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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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
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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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107
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
u
3
---
0
I
: o o D
V
t; a
-
X
- ) D @
W
'
- +%
w
10-*-
--
D
.8
-----
0
0
-
D
----
0
0
0
0 :
0
8
m 8 0 D.
IO-~
I
I
1
I
I
I
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)
-09
-I?
4-
C
.
v
-08
-t
a
4-
I
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-06
2-05
0
-04
m
-03
P(T) -02
(right s a l e )
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4 2-
loo7 6LL
a
f
a ln
2-
:=+:10'6 rw
ga A
W
Y
2
0
t
r 64-
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z
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2
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a
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c
e
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2-
-
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g
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-01
(left scale) I
I
I
I
I
I
I
I L I ' I
0
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-
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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-' -
(110)
.
NEA GaAs
- .- - -
/--
u c
c
-
c a c
fc
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l
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e
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(110) ERBUDAK AND REIHL
2 n
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1.7
I
I
1.9
I
I
2.1
.
. 2.3
I
2.5
.
2.7
PHOTON ENERGY (eV)
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
04
y
\
03
0 2
z
0 I-
; a
01
I oo
GaAs t C s (PEA) 1 x 10j9cm-3 Z n
80K
lo-'
10'2
0 I
a
I-
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2 0
Y
a m
a
zn
;
z
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W
i03 10-2
2r z U
02
3
d
10.~
01
0
I 15
20
25
10.~
PHOTON ENERGY ( e V )
FIG. 16. Polarization (left scale) and quantum yield (right scale) as a function of photon energy for two different activations of GaAs(l10) to PEA with photothresholds of (a) 1.59 eV and (b) 1.7 eV. From Reihl et al. (1979).
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)
TO 30 Ik ION
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 Torr range on the first heating of the crystal Torr but on subsequent heatings the pressure only rises to the low range. Although argon ion bombardment is a standard surface cleaning technique, it is not suitable for photocathodes a n d should be avoided; it introduces defects which decrease the photosensitivity of the cathode. Heating G a A s surfaces can cause ( 1 10) facets to form. In particular, (1 lo) facets are formed on the ( 1 1 l)B surface at temperatures usually used for heat cleaning (MacRae, 1966). The (100) surface is much more stable against faceting. If surfaces facet ( 1 lo), one might expect to observe a
SOURCES OF POLARIZED ELECTRONS
147
lower polarization a s in Fig. 15. Systematic studies of polarization a n d faceting have not been made but would be useful for source development. 4. Acrivation
The activation of the clean crystal is achieved by the controlled deposition of Cs a n d 0, a t room temperature. As a source of Cs, we prefer to use the pure metal in a molecular beam source described by Klein (1971). A glass ampul containing distilled C s is crushed in a copper tube in the U H V chamber. A valve controls the flow of Cs. In operation, the copper tube is at 140 "C so n o Cs condenses on it. Alternatively, one can obtain Cs from metal dispensers containing cesium chromate which emit Cs when heated. Such dispensers are potential sources of contamination a n d must be thoroughly outgased before use. Cs dispensers are used in the sources at ETH a n d Mainz. As a n oxygen source, we use research grade (99.99% minimum purity) 0, in a I-atm. liter flask attached to the chamber through a shutoff valve and a variable leak valve. Oxygen can also be obtained from a thin-walled silver tubing which a t temperatures above about 400 "C is permeable to 0,. The activation is monitored by watching the photocurrent resulting from a white light source incident on the crystal. T h e photoelectrons are collected by biasing a nearby electrode sufficiently positive to collect all the electrons, typically 100 V . A typical activation sequence is shown in Fig. 19. The sensitivity is plotted in microamperes/lumen which is standard in the photocathode industry even though it is a n unusual unit for a cathode that is to be used in the near infrared. In Fig. 19, photocurrent is observed about 4 min after the Cs valve is opened. The sensitivity obtained with Cs alone is usually 15-40 pA/lumen. At this point 0, is let in, a n d the Cs and 0, are deposited in such a way as to maximize the rate of increase of photocurrent. As is evident from Fig. 19, it is not always possible to control the 0, sufficiently to obtain a smooth increase in photocurrent. A plateau is reached a t 300-600 pA/lumen, a n d it is hard to maintain the Cs a n d 0, balance. Since our chamber tends to be Cs deficient, we usually overcesiate slightly which temporarily decreases the sensitivity; a n increase follows when the cathode comes to equilibrium. We find simultaneous deposition of C s and 0, fast a n d convenient. Similar results can be obtained by applying Cs a n d 0, alternately. The Cs balance in the chamber influences the cathode after activation. I n a freshly baked U H V chamber. the cathode becomes Cs deficient in a
-
-
R. J. Celotta and D. T. Pierce
148 loor
I
Open 0 , Valve
Cs Only Peak Open Cs Valve
2
0
6
4 Time (min)
500 -
stop I
8
I
10
I
I
12
I
I
14
I
cs I
16
I
I
18
I
I
20
Time (min)
FIG. 19. A typical activation sequence of a GaAs cathode. When the Cs-only peak in the photocurrent is reached, 0, is let in and the flow rate is adjusted to maximize the rate of increase of photocurrent.
