained by iteration, since the various Y k functions are defined in terms of the solutions of the equations; iteration continues until self-consistency is achieved to some preassigned accuracy. Estimates of all the orbitals are required to set up the equations at each iteration cycle. Initial estimates may be screened hydrogenic orbitals or orbitals obtained by scaling from other similar systems. Full details of the numerical procedures may be found in the monograph of Froese Fischer (1977), which also contains references to many of the computer programs available for atomic structure calculations. We merely observe that it has been found possible to obtain solutions of the H F equations such as (28)-(30) for very many ground and excited

12

M . Cohen and R. P. McEachran

states of most light atoms; these solutions satisfy the usual bound state boundary conditions P,,(O)=O,

P,,,(r)+O

as r + o o

(38)

and the phase may be chosen so that each P,,, is positive for small r. Each radial function has nodes, or zeros, in addition to those at 0 and 0 0 ; as in hydrogen, P,, has ( n - I - 1 ) such nodes. In fact the nodal structure serves to distinguish two different orbitals, P,,, from P,.,. This is of great importance in numerical calculations involving highly excited states. The equations for the state ls22s2n’I 2L are identical in form with those for ls22s2nI 2L in Eqs. (28)-(30), merely having n’l in place of nl throughout. However, this small change has the consequence of making the converged 1s and 2s orbitals slightly different for the two states, on account of the coupling to P,, in one case and to P,,, in the other. Thus the equations for P,, and P,,,, also contain slightly different kernel operators, and this has some undesirable results. The major terms appearing on the right-hand sides of Eqs. (28)-(30) are the “exchange” terms which arise directly from the antisymmetry of the trial wave functions. A term such as Y o ( l s , 2 s ) P 2 ,in (28) may be rewritten, somewhat artificially, with the aid of an “exchange operator” J ( 1 s, 2s) such that J ( Is, 2s)P,, = YO( Is, 2s)P2, (39) and it is clear that a term such as E ~ ~may . be ~ treated ~ P similarly, ~ ~ so that it is possible to rewrite Eq. (28) formally as

{ HO + v,.,ff- E , , } P l s= 0 with similar forms for Eqs. (29) and (30). The equivalence of Eqs. (2) and (28)-(30) is thus made explicit.

111. Properties of Hartree-Fock Wave Functions A. IONIZATION POTENTIALS

When the E,, orbital energy of Eq. (33) is subtracted from the total energy E(ls22s%I; ’L) of Eq. (27). the difference is found to be E(ls22s2; IS): ~ ( l s ~ 2 s ~2 n~ 1->; E,, = 2 ~ ( 1 s+ ) 21(2s) + ~ ‘ ( l sI ,S ) + F ’ ( ~ s , ~ s )

+ 4[ F 0 ( l s ,2s) = E(ls22s2: 1s)

Go(IS, 2 ~ ) ]

(4’)

ATOMIC HARTREE-FOCK THEORY

13

Similarly, when the c2s orbital energy of Eq. (32) is subtracted from the total energy 'L), the difference is found to be the average __ E(ls22s2nl; . energy of the configuration ls22snl: E ( ls22s2d;2L) - cZs = 2Z( IS)

+ Z(2s) + Z(d) + Fo( IS, IS) + 2 [ Fo(

+ 2{F0(ls,nl)-[2(21+

l)]-'G'(ls,nl)}

+ { F0(2s, n l ) - [ 2(21+

l)]-'G'(2s, nl)}

S,

2s) - 4 Go( s, 2 s ) ]

= Eav( 1s22snl) There is a similar result involving cis. These are examples of a general result first discussed by Koopmans (1933) who identified the orbital energies of the H F procedure with ionization potentials for removal of a particular nl electron from an atom. It should be stressed that the energy difference (42), relative to the average energy of the four-electron parent ion, is hardly a physical entity. Furthermore, the energy difference of Eq. (41), which does refer to well-defined physical configurations, assumes that the 1s and 2s orbitals of the parent ion and the atom are identical. But when separate H F calculations are carried out for the ion and the atom, these orbitals will differ slightly, so that the difference of total energies [ E (ls22s2n1;'L) E( ls22s2;IS)] and the nl orbital energy E,, are no longer precisely equal. In spite of these difficulties, the orbital energies of the H F procedure frequently provide reasonable approximations to the appropriate experimental ionization potentials; we show in the following section that the FC procedures yield more reliable results.

OF EXCITED STATEFUNCTIONS B. ORTHOGONALITY

The entire H F procedure is based on the variational procedure, applied to EN of Eq. (6); and for the ground state of any system or for the lowest state of any symmetry, EN is a rigorous upper bound to the true energy. However, for a general excited state, EN will not provide an upper bound to the energy unless the trial function Q,, is taken orthogonal to all lower-lying exact eigenfunctions 9, ( m = 0,1, . . . , n - 1). These \k, are, of course, unknown so that the orthogonality requirements can be met only approximately, by constraining a,, to be orthogonal to all lower-lying approximate (HF) eigenfunctions Q,,, (rn = 0,1, . . . , n - 1) of the same symmetry. In practice, H F excited state calculations have usually ignored the orthogonality requirement. The resulting EN is not a rigorous upper

14

M . Cohen and R. P. McEachran

bound, although in practice it will often provide an effective bound. (An appa.rent exception is the lsns 'S series of the helium atom whose total HF energies lie below experiment; we consider this case further later.) A rigorous upper bound to an arbitrary excited state may be obtained, provided that the trial functions satisfy (MacDonald, 1933)

Thus, the H F wave functions of different states of the same symmetry should, in principle, be constrained to satisfy Eqs. (43). This might be achieved through inclusion of some additional Lagrange multipliers (Cohen and Kelly, 1966), but the procedure is cumbersome and has not been applied generally. However, some of the conditions (43) may be satisfied accidentally without imposition of explicit constraints; this can be seen most easily for the overlap integral, (@,, I @,), as follows. The energy expression and H F equations of an excited state ls22szm/*L of the boron isoelectronic sequence can be written down from (27) by replacing n/ by m/ throughout. The two pairs of Is and 2s orbitals will be slightly different for the two states, but if we ignore these small differences, the overlap integral reduces to

I @),


=

(nl I m/>

(44)

Furthermore, if the 1s and 2s orbitals are taken to be identical, then it may be shown from Eq. (30) and its analog for P,, that (en,

- em,)(ml

In/> = 0

(45)

Thus, the desirable orthogonality property follows automatically from keeping the inner "core" orbitals the same in both states. This is the basis of the FC procedures to which we return later. THEOREM C. BRILLOUIN'S The H F eigenfunctions may be regarded either as approximate solutions of the nonrelativistic Schrodinger equation with Hamiltonian H , as in Eq. (I), or as exact solutions of a system described by some suitable effective Hamiltonian Herr,as in Eq. (2). The difference, ( H - Herr),should be quite small, and it is reasonable to suppose that perturbation theory can be applied so as to improve the results of the H F approximation. The first-order correction to any particular H F eigenfunction @,, is a solution of the equation (He"

-

+

E,HF)@i') ( V - Jq'))@,,

=0

(46)

ATOMIC HARTREE-FOCK THEORY

15

where we have written V = H - Herr

(47)

Since @, is the normalized HF eigenfunction, it is clear from Eq. (6) that

(a,, I H I @,,)

= (@,,

I HerrI a,,) = EPF

(48)

so that En’’) = 0. The formal solution of Eq. (46) may thus be written

are the complete set of orthonormal eigenfunctions of Herr where the {L) which satisfy Herr&= ck& and @,,,E,HFare identified with x,,,~,,.The sum in Eq. (49) includes contributions from the continuum, and by virtue of Eqs. (47) and (50), we see that (~lVI@n)=(~lHI@n)

(51)

Now the ( x k ) may be built up out of spin orbitals, some of which occur in the HF approximation Qfl, while others do not. It is convenient to distinguish between “occupied” orbitals which occur in @, and all other “virtual” orbitals, defined as all other orbital eigenfunctions of H:“. It is important to realize that the ( G ) are not, in general, HF approximations to other bound states of the system. In fact most of the “virtual” orbitals lie in the continuum (Kelly, 1963). Nevertheless, if d denotes any xk which differs from @, by replacement of a single occupied orbital by a virtual orbital, then (Brillouin, 1932)

(d IH I

Qn)

=0

(52)

showing that the “states” described by such >ol make no first-order contribution to any property of the system. The original result of Brillouin [Eq. (52)] and its generalizations to charge densities (Merller and Plesset, 1934) and to expectation values of single-electron operators (Cohen and Dalgarno, 1961) were derived rigorously for a ground state which may be described by a single determinant; there may be difficulties in other cases (Bauche and Klapisch, 1972). But whenever Brillouin’s theorem applies and the d are representations of other bound states of the system, it is clear that conditions (43) will be satisfied, so that the HF energy is a rigorous upper bound. We show in the

16

M . Cohen and R. P. McEachran

next section that these conditions are met by the FC approximation eigenfunctions.

IV. Properties of the Frozen Core Approximation The early H F calculations, performed with the aid of hand calculators only, inevitably used the results of one calculation as input data for another closely related application. It was soon found from experience that, in going from the ground state ls22s2 'S of beryllium to the lowest excited states ls22s2p 1.3P, the inner 1s orbitals changed hardly at all, whereas the 2s orbitals differed much more. This empirical result suggests (Fock, 1933) that a satisfactory approximate eigenfunction may be obtained by varying a smaller number of orbitals than in the standard H F approximation. Specifically, in the beryllium 1s22s2p 'v3P example, Hartree and Hartree (1936) set up the trial function precisely as in the H F procedure, but varied the energy expression only with respect to the 2s and 2p radial orbitals, keeping the Is radial orbital fixed. The success of such a procedure can be judged only empirically, by comparing with experimental results, energies and other properties calculated with the resulting F C wave functions. There remains the question of which (and how many) orbitals to vary, and how to choose the fixed orbitals. For convenience, it is obviously desirable to calculate as few new orbitals as possible for each new state of a system considered. This is particularly important when the aim is to calculate the properties of many excited states. For atoms with genuinely one-electron spectra, such as the alkali atoms, it seems reasonable to perform the variation with respect to the single valence electron orbital only. There are a large number of excited states of atoms to which this elementary form of the F C approximation can be applied directly, but we have chosen to give a description for the ls22s2nl 'L states of the boron sequence treated in earlier sections on the H F approximation. A. IONIZATIONPOTENTIALS

In this example, the 1s and 2s orbitals are kept fixed. In order that the nl orbital energy be identified with the ionization energy according to [cf. Eq. (4 I )I c,,, = ~ ( 1 s ~ 2 s ' n'L) l ; - ~ ( 1 ~ ~'s) 2 s ~ ;

(53)

ATOMIC HARTREE-FOCK THEORY

17

we now choose the Is and 2s "core" orbitals to be the HF orbitals of the ls22s2 'Sparent ion ground state. These orbitals are obtained by varying the HF energy expression: E(ls22s2;

1s)= 2Z(ls) + 21(2s) + FO(Is, 1s) + F0(2s,2s)

+ 4[ Fo( IS, 2s) - + Go( IS,2 ~ ) ]

(54)

from which we obtain the standard HF equations: { I f 0 + YO(ls, ls)+2Y0(2s,2s)- q s ) P I , = Y0(ls,2s)P2,

(55)

{ H o + 2 Y O(

(56)

IS, IS)

+

Yo(2s, 2s) - E , , } P,,

=

Yo(IS, 2s)PI,

Note that these equations are slightly different (and simpler) than Eqs. (28) and (29). The nl valence orbital is obtained by varying the HF energy expression, Eq. (27), leading to the orbital equation (30) for P,,!. However, since P I , and P,, now satisfy Eqs. (55) and (56) rather than (28) and (29), the procedure employed earlier may be used to show that we may choose €ls,ns

= €2s,ns = 0

(57)

in the FC procedure, whereas these off-diagonal multipliers are definitely nonzero in the HF procedure [cf. Eqs. (36) and (37)]. Thus, the HF valence orbital equation is slightly nonlinear, whereas the corresponding FC equation is linear, a result of considerable computational significance. It seems probable that off-diagonal multipliers may often be eliminated in the FC model (McEachran et al., 1968). B. ORTHOGONALITY OF FROZEN COREFUNCTIONS The radial equation for a valence m/ FC orbital is thus identical with the HF equation (30), but with ml replacing n l and the off-diagonal multipliers set equal to zero:

{ H ' + 2Yo(ls, IS) + 2Yo(2s,2s) - E,,,~}P,,,~ From the FC equations for P,,' and Pn/, it is easily shown that (En/

- %d)
=0

(59)

while the overlap between the total FC eigenfunctions for two states Qn( 1s22s2nl;,L) and ~,,,(ls22szm/; 'L) with identical 1s and 2s orbitals reduces precisely to the ( n l I m l ) overlap integral. Thus, these excited FC

M . Cohen and R. P. McEachran

18

eigenfunctions are mutually orthogonal, which is also a particular example of a general result. Now it is not difficult to show that

(a,, I H I am> = I(mL n l ) + 2 { R o ( l s , m l ; ls,nl) -[2(21+ l)]-'R'(ls, l s ; m l , n / ) )

+ 2{ R '(2.3, m ~2.9,; n

~-) [ 2(21+ I ) ] - ' R ' ( ~ s ,2s; ml,n/)>

(60)

which may be obtained by analogy with the diagonal energy expression, Eq. (27). But, on multiplying Eq. (58) by P,,' and integrating, we find that the right-hand side of Eq. (60) is equal to cm,(nl I m/)which vanishes on account of the orthogonality condition, Eq. (59). Thus, Brillouin's theorem is satisfied automatically for these FC eigenfunctions, with the result that the total energy for each excited state is a rigorous upper bound.

C. SOME RESULTS OF FROZEN CORECALCULATIONS We now consider some results of calculations using the FC procedure. First, in Table I, we present a comparison of orbital (ionization) and total energies of a number of low-lying ls22s2np'P states of the boron atom, for which we have given the detailed equations in earlier sections. This table compares FC results with the regular H F values and, where these are available, with experimental values (Odintzova and Striganov, 1979). It is seen that the calculated FC and HF orbital and total energies differ TABLE 1

ORBITAL (IONIZATION) AND TOTALENERGIES OF BORONls22s2npzP STATES" Atomic state (ls22s2np) n=2

HF

CIS

c2s

enp

FC Expt.6

cp cp

FC HF Expt.6

-E

-E -E

3

4

5

Ionic state ( 1s22s2)

1.69534 0.49411 0.30986 0.21590 0.30492

8.03815 0.13593 0.01869 0.01863 0.08345

8.10913 0.80089 0.03992 0.03991 0.04165

8.13924 0.82866 0.02420 0.02420 -

-

24.51349 24.52908 24.65901

24.31622 24.31628 24.43151

24.21150 24.21750 24.39519

24.26119 24.26119

24.23159 24.23159 24.35414

OIn atomic units. *Odintzova and Striganov (1919).

-

8.18592 0.81382 -

19

ATOMIC HARTREE-FOCK THEORY

appreciably only for the ground state, but for the excited states, they are virtually indistinguishable. The FC procedure is thus seen to be particularly suitable for excited Rydberg states. But, even for the ground state, the difference between the FC and HF energies is very much smaller than the differences between either calculation and experiment. The total FC energies are quite accurate and differ from experiment by less than 0.5% both for the atom and for the ion. By contrast, the FC ionization energies (given formally by cnp) show relatively large percentage errors of 9 3 6 , 5.896, and 4.2%, respectively, for the 2p, 3p, and 4p valence states. Nevertheless, the absolute errors in the total energies are due almost entirely to the error in the “core” energy, which is simply the H F total energy in this case. Although no significance should be attached to the FC 1s and 2s orbital energies (which are held fixed at their ionic values), it is clear that the HF values are steadily approaching these limiting values as n increases, In Table 11, we present some one-electron expectation values for these states. Here also, the HF values for the 1s and 2s orbitals are steadily approaching their ionic limiting values, the inner Is more rapidly than the outer 2s, as expected. Apart from the ground state, none of the differences between the valence FC and HF expectation values is significant. This confirms that the FC and HF orbitals are themselves very similar, any TABLE I1 ONE-ELECTRON EXPECTATION VALUES OF BORONls22s%p 2P STATES“ ~______

~

_

_

_

_

_

~~

~

Atomic state (ls22s%p) Operator

n =2

3

HF 1s

r--I r

4.6743 0.3259 0.1434

4.6787 0.3253 0.1427

4.6790 0.3253 0.1427

4.6791 0.3253 0.1427

4.6792 0.3253 0.1427

0.7129 1.977 1 4.709 1

0.7764 1.8067 3.8638

0.7788 1.8012 3.8390

0.7795 1.7996 3.8319

0.7802 I .7982 3.8256

0.7756 0.6050 2.2048 6.1461

0.0700 0.1540 8.7509 87.6537

0.025 1 0.0786 17.9392 364.222

0.01 18

0.0478 30.1 102 1019.82

0.6947 0.5785 2.3199 6.8363

0.0700 0.1539 8.7632 87.9039

0.025 1 0.0786 17.9452 364.464

0.01 18 0.0478 30.1 137 1020.06

r2

HF 2s

r-I

r r2

HF np

r-’

r-I r .2

FC np

-’

r r-’

r r2

” In atomic units.

4

Ionic state (1 S22S2)

Orbital

5

M . Cohen and R. P. McEachran

20

slight differences occurring in the asymptotic regions which contribute mainly to ( r 2 > . The relatively poor results for the ground state of boron are not entirely unexpected, indicating that there is a strong interaction between a core 2s electron and a valence 2p electron, these having quite similar energies. A more suitable example for this elementary FC treatment of the ground state is provided by the sodium isoelectronic sequence, with a single valence electron outside completely filled Is, 2s, and 2p shells. In Table 111, we have reproduced (from McEachran et a/., 1969) a comparison of FC and experimental ionization energies for the three lowest members of this sequence. The percentage error here is only 3.7% for Na I, reducing to 2.2% for Mg I1 and to 1.5% for Al 111 for the ground 3s 2S state. The errors for all the excited states are smaller, and moreover, they decrease steadily both with increasing excitation (higher n) and degree of ionization (higher 2). The quality of the FC wave functions themselves (as opposed to the ionization energies) may perhaps be judged by comparing electric dipole TABLE 111

IONIZATION ENERGIES, en,, OF THE SODIUM ISOELECTRONIC S E Q U E N C ~

2S

3s 4s 5s 6s

0.18180 0.07011 0.03704 0.02287

0.18886 0.07 158 0.03759 0.023 13

0.54059 0.23128 0.12855 0.08 174

0.55255 0.23448 0.12977 0.08233

1.02974 0.466 17 0.26609 0.17201

1 .@I549 0.47064 0.26800 0.17303

2PO

3p 4p 5p 6p

0.10944

0.05032 0.02893 0.01878

0.1 1155 0.05094 0.02920 0.01892

0.38374 0.18328 0.10765 0.07084

0.38974 0.18511 0.10846 0.07125

0.79044 0.38762 0.23090 0. I5330

0.80018 0.39080 0.23237 0.15410

2D

3d 4d 5d 6d

0.05567 0.03132 0.02004 0.01391

0.05594 0.03 144 0.0201 1 0.01395

0.22485 0.12649 0.08085 0.05607

0.22680 0.12738 0.08 132 0.05635

0.51204 0.28788 0.18373 0.12726

0.5 1715 0.29010 0. I8489 0.12794

2p

3f 4f 5f

0.03125 0.02000 0.01389

0.03 126 0.02001 0.01390

0.12501 0.08001 0.05556

0.12515 0.08009 0.05561

0.28132 0.18006 0.12504

0.28 178 0.18034 0.12523

atomic units. FC calculations from McEachran er al. (1969). (2) Experimental values from Moore ( 1949). '(1)

21

ATOMIC HARTREE-FOCK THEORY TABLE IV ELECTRIC DIPOLE OSCILLATOR STRENGTHS FOR THE SODIUM SEQUENCE

3s-3p 4P 5P 6P

0.977 0.0 122 0.00172 0.0005 1

0.982 0.0142 0.0022 1 0.00073

0.943 0.00046 0.00 126 o.Ooo90

0.940 0.00023 0.0010

3p-4s 5s 6s

0.168 0.0141 0.00447

0.163 0.0137 0.00437

0.146

3p-3d 4d 5d 6d

0.877 0.0972 0.0298 0.0133

0.83 0.106 0.031 1 0.0140

3d-4p 5P 6P

0.127 b b

0.1 17

3d-4f 5f 6f

1.012 0.157 0.0544

1 .00 0.159 0.055

-

0.875 0.01 1 0.0068

-

0.873 0.0 123 0.007 19 0.00397

0.139 -

0.129 0.0181 0.00643

0.129 -

0.974 0.0397 0.00733 0.00238

0.920

0.94 1 0.00417 b 0.00029

0.937

0.183 0.00476 0.00135

0.178 0.0047

0.174 0.0100 0.00303

0.174 -

0.986 0.160 0.0565

0.950

0.956 0.164 0.059 I

0.96 0.169 0.06 I

0.0177 0.00608

-

__

0.164

0.057

-

-

-

“(1) Geometric mean of FC values from McEachran er al. (1969). (2) From Wiese el al. (1969). bVery weak transitions ( f < lo-‘).

oscillator strengths calculated with them. We defer to a later section our discussion of the detailed forms of the f-values actually calculated, but in Table IV we present a selection of FCf-values which are geometric means of dipole length and dipole velocity forms presented earlier (McEachran et al., 1969). The geometric means are not explicitly energy dependent, but involve two quite different radial transition matrix elements, and should provide a sensitive test of the wave functions. For comparison, we have listed some tabulated recommended values, not all of which are of very high accuracy (Wiese et al., 1969). Nevertheless, the overall agreement is impressive. Although the FC procedure may be expected to improve with increasing Z (the limiting forms of both FC and H F orbitals are ultimately hydrogenic as Z + 03). a nonrelativistic description of heavy atoms and ions based on the Hamiltonian of Eq. (1) must be inadequate. It is therefore necessary to consider relativistic effects, and this may be achieved either

22

ckt. Cohen and R. P. McEachran TABLE V

DIPOLE TRANSITION WAWLENGTHS FOR Fe XVI AND Ni XVIII" Fe XVI

Ni XVIII

Multiplet

Line

(Ub

(2)

(1)

3s-3p

1/2- 1/2 1/2-3/2 1/2-1/2 1/2-3/2 1/2-3/2 1/2-3/2

360.58 336.93 50.60 50.41 36.77 32.19

360.75' 335.3Y 50.55 50.35 36.12 32.16

320.62 293.83 41.23 41.04 29.80

32 1.96d 293.26d 41.23 41.05 29.80

1/2-1/2 3/2- 1/2 1 /2- 1 /2 3/2- 1/2 1/2-1/2 3/2-1/2

62.94 63.12 41.95 42.29 35.76 36.01

62.88 63.72 41.91 42.30 35.11 36.01

50.28 51.01

5 1.02

1/2-3/2 3/2-3/2 3/2-5/2 1/2-3/2 3/2-5/2 1/2-3/2 3/2-5/2 1/2-3/2 3/2-5/2

250.36 269.19 260.9 1 54.1 I 54.75 39.85 40.15 34.88 35.12

25 1.06' 265.W 262.9gE 54.14d 54.73d 39.83 40.13 34.85 35.09

219.89 234.55 23 I .73 43.82 44.33 32.06 32.34 28.00 28.22

219.25d 234.91d 232.4@ 43.81d 44.35d 32.04 32.35 28.00 28.22

3d-5p 6P

5/2-3/2 5/2-3/2

49.04 41.21

48.91 41.17

3d-4f

3/2-5/2 5/2-1/2 3/2-5/2 5/2-1/2 5/2-1/2

66.36 66.48 46.12 46.79 40.30

66.26d 66.37d 46.66 46.72 40.20

52.65 52.78

52.6Id 52.72d

37.09 31.93

37.04 3 1.87

4P 5P 6P 3p-4s 5s

6s

3p-3d

4d 5d 6d

5f 6f

,

I

(2)

50.27

'In angstroms. b ( l ) FC calculations with relativistic corrections (Tull er al., 1971). (2) Experimental values (Fawcett er al., 1966) except where otherwise indicated. 'From Fawcett er al. (1967). dFrom Moore (1952).

ATOMIC HARTREE-FOCK THEORY

23

by means of a suitable fully relativistic theory, or more simply, by application of first-order perturbation theory based on the nonrelativistic FC wave functions. We have followed the latter course and have calculated relativistic corrections to the energy levels of three heavy ions of the sodium sequence, Fe XVI, Co XVII and Ni XVIII (Tull et al., 1971). These calculations involve matrix elements of a first-order perturbation Hamiltonian which is a generalization of the Pauli approximation to the Breit equation for the two-electron atoms (Bethe and Salpeter, 1957). We defer to an Appendix our discussion of the reduction of the calculation to radial matrix elements (Blume and Watson, 1962; Hartmann and Clementi, 1964). Here, it is sufficient to note that for a single electron outside closed shells, the mass variation, Darwin term, and spin-spin interactions together produce a relativistic level shift, while the spin-orbit and spin-other-orbit interactions split the doublet degeneracy (provided that I # 0). In Table V, we have reproduced (from Tull et al., 1971) F C and experimental transition wavelengths for transitions in Fe XVI and Ni XVIII. The success of the FC procedure with relativistic corrections is impressive for these ions.

V. The Extended Frozen Core Approximation In spite of its evident success, the elementary F C procedure is not always directly applicable, particularly for configurations containing equivalent electrons. The simplest example is provided by the ground state of the helium isoelectronic sequence described in the standard H F procedure as 1s’ IS, with a pair of equivalent electrons, both described by a single radial function ( l / r ) P l s , When we identify one of the electrons as a “core” 1s electron and the other as a “valence” (Is)’ electron, they require different radial functions. Clearly it makes no sense to demand that these two 1s-type orbitals be chosen orthogonal, so that it becomes necessary to set up an energy expression based on nonorthogonal orbitals. The extended Hartree-Fock (EHF) procedure (Pratt, 1956; Froese, 1966; Jucys, 1967) leads to equations considerably more complex than the standard H F procedure, but EHF calculations have been performed on a few systems involving open s and p shells. The particular case of a shell of two nonequivalent s electrons nsn’s IS has been treated in detail by Froese (1966); the treatment of Jucys (1967) is more general and applies equally to shells of p, d, f, . . . electrons. As we shall see, its application within the framework of the frozen core model leads to a valence orbital equation not

M. Cohen and R. P. McEachran

24

significantly more complicated than the regular FC valence orbital equation. Furthermore, the convenient properties of the FC procedure are maintained in the extended frozen core (EFC) approximation. A. DERIVATION OF THE ENERGY EXPRESSION

We follow the procedure of Jucys (1967) and derive the E H F energy expression from the H F energy by means of two “correspondences.” These are to be applied whenever a shell of q equivalent nl electrons, all of which are described by a single radial orbital In/) in the H F procedure, are to be treated individually, each with a different radial orbital, denoted by Idi), i = 1, . . . , q in the EHF procedure. The EFC procedure emerges as a special case, since ( q - 1) of the nl electrons remain part of the “core” and will be treated as equivalent electrons, while the remaining single valence nl electron requires a different (nI)’ orbital; the nl and (nl)’ orbitals are not mutually orthogonal. Following Jucys (1967), we have to make the following substitutions in the HF energy expression: 9

2

q R ( l I ) + Sq-’(/) Sq(i,t ; I ) I

I=

and

2 ’2 Sq(i,j , t ; I ) a

(61)

1-1

i [ q ( q - I)]R(/l,ll)-+S[’(I)

J=2

(62)

I = ‘

Equation (6 1) is to be applied to one-electron integrals,

‘(‘1)

= (Pn/(ri)

I t(ri) I ‘n/(ri))

(63)

while Eq. (62) applies to two-electron self-interaction integrals,

‘(‘1,

“1 = (Pn/(ri)Pn/(5)I t(ri*5 ) I Pn/(riY’n/(5))

(64)

S,(I) is a permanent (i.e., a determinant in the expansion of which all the minus signs are reversed) of overlap integrals:

Sq(4=

c 9{(/1

I~,)(/z

I Ip) . *

*

>

(65)

where 29 represents a sum over the Greek subscripts of all permutations of ( L 2 , . . . , 4). Sq(i, 1 ; I ) is obtained by replacing all the overlap integrals from the ith row of S q ( I ) according to I’A)j(ll

1 t I /A)?

A = 1, . . . >

(66)

and Sq(i, j , t ; I ) is obtained from Sq(I) by replacing each product of overlap integrals, one from the ith row and one from the j t h row,

25

ATOMIC HARTREE-FOCK THEORY

according to


I 'A><$ 1 ' p ) j ( ' # $ 1 t I /),Ip),

h,

= 1,

...

9

(67)

Now the standard H F energy expression contains the radial integrals I ( n l ) , F k ( n f ,n / ) , F k ( n l ,n'l'), and G k ( n l ,n ' / ' ) defined in Eqs. (9), (12), and (13). Only the self-interaction integrals of the type F k ( n / ,n / ) require the substitution (62), whereas all other terms in the H F energy transform according to the substitution (61). As an example of this procedure, we calculate the energy of configurations ls22s22p2 3P, ID, IS of the carbon isoelectronic sequence. First we require the H F energy, which in this case contains contributions from both E,, and AE( L S ) ; here, E,"( ls22s22p2) = 21(ls)

+ 21(2s) + 21(2p) + FO( Is, 1s) + F0(2s, 2s)

+ [ F0(2P, 2P) - & F2(2P,2P)]

+ 4[ Fo( IS, 2s) -

Go( IS, 2 ~ ) ]

+ 4[ Fo( IS, 2p) + 4[ F0(2s, 2p) -

G I( IS, 2p)] G ' ( 2 ~2p)] ,

(68)

and

A E ( L S ) = CF2(2p,2p) with 3 C = - - (3P), 25

3 12 25 (ID), 25 (IS)

Since only the two 2p orbitals are to be treated by the extended procedure, it is more convenient to transform the energy difference A&, given by

At;

[ E(ls22s22p2;3P, ID, IS) - E(ls22s2;'S)] = 21(2p) + [ F0(2p, 2p) & F2(2p,2p)] + 4[ Fo( IS, 2p) + 4[ F0(2s, 2p) G '(2s, 2p)] + C F (2p,2P) =

-

-

G I ( IS, ZP)] (71)

Denoting the two normalized 2p orbitals by 2p,, 2p2 and their overlap integral by S , we find from Eq. (65) that S2(2p) = 1 + S 2

(72)

and from (65) and (66) for the one electron operator H',

S2(1, H ' ; 213) = 1(2pI)+ S1(2pI,2p2)

(73)

There is an analogous result for S2(2,H ' ; 2p), so that applying (61) we

M . Cohen and R. P. McEachran

26 obtain

21(2P) + S , '(2P){ 1(2Pl) + 1(2P,) + 2S1(2Pl 2P2)) 9

(74)

Similarly, for the one-electron operator Yo(ls, Is), the result derived from (61) is

2~O(ls,2p)+S,'(2p){ Fo(ls,2pl)+ F0(ls,2p,)

+ 2SR0(ls,2pl; Is,2p,)} (75)

and there are similar expressions to replace G I( 1s, 2p), F0(2s, 2p), and G '(2s, 2p). On the other hand, the operators Yk(2p,2p) must be treated as two-electron operators, and we obtain from (62) Fk(2p,2p)-, S,'(2P){ Fk(2P1JP,) + ~k(2P1JP2>}

(76)

Gathering these results, we have in place of Eq. (71): (1 + Sz) A&

2

2 { Z(2pl) + 2[ Fo( IS, 2pJ -

=

G I ( IS, 2pJ]

i= I

+ 2[ F0(2s, 2pJ - :G

2P;)]}

+ [ F0(2Pl 2P2) + G0(2Pl 2P,)] 9

+(C-

3

t)[ F2(2PIJP,) + G2(2P192P,)]

+ 2 S { 1(2pl,2p,) + 2 [ R O( IS, 2p, ; IS, 2p2) - R I( Is, 1s; + 2[ R0(2s, 2pl ; 2s, 2p2) - R ' ( 2 2s; ~ ~2p192pJ]}

(77)

This is the E H F energy expression when only the pair of orbitals 2 p l, 2p, are allowed to differ, but the Is2 and 2s2 pairs of electrons are treated as equivalent electrons. We note that if these 2p orbitals become identical so that S = 1, Eq. (77) reduces correctly to Eq. (71), when use is made of Eqs. (12) and (13).

B. THEEFC VALENCE RADIALEQUATION The E H F procedure now consists of varying the total energy with respect to each of the four orbitals ls,2s,2pl,2p, subject to the usual orthonormality constraints, but with (2pl I2p,) = S # 0. The resulting radial equations are unpleasantly coupled, although it is possible to solve for the linear combinations 12p,) + S12p2) and 12p2) + S12pI). By contrast, the EFC procedure is quite straightforward.

27

ATOMIC HARTREE-FOCK THEORY

In the spirit of the FC model, we now write 2p for the orbital which belongs to the frozen core, while 2p’ denotes the valence orbital. Then, P,, satisfies the regular H F equation derived as Eq. (30) for the configuration ls22s22p ,Po. (We recall that C I1 is a member of the boron isoelectronic sequence.) We have then, from Eq. (33) E~~

=

1(2p) + 2[ Fo( IS,2p) - i G I ( IS,2p)]

+ 2[ F0(2s, 2p) - i G ‘ ( 2 2p)] ~~ (78)

and analogously, E,,S

=

1(2p, 2p’) + 2[ Ro( IS, 2p; IS,2p’) - i R ‘(IS, 1s; 2p, 2p’)]

+ 2[ R0(2s, 2p; 2 ~2p’) , i R ‘ ( 2 ~2s; , 2p, 2p’)] -

(79)

so that Eq. (77) may be rewritten:

(1 + S2)(A& - E , , ) = E ~ , S ~1 (+2 p ’ ) + 2 [ F o ( l s , 2 p ’ ) - ~ G ’ ( l s , 2 p ’ ) ]

+ 2[ F0(2s, 2p’) - i G ‘ ( 2 ~2p’)I , + [ F0(2p, 2p’) + G0(2p, 2p’)] + ( C - &,[

F2(2P,2P’) + G2(2P, 2P’)I

(80)

This energy expression must now be varied with respect to the 2p’ orbital only, subject to normalization constraints on this orbital and on the total EFC wave function. The overall normalization integral is evidently (1 + S ’), so that we vary W = (1

+ S2)(A& - el,)

- ~ , ~ , ( 2 I&’) p’

- p(1 + S 2 )

(81)

The resulting valence electron radial equation is found to be

{ H ’ + 2 Yo(Is, 1s)

+ 2 Y0(2s, 2s) + Y0(2p, 2p)

+ ( C - b ) Y ( 2P12P) - E2p.}P2p. =f

{ Y ’( Is, 2p’)P,, +

+ (c -

Y ‘(2%2p’)P,,) -

B ) Y2(2P92P’) + E2p,2p’}P2p

{ Y0(2p, 2p’) (82)

and we have written E2p.2,. = (€2, -

PIS

for the off-diagonal Lagrange multiplier. I t may be shown in the usual way

M. Cohen and R. P. McEachran

28

from Eqs. (80) and (82) that ezPa- c ~ ~ = , (A& ~ ~-, cZp) S

+ (A& - 2c2,)SZ

combining this result with Eq. (83) and equating separately the coefficients of S 2 and S o , we identify the Lagrange multiplier p = A& - cZp= c ~ ~ ’

(85)

Comparison of Eqs. (45) and (83) now shows that E ~ is a ~ direct , measure of the deviation from orthogonality of the overlap integral (2P l2P’). We have seen in Eq. (41) that E ( ls22s22p; 2PO)

=

E ( ls22s2;

1s)+ Elp

(86)

so that Eq. (85) shows that Koopmans’ (1933) result on ionization potentials holds rigorously for this EFC case. I t has been established that a similar result holds for the ground states of N(2p22p’; ‘S@) and O(2p32p’; 3P).

An explicit integral expression for Eqs. (79) and (82):

E

~

may~ be ,obtained ~ ~as usual ~ from

so that, as before, the EFC valence orbital equation is linear in Pzp,. C. ORTHOGONALITY OF THE EFC WAVEFUNCTIONS The EFC procedure may be applied directly to excited states of the configurations (2pnp; 3P, ‘D, IS) with nonorthogonaf 2p and np orbitals. The valence np orbitals satisfy Eq. (82), with np replacing 2p‘ throughout. The overlap integral (amI @.,) between total EFC eigenfunctions of two excited states of the same symmetry, (2pmp) and (2pnp), is given by



= Nmn[ (mp

I nP> + (mp I2PX2P I .P>]

(88)

where N,, contains only normalization integrals. Now, from Eq. (82) for Imp) and Inp) in turn, we obtain (cmp

- ‘np)(mP

I ‘P)

= eZp,rnp(’~

I np> - cZp.np(2P I mp)

(89)

while the analog of Eq. (83) is ‘2p.np = (ezp

I

- cnp)(2P ‘P>

(90)

with a similar result for c2p,rnp.Combining the results of Eqs. (89) and (90),

~

~

~

29

ATOMIC HARTREE-FOCK THEORY

we have

and CJm, CJ,, are orthogonal, as required. Similarly, the ground state (2p2p’) is orthogonal to all excited states (2pnp). It is not difficult to show that Brillouin’s theorem also holds for these EFC eigenfunctions, so that rigorous upper bounds are obtained for all the EFC total energies.

D. SOMERESULTS OF EFC CALCULATIONS The simplest system that must be treated by the EFC procedure is the lsns ’S series of the helium isoelectronic sequence. This system has also been treated by the EHF procedure which is much more difficult, even for so simple a configuration (Froese, 1966, 1967), as well as by the H F procedure (with orthogonal 1s and ns orbitals). Table VI provides a comparison of total energies calculated in all three procedures with some essentially exact variational results of Pekeris (1958) and Accad et al. (1971). We have already noted that the excited lsns states calculated in the ordinary HF procedure have energies below their exact values in violation TABLE VI TOTALENERGIES, - E , FOR lsns

‘sSTATESOF THE HELIUMISOELECTRONIC SEQUENCE^

He I

Isls‘ 1s2s

ls3s 1s4s Li I1

Isls’

ls2s I s3s 1s4s

Be I I I

Isls’ 1s2s

I s3s 1s4s

2.9037 2.1460 2.0613 2.0336

2.8617 2.1699 2.0667 2.0356

2.8725 2. I434 2.0605 2.0333

2.8780 2.1435 2.0606 2.0333

7.2364 5.0992 4.7467 4.6347

7.2436 5.0360 4.7323 4.6292

7.2515 5.0364 4.7324 4.6292

4.7338 4.6298

13.6113 9.2796 8.5379 8.2963

13.6167 9.1784 8.5154 8.2877

13.6258 9.1792 8.5156 8.2878

13.6556 9.1849 8.5173 8.2885

7.2799 5.0409

“In atomic units. ‘(I) HF values from Froese Fischer (1977). (2) EFC values from Cohen and McEachran (1967), and unpublished calculations. (3) EHF values from Froese (1966, 1967). (4) Accurate variational values from Pekeris (1958) and Accad er al. (1971).

M . Cohen and R. P. McEachran

30

of the upper bound theorem. This is evidently a consequence of the imposition of the orthogonality constraint on the 1s and ns orbitals which makes the excited state H F eigenfunctions slightly nonorthogonal to the ground state H F eigenfunction, and also causes a small violation of Brillouin's theorem. When the orbitals are nonorthogonal, as in both EFC and EHF procedures, upper bounds are obtained. The differences between EFC and EHF energies are negligible except for the ground states, and the worst discrepancy is only 0.2% for the helium ground state. The extraordinary accuracy of the EFC procedure is probably a consequence of the fact that the ion core is here described exactly by the 1s hydrogenic orbital; we have noted earlier that the main source of error in the F C procedure is in describing the core electrons, but there is clearly no such difficulty in the present case. The situation for the 2sns 'S states of beryllium is quite similar. As will be seen from Table VII, the EFC total energy is slightly lower than the H F total energy for the ground state, whereas for excited states, the EFC and EHF procedures yield essentially identical results. The remaining discrepancy with experiment is only slightly larger than for helium. In Table VIII, we present a comparison of ionization energies of some ground and excited (2pnp; 'P, ID, IS) states of carbon. We note that the EFC total energy of the ground (2p2p'; 'P) state is -37.6784 a.u., slightly higher than the H F value of -37.6886 a.u. (Froese Fischer, 1977); this is mainly due to some inaccuracy in the core energy. The percentage errors in the EFC ionization energies of the three ground configurations are rather large (7.3%for 'P, 9.4%for ID and 17.5%for IS), similar to the error in the boron ground state. Nevertheless, the interval ratio (IS - 'D)/('D 'P) which has the H F value 1.43 has an EFC value of 1.37, slightly closer to the observed ratio, 1.11. It is disappointing that the EFC excited state ionization energies are generally less accurate than the corresponding FC energies, although the differences diminish steadily with increasing n. It is TABLE VII TOTAL ENERGIES, - E , FOR lS22SnS 's STATES O F BERYLLIU~.~

2s2s' 2s3s 2s4s

14.5730 -

14.5754 14.3620 14.3197

14.3622 14.3197

14.6691 14.4200 14.3718

"In atomic units. b(l) HF value from Froese Fischer (1977). (2) EFC values from Cohen and McEachran (1979). (3) EHF values from Froese (1967). (4) Observed values from Johansson (1962).

31

ATOMIC HARTREE-FOCK THEORY TABLE VIII IONlZAllON

2p"P ID

's 3p 'P ID 'S

4p'P ID

IS 5p 'P ID

IS

ENERGIES OF c I 2pnp STATES"

0.38609 0.33320 0.26074

0.41388 0.36758 0.31538

6p 'P ID 'S

0.01706 0.01676 0.01638

0.01667 0.01626 0.01569

0.01712

0.08906 0.08530 0.08068

0.08404 0.07928 0.07297

0.08880 0.08317 0.07695

7p 'P ID 'S

0.01214 0.01197 0.01174

0.01 191 0.01167 0.01132

0.01217 0.01186 0.01152

0.04328 0.04206 0.04053

0.04167 0.04005 0.03783

0.04342 0.04143 0.03924

8p 'P ID

O.Oo909 0.00897 0.00882

0.00894 0.00878 0.00855

O.Oo906 0.00888 0.00867

0.02572 0.02517 0.02447

0.02500 0.02425 0.02321

0.02583 0.02489 0.02387

9p'P

0.00705 0.00698 0.00687

0.00695 0.00684 0.00669

0.00689 0.00675

-

IS

ID

IS

0.01661

0.01605

-

"In atomic units. ' ( I ) FC values from Cohen and McEachran (1978). (2) EFC values. (3) Observed values from Moore (1970).

clear from these results that the 3P, ID, and ' S F C total wave functions of excited carbon states contain admixtures of the appropriate ground state eigenfunctions, which lower their total energies significantly toward the observed energies. These total energies do not lie below experiment only because of the inadequacy of the H F core description; otherwise, the situation in carbon would be identical with that seen earlier for helium lsns IS states. By contrast, the EFC total energies are true upper bounds, and they lie uniformly above the F C total energies. The EFC ionization energies are all bounded from above by the observed ionization energies, whereas the FC ionization energies are not bounded. These carbon results thus serve to emphasize the dangers of performing HF excited state calculations without imposing orthogonality constraints on the overall trial functions. We recently published oscillator strengths for electric dipole transitions between excited states in neutral carbon, based on FC wave functions (Cohen and McEachran, 1978). Reliable data on transitions of the type 2p2p' + 2pns, 2pnd is conspicuously absent, and the few tabulated recommended values (Wiese er al., 1966) are all subject to uncertainties of 50%. We have therefore felt it worthwhile to include our calculated f-values based on EFC wave functions for the 2p2p' ground configurations, and on

M. Cohen and R. P. McEachran

32

TABLE IX ELECTRIC DIPOLE OSCILLATOR STRENGTHS FOR 2p2p'-2pns IN CARBON TRANSITIONS

n

AND

2p2p'-2pnd

4

5

6

7

8

9

2p"P-ns3P0 3.11 (-2)" - n d 3 P 2.18(-2) -nd3D0 7.85(-2)

5.13(-3) 1.07(-2) 3.59(-2)

1.84(-3) 5.73(-3) 1.87(-2)

8.78(-4) 3.38(-3) 1.09(-2)

4.89(-4) 2.15(-3) 6.83(-3)

3.00(-4) 1.45(-3) 4.58(-3)

2.00-4) 1.03(-3) 3.23(-3)

2p"D-ns'p -nd'P" -nd'Do -nd'F'

5.11 (-2) 9.17(-4) 2.35(-2) 7.99(-2)

8.30(-3) 4.26(-4) 1.04-2) 3.63(-2)

2.98(-3) 2.24(-4) 5.28(-3) l.89(-2)

1.43(-3) 1.31(-4) 3.03(-3) l.lO(-2)

7.96(-4) 8.3 (-5) 1.91(-3) 6.97(-3)

5.00(-4) 5.6 (-5) 1.26(-3) 4.62(-3)

3.33(-4) 3.9 (-5) 8.85(-4) 3.28(-3)

2p"S-ns'PO -nd'Po

1.22(-2) 1.41 ( - 1 )

1.49(-3) 6.02(-2)

5.08(-4) 3.06(-2)

2.37(-4) 1.76(-2)

1.33(-4)

8.2 (-5) 7.30(-3)

5.2 (-5) 5.10(-3)

Transition

"3.11 (-2)=3.11

=3

1.10(-2)

X

FC wave functions for the 2pns, 2pnd excited states. Geometric means (see later) of our dipole length and velocity f-values are presented in Table IX, while Table X contains a few of our calculated length and velocity values together with results of other H F calculations (Wilson and Nicolet, 1967). In view of the success of the EFC procedure for helium and beryllium, we are confident that the present calculations will prove reliable for many transitions. For completeness, we include a brief description of our EFC calculation of f-values. TABLE X ELECTRIC DIPOLEOSCILLATOR STRENGTHS FOR 2p2p'-2p3s IN CARBON TRANSITIONS

AND

2p2p'-2p3d

2p' ' p - 3 ~ 'PO -3d -3d 'Do

3.829 (- 2)* 2.385 (- 2) 8.475 (- 2)

2.520 ( - 2) 1.995(-2) 7.275 ( - 2)

1.7 (- I) 2.9(-2) 6.3 (- 2)

1.05 (- 1) 2.60(-2) 7.67 (- 2)

2p' l D - 3 ~'PO -3d'p -3d 'Do -3d 'F'

6.620 (- 2) 1.011 (-3) 2.304 (- 2) 8.828 (- 2)

3.935 ( - 2) 8.31 (-4) 2.397(-2) 7.224 (- 2)

8.2 ( - 2) 7.0 (- 3) 1.1 (-2) 9.3 (- 2)

7.29 (- 2) 7.46 (-4) l.lO(-2) 6.25 (- 2)

1.507(-2) 1.453 (- I)

9.930 ( - 3) 1.375 ( - 1)

9.4 (- 2) 1.2 (- I )

6.76 (- 2) 7.48 (-2)

2p' Is-3s -3d

'Po 'PO

"(1) Present work, dipole length values. (2) Present work, dipole velocity values. (3) From Wiese ef a/. (1966). (4) From Wilson and Nicolet (1967). b3.829 (-2) = 3.829 X

33

ATOMIC HARTREE-FOCK THEORY

The multiplet oscillator strength is given by gmfmn

=

5 ’(-1

AEnma2

(92)

where g, is the statistical weight of the initial state, AE,, is the excitation energy ( E n - Em), and S(5%) denotes the relative multiplet strength, which arises from angular integrations and has been tabulated (Allen, 1976).* Since Koopmans’ (1933) result holds for our EFC energies, AE,,, is given simply as a difference of orbital (ionization) energies. Now the 2p2p’ EFC eigenfunctions contain nonorthogonal orbitals, so that u2 is given, in the dipole length formulation, by the expression u t = [ 3( 1

+ S ’ ) ] ’ { ( n s I r I 2p’) + S ( n s I r I2p)}’ -

(93)

for 2pns + 2p2p‘ transitions, and by u t = [ 15(1 + S2)]-’{(2p’ ( r I n d )

+ S ( 2 p J rInd)}2

(94)

for 2p2p‘ -+ 2pnd transitions. In the dipole velocity formulation, we have, corresponding to Eq. (93) for ns +2p’

and corresponding to Eq. (94) for 2p’ + nd, u; =

1 1 15(1 + S2) (AE,,,)’

((2p’I

5+ f

lnd)

+ S(2p/

5+ f

Ind))’ (96)

The EFC wave functions are, of course, only approximate solutions of Schrodinger’s equation, so that u t and u$ differ in general. The radial matrix elements appearing in u t and u: emphasize two different regions of space, and there has been a long controversy in the literature as to which form should provide the more reliable f-values. However, we observe that by choosing the geometric mean uLuv, we obtain an oscillator strength formula which is independent of the excitation energy, A En,. This form has therefore been adopted generally throughout the present work, but we wish to emphasize that in very few cases do FC or The recently published f-values for transitions between excited states of C I (Cohen and McEachran, 1978) are incorrectly normalized. The correct values may be obtained from the tables by multiplying by the appropriate S ( 3 n ) and dividing by the corresponding row sum,

CS(%).

34

M . Cohen and R. P. McEachran

EFC length and velocity f-values differ by more than 20% from these mean values. However, Table X contains some transitions for which f L and fv are almost equal, but still differ appreciably from the H F and measured values which form the basis of the comparison data, itself of unknown accuracy for carbon. It seems desirable to improve the frozen core model, particularly for the ground states, and we consider a number of possibilities in the following sections.

VI. Improved Frozen Core Approximations A. MULTICONFIGURATION FROZENCORES

Both the FC and EFC procedures yield generally good results, but we have seen that in a number of cases, such as the boron ground (2p) state, the major source of error in the total energy arises from the ion core. A logical way of remedying this defect is to improve the description of the core, and it is desirable to do this in such a way that the conceptual and computational advantages of the F C models are retained as far as possible. There are, theoretically, two basically different approaches possible. Either we may introduce explicit correlation terms, such as rV,into the core wave functions, thereby destroying the single-electron description of the core electrons, or we may employ multiconfiguration core wave functions which retain the orbital model, at least in part. The former procedure is likely to prove more accurate, but the latter is more closely related to the H F method on which the FC procedures are based. Recently, elaborate multiconfiguration Hartree-Fock (MCHF) calculations have been carried through for many atomic systems in an effort to obtain accurate estimates of correlation energies (Froese Fischer, 1977, and many references therein). The MCHF procedure is considerably more complicated than the ordinary H F method, but its success suggests that it may serve as a suitable model for improved core wave functions. A multiconfiguration frozen core (MCFC) procedure may be based on the MCHF core, in the same way as the FC procedure was based on the H F core. In the case of our earlier example of boron states, a two-configuration core wave function is suggested, of the form @(IS) = c , @ l (ls22s2; 1s)

+ C2Q2(ls22p2; IS)

(97)

The two configurations included in @(IS) form a complex (Layzer, 1959), whose wave functions are energetically degenerate in the limit as Z + co,

35

ATOMIC HARTREE-FOCK THEORY

and there is evidence (Clementi and Veillard, 1965) that including Q2 accounts for much of the correlation energy error for all values of Z . The general MCHF procedure is analogous to the ordinary H F procedure, but now the configuration weights [c I and c2 in Eq. (97)] as well as the orthonormal orbitals must be calculated self-consistently. Thus, there are two distinct solution cycles; for given c I and c2, the MCHF equations must be solved iteratively for the orbitals as in the H F procedure and when a converged set has been obtained, new c I and c2 are obtained as the solution of a 2 X 2 secular equation, and the whole process repeated. Although the calculation of weights is a very simple matter, the new iteration required to obtain orbitals at each cycle makes the MCHF process lengthy. For simple systems, such as our boron example, a much more direct approach is possible. As in the MCHF procedure, we set up the total energy expression of the B + core in terms of radial integrals: E ( ' S ) = c:E( 1 ~ ~ 2 s IS) ';

+ c$(ls22p2;

IS)

+ 2clc2(Ql('S)I HI Q2('S)) (98)

where, due to normalization, 1 = c;

+ c;

(99)

The energy expression and the radial equations of the dominant configuration ls22s2 IS have been written down previously in Eqs. (54)-(56). The energy expression for the second configuration ls22p2 'S may be obtained from Eq. (68), by suppressing all the terms which involve 2s electrons. Since it is convenient to have identical 1s orbitals in both configurations, we now vary E(ls22p2;IS) with respect to the 2p orbital only, keeping the 1s orbital fixed from the configuration ls22s2 IS. This procedure is similar in spirit to, but slightly different in practice from, the ordinary F C method. The equation for the resulting radial 2p orbital is found to be the regular H F orbital equation:

{ H ' + 2Yo(ls, IS) +

Y0(2p,2p)

+ $ Y2(2p,2p) - E

4

~ ~ } PY1(lS,2p)Pl, ~ ~ = (109

Now, with the orbitals determined from Eqs. (55), (56), and (lOO), it is only necessary to calculate the off-diagonal contribution to E 1

(QI('S) I H I @2('S))= - - G ' ( 2 ~ , 2 p )

6

and to solve a 2 x 2 secular equation for c I and c 2 . Before presenting

M . Cohen and R. P. McEachran

36

results of this procedure, it is instructive to see exactly what is involved in the complete MCHF calculation. The energy expression (98) is varied with respect to the orbitals subject to the usual orthonormality constraints so that, for any c I ,c2 which satisfy the overall normalization condition (99), the following radial equations are obtained : { HO + YO( Is, Is) + 2cfY0(2s, 2s) + 2cIY0(2p, 2p) - Els}Pls = cf{ Y O ( l s , 2 s ) + € l , 2 ~ } P 2 , + : c f Y ' ( I s , 2 p ) P , ,

{ H o + 2Yo(ls, IS) + =

{HI

( 102)

Y0(2s,2s) - C ~ , } P , ,

{ Yo(ls,2s) + cls,2s}Pls+ (c2/cld~)Y1(2s,2p)P2,

(103)

+ 2 Yo(IS, IS) + Y0(2p, 2p) + $ Y2(2p, 2p) - E ~ , } P,, =

+ Yl(ls,2p)P1, + (c,/c2d~)Y1(2s,2p)P2,

(104)

Equations (102) and (103) reduce to Eqs. (55) and (56) when c I = I , c2 = 0, while Eq. (104) reduces to Eq. (100) when c I = 0, c2 = I , as they obviously should. But the set of equations (102)-(104) is clearly more complicated, and the off-diagonal Lagrange multiplier is not zero when c1 # 1. This has the undesirable consequence of introducing a spurious node into the Is orbital (Froese Fischer, 1977). The set of equations (102)-(104) must be solved for the orbitals, the resulting energy expression (98) minimized to obtain optimal weights (for these orbitals), and the whole process repeated until convergence is achieved. In this simple example, c I is much larger than c2 and c: is not too different from unity. Consequently, the integrals of Eqs. (55) and (102) for c l S and of Eqs. (56) and (103) for c2, differ only by small terms. On the other hand, integrals of Eqs. (100) and (104) for cZp differ by a term (cl/c2dT)G '(2s, 2p) which may be large on account of the ratio c I / c 2 .The observed ionization energy cannot be related to cZpsince the spectroscopic designation is based on the single configuration ls22s2 IS, but should be approximated by c2s. Table XI contains the results of the two calculations of the BII core wave functions from which it is clear that our simplified procedure is entirely adequate in this case. The 1s and 2s orbitals of the two procedures are very similar, judging by the calculated expectation values: on the other hand, one of the 2p orbitals is more compact than the other, as is shown by the smaller values of ( r ) and (r'). Both values of c2, are reasonable approximations to the observed ionization energy of 0.9245 a.u. (Moore, 1949), and both two-configuration calculations account for roughly 50% of the core correlation energy.

ATOMIC HARTREE-FOCK THEORY

37

TABLE XI RESULTS OF TWO-CONFIGURATION CORECALCULATIONS FOR B II‘ Orbital

Property

-E

Total

CI

c2

(2) 8.1859 4.6792 0.3253 0.1427

8.1625 4.6978 0.3229 0.1401

0.8738 0.7802 1.7982 3.8256

0.9312 0.7718 1.7736 3.7021

0.5762

1.1579

1.0459 0.6794 1.9369 4.6832

1.0833 0.7350 1.6820 3.3540

24.28980 0.9585 0.2850

24.29664 0.9569 0.2906

“In atomic units. b ( l ) Solution of Eqs. (55), (56), and (100). (2) Solution of Eqs.

(102)-(104).

Thus, the core description may be improved quite simply, and we now consider the consequences of this improvement on an excited state valence electron orbital.

B. VALENCE ORBITALS FOR EXCITED STATES A two-configuration atomic wave function analogous to @(IS) of Eq. (97) may be written quite generally

@(’L)

=

cial( ls22s2n/;2L)

+ C2Q2(

ls22p’nl; 2L)

(105)

If we exclude the ground state (n/ = 2p’), the elementary FC procedure may be applied directly. Thus, we set up the energy expression, similar to Eq. (98):

E(’L)

=

c:E( ls22s2n/;’L)

+ c$(

ls22p2n/;’L)

+ ~ C , C , ( @ , ( ~I HL )I @2(2L)) (106)

M. Cohen and R. P. McEachran

38

and note that there is no difficulty in choosing In,) orthogonal to all the core orbitals. Furthermore, it may be shown that

identical with Eq. (101) so that, provided that we take the same configuration weights in (97) and (105), the difference between atomic and ionic energies is given by A E = [ E(2L) - E ( ' S ) ] =

ci[ ~ ( l s ~ 2 s ~'L) n l; ~ ( 1 s ~ 2 .IS)]s ~ ;

This energy difference is now varied, subject to the usual orthonormality constraints on the n l orbital, and we easily derive the MCFC valence orbital equation:

{ H' + 2 Yo(lS, IS) + 2c:Yo(2s, 2s) + 2c:Y0(2p,

2p) - E,,,)P,,,

Integrating this equation in the usual way, we find that

Thus, Koopmans' (1933) result holds provided that the same core orbitals are used for both atom and ion. Note that Eq. (1 10) holds for both sets of core orbitals discussed previously. The absence of cross terms in Eqs. (108) and (109) makes their interpretation particularly simple. Effectively, the valence nl electron experiences a superposition of central fields due to the separate core configurations ls22s2 IS and ls22p2 IS with weights c; and cf. In Table XII, we present a comparison of FC and MCFC ionization energies for a number of excited states of boron. The two-configuration values were calculated relative to the simpler core procedure, but are identical with results obtained using the complete MCHF core procedure (Cohen and Nahon, 1980). For ns and nd states, the differences that result from improving the core description are insignificant, and there is no consistent improvemenr of agreement with experiment. For the 3p state,

39

ATOMIC HARTREE-FOCK THEORY TABLE XI1 CALCULATED AND OBSERVED IOMZATION ENERGIES OF B I" Term

(1Ib

(2)

(3)

(4)

2s 3s

0.1 1452 0.05 180 0.02956 0.0191 1 0.01336 O.C.3987 0.00759

0.1 1436 0.05 175 0.02954 0.0 I9 10 0.01336 0.00987 0.00759

0.12252 0.05381 0.03038 0.01952 0.01360 0.01002 0.00769 3.00905

0.12252 0.05431 0.03090 0.02026' 0.01263' 0.00973' 0.00756'

0.27590 0.07863 0.0399 I 0.02420 0.01624 0.01 165 0.00877 0.00684

0.08087 0.04059 0.02449 0.0 1640 0.01 175 0.00883 0.00688

0.30492 0.082 I5 0.04104 0.0247 1 0.01652 0.01 182 0.00888 0.00691 2.97460

0.30492 0.08345 0.04165 -

0.05617 0.03160 0.02020 0.01401 0.0 1028 0.00787 0.0062 1

0.05622 0.03 162 0.02021 0.01402 0.01029 0.00787 0.0062 1

4s 5s 6s 7s 8s 9s P

'PO

2p 3P 4P 5P 6P 7P 8P 9P P

'D 3d 4d 5d 6d 7d 8d 9d

-

-

0.0554Id 0.03 I 6 0 0.02024 0.0 1405 0.01031 0.00789 0.00623

"In atomic units. b ( l ) FC values from McEachran and Cohen (1971). (2) MCFC values. (3) PFC values. (4) Observed values from Odintzova and Striganov (1979). 'Levels perturbed by ls22s2p2'S. dL,evel perturbed by ls22s2p2'D.

there is a more substantial improvement in the ionization energy and we see slightly larger shifts in ionization energies of most excited np states. Unfortunately, a proper treatment of the ground state must involve the use of nonorthogonal 2p and 2p' orbitals, so that the excited np states also require nonorthogonal 2p and np orbitals; we speculate that this would lead to slightly poorer results similar to those obtained for the 2pnp 'P, 'D, and 'S states of carbon (cf. Table VIII). Similar calculations were performed for a number of isoelectronic ions C 11, N 111, and 0 IV; and in all cases, the differences between corresponding ionization energies calculated with a single-configuration core and a

40

M . Cohen and R. P. McEachran TABLE XI11 ELECTRIC DIPOLE OSCILLATOR STRENGTHS FOR BORON (GEOMETRIC MEANS) ns 2S-mp n

3

4

5

m=2

3

5

6

0.008 0.001 0.0 10

0.001 0.003 0.001

b b b

0.606 0.533 - 0.607

1.586 1.517 1.580

0.020

0.009 0.024

0.003 0.001 0.04

- 0.060

- 1.011 - 0.932 - 1.009

2.025 1.961 2.016

0.034 0.0 18 0.038

- 0.093 - 0.094 - 0.091

- 1.409 - 1.319 - 1.407

2.454 2.392 2.444

5

6

-

(1) (2) (3)

- 0.028

-

0.027 - 0.029

-

(1) (2)

- 0.010 - 0.010

(3)

-

6 (1) (2) (3)

4

1.127 1.041 1.133

0.191

(1)o (2) (3)

- 0.182 - 0.209

-

2pTransitions

- 0.057

0.010

-

0.059

- 0.005

-

0.020

- 0.005 - 0.005

- 0.019 - 0.020

np 2P"-md 'D Transitions m=3

n

4

0.125 0.122 0.116

0.050 0.05 1 0.050

0.026 0.025 0.025

0.015 0.014 0.014

0.844 0.947 0.853

0.006 b b

0.008 0.003 0.004

0.006 0.002 0.003

- 0.247

1.217 1.294 1.238

b

0.002

0.003 0.005

b b

I .542 1.608 1.57 I

0.004 0.0 12

- 0.752 - 0.805 - 0.803

1.848 1.909 1.882

- 0.281

- 0.257 - 0.022 - 0.023

- 0.019 -

0.00'7

- 0.007 -

0.006

- 0.499 - 0.544

- 0.530

- 0.045 - 0.047 - 0.040

0.017

O(1) FC values from McEachran and Cohen (1971). (2) MCFC values. (3) PFC values. *Weak transitions (f <

ATOMIC HARTREE-FOCK THEORY

41

two-configuration core decrease uniformly both with degree of excitation (increasing n) and with degree of ionization (increasing Z ) . Thus, we see that improving the core description through a two-configuration MCFC procedure has little effect on the valence orbital energies of excited states. Nevertheless, the valence orbitals themselves are slightly different as may be seen from Table XI11 where we compare some electric dipole oscillator strengths for neutral boron calculated with various FC orbitals. For simplicity, we present only geometric means, but our length and velocity values for each set of orbitals are generally in excellent agreement. Lack of reliable comparison data makes it impossible to draw firm conclusions concerning the relative accuracy of the FC and MCFC values, and we conclude that further calculations will be required in order to make a proper assessment of the utility of the MCFC procedure for general excited states. STATEPROBLEM C. THEGROUND The analogous treatment of the ground state of boron in the framework of a two-configuration core involves a wave function ls22s22p’; 2PO) Q(2PO) = clap,(

+ c2Q2(ls22p22p’; 2PO)

(111)

and leads to an energy expression E(2PO) = c$( ls22s22p’; 2PO)

+ cfE( 122p22p’; 2PO)

Here, we are led naturally to a multiconfiguration extended frozen core (MCEFC) procedure, since it is clear that 12p) and (2p’) cannot be taken as orthogonal. Unfortunately, this prescription involves some serious difficulties, as we now indicate. First, we cannot follow the procedure of Jucys (1967) described in Section V to obtain the energy E(ls22p22p’;2P4. This is because our model requires that the 2p’ valence electron be added to a single well-defined parent ion configuration (here 1s22p2 IS); the Jucys procedure implicitly involves a linear combination of functions in which the 2p’ valence orbital is coupled to each of the possible parent ion core configurations (3P, ID, and IS in this case), with appropriate weights. Nevertheless, by constructing eigenfunctions of L2 and S 2 and carrying through the evaluation of energy matrix elements explicitly using methods described in detail by Hartree (1957) and Slater (1960), we obtain as the result of some lengthy

M. Cohen and R. P. McEachran

42

but straightforward analysis (3 - S’)[ E(1s22p22p’;’PO) - E(ls22p2;IS)] =3{I(2p’)+2[Fo(ls,2p’)- i G ’ ( l ~ , 2 p ’ ) ]

+ 2[ F0(2p, 2p’) -

G0(2p, 2p’) - & G2(2p,2p’)I) - S’C,,

( 1 13)

In writing the result (113) we have assumed that the orbital 12p) and energy cZp are determined by the solution of Eq. (IOO), and that (2p 1 2 ~ ‘ ) =

s.