few hours. This can be corrected by “peaking up” with Cs without retracting the cathode from the operating position; line of sight is not required. A fresh vacuum chamber becomes seasoned after a few activations or peaking up the cathode with Cs a few times. There is a simple check of the quality of the photocathode activation which can be made in addition to measuring the luminous sensitivity. When a red filter, which passes only wavelengths longer than 700 nm (5% and 50% transmission at 700 and 715 nm, respectively) is put into the white light, the photocurrent ratio with and without filter is 1/2 or more.
5. Electron Optics Any electron optical design is fundamentally determined by two sets of parameters: (1) the characteristics of the emitted beam and (2) the requirements on the final beam set by the experiment. We discuss here what is known about the emitted beam. We will not describe any details of electron optical design determined by the beam requirements at the target. The emitted beam parameters needed for a meaningful beam transport calculation are the beam size, the angular divergence of the beam, and the initial energy and energy distribution of the electrons. The beam size is
SOURCES OF POLARIZED ELECTRONS
149
well defined by the size of the incident light beam on the photocathode, which is about mm in diameter for the NBS source. A more highly focused light spot could be obtained if needed. An angular spread of 4" cone half-angle of the photoemitted electrons was calculated by Bell (1973) for emission from a perfectly flat surface NEA photocathode at room temperature. Pollard (1973) measured a cone half-angle at 5 " . However, this measurement has been criticized by Bradley ei al. (1977) who measure an external transverse kinetic energy of 107 18 meV which corresponds to a cone half-angle of approximately 30". The increase over the theoretical value is attributed to the roughness of real surfaces. A comparison of the electron optical calculations with the experimental results for the NBS source (Pierce el a/., 1980) are consistent with an angular spread of about 30" cone half-angle. The energy distribution of the electron beam, measured using a small retarding field analyzer in a Faraday cup. is shown in Fig. 20. Also shown is the total current collected. The resolution of the energy analyzer A E / E was determined to be 0.14%. At the beam energy of 50 eV used for the measurement of Fig. 20, the full width at half-maximum is composed of a 0.07-eV contribution from the analyzer resolution and a 0.13-eV contribution from the true FWHM of the beam. The measurements in Fig. 20 were at 120 K ; similar measurements at 300 K give beam energy distribution widths of 0.16 eV. The low-energy electrons which give rise to an asymmetric distribution are the result of electron phonon scattering in the bandbending region (see Fig. 14a). The electrons have a peak intensity at 0.25 eV. If we assume they are emitted from a $-mm diam. area into a cone with half-angle 30",
*
FWHM = 0.15 eV
FIG. 20. The beam current is plotted as a function of the retarding voltage in a retarding field energy analyzer. The derivative gives the energy distribution of the electrons. The low-energy tail on the right of the distribution is due to electron scattering with phonons in the band-bending region of the GaAs. The photocathode is at 110 K. When the resolution of the energy analyzer is accounted for, the energy spread of the beam is found to be 0.13 eV.