Second, the cross term in Eq. (1 12) must be evaluated, and we find that (3 - S2)’/’(@,(’P0) I H I @.,(’Po)) = - f [ 3 G ’ (2s, 2p) - SR ‘(2s, 2s; 2p, 2p’)] ( 1 14) This expression clearly differs from the result for the ion [Eq. (IOI)] unless S = 0. Thus, the energy difference [E(’P? - E(’S)] contains a nondiagonal term in this case, and moreover, it also involves (3 - S 2 ) ’ / ’ . The square root now precludes any possibility of obtaining a linear orbital equation for 12~’)and we have not felt it worthwhile to devise new numerical procedures capable of solving this problem rigorously. On the other hand, we may suppry any suitable 2p’ orbital and employ it in calculating one-electron properties within the two-configuration framework. A convenient choice is obviously given by the 2p orbital of the single configuration ls22s22p’Po ground state. This will not be orthogonal to the two-configuration core 2p orbital, so that the total energy is given correctly by Eq. (112) which furnishes a rigorous upper bound to the total energy although it may be far from optimal. In fact, this a d hoc procedure yields a total energy of - 24.53706 a.u., as compared with the single-configuration FC value of -24.51349. If we use two-configuration wave functions for calculations of transitions involving the ground state, the dipole length radial transition matrix element must be modified according to

+

(rns I r 12p’)-+a(rns I r 1 2 ~ ’ ) b(ms I r 12p)

(115)

where a

=

1 - 1 - 3/(9 - 3s’)‘ / 2 ] c 22 ,

[

b = - Sct/(9 - 3s’)’’’

(1 16)

with a similar replacement for the dipole velocity integral. A number of rns-2p‘ and 2p‘-nd oscillator strengths calculated in this way have been found to differ very little from the single-configuration values calculated previously (McEachran and Cohen, 197I). Only geomet-

ATOMIC HARTREE-FOCK THEORY

43

ric means are included in Table XIII, but it should be stressed that the ms-2p‘ length and velocity values differ quite appreciably for these transitions. Clearly, other means of improving the FC valence orbitals must be devised. A simple semiempirical approach involves a polarization potential and is described in the following section. D. POLARIZED FROZEN COREAPPROXIMATIONS We have seen that improving the core description through a multiconfiguration procedure has very little effect on the valence orbital, and so, provided that core and valence electron energies remain essentially additive, we may improve the treatment of the valence electron by taking account of a physical effect neglected thus far. As was pointed out by Hartree (1957) the single valence electron may be expected to polarize the spherically symmetric core of the remaining ( N - 1) electrons, inducing a dipole moment of magnitude a / r 2 at the nucleus, where a is the dipole polarizability of the core. This gives rise to an additional attractive field which is experienced by the valence electron, and at large distances may be derived from a “polarization potential”

vpol= - a / 2 r 4 Such a potential is singular at small r , but it may be modified to (Biermann and Trefftz, 1953)

vPol= - ( a / 2 r 4 ) [ 1 - exp(-xP)],

x =r/p

which is finite at small r for a n y p > 4. Including a- Vpol which has the long-range r dependence given by Eq. (1 17) in the orbital equation for the valence electron can be justified on the basis of first-order perturbation theory (Caves and Dalgarno, 1972), but the choice of p and p in Eq. (1 18) remains essentially arbitrary; the results of our calculations may provide some justification for the particular choice we have made. In the present work, we set p = 6 universally, and calculate a separate p for each LS series so as to reproduce the observed ionization energy of a single term of the series. Since Vpol is introduced into the valence orbital equation in a completely ad hoc manner (it cannot be derived from an energy expression by means of the usual variational procedure), we may consider for simplicity its effect in the framework of the elementary (single-configuration core) FC approximation, once again taking the example of the nl *L states of the boron atom. The core 1s and 2s electrons are described as usual by Eqs.

M . Cohen and R. P. McEachran

44

(55) and (56), while a valence nl orbital now satisfies the equation { H I + Vpol + 2Yo(ls, IS) + 2Yo(2s,2s) - c,,/}P,,

The off-diagonal Lagrange multipliers are now required to ensure orthogonality for ns states, in which case the standard procedure yields Cls,ns

I

= ( n s 'pol

I IS),

c2s,ns

= (ns

I 'poll2~)

(120)

It is a simple matter to verify that (nl 1 ml) = 0 for different solutions of Eq. (1 19). Comparison of Eqs. (58) and (1 19) shows that the change A€,,/ in orbital energy as we go from the FC to the polarized frozen core (PFC) description is given formally by Acn/ = (nl

I 'pol I n'>

(121)

reminiscent of first-order perturbation theory. In Table XII, we compare F C and PFC ionization energies of a number of ns and np states of boron with observed values. The dipole polarizability of B I1 was taken to be 12.55 a.u. (Adelman and Szabo, 1973). Values of p close to 3 bring the lowest 'S and 2Po ionization energies into exact agreement with observations, and improve the FC results for the excited states, even including the higher ns states which are significantly perturbed by the ls22s2p2 *S state. The 3d FC ionization energy is larger than the observed value and the 4d F C ionization energy is in precise agreement with the most recent observed value (Odintzova and Striganov, 1979), while the higher nd levels are all very close to the experimental values. Clearly, a polarization potential must have negligible effects for this series, and we have therefore not applied the PFC procedure in this case. A similar situation occurs for the 'D series of beryllium (see later). In Table XIII, we compare PFC oscillator strengths with the FC and MCFC values discussed previously. The np-md PFC results contain polarized orbitals for the np states only. All three sets of results are generally very similar, except for some very weak transitions. Except for the ns-2p series, there is good agreement between length and velocity values for each set of functions used (as usual, we present only energyindependent geometric means). Lack of accurate comparision data makes it impossible for us to substantiate our claim that the PFC values should be the most accurate ones. There is rather more comparison data available for neutral beryllium. The 2sns IS series must be treated by the EFC procedure (see Section V), but the introduction of a polarization potential causes no difficulty. For beryllium, there are both singlet and triplet series converging on the Be I1

45

ATOMIC HARTREE-FOCK THEORY TABLE XIV CALCULATED AND OBSeRvED IONIZATION ENERGIES OF Be Ia

1s 2s 3s 4s 5s 6s 7s 8s

0.29800 0.08465 0.04227 0.02534 0.01688 0.01204 0.00902

P 'PO 2p

3P 4P 5P 6P 7P 8P

0. I 1718 0.05577 0.03175 0.02035 0.01412 0.01036 0.00792

P

ID 3d 0.05517 4d 0.03106 5d 0.01990 6d 0.01383 7d 0.01017 8d 0.00779

0.34262 0.09063 0.04417 0.026 19 0.01733 0.01231 0.00920 3.26784

0.34262 0.09348 0.04532 0.02675 0.01764 0.01250 0.00932

0.14867 0.06249 0.03418 0.02 150 0.01476 0.01075 0.00818 3.28822

0.14867 0.06837 0.03717 0.0231 1 -

-

0.04905 0.0292 1

-

0.01909 0.0 1340

-

'S 3s

4s 5s 6s 7s 8s

0.1O009 0.10531

0.04845 0.02804 0.01829 0.01288 0.00956 5.02000

0.22725 0.07144 0.03726 0.02293 0.01554 0.01122 0.00849

0.24247 0.24247 0.07476 0.07420 0,03835 0.03819 0.02343 0.01581 0.01139 0.00860 4.14540

P

'p2p 3p 4p 5p 6p 7p 8p P

0.00991 0.00762

'D 3d 0.05673 4d 0.03188 5d 0.02035 6d 0.01410 7d 0.01034 8d 0.00791 0

0.10531 0.04869 0.02817 0.01837 0.01293 0.00959

0.04721 0.02754 0.01805 0.01274 0.00947

0.05987 0.03292 0.02083 0.01436 0.01050 0.00801 5.24000

0.05987 0.03304 0.02091 0.01441 0.01053 0.00803

"In atomic units. " ( I ) FC and EFC values from Cohen and McEachran (1979). (2) PFC and PEFC values. (3) Observed values from Johansson (1962, 1974).

ls22s 'S limit, and we have performed PFC calculations using a separate p for each series. The core dipole polarizability was taken to be 25.0 a.u.* (Heaton and Stewart, 1970). The entire 2snd 'D series has FC ionization energies larger than the observed values, presumably due to perturbations from the low-lying 2p2 'D state, so that the PFC procedure is inappropriate for this series. Table XIV contains F C and PFC ionization energies for beryllium, together with the observed values (Johansson, 1962, 1974). Slightly larger values of p than for boron are required to bring the lowest IS and * A recent calculation of beryllium ionization energies and oscillator strengths using a polarization potential (Cohen and McEachran, 1979) was based on an incorrect value of the core dipole polanzability. The present PFC results supersede the earlier incorrect values.

M . Cohen and R. P. McEachran

46

ionization energies into agreement with experiment, and even larger values of p are required for the triplets. The triplet 3p and 4p PFC ionization energies are slightly too large, but the polarization correction is evidently highly effective in general. Although our polarization potential is slightly spin multiplicity and angular momentum dependent (through use of a different p for each series), it is clear that a model potential based on the HF procedure and including a polarization correction for the valence electron provides a highly satisfactory description of simple spectra. Oscillator strengths, calculated with both FC and PFC procedures for singlet and triplet transitions in beryllium, are presented in Tables XV and XVI. Our PFC calculations are based on the ordinary dipole transition operators, although it is knpwn that the moment operator d, should for TABLE XV ELECTRICDIPOLE OSCILLATOR STRENGTHS FOR BERYLLIUM SINGLET TRANSITIONS (GEOMETRIC MEANS) ns IS-mp n

2

(IP

(2) 3 (1) (2) 4 (1) (2) 5 (1) (2) 6 (1) (2)

1 pTransitions

m=2

3

4

I .325 I .470 - 0.448 - 0.419 - 0.015 - 0.032 - 0.004 - 0.010 - 0.002 - 0.005

0.108 0.037 1.325 1.331 - 0.895 - 0.802 - 0.047 - 0.067 - 0.014 - 0.022

0.024 0.005 0.055 0.023 1.776 1.756 - 1.315 - 1.187 - 0.077 - 0.098

5

6

0.00 1 0.01 1 0.003 0.074 0.036 2.209 2.181 - 1.726 - 1.575

0.004 0.001 0.004 0.00I 0.0 I6 0.006 0.094 0.049 2.632 2.602

6

0.009

np 'p-rnd ID Transitions n

2 (2) 3 (1) (2) 4

(1)

(2) 5 (1) (2) 6 (1) (2)

m=3

4

5

0.717 0.497 0.005 0.289 - 0.027 - 0.083 - 0.006 - 0.013 - 0.002 - 0.005

0.113 0.120 0.573 0.270 0.104 0.551 - 0.077 - 0.200 - 0.016 - 0.032

0.050 0.136 0.092 0.492 0.191 0.214 0.792 -0.139 - 0.327

0.040

0.019 0.026 0.056 0.043 0.134 0.073 0.458 0.156 0.315 1.008

'(1) FC and EFC values from Cohen and McEachran (1979). (2) PFC and PEFC

values.

47

ATOMIC HARTREE-FOCK THEORY TABLE XVI

ELECTRIC DIPOLEOSCILLATOR STRENGTHS FOR BERYLLIUM TRIPLET TRANSITIONS (GEOMETRIC MEANS) ns 'S-mp

m=2

n

0.221 0.236 - 0.033 - 0.033 - 0.012 - 0.012 - 0.006 - 0.006

-

-

3 1.154 1.159 - 0.619 - 0.603 - 0.065 - 0.068 - 0.022 - 0.023

Transitions 4

5

0.004 0.003 I .608 1.59 1 - 1.009 - 0.975 - 0.100 - 0.102

b b

0.0 I3 0.009 2.043 2.016 - 1.393 - 1.345

6 b b 0.001 0.001 0.022 0.017 2.470 2.436

n p 3P'-md 'D Transitions n

2

(1Y (2)

3 (1) (2) 4 (1) (2) 5 (1) (2) 6 (1) (2)

m=3

4

5

6

0.245 0.287 0.634 0.570 - 0.160 - 0.140 - 0.020 - 0.018 - 0.007 - 0.006

0.088 0.09 I 0.085 0.103 0.928 0.879 - 0.327 - 0.302 - 0.043 - 0.041

0.041 0.041 0.041 0.046 0.053 0.062 1.191 1.149 - 0.498 - 0.473

0.023 0.022 0.022 0.024 0.030 0.033 0.038

0.044 1.441 1.403

O(1) FC values from Cohen and McEachran (1979). (2) PFC values. *Weak transitions (f<

consistency be modified to dmod,where (Weisheit and Dalgarno, 1971)

dmod= d{ 1 - a / r 3 [1 - exp(-x3)]},

x =r/p

p is an empirical cutoff parameter, as in V?,. However, some sample calculations with dmodand d yielded essentially identical results. Our claim that the PFCf-values should be the more accurate set is based on a number of considerations. First, the improved accuracy of the PFC energies has a direct effect; large differences between FC and PFC values are often due almost entirely to changes in the energies. Second, for the 'PO-'D transitions, the FC model predicts np+(n + 1)d as the strongest line for an initial np state, whereas the PFC model predicts np + nd, in harmony with calculations and observations on homologous systems. In Table XVII, we present our FC and PFC length and velocity values for a number of singlet transitions, compared with results of refined superposi-

48

M. Cohen and R. P. McEachran TABLE XVII FOR BERYLLIUM SINGLET TRANSITIONS IN ELECTRIC DIPOLEOSCILLATOR STRENGTHS VARIOUSAPPROXIMATIONS

Term

(ly

(2)

(3)

(4)

(5)

(6)

(7)

2p-3s 4s 5s 6s 7s

0.149 0.0046 0.0012 0.0005 0.0003

0.150 0.0053 0.00 15 0.0007

0.143 0.0117 0.0037 0.0017 O.OOO9

0.127 0.0091 0.0035 0.0020 0.0013

0.121 0.0103 0.0039 0.0022 0.0014

0.1485 0.0099

O.OOO4

0.137 0.0099 0.0030 0.0013 0.0007

3p-4s

0.302 0.0163 0.0049 0.0023

0.262 0.02 15 0.0069 0.0032

0.272 0.0232 0.0075 0.0036

0.223 0.057 0.0088 0.0036

0.207 0.022 0.0074 0.0037

0.2560 0.0262

6s 7s

0.295 0.0154 0.0045 0.0021

4p-5s 6s 7s

0.435 0.0253 0.0076

0.441

0.026 I 0.0080

0.391 0.0322 0.0104

0.400 0.0336 0.0109

0.310 0.0383 0.0137

0.306 0.0393

0.3522 0.0379 0.0169

2p-3d 4d 5d 6d

0.688 0.106 0.0372 0.0178

0.747 0.120 0.0430 0.0209

0.537 0.133 0.0547 0.0283

0.460 0.110 0.0448 0.0230

-

-

-

-

-

-

-

-

5s

0.0141

0.0041

0.0021 0.0016

O.Oo90

0.0058

0.4184 0.191 I 0.0898 0.0498

a(l), (2) FC length and velocity values from Cohen and McEachran, (1979). (3), (4) PFC length and velocity values. (5), (6) Superposition of configurations, length and velocity values from Weiss (1972). (7) Model potential length values from Victor and Laughlin (1973).

tion of configurations (SOC) calculations (Weiss, 1972) and of elaborate model potential calculations (Victor and Laughlin, 1973). The improvement due to the PFC model is generally significant, but it is quite evident that more elaborate models may be required to overcome the intrinsic limitations of the basic HF description for properties as sensitive as oscillator strengths. We note that some pairs of length and velocity values differ appreciably, and that even where there is good agreement for each individual model, the results are still well scattered. This is particularly the case for some weak transitions.

E. MOREELABORATE PROCEDURES The two-configuration treatment of boron excited states described previously is, of course, only a very simple example. The MCFC procedure is easily generalized to include many more configurations simultaneously, but it is generally simpler to carry through variational SOC calculations

ATOMIC HARTREE-FOCK THEORY

49

with orbitals obtained from single-configuration problems. Froese Fischer (1977) shows that many important correlation effects on a variety of atomic properties may be accounted for in this way. The traditional difficulty associated with SOC procedures remains that of choosing the appropriate “perturbing” configurations. Unfortunately, different properties are often sensitive to different “corrections” and only extensive calculations of many different systems can lead to significantly more accurate theoretical data. Nevertheless, careful choice of configurations yields satisfactory results for simple systems (Calvert and Davison, 1971; Seaton and Wilson, 1972; Norcross and Seaton, 1976), and use of the basic frozen core idea greatly facilitates these multiconfiguration calculations. A generally more satisfactory approach would seem to lie in the direction of more elaborate model potentials, which include additional physical effects such as quadrupole polarization of the core (Caves and Dalgarno, 1972). This procedure may be generalized so as to treat a system with fwo valence electrons (Victor and Laughlin, 1972) and may yield very reliable results. Such procedures go beyond the H F model and attempt to improve its results. Unfortunately, there seems to be no simple systematic procedure for achieving this improvement without destroying the descriptive simplicity of the elementary HF theory.

VII. Conclusions We have limited our discussion to a number of simple variants of the HF theory of atoms. All of these utilize the frozen core description of ( N - 1) of the electrons, and lead to a linear radial equation for the single valence electron. This equation contains exchange terms, which are treated exactly, so that the FC valence equations involve a nonlocal model potential. The F C procedure may be expected to be more reliable than the Coulomb approximation (CA) (Bates and Damgaard, 1949) for calculating oscillator strengths, insofar as the FC procedure is a better approximation to the H F model than is the CA. Furthermore, the CA requires accurate term values for its application; and when these are not available from experiment, they may be calculated very easily in the FC approximation, which then also yields the corresponding orbitals. The accuracy of one-electron atomic properties calculated with FC wave functions may be expected to parallel similar results for H F wave functions for valence electrons, but there will be some slight loss of accuracy for inner shell (core) electrons. For this reason, transitions that involve a valence electron well outside a frozen core should be accurately deter-

M . Cohen and R. P. McEachran

50

mined, but the well-known sensitivity of f-values to details of the wave functions makes this a severe test of the FC model. In principle, important transitions which are not strictly of this type, such as ls22s22p"-' -+ ls22s2p", might be treated in the FC framework on the basis of a ls22s2pn-' core, but it seems certain that 2s-2p correlation effects must be taken into account in such cases. The main outstanding difficulty concerns the inadequate FC description of ground states. We feel obliged to reiterate that this inadequacy is shared by the basic H F description, but it is clear that further work is needed to obtain an improved one-electron model of atomic ground states.

Appendix: Relativistic Corrections to the Energy Levels For a many-electron atom of low atomic number, the relativistic Hamiltonian may be written as the generalization of the relativistic two-electron Hamiltonian (Bethe and Salpeter, 1957). This is obtained by applying the Pauli approximation to the Breit equation and, correct to terms of order a', yields a first-order perturbation operator H::) which must be added to the nonrelativistic Hamiltonian of Eq. (1):

H:A)

= a2{Hmv

+ Hdt + Hso + H s m + H s s l + Hss2 + H m )

Here, a is the fine structure constant whose value has been taken as 1/137.03604 (Cohen and Taylor, 1973), while some of the operators now involve electron spin. They are given explicitly by .

N

c

--$ c

"1

l N Zr . Hdt= Vi*[ 4 i=l j+i r$

N

c

4s Hssz = - 3 i-1

j+i

si* sj8(r,)

ATOMIC HARTREE-FOCK THEORY

51

and 4

- -

-

i=l

j+i

1 V j Vj + rii (rii V J V j rl ! I

H,, gives the correction due to the relativistic variation of mass with velocity, while Hd,, the so-called Darwin term, is characteristic of the Dirac theory but has “no obvious physical interpretation” (Froman, 1960). The spin-orbit coupling term, H,,, arises from the interaction of spin and orbital magnetic moments of each individual electron, while the spin-otherorbit coupling term, H,,,, is due to the interaction between the spin magnetic moment of one electron with the orbital magnetic moment of another. The spin-spin coupling terms, Hssl and Hss2,describe the interactions between the spin magnetic moments of pairs of electrons, and the orbit-orbit coupling term, H,, takes account of interactions between the orbital magnetic moments of pairs of electrons. Ufford and Callen (1958) have shown that the expectation value of H,, cannot be calculated accurately with uncorrelated wave functions, and we have neglected it here. The results of Table V suggest that it must have a very small effect for FeXVI and NiXVIII. The sodium sequence ions have a single nl electron in addition to closed shells, and the total nonrelativistic wave function is 2(21+ 1) fold degenerate. The operators H,,, Hdt,and Hss2 contribute to an overall energy shift, but do not split the degeneracy. Hartmann and Clementi (1964) have shown that this energy shift may be written as a sum of one-electron radial integrals, each of which involves only a single radial function: AEshift

= %v -k

‘dt

+ ‘ss2

where

and

The sums are over all the occupied shells and qj denotes the number of electrons in the njlj shell; the first sum in (A1 1) is over s electrons only. The operators H,,, H,,, and Hssl serve to split the degeneracy of the HF energy levels, and give rise to fine structure in the atomic spectrum. For systems, such as the sodium sequence, having a single valence electron outside closed shells, the operator Hssl contributes nothing; whereas Blume and Watson (1962) showed that the contribution from H , and H , to the relativistic Hamiltonian may be written as l,I * s. Here I and s denote the

M. Cohen and R. P. McEachran

52

orbital and spin operators of the valence nl electron and 5, is the so-called spin coupling constant which is given as a sum of one- and two-electron radial integrals, the sum being over closed shells only:

5,

= ta2z
11r3

I nl> - 2 { (21’ + 1)Mo(nl, n’l’) n’/’

-

2 [ nk(Z,I’)Nk(nl,n’Z’) + uk(l,I t ) V k ( n l ,n r I t ) ] ) k

(A12)

The two-electron integrals are defined by Blume and Watson (1962) as:

while the coefficients nk(l,I’) and uk(l,1’) have been calculated generally, and are given explicitly for valence p-, d-, and f-shell electrons. The derivation of Eq. (A12) is not valid for valence s-shell (I = 0) electrons, but in this case the expectation value of I s is zero, and the degeneracy is not split. For I # 0, the expectation values depend on the total angular momentum quantum numberj, which for a single valence electron is given by I & +. The expectation value is then given by 9

AEspht = + K C ?

for j = l + i

(A161

I -f

(‘417)

and by AE,,,i, =

-

+ (I + l)S,

for j

=

The total relativistic correction is given finally as the sum of (A9) and (A16) or (A17).

REFERENCES Accad, Y., Pekeris, C. L., and Schiff, B. (1971). Phys. Rev. A 4, 516. Adelman, S. A., and Szabo, A. (1973). J. Chem. Phys. 58, 687. Allen, C. W. (1976). “Astrophysical Quantities,” 3rd rev. ed. Oxford Univ. Press (Athlone), London and New York.

ATOMIC HARTREE-FOCK THEORY

53

Bates, D. R., and Damgaard, A. (1949). Philos. Trans. R. SOC.London, Ser. A 242, 101. Bauche, J., and Klapisch, M. (1972). J . Phys. B 5, 29. Bethe, H. A., and Salpeter, E. E. (1957). “Quantum Mechanics of One- and Two-Electron Atoms.” Springer-Verlag, Berlin and New York. Biermann, L., and Trefftz, E. (1953). Z. Asrrophys. 30, 275. Blume, M., and Watson, R. E. (1962). Proc. R. Soc. London, Ser. A 270, 127. Brillouin, L. (1932). J. Phys. 3, 373. Calvert, J. McI., and Davison, W. D. (1971). J. Ph;x B 4, 314. Caves, T. C., and Dalgarno, A. (1972). J. Quant. Spectrosc. & Radiaf. Transfer 12, 1539. Clementi, E., and Veillard, A. (1965). J . Chem. Phys. 44, 3050. Cohen, E. R., and Taylor, B. N. (1973). J. Chem. Phys. Ref: Data 2, 663. Cohen, M., and Dalgarno, A. (1961). Proc. Phys. SOC.,London 77, 748. Cohen, M., and Kelly, P. S. (1966). Can. J . Phys. 44, 3227. Cohen, M., and McEachran, R. P. (1967). Proc. Phys. SOC.,London 92, 37. Cohen, M., and McEachran, R. P. (1978). J. Quant. Spectrosc. & Radial. Transfer u),295. Cohen, M., and McEachran, R. P. (1979). J. Quant. Spectrosc. & Radiaf. Transfer 21, I . Cohen, M., and Nahon, J. (1980). J . Phys. B ( i n press). Delbruck, M. (1930). Proc. R. SOC.London, Ser. A 129, 686. Fawcett, B. C., Gabriel, A. H., Irons, F. E., Peacock, N. J., and Saunders, P. A. H. (1966). Proc. Phys. SOC.,London 88, 105 I . Fawcett, B. C., Gabriel, A. H., and Saunders, P. A. H. (1967). Proc. Phys. SOC.,London 90, 863.

Fock, V. (1933). 2. Phys. 81, 195. Froese, C. (1966). Phys. Rev. 150, 1. Froese, C. (1967). J. Chem. Phys. 47, 4010. Froese Fischer, C. (1977). “The Hartree-Fock Method for Atoms.” Wiley, New York. Froman, A. (1960). Rev. Mod. Phys. 32, 3 17. Hartmann, A., and Clementi, E. (1964). Phys. Rev. 133, A1295. Hartree, D. R. (1957). “The Calculation of Atomic Structures.’’ Wiley, New York. Hartree, D. R., and Hartree, W. (1936). Proc. R. Sac. London, Ser. A 154, 588. Heaton, M. M., and Stewart, A. L. (1970). J. Phys. B 3, L43. Johansson. L. (1962). Ark. Fys. 23, 119. Johansson, L. (1974). Phys. Scr. 10. 236. Jucys, A. P. (1967). Int. J. Quanrum Chem. I, 31 I . Kelly, H . P. (1963). Phys. Rev. 131, 684. Koopmans, T. (1933). Physica (Utrechr) I, 104. Layzer, D. (1959). Ann. Phys. N . Y . 8, 271. MacDonald, J. K. L. (1933). Phys. Rev. 43, 830. McEachran, R. P.. and Cohen, M. (1971). J . Quant. Spectrosc. & Radial. Transfer 11, 1819. McEachran, R. P.. Tull, C. E., and Cohen, M. (1968). Can. J. Phys. 46, 2675. McEachran, R. P., Tull, C. E., and Cohen, M. (1969). Can. J. Phys. 47, 835. Msller. C., and Plesset, M. S. (1934). Phys. Rev. 46, 618. Moore. C. E. (1949). Natl. Bur. Stand. ( U . S . ) , Circ. 467, Vol. 1. Moore, C. E. (1952). Natl. Bur. Stand. ( U . S . ) ,Circ. 467, Vol. 11. Moore, C. E. (1970). Natl. Stand. Ret Data Ser., Natl. Bur. Stand. 3, Sect. 3. Norcross, D. W., and Seaton, M. J. (1976). J. Phys. B 9, 2983. Odintzova, G . A,, and Striganov. A. R. (1979). J. Phys. Chem. Ref: Data 8, 63. Pekeris, C. L. (1958). Phys. Rev. 112, 1649. Pratt, G . W. (1956). Phys. Rev. 102, 1303. Seaton, M. J., and Wilson, P. M. H. (1972). J . Phys. B 5, LI. Slater, J. C. (1960). “Quantum Theory of Atomic Structure,” 2 vols. McGraw-Hill, New York. Tull, C. E., McEachran, R. P., and Cohen, M. (1971). A I . Data 3, 169.

54

M. Cohen and R. P. McEachran

Ufford, C. W., and Callen, H. 8.(1958). Phys. Rev. 110, 1352. Victor, G. A., and Laughlin, C. (1972). Chem. Phys. Leu. 14, 74. Victor, G . A., and Laughlin, C. (1973). Nucl. Instrum. & Methods 110, 189. Weisheit, J.C., and Dalgarno, A. (1971). Phys. Rev. Lett 27, 701. Weiss, A. W. (1972). Phys. Rev. A 6, 1261. Wiese, W. L., Smith, M. W., and Glennon, B. M. (1966). Nail. Srand. Ref: Data Ser., Narl. Bur. Stand. 4, Vol. I. Wiese, W. L., Smith, M. W., and Miles, B. M. (1969). Narl. Stand. Ref: Data Ser., Narl. Bur. Stand. 22, Vol. 11. Wilson, K. H., and Nicolet, W. E. (1967). J . Quant. Spectrosc. & Radiat. Transfer 7 , 891.

I1

ADVANCES IN ATOMIC A N D MOLECULAR PHYSICS, VOL. 16

EXPERIMENTS A N D MODEL CALCULATIONS TO DETERMINE INTERATOMIC POTENTIALS R. DUREN Max- Planck-Institut f u r Srromungsforschung Gottingen, West Germany

I. Introduction . . . . . . . . . . . . . . . . . . . . . . 55 A. General. . . . . . . . . . . . . . . . . . . . . . . 55 B. Statement of the Problem. . . . . . . . . . . . . . . . . 56 C. Scope and Outline . . . . . . . . . . . . . . . . . . . 57 11. Electronic Model Potentials and Interatomic Potentials. . . . . . . . 58 A . T h e o r y . . . . . . . . . . . . . . . . . . . . . . . 58 B. General Behavior and Actual Forms of Model Potentials . . . . . . 62 C. Determination of Interatomic Potentials. . . . . . . . . . . . 67 111. Experimental Sources . . . . . . . . . . . . . . . . . . . 70 A. Relation to Model Potentials . . . . . . . . . . . . . . . 70 B. Specific Measurements. . . . . . . . . . . . . . . . . . 71 C. Comparison of Interatomic Potentials from Different Experiments. . . 87 IV. Interatomic Potentials Determined with Model Potentials . . . . . . . 91 V. Conclusions . . . . . . . . . . . . . . . . . . . . . . 96 References . . . . . . . . . . . . . . . . . . . , . . . 97 Note Added in Proof . . . . . . . . . . . . . . . . . . . 100

I. Introduction A. GENERAL

During recent years the determination of interatomic potentials has received new momentum by the extension of the subject to electronically excited species of interacting particles. Many phenomena can be interpreted conveniently on the basis of these potentials. Naming a few of the practical applications-multiplet transitions and energy transfer in general, state specific reactions, the field of line broadening, and excimer transitions-demonstrates sufficiently one basis for this interest. For many of these practical applications interatomic potentials of suffi55 Copyright 0 1980 by Academic Press. Inc All rights of reproduction in any form reserved. IESN 0-12-003816-1

R. Diiren

56

cient accuracy especially at large internuclear distances are readily available. But if the range of intermediate internuclear distances and high accuracy, by which we mean a few percent of the well depth, is considered, few examples remain. To these this report is devoted. This implies that a methodological study rather than a summary of potentials will be presented. The new momentum has two sources. First, general advances in experimental technology have been enhanced specifically by the development of lasers. With the unprecedented narrow linewidth and power of this light source, experiments with well-defined initial conditions and with a high degree of differentiation in the exit channel can be performed. This is not only an academic interest; the accuracy of the potentials obtained from the evaluation of such well-defined experiments makes the subject attractive again. The second source of intensified activity is the development of theoretical tools correlated to such experiments, specifically the use of model potentials in calculations of the interatomic potential. In contrast to ab initio calculations, this method also allows the calculation of interatomic potentials for heavier systems with an accuracy comparable to that which can be obtained in experiments. To a certain degree these two sources have developed independently of each other, and it is one of the goals of this article to contribute to bridging the remaining gap. Basically it will be an attempt to demonstrate how model potential calculations and experimental data can be combined to the benefit of a highly accurate determination of interatomic potentials. We hope to propose a conceptual view of the determination of interatomic potentials, which offers significant advantages over the standard phenomenological approach. B. STATEMENT OF

THE

PROBLEM

The standard determination of interatomic potentials is either to calculate a given measured quantity on the basis of a parameterized model for the interatomic potential and vary iteratively the parameters of the model until a best fit to the experimental data is obtained, or to establish an inversion procedure to obtain the interatomic potential from the measured data. In both cases the result remains unsatisfactory because of the phenomenological basis, which implies, for instance, that one measurement concerning one state of the interacting species yields one potential function. As long as only one state is involved in the interaction these methods serve their purpose. Considering however the case where many states are involved, two specific problems (at least) can be formulated which will be covered unsatisfactorily.

DETERMINATION OF INTERATOMIC POTENTIALS

57

(1) The first problem is related to the fact that even though the interaction of given atoms in different states is to be described with different interaction potentials, the approximation of the full Hamiltonian of the problem by a model Hamiltonian exhibits their common basis. At least the major part of the difference between the interaction in the ground state and in the excited state is then attributed to the known differences in the respective electronic wave functions. A fortiori this should hold for the difference between the various molecular states related to one common state of the separated atoms. ( 2 ) A similar argument applies to the problem of a series of interacting systems. Consider for instance the interaction of the alkali atoms with, say, one particular rare gas target. Again the differences in the net interatomic potential should be largely due to the known differences in the wave functions of the different alkali atoms. Both problems are increasingly being encountered in the evaluation of experimental data, and we want to point out that the model potential approach of calculating interatomic potentials seems to be a viable solution to both problems. A review of the various theoretical and experimental sources concerned with fitting the parameters of the model potential will show that such attempts are sufficiently advanced to come into widespread use. Even though a definitive solution has not yet emerged, enough contributions have been accumulated, to be summarized in a unifying concept.

C . SCOPEA N D OUTLINE Surveying the literature for systems where experimental material is available and where the theoretical treatment has come to some sufficiently established point, we find that intermediate and large internuclear distances, interactions of “good” model atoms, and the lowest excited states play a dominant role at the present time. Large and intermediate distances translate into thermal collision energies for scattering experiments and into the usefulness of spectroscopic measurements with the respective molecule. The alkali atoms obviously emerge as good model atoms, since the quality of a model potential approach strongly depends upon the possibility to separate the core and the valence electrons, for which the alkali atoms are the standard demonstration system. Fortunately these systems are also well behaved in the experiments: The generally large dipole moments yield good efficiency in spectroscopic measurements, and the detection efficiency in scattering experiments is usually good. For these reasons we will concentrate our attention on intermediati: and

58

R. Duren

large internuclear distances and on interactions involving alkali atoms, specifically the alkali-rare gas interactions. As narrow as these limitations may appear at first, it is because of both its practical aspects and as pilot systems for further developments that a thorough discussion may be useful. To discuss the various aspects of the model potentials we will first (Section 11) discuss the theoretical aspects of their relation to interatomic potentials. There we will summarize the theoretical background of the model potentials (Section 11, A), describe the general form and particular realizations of model potentials (Section 11, B), and describe in detail their use in determining interatomic potentials (Section 11, C). The next section (111) is devoted to the experimental sources for this determination. A general survey of the relation between experiments and model potentials will be given (Section III,A), followed by a description of specific measurements (Section 111, B), and a comparison for a reference system (Section 111,C). In section I V , we will compare the results for this reference system with interatomic potentials determined with the aid of model potentials. Throughout this report atomic units are used with the following conversions: length, 1 a.u.= 0.52917715 X lo-* cm; energy, 1 a.u.= 2. 194668 x lo5 cm-' = 4.35968 x l o - ' ' erg = 27.2107 eV.

11. Electronic Model Potentials and Interatomic Potentials Model potentials have found extensive applications in the calculation of atomic and molecular properties. In this section we will summarize their connection to interatomic potentials. A. THEORY

For a system of valence electron(s), core electrons, and the respective nuclei, we write the Schrodinger equation from the exact Hamiltonian for fixed internuclear distance R as

where H A and H , are the Hamiltonians for the cores A and B, H , is that of the valence electron(s), uABis the interaction of core A and core B, + ( r , R ) is the total molecular wave function depending on electronic coordinates r and the interatomic coordinate R , and E is the total energy. The intera-

DETERMINATION OF INTERATOMIC POTENTIALS

59

tomic potentials V ( R ) for the different electronic states that we want to determine are obtained as eigenvalues of this Schriidinger equation. The basic guideline for the development of the model potential approach is the observation that the interatomic potentials of the complete system can be obtained approximately from the sum of a term determined by the behavior of the valence electrons and a term available from unperturbed core properties. In particular at the large and intermediate distances to which we have concentrated our attention, this approximation should hold. The total wave function can be expanded in terms of products of core and valence functions (without explicitly stating antisymmetrization) as

where +A is the wave function for core A, @B for core B, and @, for the valence electrons. Following the basic guideline stated previously we will now assume that the core functions are independent of the valence functions. Hence the interatomic potential splits into two contributions, namely V , ( R ) from the cores and V , from the valence electron

Inserting (3) and (2) into (1) and collecting the terms that refer to the cores and the valence electrons we obtain V , and V , from the two independent Schrodinger equations

where c A , c B , e v are the asymptotic energies of core A, core B, and the valence electron, respectively. To obtain the total wave function [Eq. (2)] from the solutions of these equations the orthogonality between the core and valence functions must be maintained:

+,

where describes the state of the valence electron, and +A and GB the core orbitals to which @, must be orthogonal. To determine V c ( R )we may assume that cores A and B d o not overlap significantly and calculate this quantity by a statistical model (Gombas, 1967; Kim and Gordon, 1974): (+A+B

I uAB I + A + d

= V d R1

(7)

R. Diiren

60

On the other hand, scattering experiments of the cores can be used to determine this quantity. In this discussion we will regard this core contribution as known, even though we will see later that in actual calculations its quantitative determination either from a calculation or from an experiment poses a serious problem. To discuss the determination of V v ( R )we write the valence Hamiltonian H, as

where T is the kinetic energy operator, and veA and ueBare the potentials of the interaction of the valence electron with the cores. To introduce the concept of the model potential we note that the assumed independence of core and valence electrons is obviously not correct. In the first place the wave functions are related by the orthogonslity condition (6). In addition the effects of exchange and correlation or polarization are omitted. The central idea of the model potential approach is to attempt to replace the potential of the valence Hamiltonian by a model potential, capable of mimicking these omissions. We therefore write a model Hamiltonian H , for the valence electron

H,

=

T

+ u,,,(r, R )

(9)

instead of Eq. (8), where vm(r,R ) is the model potential, and try to obtain the valence contribution to the interatomic potential V,( R ) from this model Hamiltonian by

instead of Eq. (5). Going through all these steps with care, the restriction which we have imposed for simplicity can be alleviated and the possibility indicated by Eq. (10) can be shown to hold exactly (Weeks et af., 1969; Bardsley, 1974) but with the restriction that then urn,called the pseudopotential, turns out to be a nonlocal potential defined in terms of the core functions. Strictly speaking then the attempt to build the exact solution from Eq. (10) shows it to be a rewriting of the original Schrodinger equation. Thus the solution is not really simplified. On the other hand, a semiempirical approach can be devised. To d o so one chooses the model potential om in most cases as a local one in a suitable parameterized form and determines the parameters from experiments. To this end, perturbation theory and pseudopotential theory is employed, and at least the qualitative aspects to be incorporated in the construction of the model are established. At this point we will assume that the concept of a local model potential

DETERMINATION OF INTERATOMIC POTENTIALS

61

is valid and consider urn as a parameterized potential indicated by u m ( r ,R ) = urn(r,R ; 8 , )

( 1 1)

where 8, describes the set of parameters used in the formulation of the model. With this concept the calculation of interatomic potentials is straightforward: As mentioned previously we can assume the core contribution V , ( R ) to be known in advance. For the valence contribution we have then to solve the one-electron equation [Eq. (lo)] with the model potential [Eq. ( 1 I)] for a given set of parameters 8,. Taking both results together we obtain the interatomic potential by Eq. (3). The experimental sources to determine the parameters in urncan enter in three ways: ( 1 ) The model potential can be formulated to reproduce the asymptotic behavior correctly. Then macroscopic properties, in particular the polarizabilities which may be determined from experiments, will enter the model potential. ( 2 ) We can compare directly the interatomic potential V ( R ) obtained in the calculation with experiments where this potential is involved. The obvious sources for this type of comparison are various scattering experiments and the spectroscopy of the respective molecule. A global fit of the parameters 8, is then achieved. (3) We can further differentiate by noting that urn can be decomposed as z ; ~r (,

R ) = ceA(r , R ) + ueB(r , R )

(12)

by identifying the model and the valence Hamiltonian [Eqs. (8) and (9)]. A further decomposition into terms that depend on the interaction of the valence electron with core A in the absence of core B and vice versa with core B plus a term ulntwhich depends on the interaction yields

) = t ; e A ( r A ) + u e B ( r B ) + u~nt(rA’‘B’

(13)

where we have indicated the absence of the respective other core by dropping the dependence of R in ueA and u e B . By this decomposition we have achieved three things. First we see that the valence Hamiltonian goes to the correct asymptotic limit if we construct u I n , ( r A , r BR, ) such that lim u l n , ( r Ar, B ,R ) = 0 R+crj

Second we (hopefully) can choose uln,to be a small correction to the terms ueA(rA),ueB(rB)which simplifies the calculation. A third aspect refers to the

R. Diiren

62

experimental sources: In general the total set of parameters 0, of urncan be decomposed, according to Eq. (13), into subsets and B,(lnf) ,g = I

{ el("),@ ( W , 4 (In0 1

1

(15)

1

where the individual sets are used to parameterize ueA, ueB and qnlas indicated by

The individual subsets can be fixed in advance by related and experiments: from the atomic spectra of the valence atom (A e - ) and Bl(B) from the scattering of electrons from target B or from macroscopic properties, to name a few examples. It is only the remaining (fewer) parameters (if any) that are to be fixed in the above-mentioned experiments where the interatomic potential is involved.

+

B. GENERAL BEHAVIORAND ACTUALFORMS OF MODEL POTENTIALS The model potential urn as used in the previous section can be considered in the sense of a purely phenomenological approach. In this section we want to discuss the question of how to construct this model potential in order to include general restraints as well as possible and to reduce the phenomenological aspect. As general requirements, we have mentioned before the possibility of differentiating between the various contributions to urn.Second, the orthogonality condition of the valence wave function with respect to the core functions is to be maintained. Finally, the asymptotic behavior yields additional requirements that should be satisfied. The sources from which knowledge of these requirements is introduced are basically pseudopotential theory and perturbation theory for model potentials. Accordingly the actual forms of model potentials may have terms based upon these two approaches. The pseudopotentials have the advantage that one equation for the valence function can be formulated which is exact and which contains the orthogonality condition. In the model potential approach the orthogonality condition must be taken into account implicitly or explicitly. Pseudopotential theory has been reviewed extensively elsewhere (Weeks el af., 1969; Bardsley, 1974; Dalgarno, 1975). In the framework of pseudopotentials one can obtain urnin principle exactly without model parameters (sometimes called ab initio pseudopotentials).. In the context of our prob-

DETERMINATION OF INTERATOMIC POTENTIALS

63

lem the merit of the theory is mostly to provide qualitative information rather than to be used in practice. The obstacle to the application lies in the fact that the computational effort to obtain sufficiently accurate pseudopotentials is comparably high. In the following we will report about the asymptotic behavior of the model potentials. This discussion will be mainly a summary of the outstanding paper of Bottcher and Dalgarno (1974) who have given the basis for a differentiation of the various contributions to the model potential and for many of the model potentials actually used. [For more recent supporting work, see Laurenzi (1978) and Valiron et al. (1979).] T o take the asymptotic form as a guideline is quantitatively in accord with the range of the interaction that we want to consider, namely the intermediate and large internuclear distances. In addition, with the differentiation of the various contributions, it provides the key to the rational inclusion of other experiments into the construction of the model as discussed previously in terms of the subsets of parameters 0, [Eq. (15)]. In order to obtain the perturbation equations we rewrite the exact Hamiltonian for two cores A and B and the valence electron(s) [Eq. (l)] as

He, = H A + H ,

+ ueA+ v,, + uAB+ T

(17)

where H A and H , are the Hamiltonians of the cores; veAand u,, are the interaction of the valence electron(s) with cores A and 9, respectively; vAB is the interaction of the core electrons; and T the kinetic energy of the valence electron(s). Introducing the model Hamiltonian H , yields

He, = H A + H ,

+ H , + Av

with

H,=T+v, and

AU

= ueA

+ o,, + uAB-

U,

(20)

The zeroth-order solution to the respective Schrodinger equation is the product +‘O’(rArJv)

=+A(rA)+drd+drv)

(21)

where we have indicated by r the quantum number of the respective part. and +, to be known from As before we assume HA+A(~A)

=

eLr)+A(rA)>

~ , + B ( r e )= E F ) + B ( ~ B )

The molecular wave function to first order +(

I)

=

+(I)

(22)

can be written as

+(O)(rArsrv) + cA,~+(0)(rArBrv)

(23)

R. Diiren

64

with (24) and (25) with

and similar expressions for core B. From these the model potential is obtained to second order as 0,

= (rArB

I 0 + u G A B + uGiB[ ff,,

u]

I rArB)

(27)

To calculate this matrix element the state of the cores is assumed to be unchanged in the interaction, rAand re referring, for instance, to the ground state of the cores. Expanding u in terms of spherical harmonics, the integrals can be simplified by the introduction of the 2'-pole polarizabilities, a(') for the static contribution and p(') for the dynamic contribution. Then one collects the various contributions according to the sum

+ UeB + CAB + DAB + ucc

u, = ueA

(28)

where ueAand ueB refer to the interaction of the valence electron(s) with core A and B, respectively; GAB to the averaged core-core interaction; uAB to the polarization of the cores; o,, to a core interaction correction. Each of these contributions can be visualized to contain parameters analogously to Eq. (15). Collecting terms to the order of l / r 6 , the asymptotic limit for the special case of one valence electron is obtained as

CAB+O

[see Eq. (7)]

(29c)

DETERMINATION OF INTERATOMIC POTENTIALS

65

As we will see later the model potentials are in practice constructed to contain these terms more or less completely so that the asymptotic behavior is reproduced correctly, in particular the van der Waals interaction with c(6)

=

aL”<$ 1 [

+ ‘2(‘A)Ir2

I +)

(30)

At intermediate distances these terms are then truncated by suitable cutoff functions which contain the f i t parameters. Simultaneously the core-core contribution [Eq. (7)] which vanishes in the asymptotic limit [Eq. (29c)l becomes significant. We will now present some model potentials actually used. It will be seen that most of them are applications of the work mentioned previously (Bottcher and Dalgarno, 1974). As a classification of the various terms taken into account we will use the differentiation as given by the asymptotic expansion [Eq. (29)]. Throughout this compilation w K ( r / q )is used to describe the cutoff function wK for the variable r with a cutoff radius rj (without differentiation with respect to the shape of this function). l a . Baylis ( I 969) Model potential: ueA = t i B D (by use of Bates-Damgaard wave functions)

(pA,pB densities of core A and core

B from “simplified Hartree-Fock”

results)

Application.

Alkali atoms-rare gas atoms.

Parameter sources. By the use of Bates-Damgaard wave functions, atomic spectra (Moore, 1949) of the alkali atom are taken into account to determine ueA, the core-core interacfion is taken from Hartree-Fock results (Gombas, 1967), the dipole polarizability ah1)is taken from semiempirical calculations (Dalgarno and Kingston, 1961), the cutoff radius ro is determined by fitting the well depth of the respective ground state interaction (Buck and Pauly, 1968; Diiren el a / . , 1968).

R. Duren

66

J b. Pascale and Vandeplanque ( I 974) The same model has been used and applied to the same interaction systems using the same experimental sources, with computational improvements.

Ic. Duren (1977) Model potential: The same model has been used as in Baylis (1969) but the polarizability has been used as a second free parameter. Application.

Alkali atoms-Hg.

Parameter sources. In addition to the parameter sources mentioned previously, the cutoff radius ro and the “polarizabitity” a ( ’ ) have been determined by a simultaneous fit to ground state (Buck et at., 1972) and excited state scattering experiments (Duren and Hoppe, 1978). 2a. Bottcher el al. (J975) Model potential:

+ (cAo + c A l r A+ c A z r i )exp

+ (cBo + c B l r B+ cB2ri)exp

Application.

Li-He and Na-He.

(

(

- -

lr:

- -

1

1

DETERMINATION OF INTERATOMIC POTENTIALS

67

Parameter sources. Polarizabilities a(')and a(') are used for the atomic potentials together with the spectra to fit c o , c I , c 2 , and r , for the alkali core (ueJ and for the rare gas atom (ueb). No further fit to experimental data is used to determine the interatomic potential.

2b. Philippe et a/. ( I 979) Modelpotential. A similar model as in (2a) has been used except that for CAB various core-core potentials (experimental ones extrapolated and

calculated ones) have been applied leading to ambiguities in the interatomic potential (see Section IV). Application.

Na-Ne.

Parameter sources. The polarizability ak') is used and the parameters of ueB are determined from electron scattering; in addition the data from N a + -Ne scattering experiments are used.

We could proceed to give more examples of model potentials but we have selected these few to reflect typical applications within the scope of this article. Similar examples for the application to the excited state of He (Peach, 1978) and for the extensive use of pseudopotential theory have been omitted (e.g., Bardsley et al., 1976; Habitz and Schwarz, 1975; Melius et a f . , 1974). Surveying this compilation we notice that only in group (1) are the measured interatomic potentials used to determine the parameters of the model, while group (2) relies only upon the partial measurements of atomic properties to obtain the interatomic potential. C . DETERMINATION OF INTERATOMICPOTENTIALS

Basically three types of experiments are used as standard experimental sources of information to determine the interatomic potentials: ( a ) the direct observation of differential cross sections and (6) the extreme wing line broadening. (In addition to these the observation of macroscopic properties and the integral cross sections contribute with less resolution.) Similarly to these experiments, (c) the spectra of the respective (diatomic) molecules yield the potential. For the evaluation of these experimental sources either iterative procedures or inversion procedures are used. The straightforward phenomenological approach of the iterative procedure can be summarized as a procedure in the following way. The

68

R. Diiren

“theoretical” counterpart M Ih to a measured quantity M e x pis calculated with a phenomenological form of a potential function V ( R ,0;) depending on R and parameters 0,. Theoretical and experimental results are compared with each other and the parameters of the interaction potential are iteratively varied until the comparison yields satisfactory agreement. This can be represented in the following sketch calculation

V ( R , f ? , ) ---+

T

Mth compare with

iterative variation of

Oi

Mexp

I

Many results have been obtained this way with models ranging from the hard spheres to the Lennard-Jones potentials or the Morse potential and finally to many parameter models of all sorts (Pauly and Toennies, 1965, 1968; Fluendy and Lawley, 1973; Pauly, 1979). The remaining merit of this approach is to provide with a few numbers a simple means of communication of the results. But there are some major problems associated with it, as, for example, the application of objective criteria for the quality of the fit and of the flexibility of the model. These can be overcome with a sufficient broad basis of experimental data for the determination of many parameters in the model function (e.g., Duren and Schlier, 1967; Buck and Pauly, 1968; Duren et af.. 1968). A related problem is that the range of validity of the model mapped out by the range of significance of the underlying experimental material can in practice hardly be determined. These problems have in common that their solution can be obtained if the eventually large effort required is provided. Great improvement concerning this effort is obtained by advances in inversion procedures, in which the interaction potential is obtained directly from the experimental data. This leads to the simple scheme: inversion

Mexp-

V(R)

Besides the eventually large savings in computational effort to obtain the potential from the experimental data, a particular advantage of this method is that the potentials obtained contain the range of validity of the results as given by the experimental data used in their determination. For all three above-mentioned types of experiments probing the interatomic potential, inversion procedures have been developed and successfully realized: for differential scattering processes (Buck, 1974; Shapiro and Gerber, 1976), for line shape measurements (Behmenburg, 1972, 1978), and for spectroscopic measurements (Rydberg, 1931, 1933; Klein, 1932; Rees, 1947: Dunham, 1932; Thakkar, 1975).

69

DETERMINATION OF INTERATOMIC POTENTIALS

Both the iteration and the inversion procedure have the common disadvantage of their phenomenological basis. For one thing, this is reflected in the fact that one measurement for one system in one particular state yields one interatomic potential function; extensions to other states and/or other interacting systems are impossible. Second, these methods are strictly related to the measurement of the global interatomic potential. There is no rational way to include in the evaluation of data the knowledge of supplementary measurements which in our picture based on the model potential refers to a subset of parameters. The procedure which includes the use of model potentials for the electronic interaction appears to be more complicated at a first glance. The starting point is the model potential u,(r, R , 8,) which depends on the electronic and the internuclear coordinates and el, a set of paiameters. By a potential calculation one obtains from this model for a given set of parameters the interatomic potentials V y ( R )for various electronic states y. If the experimental interaction potential is known from an inversion procedure, the comparison of the calculation and the experiments to obtain the best fit parameters for the model potential can be performed on this level of interatomic potentials. Otherwise the calculation of the experimental data has to be performed. Summarizing these two possibilities we have the following sketches: u,,,(r, R , 8, )

T

-

potential calculation

calculation of M ( V , )

M:h comparison M,'"

V,(R) 0,

iterative variation of

I

for the case without inversion and om(r , R , 8, )

potential calculation A

inversion

Vih( R ) comparison VYPt---MYp

iterative variation of

0,

I

for the case with inversion. This approach has clearly some advantages compared with phenomenological ones. Its basis lies in the fact that the modeling is transposed to the electronic interaction. Consequently one can determine the parameters of the model potential with a set of experimental data and obtain simultaneously the potentials for many states. The reason for this larger range is that the knowledge of the electronic wave function, which is comparably well established is introduced into the evaluation of the experiment. Similarly the model may

70

R. Duren

be constructed in such a way that the extension to other systems is achieved. Again it is basically the additional knowledge from other experiments that leads to this extension. As another advantage the actual model potentials have fewer free parameters than accurate interatomic model potentials if both are used for evaluations of comparable accuracy, provided that the actual model potential is carefully designed. Finally we notice that supplementary information as provided by experiments which refer to a particular part of the model potential can be incorporated in advance by specifying the respective subset of parameters [see discussion with Eqs. (15) and (16)]. Obviously there are also some severe problems associated with this procedure. Thus we note as a technical problem that an additional effort to establish and perform the potential calculation is required. More severe than this are, however, the difficulties which result basically from the fact that this approach is relatively new. Specifically this means that there is not yet much experience accumulated with respect to the range of validity of the results. Clearly there is a risk of extrapolating the results into ranges (of states and internuclear distances) where they are not supported by the underlying experimental material. T o prevent this from occurring a second inversion procedure would be desirable, and great care must be exercised as long as this is not yet available. With this caution in mind it seems however that the advantages outweigh the disadvantages, and we will present later (in Section IV) results to support this assessment.

111. Experimental Sources A. RELATIONTO MODELPOTENTIALS

According to the decomposition of the model potential [Eq. (13)] confirmed in the asymptotic expansion [Eq. (29)] we can distinguish experimental data in three categories according to their relation to the model potentials. 1. Data that enter the model as constants. These are basically the polarizability a(’) and p (’). 2. Data that determine components of the model potential [Eqs. (15), (16), and (29)]. For this determination the atomic spectra for ueA, the electron scattering data for ueB, and the core-core scattering data for uAB are used. 3. Data that determine the total interatomic potential either by the

DETERMINATION OF INTERATOMIC POTENTIALS

71

determination of f l i ( I n t ) or the set of parameters 0, of om [Eq. (16) or (1 I)]. Such data are found from molecular spectroscopy and from scattering experiments in the ground state and the excited state. The latter experiments can be further differentiated into line shape experiments and scattering experiments (differential and integral) with atomic beams. We will describe these experiments in the following section, but we found it more convenient to order them according to more technical considerations rather than the categories that the previous discussion suggest, namely: polarizabilities, electron scattering, spectroscopy, and heavy particle scattering, Of course some of the experiments use traditional techniques, and we will confine ourself to summarizing results including the most recent ones. Other experiments are the result of more recent developments which will be discussed in greater detail. Special care has been taken to describe completely the route from the measured quantity to the data used in a potential calculation, including the inversion procedures if available.

B. SPECIFIC MEASUREMENTS 1. Polarkabilities We have seen that polarizabilities play a key role in all the model potentials presented. It was understood that the values used there are the “true” values as determined by experiments. If these are not available, semiempirical values and values obtained in ab initio calculations are sometimes to be used. In contrast to such “true” values corresponding quantities are encountered which are indeed fit parameters as coefficients of the respective power in an expansion of the potential. These are used to reproduce some measured quantity, for example, a(’) in Bottcher and Dalgarno (1975) and in Peach (1978). a ( ’ ) and a(’) in Bardsley (1974, see Table 13), or a i l ) in Diiren (1977). Such values are obviously only valid in the context of the specific model potential used and will not be considered here. The determination of polarizabilities from experiments and calculations has been reviewed by Miller and Bederson (1977) very carefully. Therefore we will not consider this aspect but we give for the systems of interest within the scope of this article in Table I a summary of presently available values for a i l ) ?a(’), and /3(’). They are collected from various sources as indicated by the respective references. Results for other atoms may be found in Dalgarno (l962), Dalgarno et al. (l968), and Miller and Bederson (1977).

R. Duren

72

TABLE I PRESENTLY AVAILABLE VALUESOF a ( ’ ) ,a ( ’ ) ,AND p ( ’ ) Error Li+ Na+ K+

Rb+ Cs+ He Ne A

Kr Xe Hg

0.190 1.00 [8-51 112-81 [2 1- 151 1.3833 2.667 11.068 16.737 27.265 35

<1 <5

u

<1 <1
b,c d, e d,e d,e j,o k /,m I, j I, j

50

n, j

<1

Error

p(’)

a(’)[ui]

(%)

Ref.

0.114 1.54 17.4 73 189 2.3260 [9-61 53

<5

u , f 3.53 X lo-’ h, c h, g h h a, f 0.706

I 10 20 20 <1

c,

50

p

P

1.27 8.33 14.5 29.2

(%)

I

Ref. i

< 20 < 20 < 20

i i i

<20

i i

< 20

The percentage errors given in most cases are conservative estimates obtained from a compilation of available data. It is seen from the table that values accurate to better than 1% are available for a ( ’ ) for Li+ and the rare gases. In some cases (heavier alkali ions), limits are given in brackets; these limits reflect lower and upper values obtained in the literature. 2. Electron Scattering

To determine the contributions veB to the model potential, electron scattering may serve as a source of information. For our special purpose the scattering data from low energy experiments are of interest. For these only a few partial waves contribute to the scattering process, which has led both experimentalists and theoreticians to represent their results in terms of the phase shifts for these few partial waves. Hence in the procedure to determine ueB (see sketch in Section 11, C) the measured quantity M is represented by a set of phase shifts 77, at some energies EJ and one has to:

(I)

obtain the phase shifts from the experiment;

DETERMINATION OF INTERATOMIC POTENTIALS

73

(2) obtain the phase shifts from their calculation for the model potential [Eq. (16)]; and (3) compare the results and vary the parameters O,(B).

u,B(TB,O,(~))

Two useful inversion procedures could be thought of, namely one to obtain the phase shift from the measured cross sections (Gerber and Shapiro, 1976) and a second one to obtain the potential from the phase shift (Bottcher, 1971; Shapiro and Gerber, 1976). The first step should not be complicated, whereas in the second step the present solutions are restricted to local potentials (see later). To the author's knowledge not even the first step has been applied in electron scattering. Experiments to determine the phase shifts have been reviewed extensively (Andrick, 1973; Golden, 1978) and a brief discussion will be sufficient. Due to the low mass of the electron target system and the low energies considered (in contrast to heavy particle scattering) from all the partial waves in the sums for the cross sections only the lowest ones, namely I < 2 are important, whereas the higher ones can be calculated in the Born approximation as corrections, indicated by the summation sign 2' . . . . . C;"=3. . . with the second term calculated in the Born = C:=, approximation. There are three basic experiments that lead to the phase shifts.