150
R. J . Celotta and D. T. Pierce
cm radwe calculate for the emittance invariant E,,,= &!? ra = 6.5 X eV'/'. For a PEA cathode there is a barrier; electrons lose energy on emission and some will get over the barrier with near-zero momentum normal to the surface but with considerable transverse momentum. The electrons are emitted into 27r solid angle, that is, a cone half-angle of 90". A PEA emitter is not as electron optically bright as an NEA emitter. The electrons are emitted with longitudinal polarization. The 90" electrostatic deflector in Fig. 17 changes the direction of the electron momentum but not of the spin so that the beam is transversely polarized as required for our experiments. In the SLAC source, the electron beam is bent 90" by a magnetic field in order to preserve the longitudinal polarization of the electron beam.
OF THE GaAs SOURCE C. CHARACTERISTICS
The GaAs polarized electron source is capable of producing an intense beam of polarized electrons. The intensity depends on the yield and the intensity of the light source. For a 3% yield, as in Fig. 15, an incident light intensity of 1 mW produces a photocurrent of 20 PA. The SLAC source produces pulsed beam currents of several hundred milliamperes. The figure of merit for a polarized electron source independent of the light intensity is P 2 Y . Comparing the yield and polarization of the NEA GaAs(100) source with the yield and polarization of the PEA cathode of Fig. 16b, we have P 2 Y = 6 X and I X lop3,respectively. Conrath et a/. (1979) reported a P 2 Y = 1.7 X for their GaAsP source. The ultimate beam intensity has yet to be determined. Photocurrent densities up to 3 A/cm2 have been obtained from NEA GaAs (Schade et al., 1971). This implies that beams up to the space charge limit of the cathode structure could be obtained. We have observed a decrease in the lifetime of the photocathode intensity as the electron current in the beam is increased. We believe this is due to the electron-stimulated desorption of ions and neutrals from electrode structures struck by the beam. Ions can be accelerated back to the cathode and can actually bombard it; neutrals can find their way to the cathode and adsorb on its surface. For highcurrent sources, the electron-stimulated desorption will have to be minimized by carefully engineered electrode structures. The adsorption of impurities on the photocathode at low temperature was an important factor affecting the lifetime of the SLAC source. By cooling the electrode structures seen by the cathode to liquid nitrogen temperature, the lifetime could be increased to 24 hr.
SOURCES OF POLARIZED ELECTRONS
151
The intensity of our source decays to l / e of its initial value in 4-12 hr: the polarization, however, remains constant. The photocathode intensity can be returned to its original value by warming i t to room temperature or by warming to room temperature and adding Cs. This procedure is so reliable that a cathode can be used for weeks or even months without reactivation. Care has been taken to avoid motion of the light spot on the cathode and to minimize electrical noise on the lens elements and deflection plates. We can measure spin-dependent scattering intensities at a noise level below 5 X of the spin-independent scattering intensity: this represents an upper limit on spurious fluctuations in the beam intensity at the signal frequency. The polarization from NEA GaAs(100) is 43 2 2% at 110 K and 36 ? 2% at room temperature. This compares to P = 35% for the ETH PEA source at 80 K and 35-42% reported for the Mainz GaAsP source. As discussed previously, the polarization can be reversed rapidly with arbitrary time structure without affecting the beam intensity. Electron optically, the GaAs source is very bright, second only to the EuS/W field emission source. The angular spread of the emitted electrons approximates a 30" cone half-angle. The energy distribution of the emitted beam is peaked at 0.25eV and has a width of 0.13eV (FWHM). For an emitting area of t mm, we have clnv= 6.5 x lop3 rad cm eV'I2. As we have mentioned throughout this section, there are several questions which require further investigation that could lead to a better understanding of or improvement of GaAs polarized electron sources. A very interesting prospect is that of increasing the polarization. D'yakonov and Perel (1974) predicted that a uniaxial stress of the crystal would create a preferred axis along the deformation axis, lift the degeneracy at the valence band maximum, and in a favorable configuration lead to P > 50%. Berkovits et a/. (1976) have observed Pi = 80% from a stressed GaAlAs crystal. Recently, Miller et a/. (1979) have measured the polarized luminescence from multilayer structures. These consist of alternating thin (-50 A) GaAs wells and Al,Ga, -,As barriers grown by molecular beam epitaxy. The confinement of the electrons and holes in these one-dimensional potential wells causes a splitting of the bands at the valence band maximum. An optical orientation of the electrons of P, > 80% was found for incident circularly polarized light of photon energy 1.60 to 1.62 eV. Excitation from only the heavy hole band, along a specific direction, gives a theoretically possible polarization of 100%. Preliminary work by the SLAC group indicates that these multilayer structures can be activated to obtain NEA. In the chalcopyrite-structure semiconductors, the degeneracy at the valence band maximum is lifted by the crystal field splitting which
152
R. J. Celotta and D.T. Pierce
can lead to higher polarization. Zurcher and Meier (1979) observed P = 50% for ZnSiAs,. CdSiAs, with a direct gap of 1.74 eV, appears to be promising for future investigation. In conclusion, the GaAs source has many outstanding characteristics that offer many advantages for a wide range of applications. In addition, there is potential for the further improvement of these characteristics.
IX. Summary As pointed out in Section 11, the choice of the optimum source for a given application depends upon a great many interrelated factors. Most of the source characteristics to be considered are summarized in Table I for the devices we have discussed. Although the table entries correspond to devices at varying levels of development, it is possible to gain some perspective on their relative advantages as they exist today. The polarizations range from a high of 85% to a low of 23%. The higher polarization may be necessary in experiments where the beam current is limited; the low polarization can usually be compensated for by the enhanced signal resulting from a larger current. The ability to optically reverse the polarization direction is a very significant advantage for most applications, and the majority of the sources use this technique. The obtainable currents show a wide variation from 0.01 to 20 pA for the continuous sources and from 2 X lo9 to 10” e/pulse for the pulsed versions. The high currents given for the continuous GaAs sources represent what is typically obtained presently using 1 mW of incident light. I t is presumed that substantially larger currents may be obtained from GaAs devices in the future. The pulsed GaAs gun produces a factor of 50 times the number of electrons per pulse as the alternate methods, but its polarization is presently somewhat lower than the other pulsed sources. Until such time that the polarizations become comparable. there may be applications where the lower current pulsed source is preferred. The energy distribution widths of the sources considered range from a low of 0.1 to a high of 3 eV. Those clustered about 0.1 eV will be suitable for many atomic and molecular electron spectroscopy applications without monochromatization. The other sources will find only limited application in this area. The energy distribution produced by the LEED source will depend upon the incident beam energy distribution, as well as inelastic processes in the diffraction process. Any attempt to monochromatize the incident beam by conventional means will severely limit the current;
153
SOURCES O F POLARIZED ELECTRONS TABLE I
SOURCESOF POLARIZED ELECTRONS
Source
Group
Chemi-ionization of metastable He Photoionization of polarized atoms Fano effect
Rice (Hodge ef of., 1979) Yale (Alguard er ol., 1979) Bonn (von Drachenfels et al., 1977) Yale (Wainwright el ol., 1978) Bielefeld (Kisker ef of., 1978) NBS (Wang ef al., 1980) NBS (Pierce er of., 1980) SLAC (Gamin el of., 1980) ETH-Zurich (Reihl er af., 1979)
Fano effect
Field emission, EuS/W LEED
NEA GaAs
NEA GaAs
PEA GaAs
Polarization
Polarization Ipulsed reversal (PA) (e/pulse)
AE (eV)
(rad cm eV1/')
0.15
0.04
€1""
0.40
Optical
0.85
Magnetic field
2 x 109
1500
0.4
0.65
Optical
2.2 x 109
< 500
0.2
0.63
Optical
0.01
3
< 0.6
0.85
Magnetic field
0.01
0.1
1 x 10-6
0.44 0.23
Electron optical
0.024 0.22
b
0.43
Optical
20"
0.13
0.37
Optical
0.35
Optical
2
10"
4"
b
6.5 x 10-3
0.13
0.3
"For 1 mW of incident light. *Depends upon incident beam.