+

1. Differential cross sections for direct scattering. The measured quantity at a fixed energy for the scattering angle 19 is given in these experiments by

with

and where k is the wave number and 9, are the phase shifts. For the evaluation of such measurements usually trial values for the phase shifts q,, 0 < I < 2 (taking the above-mentioned Born corrections into account) are independently varied to obtain a best fit checked with a X2-test. If the measured cross section is sufficiently structured, a unique f i t is obtained. Such measurements and evaluations are performed at various energies yielding in summary a set of phase shifts for each energy.

2 . Differential cross sections for resonance scattering. Usually at higher energies the electron target system shows more or less well pronounced resonances which can be attributed to one of the phase shifts which as a function of the energy varies rapidly in comparison to the others. Accord-

74

R. Duren

ing to Fano (1961) this particular phase shift at I = I, can be written as a function of the energy as

with

s,,

= arccot[ 2

( ~ ,- E ) / T ]

(34)

where qpR is the nonresonant part of the phase shift, E , the resonance energy, and r the width of the resonance. Introducing Eqs. (33) and (34) into (32) and (31) yields the resonance structure of the differential cross section which again may be evaluated to yield the set of phase shifts as before. 3. Integral cross sections. Integral cross sections can also be used for the determination of phase shifts. Both the total cross section uT measured by attenuation

and the momentum transfer cross section measured in drift experiments

can be fitted by trial sets of phase shifts as a function of the energy. Complications arise concerning the uniqueness of the phase shifts. Since these problems are absent or less severe in differential cross sections, the evaluation of integral cross sections plays a minor role. In summary, from these experiments the phase shifts for the lowest /-values are obtained at various discrete energies with comparably high precision. To complete this information to be applied for comparisons with theoretical values, interpolation formulas derived from the effective range theory (O’Malley, 1963) or from corresponding ( a b initio) calculations (Nesbet, 1979, and references therein) are available. Given this set of experimental values the other step is to obtain the theoretical phase shifts. Assuming the potential to be local, this reduces to the well-known analysis of the asymptotic behavior of the radial wave function as solution of the Schrodinger equation

- , , [

d2

I(/+

1)

1

+ k 2 G,(kr,) = 0

2u,,(r,,19!~’)

(37)

DETERMINATION OF INTERATOMIC POTENTIALS

75

As boundary condition, regularity of the solution at the origin

is required and the phase shift is obtained by comparison of the calculatcd GI with the asymptotic behavior defined by

lim G,(kr,) = sin[ k r , -

rg+m

(39)

Each set of parameters Oi(B) in the model potential yields the phase shifts and by variation of the Oi(,) these are fitted to the experimental ones. However, Valiron et a/. (1979) have shown that the orthogonality constraint of the wave function with respect to the core may play a role. The solution to this problem may occur incidentally because the model potential is deep enough to contain bound states to which the free solution is orthogonal. Then this state induces approximate orthogonality to the core and the local model potential yields the correct result. Otherwise a special choice of the generalized pseudopotential may be used (Bardsley, 1974). To the Schrodinger equation (36) an inhomogenous term with Lagrangian multipliers A, is added

+,

where the on the right-hand side are the orbitals to which the scattering solution is required to be orthogonal. By variation of A, orthogonality is achieved and 0, is then a purely local potential. Except for the additional variation of A which must be determined for each energy, the calculation of phase shifts and the variation of the parameters is the same as discussed previously for the local potential. Experimental results from electron scattering are available from various sources. Most work has been done on He scattering and the results for this target are well established. In addition to the analysis of Andrick and Bitch (1975) from direct and resonance scattering, Steph et al. (1979) have given a uniform evaluation of several types of measurements (see references in Steph et al., 1979). These determinations for He have been confirmed recently by Williams (1979) but with a strong reduction of the error limits and an extension to Ne and Ar. Compiled in the extensive tabulation of s, p, d, and (for Ar) f-wave phase shifts for the energy range from 0.58 to 20 eV, a very accurate data set for model potential calculations is available in Williams' paper for these systems. Errors range from 1.4% to 8.4% for the

R. Diiren

76

lower I-values and are 23% for the f-wave phase shift of Ar. Measurements for heavier rare gases are rare, and results are only known with less accuracy (Kr: Weingartshofer et aI., 1974).

3. Spectroscopy Spectroscopic data enter in two ways in the determination of the parameters of the model potential: as atomic spectra to determine the contributions weA and as molecular spectra to determine the resulting interatomic potential. All the atomic spectra for the atoms considered here are well known and summaries are available in the tables of, e.g., Moore (1949) and of Bashkin and Stoner (1975). The complementary theoretical values to determine the parameter in ueA are obtained by calculating the stationary levels in the respective potential. With regard to molecular spectroscopy we have to consider within the scope of this review only that of alkali-rare gas molecules. The technique to obtain these spectra has only recently been developed and results are available for relatively few systems. Due to the weak bond induced by the van der Waals interaction the basic difficulty is to produce a sufficient amount of these molecules. If this is achieved, the well-developed technique of laser induced fluorescence eventually with dispersion of the fluorescence can be applied. The production of the alkali-rare gas molecules is achieved in a supersonic beam of a carrier gas, to which the atomic species to form the molecule are added in a small fraction. The properties of the carrier gas are given by its expansion from an oven at high pressures into the vacuum. The temperature T drops to T = To[1

+ i(y - l)M*] ‘ ~

where To is the temperature in the oven, y is the specific heat ratio c,/c,., and M is the (local) Mach number (Liepmann and Roshko, 1957). M varies with the distance X measured from the orifice along the beam in units of D , the diameter of the orifice, as M = 3.26(X/D)2’3

(42)

(Becker, 1968) for monatomic gases. The temperature is seen to decrease along the beam roughly with X . Relation (42) holds only as long as collisions in the beam are sufficiently frequent. As the rate of collisions decreases M approaches a limit M , which by Anderson and Fenn (1965)

DETERMINATION OF INTERATOMIC POTENTIALS

77

has been obtained as

where &, is the mean free path and the constants are for monatomic gases. With these relations, sufficiently accurate estimates for most cases can be obtained. For very low temperatures (high Mach numbers) Toennies and Winkelmann (1977) have given a careful study revealing the importance of the correct quantum-mechanical treatment of the collisions. This effect is particularly well visible for He where it has been verified experimentally by the same group (Brusdeylins et a/., 1977) obtaining 8.4 x K as the final temperature. In these studies it is found that M , rises more rapidly with the product p o x D (,po for the oven pressure) than suggested by Eq. (42). Returning to the formation of alkali-rare gas molecules, the carrier is loaded with a small amount of the respective atomic species of the molecules [e.g., 2% Ar and lo-’% Na in 98% He in the Na-Ar experiment by Smalley et a/. (1977)l. Once the molecule is formed, the high pressure in the oven (10-100 atm) and along the early stage of the beam guarantees such a high collision frequency that in general rotational relaxation leads to stable and “cold” molecules. Parenthetically we may remark that this technique is not restricted to these peculiar molecules. Stable molecules introduced into the carrier gas or dimers of the carrier can be rotationally cooled as well (e.g., Borkenhagen et a/., 1975; Wharton et a/., 1978; Bergmann et a/., 1976). The principle and realization of such experiments seem relatively easy and the few experiments obtained so far, namely with Na-Ar (Smalley el a/.. 1977) and Na-Ne (Ahmad-Bitar et a/., 1977), look promising. As minor problems associated with this method, competing dimer formation and successive inelastic collisions or the presence of atomic fluorescence may be mentioned. A quite serious experimental problem seems to be to achieve the necessary stability of the beam for a sufficient signal-to-noise ratio in the fluorescence signal. The evaluation of such experiments from the given spectral distributions to the interatomic potential is quite involved. In the forward procedure the ground state and the excited state term energies are written qualitatively along the general rules discussed by Herzberg (1971) and the spectroscopic constants are obtained by a f i t to the observed intensities. From there the complete potential can be obtained with a suitable model. Inversion procedures basically derived from Dunham ( 1 932) and Thakkar ( 1 975) are available. On this basis an evaluation of the above-mentioned measurements has been given by Goble and Winn (1979). In their paper a

-

78

R. Diiren

comparison with a long-range analysis (Le Roy, 1973; Stwalley, 1975) is also given. Both methods require a high computational effort, and problems arise from insufficient data in the inversion and from the model dependence in the forward procedure. 4. Heavy Particle Scattering

We will now turn to the experiments of heavy particle scattering which as mentioned earlier may be used to determine either the contribution V , to the model potential or the interatomic potential V . Let us first consider the core-core scattering experiments to determine V , . Within the scope of this article only the alkali ion-rare gas interaction needs to be considered. There are two basic problems for experiments relevant to our purpose. The first one is that, in general, low energy data are required probing the intermediate range of internuclear distances. Second, a comparably high accuracy is required since in the range of the equilibrium distances of the interatomic potentials the core-core contributions are of the same magnitude as the well depth itself. High energy data are available ( e g . Kita et al., 1975); for moderately high energies a semiempirical set of potentials for some of the interesting systems has been given (Sondergaard and Mason, 1975). But the extrapolation of such data to larger internuclear distances leads to serious errors. Concentrating on low energies, the number of experiments and of results is found to be very small ( 2 ) ; and these are restricted to the investigation of Li+ with the rare gases. Integral cross section measurements with beams (Powers and Cross, 1973) will not be considered here due to their insensitivity to details of the potential. Another integral cross section experiment is the study of the ion mobility in gases. The theoretical treatment (Viehland and Mason, 1975) and an iterative inversion procedure have been developed (Viehland et al., 1976). On this basis specifically the potential for the Li+ -He interaction has been obtained with high accuracy (Gatland er al., 1977). In differential cross section experiments the potentials at intermediate internuclear distances have been determined in two ways. Bottner e f al. (1975) have measured the cross section in experiments with slow (3 eV < E < 9 eV) ions. The well-known rainbow phenomenon (e.g., Pauly, 1979) has been observed by them for the interaction of Li+ with Ar, Kr, and Xe. At these energies the rainbow angle 9, (in radians) is given to a good approximation by

DETERMINATION OF INTERATOMIC POTENTIALS

79

where c is the well depth of the potential, and E the collision energy. This relation (obtained in the classical approximation) indicates that a measurement of the rainbow angle for a given energy yields the well depth of the potential. This simple relation quoted here to demonstrate the main sensitivity of this experiment is obviously not sufficient for the actual evaluation. This has been performed by Bottner et al. (1975) in a quantummechanical treatment. At higher energies the rainbow phenomenon is no longer visible, but forward diffraction oscillations have been observed (Wijnaendts van Resandt et al., 1976). These oscillations reflect the dimensions of the obstacle and their period is given in a semiclassical approximation by

A8x2a/kR,

Ap

(45)

where k is the wave number of the collision, R , describes the “size” of the obstacle, and A D the difference of the reduced impact parameters leading to this particular interference. Again we have given this approximate relation to demonstrate the main sensitivity of this experiment, namely to the size R , of the interatomic potential, whereas the actual evaluation by Wijnaendts van Resandt el al. (1976) has been performed on a quantummechanical basis of elastic scattering. In any case, both experiments are seen to be complementary to each other in that the well depth (c) and the equilibrium distance ( R , ) can be obtained from both together. In both experiments Li+ has been used as the only alkali ion in collision with various rare gases. Our search for experimental data for the core-core interaction appears to be unsuccessful. However, calculated potentials are available. In particular the work of Kim and Gordon (1974) is to be mentioned, who have given potentials for Li+, N a + , K + interacting with He, Ne, Ar, and Kr. Their accuracy has been confirmed by more recent calculations (Hariharan and Staemmler, 1976), and improvements which might be helpful for heavier systems have been developed (Gianturco and Lamanna, 1978). Thus the Kim-Gordon potentials may be used as a “substitute” for experimental values. We want to try to estimate their accuracy with reference to experimental ones, in particular at intermediate internuclear distances. To do so we have compiled in Table 11 a comparison of their well depths c and the equilibrium distances R , with experimental ones as far as available. For a differentiated analysis of the errors the experimental background should be improved. The work of Wijnaendts van Resandt et a/. (1976) indicates a shift of the radial coordinate of the calculated potential by 0.75 a.u. is necessary to reproduce their measurements (see their values

-

R. Diiren

80

TABLE I1 EQUILIBRIUM DATAFOR ALKALI+ -RAREGASPOTENTIALS

Li' -He

Kim and Gordon (1974)

Bottner et al. (1975) -

4.1 t 0.3

-

-

-

4.35 t 0.4

-

-

-

-

5.0 t 0.4

-

1.11-2 t 2.-4

-

-

-

5.1 t 0.4

-

€

3.68 2.54-3 3.76 4.56-3 4.25 1.09-2 4.48 1.32-2

1.44-2 t 2.-4

-

-

R,

-

-

5.4' 0.4

-

€

-

1.87-2 t 2.-4

-

-

R, €

Li+ -Ne

R, €

Li+ -Ar

R,

Li+ -Kr

R,

€

Li+ -Xe

Wijnaendts van Resandt Gatland et al. (1976) et al. (1977) 3.65 2.75-3 -

for R,,,). The experiments of Bottner et al. (1975) require a deeper potential well (by about 7%) than the calculated one. In the ion mobility experiments of Gatland et a/. (1977) a similar deviation is obtained in the well depth. but an intriguing agreement with respect to R , is observed. Despite these differences the agreement can be said to be reasonably good. We dare to make this statement particularly in the light of the usually much more coarse approximations for the determination of V , in model potential calculations. The experiments described in the remaining part of this section are used to determine the total interaction potential. In principle, scattering experiments of ground-state atoms and of excited atoms and collisional excitation experiments can be used for this purpose. Of these the excitation experiments are usually less sensitive to our particular interest in the intermediate range of internuclear distances. For instance the alkali-rare gas interactions have such high thresholds for the excitation (several 10 eV) that all detailed information about, for example, the well depth of the excited state is lost (Duren et al., 1974; Mecklenbrauck et a/., 1977; Ostgaard Olsen et a/., 1977). With one exception we will therefore discuss experiments with thermal collision energies. As the first type of experiments we will consider the line shape experiments, which can yield both the interaction potential in the ground state and the excited state. This topic has been reviewed recently (Gallagher, 1975; Behmenburg, 1978); therefore a brief summary confined to our specific interest will be sufficient.

DETERMINATION OF INTERATOMIC POTENTIALS

81

The principle of these experiments is to measure the dispersed fluorescence from a cell which contains the two atomic species, where the exciting light is “tuned” to the atomic line of one of the atoms. To a good approximation the light emitted at a certain frequency o off from the line center is given by the energy difference between the upper and the lower state of the transition both distorted by the interaction, that is,

where V , and V/ are the interatomic potentials for the interaction in the upper and the lower state, respectively, and R is a certain constant internuclear distance corresponding to this w. For such an experiment the apparatus is in principle comparably simple: A gas cell contains mainly the perturber gas and a small fraction of the species to be excited. Excitation is achieved with a discharge lamp for the latter material or a laser tuned to a particular transition thereof. Fluorescence light is observed perpendicular to the light beam and dispersed by appropriate means. The requirement with respect to the wavelength resolution of this element is not very high. But special care is needed to guarantee sufficient suppression of the central line. This is crucial since the intensity of the spectrum in the interesting range is 10-6-10-8 of the central line intensity. For the same reason the parameters of the cell must be chosen to avoid radiation trapping as well as possible. As parameters of the measurements, the temperature of the cell or the density of the perturber gas is varied. In the observed spectral distribution (of the fluorescence) one can according to the basic equation (46) distinguish three regions: 1. The impact region which typically stretches by AA x -+ 1 A around the line center: According to our relation above, far this region the interaction is governed by small distortions, which means that the interaction potential at large internuclear distances is involved. I t can be related to the width and the shift of the Lorentzian profile near the center, but we will not discuss this further (see, e.g., Lewis er at., 1971; Roueff, 1972, and references therein). 2. The far wing region with lAAl x 1-50 A: I n this region the distortion is larger but large internuclear distances are still involved. In this domain no temperature and density dependence of the intensity other than direct proportionality to the density is observed, which makes i t less attractive for our purpose. 3. The extreme wing region [MI2 50 A: Here the distortion reflects a comparably strong interaction (within the thermal limit) and it is this region in which we are mainly interested. We will see later how the

82

R. Diiren

temperature and pressure dependence of the intensity can be used to determine the interatomic potential here. The theory of line shapes is well developed especially for the binary collisions which we are interested in (see, e.g., Cooper, 1967; Reck, 1969; Szudy and Baylis, 1975; Herman and Sando, 1978; West and Gallagher, 1978). We will discuss the evaluation of these experiments by means of the quasi-static theory, of which we have anticipated the central idea earlier by Eq. (46). This states that transitions of the perturbed oscillators occur with unchanged internuclear distance for the upper and lower state. I t applies quantitatively in the region of our interest (region 3) and it allows a proper discussion of the experiments and their relation to the interatomic potential. To determine the spectral distribution we first consider the number N, of transitions which lead to an emission with a frequency between w and w + dw. By the correspondence between w and R established by Eq. (46) this is given by N,dw = NO4rR2 dR P ( R ) d o do

(47)

where No is the average density of the perturber, and P ( R ) the probability for the occurrence of internuclear distances between R and R dR. The usual expansion of the density yields for this probability the pair distribution

+

g,

P ( R ) = - exp[ V,(R)/KT] ga

(48)

where V , denotes the initial state of the experiment which is the lower one in an absorption experiment and the upper one in an emission experiment [the asymptotic value V,(co) being always subtracted]. g,/g, describes the ratio of the statistical weights for the molecular and the corresponding asymptotic atomic states. In this expression for P , equilibrium for all possible states is implied. To convert the number of transitions into intensity we have

Z(O) d w = N, . hwA (R ) d w

(49)

where A ( R ) is the transition probability, which can be written as Z(w)do= N o 4 r R d2w~ h 4 w 4 P ( RD ) .( R ) d w

(50)

where D ( R ) is the local dipole moment of the transition. In the following we will assume this to be independent of R and equal to its asymptotic value Do. This assumption should hold within 10% for the internuclear

DETERMINATION OF INTERATOMIC POTENTIALS

83

distances considered here, in particular for the alkali-rare gas interaction. Normalizing this expression to the total intensity, one obtains " "

where w, is the central frequency. For simplicity we will normalize the intensity ratio above with respect to the trivial factors on the r.h.s. of Eq. ( 5 I ) and define as normalized measured intensity

I*(o)dw = 4 a R 2 dR P ( R ) d w dw

As we have defined the far wing region (region 2) by small distortions V , is in general negligible there. With P ( R ) 2~ 1 the expression (52) for I * becomes independent of the temperature and the density. In contrast to this the influence of P ( R ) becomes visible in the extreme wings. If the spectral distribution is measured as a function of the temperature, one obtains the potentials in the following way (Hedges et a / . , 1972).

1. The plot of log Z*(o) versus 1 / T at different values w yields the initial potential, say, V , for emission versus w and since Aw = V , - V! the lower potential V,(w). By using a value for V/ determined from other sources (e.g., scattering experiments) on the w scale for one point, say wo. the corresponding R value R, can be fixed. 2. To obtain the scaling of the w-axis in R the spectral intensities I: obtained from extrapolation of I * to 1/ T = 0 are integrated to yield

s::

I*(o)do = (4a/3)[R 3

( ~ )-

R '(wg)]

(53)

By inversion of this (measured and integrated) function the scaling in R of the potentials V , and V/ obtained in step 1 is fully established. If the spectral distributions are measured for a fixed temperature at a high and at a low density of the perturber, the evaluation to obtain the potential is similar. I t is complicated by the fact that the distribution as given by Eq. (48) is valid in the high density limit (where all states are populated), whereas in the zero density limit contributions from inside the rotational barrier, that is, all bound states, must be excluded (York et a/., 1975). Clearly the calculation of this latter distribution requires the knowledge of the initial potential. In the above mentioned publication, however, this difficulty has been solved by an iterative procedure which obtains its starting potential from the assumed classical distribution.

84

R. Diiren

In some cases satellites are observed in the far wings of the spectral distribution. Usually they are found as a shoulder or a maximum followed in some cases by small regular oscillations in the distribution. They occur at frequencies w where d w / dR vanishes leading in the quasi-static theory to a singularity in the spectral distribution [see Eq. (52)]. For a quantitative treatment of such observations the quasi-static theory is therefore insufficient. The theoretical work on this subject is well advanced (Sando, 1974; Szudy and Baylis, 1975, and references therein). Since such calculations are complicated, especially in relation to the information about the potential to be expected, we need not consider this special feature here. As mentioned previously these experiments yield the interatomic potential. Therefore we will postpone the discussion of results to the following section. The second type of experiments to be considered here are beam experiments again at low energies. We will first consider experiments to determine the differential cross sections. A special feature of the alkali-rare gas interaction which we have already mentioned is a separation between the energy of the excited states and the ground state which is large compared with thermal (- 100 mecollision energies. As a consequence such experiments and their evaluation can be performed independently for the excited states and for the ground state. Of course, the ground state potentials play a central role in our comparison of experimental and theoretical results. On the other hand, techniques and the evaluation of such experiments are well known and reviewed (Pauly and Toennies, 1965, 1968; Fluendy and Lawley, 1973). In particular results for the alkali-rare gas interaction have been obtained by two independent investigations (Buck and Pauly, 1968; Duren el af.,1968) for most alkali-rare gas combinations which agree within a few percents in the minimum of the potential. For the excited states one expects similar accuracy from similar experiments, but it is only in recent years that such results have become available. We have reported recently the experimental aspects and the evaluation of such experiments in detail (Duren and Hoppe, 1978), so a brief summary will be sufficient. Basically in such an experiment the differential cross section for the excited state is measured as usual in a crossed beam experiment where the excitation of the primary beam is achieved by a laser tuned to a specific hyperfine structure level of this atomic species. The difference A of scattered particles measured for laser on and off reflects the difference between the respective cross sections:

A

= (da*/du) - (da/dw)

(54)

DETERMINATION OF INTERATOMIC POTENTIALS

85

where du*/dw stands for the effective cross section in the excited state, and d u / d o for the one in the ground state. d u * / d w is the total differential cross section; that is, it is the sum over the cross sections for the various exit channels of the collision. In the examples investigated so far the initial state is one of the doublet components of the resonance state of Na and the sum for du*/dw is over the fine structure components. In many cases th.: well depth for the ground state and .rr-components of the excited states can be expected to be quite different from each other. In particular the cross section for the ground state may be monotonic in the range of large angles. Then a structure in the difference A can be attributed to the excited state and can be evaluated accordingly. The use of the laser for the excited process will produce a selective preparation of the initial state. Only a particular fine structure level ( j ) is excited, which by the polarization of the exciting laser light has a welldefined distribution of the projection quantum number (m,).From the stationary distribution in the total angular momentum ( F , M ) (Hertel and Stoll, 1974; Macek and Hertel, 1974) one obtains with linearly polarized light for the j = 3/2 component the weights W ( j ,m,) as j=3/2,

mJ= -3/2

W(j,m,)=

1/12

-1/2 5/12

+1/2 5/12

+3/2 1/12

(55)

The knowledge of this distribution is desirable since it enters the evaluation of the data. The theoretical background for this evaluation derived from the general quantum-mechanical scattering theory has been given by Mies (1973) and Reid (1973). Compared with the general case it is simplified by the above mentioned separation of ground state and excited states such that only the excited states must be taken into account. The scattering amplitude for transitionsj'mJ' to j , m, is given as

Aa) = 2 / ( k ' k ) ' / 2 C i ' - ' ' Y : P ( r i ) G ~ ; Y ~ ' ( a ) (56) where i? is the beam direction, and a is the direction of the detector. Y are j(j'mJ',j,m, 1

the spherical harmonics and the sum is over all angular momenta of the relative motion. The matrices G are obtained from the usual T matrices by GF;p=

2 (jlm, I J M ) T J ( j ' l ' j l ) ( j ' l ' m ~I ~J M' )

(57)

JM

where the ( ) are the Clebsch-Gordon coefficients. To obtain the T matrices the coupled Schrodinger equations for each J

86

R. Duren

are integrated

to the asymptotic limit. The potential matrix V for the coupled equations can be expressed by the adiabatic potentials involved. The differential cross sections for individual j‘m; +jmj transitions are then given as the squares of the scattering amplitude. Averaging of these cross sections with respect to the initial distribution of mi described previously and summing over the final state quantum numbers yields the measured cross section as

No inversion procedure is available but the above mentioned steps must be calculated for each iteration in the determination of the potential parameters. Since these experiments are moderately complicated, only a few results are available at the present time: Carter et al. (1975), Hertel et al. (1976, 1977), Diiren et al. (1976), Diiren and Hoppe (1978), and Diiren and Groger (1979). We will discuss of these the alkali-rare gas systems later in the section on the comparison between experiment and theory. The most recent development of these experiments has been the use of Doppler fluorescence for the resolution of the fine structure of Na in the exit channel (Phillips et al., 1978). Results have been obtained for the differential cross section for fine structure transitions from Na (32P,,2) in collisions with Ar. The angular resolution of these measurements is rather low. But the future development may see an interesting source of information emerge from this type of experiment. In addition to the differential cross section experiments described so far, a large number of integral cross section experiments of this type have been reported. In particular, cross sections for fine structure transitions and depolarizations have to be mentioned (for recent reviews, see Baylis, 1979; Elbel, 1979). Such experiments are usually performed in cells, but beam experiments have also been reported (e.g., Anderson et al., 1976; Phillips et al., 1977). Clearly the cross sections measured in such experiments depend mainly on large internuclear distances and not that much on the behavior at intermediate distance. In reference to a very accurately measured energy dependence of a fine structure transition cross section (Phillips et al., 1977), potentials differing in the well depth by as much as a factor of two can hardly be distinguished (Saxon et al., 1977). Thus for our special

DETERMINATION OF INTERATOMIC POTENTIALS

87

interest such experiments usually will not be sufficiently differentiated to determine the interatomic potential. C. COMPARISON OF INTERATOMIC POTENTIALS FROM DIFFERENT EXPERIMENTS

Before we enter the comparison of experimental and theoretical results we want to compare the results from different experiments yielding the interatomic potential. This comparison can be given in detail only for one particular system, namely Na-Ar. For this system all the experiments mentioned earlier have been performed: the spectroscopy of the Na-Ar molecule (Smalley et al., 1977), line shape measurements in the extreme wings (York et al., 1975), and various molecular beam experiments (von Busch et al., 1967; Diiren and Groger, 1975, 1979). In some cases evaluations of the same data set have been given, which will be also quoted later. We want to use this comparative study to establish error bounds for the following comparison with theoretical results. The comparison of the methods which is given simultaneously with this study should be seen as definitely restricted to results and their accuracy. We will not compare the methods in terms of feasibility and the like. We do not have a preference for any one of these methods. To the contrary we have to stress the point that the available results show the different experiments to yield complementary information. Let us first consider the ground state of Na-Ar for which we have compiled the available equilibrium data in Table 111. The inspection of this table reveals by the evaluations of the same experimental data set by different groups the dependence on the model (see entries 1 , 2 and 3,4). In this respect the spectroscopic data and molecular beam data show no difference. (Note that we have chosen from the multiple choice results of the spectroscopic values the one closest to differential scattering experiment.) For these reasons it seems to us appropriate to give the unweighted average value as a recommended value with error as given by the standard deviation. Figure 1 gives a plot of the potential obtained from the differential cross section experiment mentioned earlier together with the equilibrium points as given in Table 111. The shape of this function has been given previously (Diiren and Groger, 1978, and references therein), and it compares well with the spectroscopic result in a fairly large range of internuclear distances around the minimum. A similar compilation for the h,,: state is given in Table IV. We will not discuss this table in detail because similar arguments apply as before. With

R. Duren

88

TABLE 111 EXPERIMENTAI. EQUII.IRRIUM DATAFOR

THE

Na-Ar

(GROUND STATE)

INTERACTION

Evaluation

Experiment Type"

Ref.

Year

MBI MBI SP SP MBD

h h e e g

1967 1967 1977 1977 1978

a.u.)

c (

R , (a.u.) 9.5 9.1 9.4 9.4 9.5

0.204 0.188 0. I76 0.190 0.195 0.190 t 2%

Average value"

9.4

Ref.

Year

c

1968 1968 1977 1979 1978

d e

1 g

* 2%

"The type of experiment is denoted by: MBI. molecular beam experiments, integral cross sections (glory undulations): SP, spectroscopy of van der Waals molecule; MBD, molecular beam experiments, differential cross sections. hvon Busch ef al. (1967). 'Buck and Pauly (1968). dDuren er al. (1968). 'Smalley et al. ( 1977). 'Goble and Winn (1979) (their potential TI). KDuren and Groger (1978). "Unweighted average with percentage standard deviation.

0:

0'

3 0

T

0

O

I

>

-0 1

-0

: I

I

1

1

1

1

8

10

12

11

16

18

R

[a u.1

FIG. I . Interatomic potential for Na-Ar (ground state). Equilibrium data: 0, Buck and Pauly (1968); 0 , Duren e? al. (1968); +, Smalley e? al. (1977); +, Goble and Winn (1979); solid line, Duren and Groger (1978).

DETERMINATION OF INTERATOMIC POTENTIALS

89

TABLE IV EXPERIMENTAL EQUILIBRIUM DATAFOR

THE

?TI/*

STATE OF THE Na-Ar SYSTEM

Experiment

Evaluation

Type*

Ref.

Year

LS SP SP MBD

b

1915 1977 1977 1979

Average valueg

c

c e

c

(lo-’ a.u.)

R , (a.u.)

Ref.

Year 1975 1977 1979 1979

2.56 2.58 2.6 1 2.55

5.99 5.49 5.49

b

(5.75v

e

2.57 2 1%

5.7 & 2%

c

d

“The type of experiment is denoted by: LS, line shape measurements; SP, spectroscopy of van der Waals molecule; MBD, molecular beam experiment, differential cross sections. bYork et al. (1975). ‘Smalley et al. (1977). dGoble and Winn (1979). ‘Duren and Groger (1979) (the experiment is primarily sensitive to c). /The particular evaluation in reference (e) yields this value from a calculated potential. g Unweighted average with percentage standard deviation.

respect to the average value we note that a similar error limit as before has been obtained. A figure of the potential will be presented later together with theoretical results. In summary then for a system studied as well as Na-Ar. the equilibrium data can be obtained with an accuracy of approximately 2%. But unfortunately there are not more systems studied that well. Similar tables can be compiled for Na-Ne [Diiren et al., 1972 (ground state MBI only); Carter er a/.. 1975; Ahmad-Bitar et al., 19771. But for this system due to various reasons, molecular beam experiments provide data with errors up to 30%. In this case the most reliable results are certainly obtained from spectroscopy of the van der Waals molecule with quoted error limits of about 2% for the equilibrium data of the ground state and the excited state (’T~,’) (Ahmed-Bitar er a/., 1977). In the light of an independent evaluation of the same data (Goble and Winn, 1979), we are rather inclined to estimate a (still sufficiently impressive) limit of 5%. To close this section on the experimental potentials we want to give a direct comparison between the measured quantities and the ones calculated from the potentials, namely the differential cross sections for Na-Ar in the ground state (Fig. 2 ) and the excited state (Fig. 3) (Diiren and Groger, 1978, 1979).

0

200

100

300

400

500

600

700

a,,,,

800

9 0 0 1000

1400 1 2 0 0

00

1300 1400

[degl

FIG. 2.

n'

Y!OO

'

0.

no

1I.d

1b.d

2b.

00'

24. u d

2b

JcM

od

3'2

od

[degl

FIG. 3.