operated near its current limit, an energy distribution of order 0.3 eV FWHM can be estimated for the LEED source. The emittance invariant qnv also shows a wide range, from 0.6 to 1x radcmeV"'. This parameter is crucial to any planned experimental application of a polarized electron source. The electron optics can d o no more than preserve qnv,so that a calculation of qnvfrom the final desired beam parameters, E, r, and a, is a n important step in source selection. A value of qnvthat is lower a t the experiment then a t the source will of necessity result in beam loss. The source may, however, produce
154
R. J . Celotta and D. T. Pierce
sufficient current so that even with some beam loss the total current a t the target is more than adequate. The field emission source, owing to the small initial value of r, has a substantially lower value of E , , , than any of the other devices. Whereas the importance of this parameter depends upon the specific application, it is probable that this will be the source of choice in applications where conventional field emission electron guns would normally be used because of their high brightness. T h e combination of high current a n d moderately low einv makes the GaAs source a n extremely attractive choice for the majority of applications. We have presented a review of the current state-of-the-art technology for producing beams of polarized electrons. T h e problem has been specified for many years: “Produce a n electron beam with all of the characteristics of a conventional electron beam, but with a spin polarization direction that is changeable at will.” Progress toward this goal has been steady and in the last few years highly accelerated. We have now reached the point where the G a A s source is being used to replace a conventional (commercial) electron gun (Wang el ul., 1979) a n d to produce currents of polarized electrons equal to the original unpolarized currents with the bonus of higher energy resolution. Thus, for a large number of experiments the challenge has been met. Research will continue toward increasing the current a n d polarization obtainable from this new class of devices. High polarization is needed most by the high energy community a n d there are experiments that require substantially larger currents. W e expect that source development will be enhanced by the increase in the number of people working in this area, but tempered by the fact that there is so much good science possible with the new technology we now possess.
REFERENCES Alguard, M. J., Ash, W. W., Baum, G., Clendenin, J. E., Cooper, P. S., Coward, D. H., Ehrlich, R. D., Etkin, A., Hughes, V. W., Kobayakawa, H., Kondo, K., Lubell, M. S., Miller, R. H., Palmer, D. A., Raith, W., Sasao, N., Schuler, K. P., Sherden, D. J., Sinclair, C. K., and Souder, P. A. (1976) Phys. Rev. Lerr. 37, 1261. Alguard, M. J., Hughes, V . W., Lubell, M. S., and Wainwright, P. F. (1977). Phys. Rev. Lett. 39, 334. Alguard, M. J., Clendenin, J. E.. Ehrlich, R. D., Hughes, V. W., Schuler, K. P., Baum, G., Raith, W., Miller, R. H., and Lysenko, W. (1979). Nucl. Instrum. & Methods 163, 29. Alvarado, S. F., Eib, W., Meier, F., Siegmann, H. C., and Zurcher, P. (1978). I n “Photoemission and the Electronic Properties of Surfaces” (B. Feuerbacher et a/.,eds.), p. 437. Wiley, New York. Baum, G., and Koch, U. (1969). Nucl. Instrum. & Metho& 71, 189. Baum, G., Lubell, M. S., and Raith, W. (1972). Phys. Rev. A 5, 1073.
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Baum, G., Kisker, E., Mahan, A. H., and Schroder, K. (1976). J. Magn. Magn. Muter. 3, 4. Baum, G., Kisker, E., Mahan, A. H., Raith, W., and Reihl, B. (1977). Appl. Phys. 14, 149. Bell, R. L. (1973). “Negative Electron Affinity Devices.” Oxford Univ. Press (Clarendon), London and New York. Berkovits, V. L., Sararov, V. I., and Titkov, A. N. (1976). Izv. Akad. Nauk SSSR, Ser. Fiz. 40, 1866.