3b.00

h o d

Yq

on

Y A . 00

82.

no

56 00

I

nu

DETERMINATION OF INTERATOMIC POTENTIALS

91

In the ground state experiment both rapid oscillations and one rainbow angle are resolved, which leads to an accurate determination of both E and R,. In the experiment for the excited state, rapid oscillations are not resolved but three rainbow maxima are visible.

IV. Interatomic Potentials Determined with Model Potentials In this final section we will discuss the comparison of experimental and theoretical results. Having established in the last section error limits of experiments, this will lead to an estimate for the reliability of the theory in its present status. Discrepancies will predominate even if the most recent development of the theory is considered, as we will demonstrate with few examples. We will begin our discussion with the previously mentioned Na-Ar interaction. For this system as for many others the potentials have been calculated by Baylis (1969) and Pascale and Vandeplanque (1974). Their procedure for determining the parameters of the model potential has been described previously. We recall in particular that one parameter is varied to be determined from the experimental interaction potentials in the ground state, represented by c, R , , the equilibrium data. However, agreement of both these values with the experimental ones is not obvious. We note that in the theoretical results both values E and R , are correctly correlated, such that a fit of both quantities can be achieved with the variation of only one parameter in the model potential. This leads us to an interesting feature of this model potential, namely that it reproduces the shape of the ground state potential quite well. This determination of the correct shape of the alkali-rare gas interaction from experiments has been a longstanding problem. Finally it has been resolved by the simultaneous evaluation of many independent measurements for the same system (Buck and Paul, 1968; Duren et al., 1968). The claim of these papers was a universal shape function, practically deduced from the fit to few systems with particularly many measurements and generalized to the others. Recently we have confirmed its use for Na-Ar (Duren and Groger, 1978), and recent spectroscopic work seems to support it (Goble and Winn, 1979). In Fig. 4 this shape function is given together with the LennardFIG. 2. Differential cross section for Na-Ar (ground state). Measured points and fit with interatomic potential of Fig. 1.

FIG. 3. Differential cross section for Na(3’P,,+Ar. interatomic potential of Fig. 5.

Measured points and fit with

92

R. Duren Reduced Distance X = R / R m

FIG. 4. Reduced potentials for the ground state alkali-rare gas interaction: Lennard-Jones (12.6) potential; . . . ., model potential calculations; , Duren ~

-el

-, a/.

(1968).

Jones (12,6) potential, the one deduced from Baylis (l969), and the one from Pascale and Vandeplanque (l974), the latter ones differ within the resolution of the drawing only for internuclear distances smaller than R , . (The same result is obtained from the reduction of heavier alkali-rare gas pairs.) In the critical region of R / R , x 1.2, V ( R ) / V ( R,) x 0.5 the calculated potential lies midway between the Lennard-Jones (12.6) potential and the modified one. This agreement (in absolute units, values of about 2x a.u. are involved) excellently supports the model potential used in the calculation. Before we discuss the results for the excited states we note again that the ground state data are the only ones used in the fitting procedure of Baylis (1969) and Pascale and Vandeplanque (1974) and they were the only ones available at that time. The inherent assumption was that this fit would determine the model potential sufficiently to yield the potentials for the excited states simultaneously. The experiments to test the excited state and this assumption have been discussed in detail previously and we compare in Table V the calculated results with the average values. The results of a recent CI calculation (Saxon et al., 1977) (deviating by 10%in 6 ) are also given. Figure 5 shows these data together with the potential determined from differential cross section measurements (see Fig. 3). Inspection of this table yields discrepancies by roughly a factor 2 for the well depth and a deviation of 10% for

DETERMINATION OF INTERATOMIC POTENTIALS

93

TABLE V COMPARISON OF CALCULATED AND EXPERIMENTAL VALUESFOR THE f , pSTATE OF Na-Ar

a.u.)

c

Experimentalu Model potential* C I calculation‘

2.57 2 I% 1.30 2.24

R, (a.u.1

5.7 2 2% 6.35 5.75

~~

~~

“See Table IV. hPascale and Vandeplanque (1974) (see Note Added Proof. p. 100). Saxon el a/. ( 1977).

5.0

7.5

10.0

R

in

12.5

15.0

[J.u]

+,

FIG. 5. Interatomic potentials for Na(32P,/3-Ar. Equilibrium data: Pascale and Vandeplanque (1974); 0 , Saxon et u/. (1977); solid line, Duren and Groger (1979).

94

R. Diiren

R , . Unfortunately more recent model potential calculations are not available for Na-Ar (see Note Added in Proof, p. 100). From the study of this particular case we think that two general conclusions can be drawn. First one finds that the model potential discussed previously quite accurately reproduces the shape of the ground state potential for the alkali-rare gas interaction. It badly fails for the excited states. The second conclusion refers to the use of experimental data: I t is seen that the reliability of a model cannot be established with ground state data only. As a second example we would like to discuss the Na-Hg interaction where we have used the model and algorithm of Baylis (1969). Our starting point however was quite different in that we had the data for the ground state and for the excited state available. (A second difference was our more phenomenologically oriented interest, to obtain the interaction potential.) Using the model as it stands, it turned out to be impossible to obtain a simultaneous fit to the results for the ground state and the excited state by the variation of the free parameter r o . So we used the dipole polarizability a(1 ) as a second free parameter. The fit to both data sets was then obtained by finding in the r o ,a(')-plane the common overlap of the areas compatible with the individual data sets. The results have been given elsewhere (Diiren and Hoppe, 1978) and a brief summary may be sufficient. Concerning the ground state interaction one pair of parameters ro, a(') was found to fit the ground state interaction for the systems Na, K, Cs interacting with Hg measured by Buck et al. (1972). Good agreement was found for the shape of these potentials determined by the inversion of differential cross sections (Buck, 1974). (See Fig. 1 of Diiren, 1977.) Comparison with differential cross sections for the excited state, at the present time only possible for Na-Hg, is also satisfactory. The *?r states exhibit a deep potential well (- 11.1 X l o w 3 a.u.) leading to orbiting collisions at low energies. Due to this effect a note of precaution has been given in Diiren and Hoppe (1978) due to some uncertainty in the indexing of the maxima which could be verified only partly in this work. Unfortunately this problem has not yet been completely settled, even though we have a new confirming piece of evidence from the direct observation of the orbiting backward peak (Diiren et af., 1979). We are well aware of the limitations imposed by the phenomenological basis of this model, which has been seen to work well for one particular system only. So we are not inclined to draw generalizing conclusions concerning this model. However, concerning the method of the determination of the potentials in general the example demonstrates that a simultaneous evaluation of ground state and excited state data is urgently required.

DETERMINATION OF INTERATOMIC POTENTIALS

95

TABLE V1 EQUILIBRIUM DATAFOR THE GROUND STATEINTERACTION OF Na-Ne a.u.1

t

Experimento Model potentialb

R, (am)

0.037 0.035

10.0 10.0

‘?Ahmad-Bitaref a/. (1977). ’Philippe ef a/. (1979).

Recent results for our final example, namely Na-Ne, obtained by Philippe er al. (1979) can be interpreted to support our conclusions. Unfortunately we have not obtained sufficiently detailed data on these results to go through the same analysis as with Na-Ar and are restricted to the published values. Again the experimental data from scattering (Carter el a/., 1975) and from spectroscopy (Ahmad-Bitar er a/., 1977) demonstrated that neither the ground state nor the excited state could be interpreted by various model potentials. By use of the model potential of Bottcher and Dalgarno (1974) with parameters fitted to e--scattering data the above mentioned calculation of Philippe et a/. (1979) obtained potentials without reference to the measured interatomic potential. For the ground state this leads to excellent agreement with the experiment as shown in Table V1. For the excited state there is the problem that the currently available core-core potential ( N a + -Ne) is not sufficiently well established. Each of the various available choices leads to values of the equilibrium data for the A27i-state outside the error limits of the experiments. Out of these choices the spectroscopy of .the van der Waals molecule is able to select two which lead to results close to the experimental findings as compiled in Table VII. It should be noted however that this particular choice of the core-core repulsion is not supported by other reasons.

TABLE VII EQUILIBRIUM DATAFOR THE %-STATE INTERACTION c

Experimento Model potentialb

a.u.)

0.638 2 2% 0.601

“Ahmad-Bitar el al. (1977). Philippe ef al. (1979).

OF

Na-Ne

R , (a.u.)

5.1 ? 2% 5.0

96

R. Diiren

In contrast to the philosophy advocated in this article, namely to use the experimental interaction potential to determine the parameters of the model, Philippe et a/. (1979) have used the determination of subsets of parameters only. Because of this their work is especially valuable as a study of the limitations in accuracy imposed by that procedure. Of course it is difficult to assess the validity of consequences for general use from this work, but a few suggestions may be allowed. Concerning the construction of the model potential the expansion of Bottcher and Dalgarno (1974) [see Eq. (29)] is confirmed as a most valuable tool. Concerning the use of experimental data a serious lack of accurate values concerning the core-core interaction is made visible. We want to close the comparison of experimental and theoretical results here being aware that many valuable contributions had to be left out. We hope that the main roots for future work could be described by these examples.

V. Conclusions Great advances in experimental techniques concerning the atomic interaction at intermediate and moderately large internuclear distances for excited states in addition to the ground state have been obtained in recent years. Various different experiments-heavy particle scattering with laser excited atoms, molecular spectroscopy with laser induced fluorescence, line shape broadening, and electron scattering4an contribute to the knowledge of the interatomic potentials involved in such interactions. To determine the potentials quantitatively from the original experimental data with an accuracy comparable to the high accuracy of the original experimental data some a priori knowledge must be introduced. Ab initio calculations are usually too expensive and they are, on the other hand, not as accurate as experiments can be. The model potential calculations present an alternative. First, they provide some qualitative insight quite easily but much more importantly they are from the beginning ready to be combined with experimental results in a natural way. Compared with the traditional phenomenological determination of the interatomic potential, this combination has the advantage that the modeling is shifted from the eventually many interatomic potential functions involved to one function, namely the model potential, and that the additional knowledge of the electronic wave functions is included. In this way the interatomic potentials for various states and eventually for various interacting systems of a series are obtained with one model potential. An additional less obvious

DETERMINATION O F INTERATOMIC POTENTIALS

97

advantage is found in the analysis of the individual contributions to the model potential. From there one finds a rational way to combine experiments as diversified as electron scattering from the target and atomic spectra of the projectile and investigations of the molecule (by scattering or by molecular spectroscopy) in a unified evaluation. As a matter of experience then the interaction contribution is observed to be described with fewer parameters than needed in the phenomenological approach, provided of course that the model potential is designed with care. However. this combined evaluation has until now been demonstrated with only a few examples. These examples lead to an accuracy of the order of lop5a.u. for the well depths (corresponding to some c m - ’ or better) and of 0.1 a.u. for the equilibrium distances. It seems fair to identify this as a satisfying achievement. Of course severe problems have been brought to light with these more recent experimental results, concerning the flexibility and reliability of currently available models. More research in this direction will be necessary to elaborate the present status of their development. To this end the combination of experimental data and model potential calculations proposed in this article may present a helpful concept.

ACKNOWLEDGMENTS

I wish to thank W. E. Baylis, F. Gianturco, and H. Pauly for fruitful discussions concerning this article and Mrs. Ch. Schneider-Rogier and Ms. B. Jung for their help in the preparation of the manuscript. Special thanks for the many discussions in the course of the experimental work with scattering experiments with laser excited beams are due to my co-workers W. Groger, H.-0. Hoppe, R. Liedtke, and H. Tischer.

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NOTE ADDED IN PROOF Some investigations related to the subject of this article that have been published recently need to be mentioned. Concerning experimental results. a new evaluation of the spectroscopic data for Na-Ar has been published (Tellinghuisen et a/., 1979). It confirms the equilibrium data used in this article and gives a refined shape function for the respective potentials. A central question concerning the various model potentials proposed. namely the accuracy of their numerical treatment, has been investigated for the Baylis model in two papers (Czuchaj and Sienkiewicz, 1979: Duren and Moritz, 1980). I n these more accurate treatments the model yields good results concerning the excited states (with the free parameter fitted to the ground state). Specifically, for the A2v,,, state of Na-Ar, c = 1.85. lo-’ a.u. and R , = 5.55 a.u. (Duren and Moritz, 1980) are obtained, which are to be compared with the values of Table V.

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I I ELECTRONS* R . J . CELOTTA and D . T. PIERCE United States Department of Commerce National Bureau of Standards Washington. D . C. 1. Introduction . . . . . . . . . . . . . . . . . . . I1 . Source Characteristics . . . . . . . . . . . . . . . . 111. Chemi-ionization of Optically Oriented Metastable Helium . . . A. Principle . . . . . . . . . . . . . . . . . . . 9 . Apparatus and Procedure . . . . . . . . . . . . . . C . Discussion of Source Characteristics . . . . . . . . . . I V . Photoionization of Polarized Atoms . . . . . . . . . . . A. Principle . . . . . . . . . . . . . . . . . . . 9 . Apparatus and Procedure . . . . . . . . . . . . . . C . Discussion of Source Characteristics . . . . . . . . . . V . The Fano Effect Source . . . . . . . . . . . . . . . A. Principle . . . . . . . . . . . . . . . . . . . B . Apparatus and Procedure . . . . . . . . . . . . . . C . Discussion of Source Characteristics . . . . . . . . . . V1 . Field Emission from Ferromagnetic Europium Sulfide on Tungsten A . Principle . . . . . . . . . . . . . . . . . . . 9 . Apparatus and Procedure . . . . . . . . . . . . . . C . Discussion of Source Characteristics . . . . . . . . . . VII . Low-Energy Electron Diffraction . . . . . . . . . . . . A. Principle . . . . . . . . . . . . . . . . . . . 9 . Realization of a LEED Source . . . . . . . . . . . . C. Discussion of the LEED Source . . . . . . . . . . . VIII . Photoemission from GaAs . . . . . . . . . . . . . . . A . Principle of Polarized Photoemission from GaAs . . . . . . 9 . Apparatus and Procedure . . . . . . . . . . . . . . C . Characteristics of the GaAs Source . . . . . . . . . . IX . Summary . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . .

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I. Introduction It has been more than half a century since Goudsmit and Uhlenbeck (1925, 1926) added the concept of the electron spin to the world of the physicist. By 1928 Oppenheimer had written a classic paper on the necessity of including electron exchange when calculating electron-hydrogen interactions. In 1929 Davisson and Germer were attempting to observe the polarization of a beam of electrons upon reflection from a metal surface. Why is it that experimental investigations along these lines are currently still topics of great interest? The reason lies in the fact that it has always been experimentally very difficult to use the electron spin as a controlled parameter in electron beam experiments. There exist no filters for electron beams exactly analogous to Stern-Gerlach magnets. Beams of electrons with their spins preferentially oriented in some direction must be formed by a highly spin-dependent process, and eventually the degree of polarization determined by a second spin-dependent process. In general, the techniques used in the past have been inefficient and experimentally cumbersome, which accounts for the paucity of experimental studies involving spin-polarized electron beams. A wide variety of experiments are possible, however, and they generally fall into two categories. The first type uses the capacity to control the spin direction to measure the degree to which an interaction depends on this variable. Conventional experiments always produce spin-averaged information, from which the exact role of the electron's spin is difficult to deduce. The second possibility is to use the spin direction as a label for a particular subset of the electrons. This assumes a knowledge of the spin conservation properties of the entire process. The fields of science that benefit from improvements in polarized electron technology are extremely diverse as a result of the ubiquitous nature of electron-based measurement techniques. In high energy physics, polarized sources have been used to test quark-parton models of the proton by deep inelastic scattering of electrons on protons with both particles polarized (Alguard et a/., 1976). More recently the unified field theory, for which Weinberg and Salam received the Nobel Prize, was confirmed by Prescott et a/. (1978) via accurate measurements of the spatial asymmetry of the high energy scattering of polarized electrons from hydrogen and deuterium. Measurements of the polarization of electrons photoemitted from solid surfaces (Alvarado et a/., 1978; Campagna et a/., 1976) are adding to our knowledge of the band structure of magnetic materials. Spin-polarized low energy electron diffraction (PLEED) is now being explored as a way to determine the structure of solid surfaces (Wang et al., 1979; Kalisvaart et al., 1978; Feder el a/., 1976), and recently direct

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observations of surface magnetism have been made via the exchange interaction between a polarized electron beam and a magnetized surface (Celotta et a/., 1979). The testing of first-principle theories of electron atom scattering is an obvious area for the application of polarized electron beams. Current efforts include the interaction of polarized electrons with hydrogen (Alguard et a/., 1977) and lithium (Raith et a/.. 1980). Studies have now been made of inelastic exchange excitations in mercury (Hanne and Kessler, 1976; Hanne, 1976). Other work has shown that even photoelectrons ejected from unpolarized atoms with unpolarized light can be polarized (Heinzmann et a/., 1979). There are also biological applications. I t has been proposed that, via the electron polarization of beta rays, the weak interaction is responsible for the predominance of chiral molecules of one-handedness in living matter. This has recently been tested (Hodge et a/., 1979; Bonner et a/.. 1975), using intense polarized electron sources to compress the evolutionary time frame to a tenable length. The future application of polarized electron technology is expected to continue in the areas just mentioned and to find new application elsewhere. The field has now reached the level of development where beams of polarized electrons are available for many applications with the same characteristics as ordinary electron beams. The quantitative advances in beam characteristics bring with them a qualitative change in experiments dealing with electron polarization. The “tour de force” nature of the previous experiments will not be necessary in the future, and the number of future polarization experiments should reflect this change. Over the past decade the topic of electron polarization and the production of polarized electron beams has received considerable attention in the literature. The most comprehensive treatment has been given by Kessler (1976). Other reviews include Farago (1965. 1971), Kessler (1969, 1973). Lubell (1977). Raith (1969), and Tolhoek (1956). New ways of producing polarized electron beams have been suggested regularly, because many times the existing technology left much room for improvement. Frequently, these clever ideas were the victim of scaling laws or engineering difficulties that prevented them from being competitive with existing designs. Aside from a diversion toward laser multiphoton ionization (Van der Wiel and Granneman. 1978) a few years ago, the general trend has been toward solid state sources. If comparable schemes can be used with solids, the number of electrons within a given cathode area is far larger. Generally, the schemes fall into one of two categories. In the first, a spin-selective mechanism is used to free the electron from its bound state. The ensemble of electrons in the cathode region have no net spin orienta-

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tion, but only a spin-directed subset is extracted. In the second general scheme, the cathode electrons are prepared with net polarization and then liberated in a way that does not affect their polarization. In this article, we shall describe the current state of the art of polarized electron sources. N o attempt will be made to include all of the suggestions for possible sources, or devices that for one reason or another we feel have been superseded by new designs. What remains is a collection of polarized electron sources diverse enough to fulfill most needs. Each source will be described in detail following a general discussion, in Section 11, of source characteristics. Finally. the sources will be compared and their various merits discussed.

11. Source Characteristics Given the wide variety of possible applications, it is clear that no single design could satisfy all of the requirements. Our purpose in this section is to set down the collection of specifications that are necessary to optimize the match between source and experiment. Most prominent is the degree of polarization produced in the emergent beam. Given a beam of electrons and a quantization axis, the polarization along that axis, P , is defined as P = ( N , - N l ) / ( N , N 1 ) , where N , and N , are the number of electrons with spins parallel and antiparallel to a quantization direction. Thus P may range from + 1 to - I , with 0 corresponding to an unpolarized or conventional electron beam. All else being equal, the highest degree of polarization is sought after in a source. The application of the device also determines the choice of the relative orientation of the quantization axis and the momentum or electron beam direction. If the two coincide, it is referred to as a longitudinally polarized beam; whereas if the polarization direction is normal to the beam direction in the electron’s rest frame, it is referred to as a transversely polarized beam. In some applications, it may be necessary to switch between longitudinal and transverse polarization. One example of such a requirement occurs when the experiment requires a longitudinally polarized beam of known polarization and the polarization is to be measured using Mott scattering, which requires a transversely polarized beam. The transition can be affected by electron optical means. Electrostatic fields can be used to rotate only the momentum direction while leaving the spin direction unchanged; or the crossed electric and magnetic fields of a Wien filter (Galejs and Kuyatt, 1978) could be used to accomplish the opposite. If

+

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such changes in the polarization direction are required, a source that accomplishes this without external electron optical devices can significantly simplify the apparatus. Spin-dependent effects are usually observed by modulating either the size or direction of the incident electron beam polarization and observing the in-phase changes produced in an experimental parameter. Phasesensitive detection can be employed to enhance the observation of these spin-dependent changes, which may be quite small. This makes the ability to modulate the polarization an important source characteristic. The ability to change the polarization without affecting any other beam property is very important. If, for example, an intense magnetic field must be reversed to reverse the polarization, it will be difficult to avoid changing the electron beam characteristics. Then, an observed asymmetry could not be uniquely ascribed to a spin-dependent interaction, but instead could be due to a change of the interaction geometry, beam intensity, and so on. I t is very difficult to make accurate comparisons between sources in this regard, but a consideration of the method of polarization modulation employed will show that some of the devices have an obvious advantage. The time structure of the polarization modulation may also be dictated by the application. Sine-wave or square-wave modulation may be most suitable for phase-sensitive detection as described previously. A random time structure could be most advantageous to guard against any chance of systematic error in measuring small asymmetries. The frequency of modulation should be variable over a wide enough range to allow for selection of an operating frequency at a minimum of the noise spectrum of the apparatus. In order to understand completely the meaning of characteristics relating to source intensity, a short digression into electron optical design theory is necessary. All designs are subject to, and therefore conditioned by, the law of Helmholz and Lagrange (Sturrock, 1955). It states that for any two points along a beam path, the energy E , solid angle dS2, and cross-sectional area dA form a conserved product,

E, d A , dQ, = E2dA,d02 Hence, for example, reducing the energy of the electrons will increase the beam area, or its divergence, or both. The same principle is frequently stated in terms of conservation of the invariant emittance, z,,, = radi?, where r is the radius of the beam, a is the half-angle of the beam divergence, and E is again the energy. The emittance, z = ra, is used to characterize a device and usually takes the form of a closed figure plotted on the orthogonal axes r and a . The boundaries of the figure represent the

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limits for the two parameters and their relationship. The task of electron optical design is then to produce a beam with the desired specifications within the limitations imposed by the law of Helmholz and Lagrange, and taking into account the fact that the phase space emittance diagrams must be carefully matched at the device interfaces or there will be beam current loss. Some sources are based on the emission of polarized electrons from targets aligned by a magnetic field. In the presence of external magnetic fields, there is an additional term to be considered in the invariant emittance,

where e is the electron charge, rn its mass, and B, the magnetic field strength along the beam axis. Thus the emittance invariant is increased by electrons that originate some distance r off axis in a magnetic field. These electrons have a generalized angular momentum which will cause them to spiral out to larger diameters when the beam is brought to a region of low magnetic field. Therefore, the existence of a magnetic field in the source can cause the invariant emittance to be so large as to make the device impractical for many applications where low magnetic field strength is needed in the region of the experiment. An example and discussion of this type of beam calculation can be found in Pierce et ul. (1979b). The current that a source is capable of producing is a very important characteristic, although we must keep in mind that the emittance invariant will determine whether or not we can use it for the intended purpose. Frequently, the quantity known as the electron optical brightness, or Richrsrrahlwert, is used to combine the intensity and emittance into one figure of merit, B = I / r 2 r 2 a 2 .The quantity B / E will be conserved if no beam is lost by the device. A figure of merit useful for comparing polarized electron sources is P’I. where P is the polarization and I is the total current (Kessler. 1976). The basic assumption made in arriving at this formulation is that in a shotnoise limited situation, one can make up for a lack of polarization by counting longer. This is true in many cases, but there are a few important exceptions. In some cases the limitations of the subsequent apparatus may limit the current, for example, if space charge limiting conditions are reached or if a fragile target which will be degraded by high currents is used. At the limit of very small beam polarizations the point will eventually be reached where the dominant error will no longer be generated by shot noise but will be a systematic error that is not reduced by increasing the beam intensity.

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For some experiments the energy spread in the electron beam must be below a certain level. Both the measured energy full width at half the intensity maximum (FWHM) and the shape of the energy distribution can be important, as is the suitability of the beam for further monochromatization. Other parameters include the long- and short-term stability of the current and polarization, the size, operating pressures, and the cost. Many of the parameters are interrelated, some in ways that are not obvious. For example, the beam from a bright source can be focused through small apertures facilitating differential pumping and thus allowing large pressure differences between source and experiment. Only a careful analysis of the experimental requirements and study of source characteristics can lead to the optimal choice of a polarized electron source.

111. Chemi-ionization of Optically Oriented Metastable Helium Here we find a clear example of the class of sources where electrons are first polarized in a bound state and then, in a second step, make a transition to free space via a process that does not destroy their polarization. In the first step a discharge produces metastable helium atoms that then have their electrons oriented by optical pumping. In the second step, these atoms ionize upon collision with others. The liberated electrons will be polarized if there is conservation of electron spin angular momentum during the reaction. The application of this process to the production of polarized electron beams was first demonstrated by McCusker et a/. (1969). They envisaged using the polarization as a technique for studying the reactions that produce ionization in gas discharges as well as constructing a high intensity source of polarized electrons. In 1969 the source utilized a weak electrical discharge and 0.01-pA beams were produced with 14% polarization. Hill el a/. (1972) confirmed that the reaction between two metastable helium atoms. each in the Z3S, state, is strictly spin conserving. This permitted the spin-polarized electrons to be used as labeled particles to demonstrate the importance of collisional ionization between metastable z3S, helium atoms as a source of electrons for the discharge (McCusker et a/., 1972). These measurements also showed that the electrons extracted from the afterglow following a pulsed discharge were even more highly polarized. By this time the source achieved 8% polarization for a continuous 4-pA beam or 17% polarization in the pulsed mode. Shearer (1974)

108

R. J . Celotta and D. T. Pierce

studied optically the resulting ion polarization following metastable-metal atom collisions. Keliher ef at. (1975) studied the chemi-ionization processes that occurred between aligned He(23S,) atoms a n d Ar, H,, N,, CO, CO,, N,O, or a brass surface a n d found that spin was conserved in these processes. Polarizations of 30% were observed in what by then had become a flowing system with electrons extracted downstream from the discharge. The most recent refinement of this device (Hodge et a / . , 1978) will now be discussed. A. PRINCIPLE

A microwave discharge is used to populate the z3S1 state of helium which lies 19.8 eV above the 1 IS, ground state. Keliher ef al. (1975) point out that in a collision with a molecule, AB, the following reactions are possible: He(23Sl)

+ AB+He(l

IS,)

+ H e ( l IS,)

+ AB+ + e -

+ A + B+ + e -

+ HeAB+ + e -

+HeA+ + B

+e-

In general the spin angular momentum is conserved so that the optical alignment of the triplet metastables will lead to polarized free electrons. The 2'S, metastable. lying 0.8 eV higher in energy than the Z3S,. is also excited in the discharge and has a deleterious effect since it can only yield unpolarized electrons. The discharge is usually operated in such a way as to emphasize the production of triplets versus singlets by more than a n order of magnitude. Because of competing processes in the discharge it is impossible to fully polarize the triplet metastable population, although polarizations of order 50% are obtainable. The electrons generated in the region where collisions occur between helium metastables and reactant molecules result primarily from one of the above-listed reactions. Other sources of electrons would tend to degrade the polarization of the final electron current. One example would be the electrons present in the active discharge. These are essentially eliminated with a flowing system, where the active discharge is spatially separated from the electron extraction region.

SOURCES OF POLARIZED ELECTRONS

B. APPARATUS AND

109

PROCEDURE

The most recent version of this source (Hodge et a/., 1978) is shown in Fig. 1. Helium flows downward to be exhausted by a large mechanical pump (500 I/sec) at the bottom, with flow rates of 25-55 Torrl/sec. A microwave discharge is utilized to produce a metastable 2 3S, population with a minimum population of the 2 IS, metastable. In the optical pumping region, the metastable triplets (23S,) are irradiated with circularly polarized 1.08-ym light. Transitions then occur between various m, sublevels of the 23S, state and the 23P0.,., states. The circular polarization of the incident light imposes a selection rule of Am, = 1 ( - 1) for positive u + (negative u ) helicity, circularly polarized light on absorption, but the higher state may radiate with Am, = 0, ? I . Thus, each cycle of absorption and emission increases m, by one, on the average, if u + light is used, and eventually m, = 1 becomes the only populated sublevel of the 3S, state. A more complete discussion of the pumping process has been given by Schearer

+

~

HEL INP

REACTANT GAS

OPTICAL MASK

RING INJECTOR

1-10 cm 4

FIG. I . Schematic diagram of flowing afterglow polarized electron source showing the microwave cavity, the optical pumping region, the ring injector, and the extraction optics. From Hodge er al. (1978).

R. J . Celotta and D. T. Pierce

110

(1961). T h e preferential direction here is the photon direction. Hence, with

the electrons extracted to the right in Fig. I , a photon beam into the figure would produce transversely polarized electrons, whereas a photon beam traveling from left to right would produce a longitudinally polarized beam. The circular polarization of the 1.08-pm radiation is obtained with the combination of a quarter-wave plate and a linear polarizer. Rotation of one with respect to the other causes a modulation of the polarization from right circularly polarized, to linearly polarized, to left circularly polarized, to linearly polarized, and so on: two full cycles of polarization reversal occurring with each rotation. A perforated ring structure is used to inject a second gas into a n d counter to the flow of metastable helium atoms. Best results have been obtained with CO, at flow rates a few percent of that of the helium. This gas was chosen because of its high cross section for chemi-ionization coupled with large vibrational excitation cross sections. The latter ensures that the electrons quickly thermalize and thus reduces the energy spread of the final beam. Figure 2 shows the relationships between current, pressure, a n d reactant species.

i z W

rz

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---

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-

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- ) D @

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SOURCES OF POLARIZED ELECTRONS

Ill

Extraction is accomplished by biasing the flow tube and extractor housing at -400 V with respect to the extractor first anode. normally held at ground potential. A filter lens system is incorporated in the design to permit measurements of the beam energy distribution by the retarding field method. Finally, provision is made for a Faraday cup collector, a phosphor screen, and a Wien filter. The phosphor screen allows the beam to be visualized for easy focusing, while the Wien filter can be used to rotate the polarization direction for different experimental applications.

C. DISCUSSION OF SOURCE CHARACTERISTICS Beam currents of 10 pA have been obtained from this device, but the polarization decreases slowly with output current above 0.2 pA. as shown in Fig. 3. The increased current is obtained by increasing the flow tube pressure, and it is thought (Keliher et a/., 1975) that the ionization of the background gas in the extraction optics provides an unpolarized component causing the degradation of the polarization. Hodge el a/. (1978) give a figure of merit of 3 x lo-’ for P21 which corresponds to a beam current of 2 pA at 40% polarization. The stated emittance of 2 X rad cm a t 400 V leads to an emittance invariant cinv of 0.04 radcmeV’/*. which is moderately low. Energy analysis of the beam has placed the full width at half maximum at < 150 meV, a value useful for many experiments that require “monochromatic” beams. The source is quite stable, with variations of less than 5% in 8 hr and a reproducibility of 10% from one day to the next. The

a

0.It 0 10-8

10-7

10-6

EXTRACTED CURRENT, A

FIG. 3. Dependence of electron polarization on extracted beam current using CO, as the reactant gas. From Hodge er al. (1978).

R . J . Celottu and D. T. Pierce

112

polarization is modulated externally by optical means, a technique that minimizes the possibility of causing experimental artifacts not related to the observation of genuine spin-dependent effects. The polarization may be fixed, modulated at frequencies up to a few hundred cycles, or given a random time structure. Essentially a continuous source, pulsed operation is limited to relatively long pulses of order 10 msec in duration. The polarization of the final beam may be either transverse or longitudinal, depending on the orientation of the incident photon flux. In some cases this may obviate the need for a Wien filter. The operating pressure in the helium flow tube was typically 65- 100 pm, necessitating a differential pumping design if this source were to be used in an experiment that required low pressures. Additionally, the requirement for exhausting the gas with a 500 I/sec mechanical pump adds significantly to the overall size of the device, and may make i t cumbersome for some applications.

IV. Photoionization of Polarized Atoms The driving force for the development of pulsed sources of polarized electrons has been to permit injection into high energy accelerators. Two such sources (Baum and Koch, 1969: Alguard et a/., 1979) have been built based upon a principle first suggested by Fues and Hellmann in 1930. The photoionization of polarized atoms was studied experimentally beginning in 1961 (Friedmann, 1961; Long et al., 1965). The source. PEGGY, developed by Alguard et a/. (1979). has been used at the Stanford Linear Accelerator Center (SLAC) to study polarized-electron-polarized-proton scattering (Alguard et al.. 1976). We shall base our discussion on this device. A. PRINCIPLE Conceptually. this technique is straightforward. A beam of alkali atoms is formed such that the spins of the valence electrons are oriented in the preferred direction. The electrons are liberated, each by a sing4e ionizing ultraviolet (UV) photon. These electrons remain polarized and can be formed into a beam. Since high degrees of polarization are obtainable in the initial atom beam. this source could ideally produce highly polarized electron beams. A schematic representation of this type of source is shown in Fig. 4. The atom beam, in this case Li, is passed through a six-pole magnet to obtain a

I I3

SOURCES OF POLARIZED ELECTRONS

LONGITUDINAL MAGNETIC FIELD COIL I

MECHANICAL CHOPPER

’ULt

NET

/ /

FIG. 4. Schematic diagram of the polarized electron source PEGGY showing the Li oven, six-pole magnet, flash lamp UV source, and interaction region. From Alguard et al. (1979).

polarized atom beam. This type of magnetic field geometry (Christensen and Hamilton, 1959) acts to focus those atoms with m, = + 4 with respect to the local magnetic field and to defocus those with mj = - i , thus removing them from the beam. Although there is no one preferred field direction within the six-pole magnet, a transition can be made to a single. external, axial magnetic field which serves to define the quantization axis in the interaction region. This transition must be adiabatic: spins will be able to maintain their orientation with respect to the magnetic field only if the transition occurs slowly on the scale of the Larmor frequency. A fundamental difficulty arises at this point. In a region of low magnetic field the nuclear spin I couples with the electron spin s to give the total angular momentum F = I + s, and it is F that will be aligned with the magnetic field direction. The electron polarization will be degraded (Long el ul., 1965) from the high field atomic polarization by a factorf(H),

where X = ( g , - g , ) p o H / A w . Here g, and g, are the electronic and nuclear g values, po the Bohr magneton, and Aw the energy splitting of the hyperfine structure. Figure 5 shows this function for two isotopes of Li. Note that for the limit of H +O, the electron polarization approaches

I I4

R. J . Celotta and D. T. Pierce 1.0

0.8

0.6

--I

0.4

/

0.2

0

0

400 500 600 (G) FIG. 5 . Dependence of f ( H ) , the hyperfine structure coupling function, on external magnetic field for 6Li and 'Li beams. This function gives the degree of polarization obtainable for a given applied field if the beam was initially 100%polarized in a high field region. From Alguard er al. (1979).

100

200

300

H

1 / ( 2 1 + I), so that even for the favorable case of Lib. with I = I , the effective electron polarization is reduced to 0.33. To minimize this effect, a moderate magnetic field is applied to a target atom selected for its small hyperfine structure splitting (for example, 6Li). Two additional effects can lead to a lower polarization of the resultant beam. First. if Li, molecules make their way into the photoionization region, they will contribute a n unpolarized background current. Second. i t is possible to cause resonance transitions to the 2 P state if a broadband light source is used, and the polarization of the atom beam will be degraded when these states decay.

B. APPARATUS AND PROCEDURE The favorable hyperfine structure characteristics just described a n d the fact that it has the highest photoionization cross section of the alkalis, makes 6Li the clear choice for this source in spite of the consequential

SOURCES OF POLARIZED ELECTRONS

1 I5

experimental difficulties. Lithium is corrosive, so special materials and techniques are required to fabricate the oven assembly. Furthermore, the Li ionization limit lies at 230 nm requiring the use of a pulsed UV source and appropriate UV optical elements. The source (PEGGY) discussed by Alguard et a/. (1979) used a Li oven of conventional design with multiple heat shields necessitated by the high (875 “ C )operating temperature. A charge o f 750 g provided a running time of 175 hr. The beam was collimated by heated apertures. The first of these had an associated reservoir to catch the collected Li. The problem of Li buildup inside the six-pole magnet was minimized by operating a mechanical chopper immediately before the magnet. The chopper was synchronized with the flash lamp so that the atom beam would impinge upon the magnet only when necessary. Two different six-pole magnets were used. A standard “atomic clock magnet” gave high atom beam polarization. but a second special large aperture design produced high beam intensities with only a small degradation of the polarization. The magnet does not focus the troublesome Li, molecules, so that the few percent Li, concentration at the oven is further reduced at the photoionization region. A special type of flash lamp was developed for use with this source. The requirements were for operation in the range of 170-280 nm with a pulse width of 1.6 psec, a repetition rate of 180 Hz. and a lifetime in excess of lo7 pulses. To accomplish this a vortex-stabilized argon flash lamp was designed using a pressurized argon flow to improve both the pulse-to-pulse stability and the lifetime of the tube by using the gas flow to carry away sputtered material. The 0.5 J per pulse output was sufficient to photoionize 2%) of the 10” atoms in the interaction region. Excitation of the 2P level was effectively reduced by using a broadband UV interference filter. The direction of polarization of the photoelectrons is specified by an axial 2 1 5 4 field in the photoionization region. The polarization may be reversed by reversing this magnetic field. a process which takes about 1 sec.

-

C. DISCUSSION OF SOURCE CHARACTERISTICS

PEGGY produces a n 85 2 7% polarized electron beam of 2 X lo9 electrons/pulse with the required pulse width and repetition rate for use at the SLAC linear accelerator. I t has been operated in excess of 3000 hr and has made possible a number of polarized scattering experiments. The relatively high polarization of 85% is an important factor in many high energy scattering experiments. The axial polarization can be reversed.

R. J . Celotta and D. T. Pierce

116

although relatively slowly, and only with the introduction of a beam intensity change of up to 5%. PEGGY is the product of a continuous, sophisticated development program and probably lies very close to what is achievable by this technique.

V. The Fano Effect Source Fano (1969) suggested that as an alternative to photoionizing a polarized atom beam with unpolarized light, one could produce polarized electrons using an unpolarized beam and circularly polarized light. The predictions were rapidly confirmed for Cs by direct polarization measurements (Kessler and Lorenz, 1970; Heinzmann et a/.. 1970a. b) and for K, Rb, and Cs by observation of the ionization rate as a function of the incident photon polarization and wavelength (Baum et a/., 1972). Conditions were found (Heinzmann et a/., 1970a, b) where a 100% polarized electron beam could be produced. Both pulsed (von Drachenfels et a/., 1977) and continuous (Wainwright et a/., 1978) versions of this source have been developed and will be discussed here. A. PRINCIPLE

The n 2 S , / , alkali atoms in the unpolarized atomic beam are in the two degenerate ground states m, = 2 1 /2. Photoionization with u + circularly polarized light causes transitions to the continuum final states 'P,/, (mj = 3/2), 2P3/2 (m,= 1/2), and ,PI/, (mj = 1/2), which we will refer to as 1. 2, and 3, respectively. The applicable selection rules for electric dipole transitions and u f light limit the number of transitions to three, which can be labeled by the final state designation, that is. 1, 2, or 3. The fact that the incident 0' light carries angular momentum does not automatically ensure that there will be a net electron spin polarization. I t is necessary to have the spin-orbit coupling in the final state for this to occur. The electron polarization obtained depends upon the relative intensities of the three possible transitions, and these transition probabilities can be reduced to their radial matrix elements R , , R,, and R,. Specifically,

P=

2(R, - RJ(2R3 + R,) + (R, - R J 2 (2R3 + R,)'+ 2(R3 - R,)'

(5)

where the polarization P can be seen to vanish in the absence of the

1 I7

SOURCES OF POLARIZED ELECTRONS PI%I 100

80

60

40 20

0 -20 -Lo -50 *

40.

20. 310

FIG 6

300

290

280

7101 260

2%

2Lo

W

Alnml

(a) Calculations of the expected polarization due to the Fano effect, as a function

of incident wavelength, for Cs, K. and Rb (b) The figure of merit, P2u, for Cs, K, and Rb as

a function of wavelength From von Drachenfels ei a/ (1977).

spin-orbit interaction, which corresponds to R , = R,. Hence, the polarization is a function of the wavelength-dependent radial matrix elements, and as in other polarization-dependent phenomena the most dramatic changes in polarization occur where the cross section for producing one spin state approaches zero while that for the other remains finite. This corresponds to the Cooper minimum in the alkali photoionization cross section. Figure 6 displays both the expected polarization and the square of the polarization times the photoabsorption cross section as a function of wavelength for Cs. K , and Rb. Each polarization curve has a zero in the vicinity of the associated cross section minimum corresponding to small but equal cross sections for the two final spin states. T h e extremes in polarization to either

R. J . Celofta and D. T. Pierce

118

side result from the large ratios, albeit small differences, between the two cross sections. The Fano effect provides a strong theoretical foundation for a polarized electron source that can produce electron beams with a polarization of up to 100% which can be readily modulated. The intrinsic drawback is that the effect occurs near a cross-section minimum. and an experimental realization of a n intense source must address this difficulty.

B. APPARATUS A N D PROCEDURE The salient features of any practical Fano source are the atom beam system and the light source used, a consequence of the need to counteract the effect of small photoionization cross sections. In the device shown in Fig. 7 (von Drachenfels el al., 1977), rubidium atoms are photoionized by a n intense pulse of light at 266 nm. The interaction region is located inside a n axial magnetic field of 120 G as maintained by Helmholtz coils HC. Electrons are extracted at EF. accelerated at AL, a n d finally scattered at D. where their polarization is determined by Mott scattering. The cross section for the photoionization of R b can be seen from Fig. 6 to be considerably smaller than that of C s for regions of high polarization. However, it is possible to obtain very high-intensity, short-duration pulses of light at a wavelength suitable for R b targets. A Nd-laser is frequency quadrupled to produce 8-mJ, 12-nsec pulses with a wavelength of 266 nm. In order to provide a sufficient number of target atoms. 20 atomic jet HC

L

CT

HC

CT

FC

D

TE ff0 F AL BC BP SP D CO FW T FIG. 7. Schematic of Fano effect source showing laser beam (L), Helmholtz coils (HC), cold traps (CT), focusing coil (FC), test electron source (TE), extraction field (EF), oven system (0),Faraday cup (F), accelerator lens (AL), beam-steering coils (BC), beam-steering plates (BP), spin precessor (SP), detectors (D), collimators (CO), foil-carrying wheel (FW), and telescope (T). From von Drachenfels el al. (1977).

SOURCES OF POLARIZED ELECTRONS

I I9

beams are arranged to intersect the laser axis. A ring structure creates a uniform electric field along this same axis to accelerate the electrons and form a beam. Care must be exercised in the electron optical design to discriminate against unpolarized electrons photoemitted by the intense UV light from chamber walls or electrodes. The continuous Fano source (Wainwright et d.,1978) contains the same essential elements but uses a single two-stage Cs oven. The light source is a 1000-W, high pressure Hg-Xe arc lamp which. after filtering and polarizing, produces 120 mW of light in the wavelength range of 250-318 nm. Both sources use a rotating quarter-wave plate to convert incident linearly polarized light alternately to right and left circularly polarized light.

-

C. DISCUSSION OF SOURCE CHARACTERISTICS The pulsed source produces 12-nsec pulses of 2.2 X lo9 electrons each, at a repetition rate of 50 Hz, with a polarization of 65%. I t has also been used with a different light source and a Cs beam (von Drachenfels et a/.. 1974) to produce 3 x lo9 electrons/pulse with 90% polarization. with a repetition rate limited to 3/min. The continuous source can produce 10-nA electron beams that are 63% polarized. Both sources require that the alkali be replaced periodically. The pulsed R b sources can run continuously for 90 min, at which time the R b must be recirculated in the oven assembly for 10 min. This may be repeated up to 24 hr before a complete refilling is necessary. The continuous source can operate without interruption for 75 hr before its oven must be refilled. In both cases. the electron beam is formed from an extended region where electrons are accelerated in a uniform electric field. As a consequence, there is a position-dependent distribution in the energy of the photoelectrons. The breadth of the resulting energy distribution therefore depends directly on the accelerating field strength. Unfortunately. if small fields are used, the polarization is found to degrade substantially. This is presumably due to low energy e-Cs spin-exchange collisions. The field strength used in the continuous source causes a broadening of the electron energy distribution to the point where the full width at half maximum is 3 eV. This is an order of magnitude larger than that of normal thermionic cathodes and limits the application of the source in electron spectroscopy. The polarization direction is longitudinal and may be rapidly and easily reversed or modulated by remote optical means. Asymmetries in beam intensity with polarization reversal may be kept to very low levels, so that experiments with small spin-dependent effects may be investigated.

R. J . Celotta and D. T.Pierce

120

VI. Field Emission from Ferromagnetic Europium Sulfide on Tungsten Fues and Hellman (1930) also suggested the possibility of obtaining spin-polarized electrons in field emission from ferromagnets. Field emission is attractive because the field emitter tip with a radius of order 1000 A is essentially a point source of electrons. Field-emitter cathodes are used in some electron microscopes where very high electron optical brightness is required. Mueller et al. (1972) discovered that electrons with a polarization of 89% were produced by field emission from a W tip coated with a thin film of ferromagnetic EuS. The EuS acts as a spin filter for electrons from W: the exchange split EuS conduction band provides a path through the EuS for electrons of one spin direction. This mechanism is different from the direct emission of ferromagnetically aligned electrons envisioned by Fues and Hellmann. An example of this type of emission would be field emission from a ferromagnetic iron tip. Using polycrystalline ferromagnetic metal tips or evaporated metal films, Chrobok et a/. (1977) were unable to obtain stable or reproducible polarized field emission. Landolt and Yafet (1978) recently reported measurements of single crystal iron emitters and found a polarization of 25% for emission along the (100) direction. Compared to EuS-coated W, however, an Fe emitter is unattractive as a source of polarized electrons due to its lower polarization and the difficulty of preparing single crystal tips. Mueller et al. (1972) found that the intensity and polarization of the emitted electrons were sensitive to the thickness of the EuS films and to the way they were annealed. Further investigation of W-EuS emitters including a determination of the optimum conditions for a polarized electron source has been carried out by a group at the University of Bielefeld (Kisker et a/., 1976, 1977, 1978; Baum e t a / . , 1976. 1977; Raith et al.. 1980). This is the work we shall discuss. A. PRINCIPLE

Europium sulfide is a ferromagnetic insulator with a Curie temperature T, = 16.6 K. The positively charged Eu2+ ion has a localized spin-only moment due to the half-filled 4f shell configuration). The occupied valence band is formed from the 3p wave functions of the Sz- ions and lies about 3 eV below the lowest conduction band made up of the 5d wave functions of the Eu2+ ions. The localized 4f levels lie 1.6 eV below the bottom of the conduction band. Below T,, the conduction band is split by ferromagnetic exchange such that those states with spin parallel to the 4f7

121

SOURCES OF POLARIZED ELECTRONS

-W-EuS

LAYER 0

100

200

-w NORMAL COORDINATE, x ( A ) 300 400

I N T E R N A L BARRIER

FIG.8. Band model for the W-EuS field emitter. Electrons of predominantly only one spin direction tunnel from the W into the lowest-lying exchange split EuS conduction band. From Kisker er al. (1978).

electron spins lie lower in energy than those with spin antiparallel. This splitting of the conduction bands causes the internal barrier for emission from W into ferromagnetic EuS to be spin dependent as shown in Fig. 8. Kisker e/ a/. (1978) find that the polarized field emission results can be well understood by considering the action at this internal barrier a n d neglecting the effect of the external barrier at the EuS-vacuum interface. The external barrier is high enough so there is n o tunneling from the magnetized 4f levels. T h e EuS energy bands decrease in energy with distance x from the W surface in Fig. 8 because the energy is lowered approximately as eFlx. where F, = F e / c is the internal electric field caused by the penetration into the EuS of the applied external field F,, and e = 1 1 . 1 is the static dielectric constant. For large enough EuS film thickness and applied field strength, the conduction band of the EuS at the vacuum interface can be lower than the Fermi level of the W. T h e conduction band of the EuS splits a s the EuS is cooled below T,. Then electrons of one spin can tunnel from the t o the lower conduction band and tungsten through the barrier of height

R. J . Celotta and D. T. Pierce

I22

proceed from there into vacuum. Because of the exponential nature of tunneling, the tunneling current into the higher lying of the split conduction bands is negligible compared to that going to the lower. The tunnel current density into the EuS is j ( T ) = [ AF,2/+,(T)] exP[

-

w(T)3'2/F,]

(6)

+,

where A and B are constants. Noting that changes only about -0.14 eV Kisker et a/. (1976, 1978) show that the logarithm of the field-emitted current is related to by I n I ( T ) - InI(T,) %(Tc) - + i T ) x In I ( 0 ) - In I ( T J +dTJ - +,PI

(7)

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4-

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.

v

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0

-04

m

-03

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(right s a l e )

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(left scale) I

I

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10 11 12 13 14 15 16 17 18 19 T I P TEMPERATURE, T ( K )

FIG. 9. The field-emitted current going through the probe hole (left scale) and the field-emitted electron spin polarization (right scale) as a function of the W-EuS tip temperature. The solid line gives M , = P(4f7), which was calculated from Eq. (7) using the I ( T ) measurements of the dashed curve. From Kisker el al. (1978).

123

SOURCES OF POLARIZED ELECTRONS

The right side of Eq. (7) is the ratio of the splitting of the conduction band at a temperature T to the maximum splitting at T = 0. T h e splitting is proportional to the magnetization as observed in optical measurements of Busch a n d Wachter (1966) and tunneling measurements of Thompson el a/. (1971). Thus the current ratio of Eq. (7) gives the relative magnetization M A T ) = M(T)/M(O). The emission current a n d the polarization measured as a function of temperature are shown in Fig. 9. The current rises steeply as the tip is cooled below T, because the internal barrier is lowered. T h e solid curve M I ( T ) , calculated using Eq. (7), fits the measured polarization points quite well. Since the magnetization is due to the 4f7 electrons, M , ( T ) = P(4f7).I t is surprising that the measured polarization P is about the same as the polarization of the 4f7 electrons since n o f electrons are emitted. In fact, as soon as the bands split slightly, one might expect a perfect filter giving unity polarization a t all lower temperatures. I t has been suggested (Kisker er al.. 1977) that the emitted electrons have about the same polarization as the f electrons because of spin-exchange scattering between the emitted electrons a n d the f electrons during the emission process. T h e temperature dependence of P can also be explained by a model of Nolting a n d Reihl (1979) in which the conduction bands consist of mixed spin states. B. APPARATUS A N D PROCEDURE A schematic of the apparatus is shown in Fig. 10. This apparatus was used to measure emission current, spin polarization. energy distribution, and energy-selective spin polarization, all as a function of emitter temperaFLUORESCENT

SCREEN

FARADAY CUP

\“IN
FLUORESCENT

,FILTER

LENS

ANODE

/

\ L E ~ S 1 U/ OVEN

1

\

ITUBE LENS LWIEN FILTER

E L E C T R I C L~~~~~~ E L E C T R I C DEFLECTION DEFLECTION PLATES MIRROR PLATES

MULTIPLIER

1

FARADAV C U P

L co LLI M A T O R S

+HIGH

VOLTAGE4

FIG. 10. A schematic drawing of the polarized Field emission apparatus. Only the part on the left up to the Faraday cup is required for a polarized electron source. From Kisker el 01. (1978).

124

R. J . Celotta and D.T. Pierce

ture and for different annealing conditions. We will discuss only those parts needed for a source of polarized electrons. A polarized electron source can be realized with a much simpler apparatus. The characteristics of the W-EuS field emission change when residual gases from the vacuum adsorb on the tip. This requires that the tip be housed in a bakable ultrahigh vacuum (UHV) chamber and maintained at a pressure of less than 1 X lo-'' Torr. The emitted beam parameters were found to be stable for several hours at pressures of 0.5 to 1 X lo-'' Torr but to change within half an hour at a pressure of 5 X lo-" Torr. If the polarized electron beam is introduced into an experimental chamber that has poorer vacuum, such as would be the case in scattering from an atomic beam, differential pumping can be used to separate the U H V chamber from the region of poorer vacuum. The tip is mounted on a cold finger that can be detached from the vacuum chamber when tip replacement is required. The cold finger is electrically insulated from the chamber and in operation is at a potential of - 2 kV with respect to the chamber which is at ground. The W tip can be heated to -3000 K by passing current through the bow that holds it. The temperature is determined by measuring the resistance across the W bow. From Fig. 9, it is evident that the desired operating temperature is 10 K or less. Kisker el al. (1978) employed a dynamic cooling system in which the rate of the flow is adjusted to vary the temperature from 9 to 40 K. Since temperature-dependent measurements are not required for operation only as a source, a simple static He dewar system would be adequate. A weak magnetic field of 25-250 Oe was produced by the solenoid which surrounds the tip. Combined with the radial electric field at the tip. this longitudinal magnetic field helps focus the field emission image onto the fluorescent screen. For the apparatus of Fig. 10 the first focus is at 70 Oe; the electrons are focused to a spot < 1 mm in diameter. Without the longitudinal field, only about 1/50 of the current gets through the probe hole. In this weak longitudinal magnetic field. the magnetization of the EuS is along the axis of easy magnetization parallel to the surface. The direction of this easy axis cannot be controlled. The magnetization can point in either direction along this axis and has a probability of reversing to the opposite direction when the tip is heated momentarily above T, and cooled again. I t is desirable to have a pair of coils (not shown in Fig. 10) capable of producing a transverse field of - 2 kOe. The coils allow the tip to be cooled below T, in an applied transverse field, thereby determining the direction of the magnetization along the easy axis. The transverse field can then be removed. The polarization reversal cannot be accomplished rapidly, but it can be achieved with very little change in the electron beam. The Larmor precession of the polarization in the weak longitudinal field

SOURCES OF POLARIZED ELECTRONS

125

can be controlled by appropriately choosing the operating conditions. If the easy magnetization axis does not lie in the desired plane, the longitudinal field can be changed from the best focusing condition so that the Larmor precession rotates the polarization into the desired plane. A Mott detector is required for these adjustments. If the longitudinal field is greater than about 2 kOe, the magnetization of the EuS is along the field and the polarization of the emitted beam is longitudinal. The potential of the anode that surrounds the emitter in Fig. 10 can be varied from 0 to 2 kV. The anode contains a set of deflection plates to steer the beam to the probe hole in the fluorescent screen. The tube lens shields the beam electrostatically and can be used to change the magnification of the field emission pattern. The pattern can be viewed in the mirror from a window. Once the beam is through the probe hole of the fluorescent screen. there will be a number of lenses and deflection plates required to transport the beam to the experimental area for a particular application. Suitable W tips with the [ 1 lo], [ 1 1 I], or [ 1121 direction parallel to the tip axis are available commercially. The tip is mounted such that it is centered on the axis of the cold finger. After bakeout of the apparatus, the W tip is cleaned by flashing to about 3000 K and a characteristic W field emission pattern is obtained. The EuS is evaporated from a small oven shown in Fig. 10, which can be inserted in front of the tip and is located 7 cm from the tip. The oven can be heated by electron bombardment to 2300 K. The deposition rate was determined in a separate measurement using a quartz crystal microbalance. In this way the approximate thickness of the layers evaporated on the W tip could be determined. The EuS film thickness varied between 350 and 750 a. Old EuS films can be flashed off the tip and new ones deposited as required. The tip is grounded during EuS deposition. Kisker el al. (1978) report that tips are easily destroyed if the potential is applied before the tip is annealed and cooled. It was also established empirically that a tip should produce a specified current below a certain extraction voltage: otherwise more annealing is required. The annealing of the EuS film is extremely critical. For example, films annealed to temperature T , over 1100 K sometimes show a decrease in emission current on cooling rather than a n increase as in Fig. 9; the polarization from such films is zero. There is a critical annealing that is optimum for a polarized electron source, which is at 840 2 10 K for about 1 sec. I t is thought that at this temperature the EuS crystallizes and is stoichiometric. After a critical anneal, the energy distribution is a very narrow peak, with 100 meV FWHM, located at the W Fermi level. The critically annealed emitter does not produce a symmetric field emission pattern. There are one or two bright spots and the rest of the screen is

-

126

R. J . Celotta and

TA

- 1200

,

,

0

-1

D. T. Pierce

K

-2

-3

ELECTRON ENERGY,E(eV)

FIG. 11. Energy distribution curves of the field-emitted electrons for different annealing temperatures. The curve (a) represents the desired annealing condition for a polarized electron source. From Kisker el af. (1978).

-

dark. On annealing to higher temperatures up to T , 1100 K, the polarization remains 85%. and the behavior of the emission is as in Fig. 9. The field emission pattern shows emission from the ( 1 12) planes. There are, however, several changes in the energy distribution. and it is for this reason that a critical anneal is desirable. The changes that take place in the film, which are manifested by the preceding effects, are described by Kisker et al. (1978). Kisker et al. (1978) note that the absolute values of the T , they report could be off by as much as 150 K due to the difficult and indirect measurement. In building a polarized electron source of this type, one would have to find the critical temperature empirically. A convenient way to do this is to have a Faraday cup which can be inserted in the beam after the probe hole. By equipping the Faraday cup with a retarding electrode and varying its potential, the electron energy distribution can be measured. Energy distributions typical for four different T A are shown in Fig. 11.

SOURCES OF POLARIZED ELECTRONS

127

When the distribution of Fig. I la is obtained. one has achieved optimum conditions. C. DISCUSSION OF SOURCE CHARACTERISTICS

-

The current from an individual field emission spot selected at t h e probe hole is typically lo-' A and the polarization is 85% transverse to the electron beam axis. The polarization direction can be reversed by raising the temperature above T, and cooling in an applied transverse magnetic field of appropriate direction, Even though the electron beam is disturbed very little by the polarization reversal, this source does not have the desirable characteristic of being able to modulate rapidly the polarization direction as do some other sources. The ferromagnetic EuS on W field emitter can be operated at a temperature < 10 K in an ultrahigh vacuum < 1 x lo-" Torr. The beam parameters are stable for several hours after which time the EuS film must be evaporated off the W tip and a fresh film deposited. Although this tip renewal can be made into a routine process, each tip is slightly different. Also, the experiment must be interrupted when the new tip is prepared, which can be an inconvenience or disadvantage for some experiments. The field emission source of polarized electrons is much brighter electron optically than any other source, which is a great advantage for experiments that can exploit this property. The high brightness comes from the very small emitting area. The emittance is estimated to be 0.8 X radcm at a n energy of 3 eV. This corresponds to an emittance invariant rad cm eV'/'. qnv= 1.4 x

VII. Low-Energy Electron Diffraction Mott (1929) predicted that polarized electrons could be produced by elastic scattering of high-energy (- 100 keV) electrons from heavy nuclei, such as gold. Shull et al. (1943) provided the First experimental evidence for this. The spin-orbit interaction, the interaction of the incident electron's spin with its own orbital angular momentum about the scattering center, causes the differential cross section for an electron with spin parallel to the orbital angular momentum to be different from that of an electron with antiparallel spin. This effect can be exploited to produce polarized electron beams. Also, if the incident beam is polarized, the angular distribution of the scattered intensity will show a spin-dependent asymmetry; this is the basis of the Mott detector which is widely used to measure spin polariza-

128

R. J . Celotta and D . T. Pierce

tion. Mott scattering from thin Au foils has, for example, been used as polarizer and detector in the famous series of 8-2 experiments (Wesley and Rich, 1972). Deichsel (1961) was the first to show that electrons with energy of order 1 keV could be polarized by scattering. in this case from a beam of atomic Hg. The polarization is still caused by the spin-orbit interaction. but from the atom rather than the nucleus. In the following decade, there were many measurements of electron scattering from Hg beams at energies ranging from several electron volts to about 2 keV; this work has been reviewed by Kessler (1969). From the standpoint of a source of polarized electrons, low energy scattering from Hg beams can yield polarized electron beams with currents up to about lo-* A and polarizations of about 20% (Steidl et a/., 1965; Deichsel and Reichert. 1965: Jost and Kessler. 1965, 1966). Polarizations up to about 85% have been observed, but at currents several orders of magnitude lower. Maison (1966) suggested that electrons could be polarized by diffraction from single crystals. The intensity should be increased due to ( I ) the higher density of a crystal compared to an atomic beam and (2) the coherent scattering of electrons into well-defined diffraction beams. The first observation of polarization in low-energy electron diffraction (LEED) was by O’Neill et al. (1975). The possibility of generating polarized electron beams in LEED from crystals of materials composed of high-2 atoms has been noted by many authors (O’Neill et a/.. 1975; Jennings, I97 I ; Jennings and Sim, 1972; Feder, 1974, 1976). We will first review the principles involved in the LEED polarized electron source. From our recent polarized lowenergy electron diffraction (PLEED) data from W(100) (Wang et a/.. 1980). we can estimate the characteristics of a LEED polarized electron source, describe how such a source can be constructed, and appraise its advantages and disadvantages. A. PRINCIPLE

The polarization induced in an electron beam scattered by an unpolarized target is described by the Dirac equation. There is a relativistic term in the energy due to the interaction of the spin of the electron with the magnetic field felt during its orbital motion. If the scattering is from a central potential V ( r ) . the spin-orbit term in the Hamiltonian can be written

SOURCES OF POLARIZED ELECTRONS

I29

where s is the spin of the electron and L its angular momentum. A partial wave solution to the Dirac equation is outlined in the book by Kessler (1976). In general, a n electron beam is not in a pure spin state but is a statistical mixture of spin states which is best described by the densitymatrix formalism. Kessler (1976) uses this formalism to show that the polarization induced in an unpolarized electron beam scattering from a n atom or nucleus is

P’ = S ( 8 ) A

(9)

where A is the normal to the scattering plane and S ( 8 )= i

fs*- s c g Ifl2

+ I 812

where f and g are the direct and spin-flip scattering amplitudes. If a n incident beam of polarization P normal to the scattering plane is scattered from an atom or nucleus, the differential cross section is spin dependent: u ( 8 ) = uo(8)[ 1

+ S(8)P- A ]

This equation is the basis for Mott scattering used to measure spin polarization. S ( 0 ) is sometimes called the Sherman function after Sherman (1956) who made calculations of S for Mott scattering. In the case of scattering from a solid, there is a n array of scattering potentials and the spin-orbit interaction of Eq. (8) becomes

Because electron atom scattering cross sections are large and there is a high density of atoms, there is a very high probability of multiple scattering. Quantities such as S ( 8 ) can n o longer be calculated in a straightforward way. I f one has a beam of electrons of known polarization P , it is possible to measure S ( 8 ) according to Eq. ( 1 1) by measuring the spin dependence of the cross section. However. because of multiple scattering, this S ( 8 ) does not, in general, equal the polarization that would be induced in a beam by scattering from a crystal as in Eq. (9). The polarization induced by the first scattering will in general be changed in magnitude and direction by the second and further scatterings. However, it has been shown by Wang er al. (1979) that when the scattering plane is a mirror symmetry plane of the crystal, the P ’ ( 0 ) induced by scattering equals the S ( 8 ) that can be measured using Eq. (11). This is an important result because it allows us to use our measurements of S ( 8 ) to determine the

R. J . Celotta and D.T. Pierce

130

characteristics of scattering from a crystal for a source of polarized electrons. Alternatively, direct measurements of P ‘ of scattered electrons could be made. Coherent scattering from a crystal produces a well-defined diffraction pattern. For the case of X rays which are only slightly attenuated by the crystal, the diffraction pattern is determined by the Laue condition K = k - k, = C,,,

(13)

where K is the momentum transfer, k, and k the wave vectors before and after scattering, and Ghkl= 2 n / d h k , , (where C,,, is the reciprocal lattice vector and d,,, is the interplanar distance). In contrast, an electron with a n energy between several electron volts and several hundred electron volts can only travel on the order of 10 A in the solid before it loses energy in collisions with other electrons and is removed from the elastic beam. Such a beam probes the outer few layers of the crystal. In this case, the Laue condition is K\I

(14)

= ‘hk

where KII is the momentum transfer parallel to the surface, and G,, is a vector of the two-dimensional reciprocal lattice. There is one diffracted beam for each G,,. For specular diffraction we have K I ,= C , = 0. A useful relation relating the wave vector k, wavelength A, and energy for the electron is

Ikl

= 2a/A = 2n[ E

(eV)/ 150.41

”’

(15)

In a single scattering approximation, the diffracted intensity is given by

I ( K ) = I F ( E , O ) I ’ ~ e x p [ i K . ( r , -r,)] J

(16)

F(O, E ) is the structure factor and contains f and g, which are different than for scattering from free atoms because the atomic potential is altered in the crystal, plus another factor if there is more than one atom per unit cell. It is straightforward to extend Eq. (16) to include the effects of electron attenuation and thermal motion of the atoms (Webb and Lagally, 1973). The exponential term is the interference function and K has the constraints of Eq. (13) for intensity maxima. The structure factor is smoothly varying but the interference function makes the scattered intensity very sensitive to angle and energy. A diffraction condition suitable for a source of polarized electrons can be expected to be limited to a narrow range of angle and energy. As discussed by Spangenberg (1948) and by Pierce (l954), space charge limits the maximum current I , in the electron beam incident on a crystal at

SOURCES OF POLARIZED ELECTRONS

131

energy E in a cone of half-angle a to

I, = 38.5 E'I2a2 (17) where E is in electron volts, I in microamperes, and a in radians. Only a fraction of the incident current will be in the polarized diffraction beam of interest; space charge is not a limitation after scattering. The space charge limited current is independent of beam diameter. However, the maximum current density of the space charge limited beam of Eq. (17) is

J,,,

= 48.6E3'2a2/d2

(18)

which is in microamperes per square centimeter if d, the beam diamter, is in centimeters. If the space charge limited beam current of Eq. (17) for the incident beam at the crystal is reached, the brightness of the electron gun sets a limit on the minimum beam diameter, that is, the current density. Multistage guns (Simpson and Kuyatt, 1963; Kuyatt, 1967) are required in the proposed realizations of the LEED polarized electron sources discussed next.