Bonner, W. A., VanDort, M. A., and Yearian, M. R. (1975). Nature (London) t58,419. Bradley, D. J., Allenson, M. B., and Holeman, B. R. (1977). J. Phys. D. 10, I 1 I. Burt, M. G., and Inkson, J. C. (1977). J. Phys. D. 10, 721. Busch, G., and Wachter, P. (1966). Phys. Kondens. Muter. 5, 232. Campagna, M., Pierce, D. T., Meier, F., Sattler, K., and Siegmann, H. C. (1976). Adv. Electron. Electron Phys. 41, 113. Campbell, D. M., Brash, H. M., and Farago, P. S. (1971). Phys. Lett. 31, 499. Celotta, R. J., Pierce, D. T., Wang, G.-C., Bader, S. D., and Felcher, G. P. (1979). Phys. Rev. Lett. 43, 728. Christensen, R. L., and Hamilton, D. R. (1959). Rev. Sci. Instrum. 30,356. Chrobok, G., Hoffmann, M., Regenfus, G., and Sizmann, R. (1977). Phys. Rev. E 15, 429. Conrath, D., Heindorff, T., Hermanni, A., Ludwig, N., and Reichert, E. (1979). Appl. Phys. 20, 155.
Davisson, C. J.. and Germer, L. H. (1929). Phys. Rev. 33, 760. Deichsel, H. (1961). Z. Phys. 164, 156. Deichsel, H., and Reichert, E. (1965). Z. Phys. 185, 169. DiChio, D., Natali, S. V., and Kuyatt, C. E. (1973). Rev. Sci. Insfrum. 45, 559. Dyakonov, M. T., and Perel, V. 1. (1974). Sov. Phys.-Sernicond. (Engl. Transl.) 7, 1551. Erbudak, M., and Reihl, B. (1978). Appl. Phys. Lett. 33, 584. Fano, U. (1969). Phys. Rev. 178, 131; and addendum Phys. Rev. 184, 250. Farago, P. S. (1965). Adv. Electron. Electron Phys. 21, 1. Farago, P. S. (1971). Rep. Prog. Phys. 34, 1055. Feder, R. (1974). Phys. Status Solidi B 62, 135. Feder, R. (1976). Phys. Rev. Lett. 36, 598. Feder, R., Jennings, P. J., and Jones, R. 0. (1976). Surf Sci. 61, 307, and references therein. Feder, R., Muller, N., and Wolf, D. (1977). Z. Phys. 828, 265. Fishman, G., and Lampel, G. (1977). Phys. Rev. B 16, 820. Friedmann, H. (1961). Sitzungsber. Math.-Natunuiss. KI. Buyer. Akad. Wiss. Muenchen 13. Fues, E., and Hellmann, H. (1930). Phys. Z . 31, 465. Galejs, A., and Kuyatt, C. E. (1978). J. Vac. Sci. Technol. 15, 865. Garwin, E., Pierce, D. T., and Siegmann, H. C. (1974). Helv. Phys. Acta 47, 393. Garwin, E. L., and Kirby, R. E. (1977). Proc. Int. Vac. Congr., 7th, I977 Vol. 3, p. 2399. Garwin, E. L., Miller, R. H., Prescott, C. Y.Sinclair, C. K., and Borghini, M. (1980). To be published. Goldstein, B., Szostak, D. J., and Ban, V. S. (1976). Sutf Sci. 57, 733. Goudsmit, S. A., and Uhlenbeck, G. E. (1925). Natunvissenschaflen 13, 953. Goudsmit, S. A., and Uhlenbeck, G. E. (1926). Nature (London) 117, 264. Hanne, G. F. (1976). J. Phys. E 9, 805. Hanne, G. F., and Kessler, J. (1976). J. Phys. E 9, 791. Heinzmann, U., Kessler, J., and Lorenz, J. (1970a). Z. Phys. 240, 42. Heinzmann, U., Kessler, J., and Lorenz, J. (1970b). Phys. Rev. Lert. 25, 1325. Heinzmann, U., Schonhense, G., and Kessler, J. (1979). Phys. Rev. Lett. 42, 1603. Hill, J. C., Hatfield, L. L., Stockwell, N. D., and Walters, G. K. (1972). Phys. Rev. A 5. 189. Hodge, L. A., Dunning, F. B., and Walters, G. K. (1978). Rev. Sci. Inrtrum. 50, 1. Hodge, L. A., Dunning, F. B., and Walters, G . K. (1979). Nature (London) 280, 250.