B. REALIZATION OF

A

LEED SOURCE

I . Performance Estimates for W (100)

To maximize the current I , in the incident beam according to Eq. (17), and hence the reflected current I, we want to work at as high an energy as possible and at a diffraction condition with minimum sensitivity to angle. In the PLEED measurements of S ( E . 0) of a W( 100) surface by Wang et a/. (1979, 1980), the maxima in S ( E , O ) were smaller above about 150 eV. An operating condition that appears favorable for a source is specular diffraction at an energy of 84 eV and an angle of incidence in the [OIO] azimuth in the range 1 1 "-22", that is, the beam incident at the crystal can be a cone centered at 16.5" with half-angle 5.5". The average reflection coefficient I/I, over this range is 8 x and the average polarization is 0.23. From Eq. (17) the incident current I, can be 273 pA so that I is 0.22 pA. This is about an order of magnitude higher than the best obtained for scattering from a H g beam a t about the same polarization (Wilmers et a / . , 1969; Zeman et a / . . 1972). Higher incident currents than calculated from Eq. (17) are found in several of the reports on scattering from Hg, presumably due to the partial neutralization of the negative space charge by positive ions in the higher pressure of those experimental chambers. As another example, a higher polarization can be obtained at slightly lower energy. 78 eV. an angle of incidence of 17". and a cone half-angle of

132

R. J . Celotta and D. T.Pierce

3". Here Pa, = 44% and the average I / I , = 3.3 x lop4. We have I , = 73 (LA and I = 0.024 (LA. This compares favorably with scattering 300 eV electrons from a Hg beam with the same cone half-angle where Steidl el al. (1965) found P = 17% and I = A.

2. Modulation of the Polarization

The modulation of P can be achieved in the LEED source by changing the scattering angle or energy. Changing the angle by physically moving parts of the apparatus is prohibitively inconvenient. The angle can be effectively changed if it is possible to use two equivalent nonspecular diffraction beams. Consider the case of a 67-eV primary beam incident normal to a W(100) crystal surface. There is a ( 1 I ) beam diffracted in a direction 42" to one side of the normal and the (IT) beam at 42" to the other side of the normal. The polarization is 23 ? 2% for the (1 1) beam when the cone half-angle of the incident beam is 2". The polarization of the (IT) beam is exactly equal but opposite in sign. If one could use an electrostatic deflection system to alternately switch one of these diffracted beams into the experimental area, the objective would be achieved. We have measured I / I , = 1.1 X which, assuming a space charge limited incident beam for the above conditions, implies the scattered current could be 0.03 (LA. The polarization of the specular beam can also be modulated by varying the beam energy as shown in Fig. 12. We show specular diffraction curves for an angle of incidence of 15" and 16" over a limited energy range. At 78.2 eV the average polarization for the curves at these two angles is -47%. At 73.5 eV it is 6%. At both energies I / I , is 3 x lop4. Thus by changing the beam energy less than 5 eV, it is possible to change the polarization by over 50% while keeping the scattered beam current constant. In practice the change in beam energy can be fine-tuned to have the current the same at the end points of the modulation, but some variation is expected during switching. A change in beam energy is required before and after the scattering so that the final energy in each case is the same. This can be accomplished with an energy add (subtract) lens; one uses a "field lens" which has the property that the imaging is nearly independent of the field strength (Kuyatt, 1967; DiChio et al., 1973). There must also be a retarding lens in the scattered beam that allows only elastically scattered electrons to remain in the beam. The current in this modulated polarized electron beam is unfortunately limited to about 0.5 nA for the conditions just given.

SOURCES OF POLARIZED ELECTRONS

I

I

I

I

I

I

I

I

I

WflOOI 1001 BEAM

I,>#>

133

i

ELECTRON ENERGY (sV)

FIG. 12. An example showing that electrons are polarized by specular diffraction from W single crystal surface. The polarization can be modulated from -47% to 6% by changin the beam energy from 78.2 eV to 73.5 eV. The scattered beam is only a small fraction of th incident beam, so the intensity of such a source is low even when the incident beam is spac charge limited.

C . DISCUSSION OF THE LEED SOURCE

Compared to scattering from a Hg beam, a LEED polarized electro source has clear advantages both because of its order of magnitude hight current and its simplicity. If one does not require modulation of th polarization. all that is required is to direct an electron beam on the crysti

134

R. J . Celotta and D. T. Pierce

surface and have a means for separating the elastically from inelastically scattered electrons. I t is not clear that the clean W(100) surface, the only one we have studied extensively, is the best for a source of polarized electrons. I t must be maintained in ultrahigh vacuum ( 5lo-'' Torr) and must be flashed frequently to avoid contamination. It must also be operated at an elevated temperature to avoid the room temperature reconstruction that occurs on this surface. We are currently studying surfaces of other crystals that may be more suitable for a source. The aim, of course, is to find a stable surface that yields a scattered beam with high current and high polarization. Feder et al. (1977) have observed a high stability in polarized LEED from Au(ll0) crystals. Riddle et af. (1979) reported that a W(100) surface exposed to CO and then annealed in a prescribed manner produced a polarization that was very insensitive to background gas and not too sensitive to the angular spread of the incident beam. They suggest the possibility of using such a surface in non-UHV conditions such as found in many experiments involving scattering from atomic beams. Otherwise, differential pumping would be required to separate the LEED source ultrahigh vacuum chamber from a higher pressure experimental chamber. In spite of its simplicity, a LEED source has some inherent disadvantages. Even though one obtains the maximum possible current in a space charge limited incident beam, the scattered polarized beam has about three orders of magnitude less current. This disadvantage is inherent in any scattering-type source. In an emission-type source, such as photoemission from GaAs, it is possible to achieve the space charge limited condition in the polarized beam. A second disadvantage is the difficulty in reversing or modulating the polarization.

VIII. Photoemission from GaAs Photoemission from GaAs is the basis of a source of polarized electrons that gives an intense beam with a polarization greater than 40%. This polarization can be modulated with an arbitrary time structure with no change in the beam intensity. For many applications, the GaAs source comes close to being the ideal spin-polarized electron gun. At the ETH-Zurich, Garwin et a/. (1974) suggested that an intense beam of polarized electrons with a polarization up to 50% should be photoemitted from a GaAs surface treated with Cs and 0, and irradiated with circularly polarized light. This was confirmed (Pierce et af., 1975a. b; Pierce and Meier, 1976) by measurements of the photoelectron spin

SOURCES OF POLARIZED ELECTRONS

135

polarization using a Mott detector. Since the potential of photoemission from GaAs as a source of polarized electrons was demonstrated, apparatus designed specifically to operate as a source have been built at the National Bureau of Standards (NBS), the Stanford Linear Accelerator Center (SLAC), the Swiss Federal Institute of Technology in Zurich (ETH), Ecole Polytechnique in Palaiseau, and the Johannes Gutenberg University in Mainz. At the time of this writing, GaAs sources are under construction at several other institutions. The intensity of the GaAs polarized electron source and its excellent polarization reversal characteristics were crucial to the success of the recent measurements at SLAC (Prescott et a/., 1978) of parity nonconservation in high-energy (16-22 GeV) inelastic scattering from hydrogen and deuterium. At NBS, the GaAs polarized electron gun has been used by Wang et al. (1979) to observe large spin-dependent effects in polarized lowenergy electron diffraction (PLEED) from a W( 100) surface. Measurements of the surface magnetization of ferromagnetic Ni(l10) (Celotta et al., 1979) were possible only because of the experimental sensitivity provided by the GaAs source. A. PRINCIPLE OF POLARIZED PHOTOEMISSION FROM GaAs The electrons are polarized in the optical excitation process for transitions from the valence band maximum to the conduction band minimum. Electrons at the bottom of the conduction band are normally bound by 4 eV. Fortunately, it is a well-established technique (Bell, 1973) to apply Cs and 0, to the surface such that the vacuum level is lowered below the conduction band minimum to achieve a negative electron affinity (NEA) and the polarized electrons can escape into the vacuum. The three-step model, ( I ) photoexcitation, (2) transport to the surface, and (3) escape into vacuum, is applicable to photoemission from NEA GaAs (Spicer, 1977). We discuss each step in turn and the accompanying depolarization that is possible.

-

I. Spin Orientation during Photoexcitation The energy levels of GaAs in the region of interest are shown in Fig. 13. At the left, the energy levels (bands) are shown as a function of wave vector k in reciprocal space (k-space). The minimum band gap E, = 1.52 eV occurs at r, the center of the Brillouin zone. The 0.34-eV spin-orbit splitting of the valence bands is essential to the generation of polarized electrons. The electron states at the r point are like an inverted alkali atom: The conduction band is an s , , ~level, the upper valence band is a

136

Ci

R. J . Celotta and D. T. Pierce

mi = - 112

mi =+1/2

E, = 1.52 eV

A = 0.34e

FIG. 13. The energy level (band) diagram of GaAs near the center of the Brillouin zone, point, is shown at the left. The band gap is 1.52 eV and the spin-orbit splitting of the valence bands is 0.34 eV. The transitions possible with circularly polarized light are shown at the right. If electrons are excited only from the valence band maximum, three times as many of one spin are excited as the other.

r

fourfold degenerate p3/2 level, and the split-off valence band is a twofold degenerate p1,2 level. The m, sublevels are displayed on the right side of Fig. 13. The possible transitions for circularly polarized u + (6) light are shown by the solid (dashed) lines. The selection rules require that Am, be 1 or - 1 for u + and u - light, respectively. The relative intensities are readily calculated and are given by the circled numbers. For u + light of energy just sufficient to excite electrons across the band gap (P,/~,+sIl2), three times as many electrons go into the m, = - 1/2 state as into the mJ. = 1/2 state. This gives a polarization, P = (n? - n J ) / ( n ? n J ) = ( 1 - 3)/ ( 1 + 3 ) = - 50%. If the photon energy is large enough to excite electrons from the level, each spin is excited in equal numbers and P = 0. As can be seen from Fig. 13 the sign of the polarization is reversed by reversing the helicity of the circularly polarized light. With care, the circular polarization of the light can be changed without changing any parameters of the electron beam other than its polarization. In particular. the intensity remains constant. This is a very important and attractive feature of photoemission from GaAs as a source of polarized electrons.

+

+

+

137

SOURCES OF POLARIZED ELECTRONS

2. Transport and Escape

Not only are the photoelectrons from GaAs polarized, GaAs is a highly efficient photoemitter when the surface is activated with Cs and 0, to obtain a negative electron affinity. NEA GaAs cathodes are manufactured commercially for a number of devices (Bell, 1973). In order for electrons to be emitted at a clean GaAs surface, they must be excited above the vacuum level which is 4 eV above the conduction band minimum. The electrons are emitted from a depth in the crystal that is limited by the mean free path for electron-electron scattering (- 10 A) even though electrons are excited in the crystal up to a depth of over lo4 A determined by the light absorption length. For an NEA photocathode, the vacuum level is lowered below the conduction band minimum in the bulk as shown in Fig. 14a. The electrons can be emitted from a depth governed by the diffusion length for thermalized electrons which is of order lo4 A; this accounts for the large quantum yield of NEA cathodes. The photoexcited electrons thermalize to the conduction band minimum in about lo-'' sec and then diffuse to the surface where there is no potential barrier impeding their escape. An intermediate case is illustrated in Fig. 14b where there is a small positive electron affinity (PEA) such as can be obtained by depositing only Cs on the GaAs surface. In this case the escape depth is limited by the mean free path for electron-phonon scattering which is of order 100 A.

EF

Cs-0

GaAs

(a)

Vacuum

CS Vacuum

GaAs

(b)

FIG. 14. (a) The valence band and conduction band of p-type GaAs bend down at the surface. The vacuum level can be lowered below the conduction band minimum in the bulk, negative electron affinity, by activating the clean surface with Cs and 0,. Very high photoyield is obtained because electrons that are excited in the bulk may thermalize to the conduction band minimum, diffuse to the surface, and escape without encountering a barrier. (b) Activation with Cs alone can produce a small barrier at the surface giving a positive electron affinity. Electrons can lose energy by scattering with phoilons and fall below the barrier.

138

R. J. Celotta and D. T. Pierce

The band bending shown in Fig. 14a, which is necessary for NEA, occurs when p-type GaAs is activated. Good cathodes are obtained at doping levels from 5 x 10l8 cm-3 to - 2 x I O l 9 cm-’. The low index planes of GaAs, (loo), ( 1 lo), ( I I I)A, and ( 1 1 I)B, can all be activated to NEA; (100) and (1 1 l)B are preferred for commercial photocathodes. The band-bending region is 100 wide. The exact nature of the Cs-0 activation layer is still an active area of research, but it is believed that the composition is approximately Cs,O. When electrons enter the band-bending region they are accelerated toward the surface. They become “hot” electrons with up to about 0.2 eV energy and can then lose energy by scattering with phonons. When an electron reaches the surface it can be transmitted and escape into vacuum or it can be reflected. If it is reflected, i t may get turned around again by scattering with a phonon for another attempt to escape at the surface. An electron can also recombine with a hole before it escapes. The probability that an electron which travels to the band-bending region is ultimately emitted is called the escape probability ?Pest. The quantum yield for an NEA cathode can be described by the equation (James and Moll, 1969)

-

Y

=

A

?Pest/[ 1 + ( a L ) - ’ ]

(19)

where here UI is the optical absorption coefficient and L is the diffusion length. This equation is valid in a photon energy range from the band gap energy E , to E , + 0.3 eV where the fraction of electrons thermalizing to the r minimum is unity and the fraction thermalizing to the L or X minima is zero. At higher photon energies. electrons can therrnalize to L or X minima where they can remain as they travel to the surface and are emitted. Equation (19) can also be extended to that case (James and Moll, 1969).

3. Depolariration

The excited electrons depolarize in a characteristic time. the spin relaxation time 7,.Neglecting for the moment the influence of the surface, there is a characteristic lifetime T of the excited electron in the bulk. By measuring the circular polarization of the recombination luminescence as a function of magnetic field, the times T and T , can be determined as discussed in reviews by Zakharchenya (1972) and Lampel (1974). The average polarization of the excited electrons P , as determined from luminescence measurements is different from the theoretical polarization P,,

SOURCES OF POLARIZED ELECTRONS

139

according to

Note that the shorter T is relative to 7,. the closer P , is to P,h. Coming back to the case of an N E A photocathode, the polarization P of the photoemitted electrons is expected to be higher than P , since, o n the average, the time the excited electron spends in the material is less than T. A quantitative calculation of P must include the details of the photoemission process such as the finite penetration of the light and the diffusion of the excited electrons to the surface. As shown in the paper describing the NBS polarized electron source (Pierce et al.. 1980).

where ( 0 ~ )=” L~, D is the diffusion constant for the electrons, and T = T,T/(T, T). As the electrons pass through the activation layer at the surface, they may suffer additional depolarization. The cross section for spin exchange scattering of low-energy electrons from alkali atoms is very large (Campbell et a/., 1971). Any atomic like Cs in the Cs,O layer, which is known to be on the Cs-rich side of Cs,O (Spicer, 1977). could cause depolarization by this mechanism. The origin of the low polarization of the photoemitted electrons from N E A GaAs(l10) observed by Erbudak and Reihl (1978) was attributed to spin-exchange scattering. They found a maximum P = 21% at hw = 1.55 eV. This value of P is much lower than PI,, = 50% or the P = 43% measured at NBS for N E A GaAs(100). The dependence of P o n the crystal face of G a A s is a n important consideration for the design of a polarized electron source a n d will be discussed more later.

+

4. Examples of N E A and PEA Photoemission

The quantum yield and the polarization measured for N E A GaAs( 110) and (100) surfaces are shown in Fig. 15. Considering the quantum yield first, both curves show the steep rise a n d sharp knee at threshold characteristic of N E A photocathodes. Such yield curves were not observed in the early measurements of Pierce a n d Meier (1976) at ETH; the higher P = 40% which they measured for GaAs(l10) was likely from a surface where negative electron affinity was n o t achieved. The yield curves from commercial GaAs photocathodes have a somewhat sharper knee than the curves of Fig. 15. and the magnitude of the

R. J . Celotta and D. T. Pierce

140 lo-' -

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FIG. 15. The quantum yield (left scale) and polarization (right scale) are shown for two faces of GaAs activated to negative electron affinity, (100) (Pierce et a/., 1979a) and (110) (Erbudak and Reihl, 1978).

yield at Aw = 1.6 eV is up to 10 times higher. Part of this higher yield can be attributed to cleaner starting surfaces and optimized activation procedures. However, the main difference is in the starting material. The GaAs of the photocathodes in Fig. 15 was bulk material, a polished bulk wafer in the (100) case and cleaved crystal surface in the ( 1 10) case. High-yield commercial cathodes use epitaxially grown material with diffusion lengths of several microns. Bulk material, on the other hand, typically has a diffusion length of about 0.5pm. Using Eq. (19) and independently measured values of a, the diffusion length of the (100) cathode of Fig. 15 was determined to be 0.4 pm (Pierce er a/.. 1980). While much higher yields can be expected for polarized electron sources using epitaxial cathodes, the effect of the longer diffusion length on the polarization has not yet been measured. Since the electrons can spend more time in the GaAs, it might be expected that according to Eq. ( 2 1) there would be a greater depolarization. This is a n area that needs further investigation. Note that the yield shown in Fig. 15 is already sufficient for many applications; at 1.6 eV, I-mW incident radiation will produce a photocurrent of 20 PA. Although the yield curves for the two faces of GaAs in Fig. 15 are quite

SOURCES OF POLARIZED ELECTRONS

141

similar, P is strikingly different. The similarity in the yield curves suggests that the diffusion lengths L = ( 0 ~ ) ‘are ’ ~similar a n d hence T is about the same. In general, T is material dependent a n d is influenced by defects and impurities. Although material or preparation differences cannot be ruled out entirely, i t is interesting to look for another cause of the difference in P for the two surfaces. Erbudak a n d Reihl (1978) compare their polarization to the P, = 30% measured by Fishman and Lampel (1977) for similar doping and temperature. I t is interesting to note that for P, = 30%. and a L = 1/2 appropriate to the yield curve of Fig. 15, the expected polarization from Eq. (21) is 42%. I f spin-exchange scattering causes the depolarization as suggested. the question arises as to why it has such a large effect on the ( 1 10) surface but not the (100) surface. T h e calculations of Burt and Inkson (1977) predict a much smaller transmission coefficient a n d surface recombination velocity for the (1 10) surface than for the (100) surface. This leads to a model of emission from the ( 1 10) surface where a n electron makes repeated attempts at transmission into vacuum. It has a high probability of being reflected at the potential step at the Cs-O/vacuum interface, traveling back into the band-bending region and getting turned around again by scattering with a phonon, and making a further attempt a t transmission. I f an electron is not transmitted a t the (100) surface, there is a high probability i t will recornbine. Thus, in spite of the difference in the transmission coefficients a t the two surfaces, the overall escape probability and quantum yield is very similar. However, the depolarization a t the ( 1 10) surface due to spinexchange scattering could be much higher if the electron passes back a n d forth through the C s - 0 layer many times before escaping (Pierce ef d.. 1979a). If the GaAs(l10) surface is used in a polarized electron source. it is advantageous to use a surface with a PEA to obtain a higher polarization (Reihl el a[., 1979). T h e height of the barrier on the PEA surface varies with Cs coverage; the yield a n d polarization for two different photothresholds, 1.6 and 1.7 eV, are shown in Fig. 16a and b. Reihl et ul. (1979) find that stable surfaces which yield P = 35% at hw = 1.95 eV as in Fig. 16b, can be achieved by activating with Cs a n d “aging” the cathode for a day. T h e electrons are believed to thermalize to the L minima. The peculiarities of point emission at the N E A ( 1 10) surface are no longer present; also the higher energy of the electrons passing through the Cs-0 layer makes them less susceptible to spin-exchange scattering. Conrath et a/. (1979) have reported the first polarization measurements from GaAs,,P,,,. Its band gap is 0.42 eV larger than GaAs. T h e value of P for room temperature cathodes ranges from 35-42% a t a photon energy of

R. J . Celotta and D. T. Pierce

142

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1.91 eV. From the data reported by Conrath et a/. (1979), it is not possible to ascertain if a NEA was achieved. These measurements are of particular interest because GaAsP is easier than GaAs to activate and the larger band gap shifts the operating energy into the range of commonly available HeNe lasers.

SOURCES OF POLARIZED ELECTRONS

143

B. APPARATUS A N D PROCEDURE The details of the apparatus, such as whether i t is run cw or pulsed, at high or low voltage, and so on, depend on the requirements of the given application. The procedure, such as heat cleaning cathodes or cleaving them, may be determined by the application or there may be latitude for choice. In this discussion, the NBS source with which we are most familiar will be emphasized, but we will show the variety and versatility of GaAs polarized electron sources by comparing to other GaAs sources where information is available. A schematic of the NBS source is shown in Fig. 17. I t is designed to produce a continuous beam of polarized electrons with energy variable from a few electron volts to several hundred electron volts for polarized electron scattering experiments from surfaces in the chamber on the right of Fig. 17. This modular, compact source replaces the electron gun of a commercial low energy electron diffraction system. A large straightthrough vacuum valve permits the source and scattering chamber to be isolated, thereby allowing the GaAs cathode to remain activated while the main scattering chamber is open to atmosphere. Details of the NBS source have been presented by Pierce et a / . (1 980).

lhu SOURCE SURFACE CHAMBER

FIG. 17. An overview of the NBS spin-polarized electron scattering apparatus. The GaAs polarized electron source can be separated from the surface analysis chamber by an isolation valve. The polarized electron gun replaces the conventional electron gun of the low-energy electron diffraction (LEED) optics and Faraday cup (FC) system.

1. Incident Rudiation

We find GaAlAs laser diodes which emit at about 1.6 eV to be suitable light sources for our cw operation. The SLAC source (Sinclair et a/., 1976; Garwin et a/.. 1980) employs a flash lamp pumped dye laser at 1.75 eV to produce 1.5-psec pulses at a 120-Hz pulse rate. The ETH PEA source (Reihl et a/., 1979) and the Mainz GaAsP source (Conrath et a/.,1979) use a HeNe laser. A low-energy source has also been built a t SLAC for polarized LEED measurements. I t is operating according to the design

144

R. J. Celotta and D. T. Pierce

specifications given by Garwin and Kirby (1977). This source employs a Kr ion laser with lines at 1.55 and 1.65 eV. In most other respects the low energy SLAC source is very similar to the NBS source. To avoid confusion in the following discussion the only SLAC source we will refer to will be the high-energy pulsed version. Circularly polarized light is obtained from linearly polarized light with a quarter-wave retarder. We use a quarter-wave plate which is mechanically rotated about the axis of the light beam to produce a modulation of the polarization from u + to u - at 30 Hz. If the faces of the quarter-wave plate are not parallel, it acts as a rotating wedge and shifts the light around on the GaAs surface. Care must be taken to avoid this since it can introduce an unwanted intensity modulation of the beam. A Pockels cell, which works well with a collimated laser source, was used to reverse the polarization from pulse to pulse in the SLAC source. As is evident from our discussion in Section VIII, A, measurements of the quantum yield are extremely valuable for characterizing the photocathode. We use a Zr arc white light source with a 0.25 m monochromator and a calibrated Si photodetector to determine the ratio of the photocurrent at each photon energy to the incident photon flux. 2. Source Chamber and Cathode Structure

The GaAs photocathode must be kept in a UHV chamber at a pressure in the low lo-'' Torr range. A bakable stainless steel chamber pumped by an ion pump is used for the source shown in Fig. 17. Standard UHV techniques must be followed in the design, assembly, and cleaning of the chamber and its parts. In the NBS source, the molybdenum block to which the GaAs is clamped can be heated radiatively by a filament in the chamber. This structure is mounted on a liquid nitrogen cold finger: the GaAs can be cooled to 100-120 K. The GaAs can be retracted 3 cm for activation with Cs and 0,. The manipulator also allows for small lateral motions and tilt for proper positioning in front of the electron optics assembly. The high-voltage SLAC source is shown in Fig. 18. The GaAs wafer is clamped to a molybdenum block which is mounted on a large insulator so the photocathode can be operated at - 70 kV as required for injection into the linear accelerator. The M o block and GaAs are heated by electron bombardment from behind. The electron bombardment heater is not in the UHV chamber, but can be inserted in the liquid nitrogen dewar which can be pumped to act as a vacuum chamber during the electron bombardment heating. Although the high-voltage source operates in principle just like its low voltage counterpart, careful engineering is required to operate at the

145

SOURCES OF POLARIZED ELECTRONS

P CERAMIC INSULATOR

GoAs CATHODE

PLATINUM-COATED CATHODE ELECTRODE ANODE ELECTRODE

i k

RETRACTABLE CESl (IN POSITION FOR CA ACTIVATION)

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LN;! COLD SHIELD

FIG. 18. A schematic of the SLAC GaAs polarized electron source. The cathode is at a potential of -70 kV suitable for injection into the linear accelerator. The Pt coating on the cathode electrode produces a high work function and inhibits high-voltage breakdown. From Gamin er at. (1980).

high voltages (Garwin et al., 1980). The problem is compounded because Cs deposits on the insulator can cause breakdown. 3. Cleaning the GaAs Crystal An atomically clean surface can be prepared for activation by cleaving in UHV as discussed by Reihl et al. (1979). In the ETH source, a blade and an anvil are built into the chamber to cleave the GaAs crystal. A 2 x 2 x 10 mm crystal is mounted with the long axis perpendicular to the cleavage plane, which is the (110) surface. The crystal is scribed with grooves 1 mm apart to facilitate the cleaving. GaAs does not always cleave smoothly; care in crystal preparation and in the cleaving is necessary to obtain consistently good cleaves. In comparison to a smooth surface, a broken surface will have a larger angular spread of the emitted electrons, thereby changing the electron optical characteristics of the photocathode. Whereas negative electron affinity was first discovered in measurements of cleaved single crystals, nowadays NEA photocathodes are made by

146

R. J. Celotta and D. T. Pierce

activating polished crystal wafers. The GaAs is first chemically cleaned and then cleaned by heating in UHV. A variety of chemical cleaning procedures are used in different laboratories, The procedure usually involves a H,SO,. H,O,, H,O etch with composition ratios 5 : I : 1 to 3 : 1 : 1 or a Br-methanol etch of from 0.5% to a few percent Br. An etching procedure based on the work of Shiota ef al. (1977) has been used successfully for the NBS a n d SLAC sources. The ultimate cleanliness of the crystal depends on attention to such details as the cleanliness of the containers, the purity of the water and solvents, a n d on the rinsing procedure (Pierce el al., 1980). Immediately after the chemical cleaning, the crystal is mounted in the chamber which is pumped down as soon as possible. The bakeout (typically 180-220 "C) required to reach UHV is a potential source of contamination. This is minimized if the crystal is held a t a higher temperature (-300 "C) during bakeout. The heat cleaning of the G a A s in UHV is a critical step in obtaining NEA. One usually monitors the temperature of the GaAs surface indirectly, for example, by measuring the temperature of the piece to which it is clamped. Owing to the difficulty of accurately measuring the absolute temperature of the GaAs. a wide variety of optimum temperatures are reported in the literature. In practice the correct temperature is found empirically. There is a range of temperatures where the G a a n d As evaporate congruently, that is, together in the same proportion. Above the maximum congruent evaporation temperature. which is 630. 663, a n d 675 "C for the ( 1 11)B. (loo), and ( I 1 l)A faces, respectively (Goldstein er al.. 1976), the As evaporates preferentially leaving little droplets of G a on the surface. This leads to a frosty appearance when viewed with obliquely incident light. Heat cleaning to a temperature that produces very slight frostiness or to temperatures 10-20" below this produces good cathodes. A variety of optimum heating times have also been reported in the literature. We typically heat a crystal to 640 "C (as measured by a thermocouple on the M o block) for 5 min a n d 650 "C for 1-2 min. The pressure may rise to the l o p 8