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James, L. W., and Moll, J. L. (1969). Phys. Rev. 183. 740. Jennings, P. J. (1971). Surf.Sci. 26. 509. Jennings, P. J., and Sim, B. K. (1972). Sur$ Sci. 33, 1. Jost, K., and Kessler, J. (1965). Phys. Rev. Lett. 15, 575. Jost, K., and Kessler, J. (1966). Z. Phys. 195, 1. Kalisvaart, M., ONeill, M. R., Riddle, T. W., Dunning, F. B., and Walters, G. K. (1978). Phys. Rev. B 17, 1570. Keliher, P. J., Gleason, R. E., and Walters, G. K. (1975). Phys. Rev. A 11, 1279. Kessler, J. (1969). Rev. Mod. Phys. 41, 3. Kessler, J. (1973). At. Phys., Proc. Int. ConJ, 3rd, 1972, p. 523. Kessler, J. (1976). “Polarized Electrons.” Springer-Verlag. Berlin and New York. Kessler, J., and Korenz, J. (1970). Phys. Rev. Lett. 24, 87. Kisker, E., Baum, G., Mahan, A. B., Raith, W., and Schroder, K. (1976). Phys. Rev. Lett. 36, 982. Kisker, E., Mahan, A. H., and Reihl, B. (1977). Phys. Lett. A 62, 261. Kisker, E., Baum, G., Mahan, A. H., Raith, W., and Reihl, B. (1978). Phys. Rev. B 18, 2256. Klein, W. (1971). Rev. Sci. Instrum. 42, 1082. Kuyatt, C. E. (1967). “Electron Optics Notes”, National Bureau of Standards, Washington, D.C. 20234 (unpublished). Lampel, G. (1974). I n “Proceedings of the Twelfth International Conference on Semiconductors’’ (M. H. Pilkuhn, ed.), p. 743. Tuebner, Stuttgart. Landolt, M., and Yafet, Y. (1978). Phys. Rev. Lett. 40, 1401. Long, R. L., Jr., Raith, W., and Hughes, V. W. (1965). Phys. Rev. Lett. 15, I . Lubell, M. S. (1977). At. Phys., Proc. Int. Con$, 5th, 1976, p. 325. McCusker, M. V., Hatfield, L. L., and Walters, G. K. (1969). Phys. Rev. Lett. 22, 817. McCusker, M. V., Hatfield, L. L., and Walters, G. K. (1972). Phys. Rev. A 5, 177. MacRae, A. U. (1966). Surf.Sci. 4, 267. Maison, D. (1966). Phys. Lett. 19, 654. Miller, R. C., Kleinman, D. R., and Gossard, A. C. (1979). Inst. Phys. ConJ Ser. 43, 1043. Mott, N. F. (1929). Proc. R. Soc. London, Ser. A 124, 425. Mueller, N., Eckstein, W., Heiland, W., and Zinn, W. (1972). Phys. Rev. Lett. 29, 1651. Nolting, W., and Reihl, B. (1979). J. Mug. Mug. Mat. 10, 1. ONeill, M. R., Kalisvaart, M., Dunning, F. B., and Walters, G. K. (1975). Phys. Rev. Lett. 34, 1167. Oppenheimer, J. R. (1928). Phys. Rev. 32, 361. Pierce, D. T., and Meier, F. (1976). Phys. Rev. B 13, 5484. Pierce, D. T., Meier, F., and Zurcher, P. (1975a). Phys. Lett. A 51, 465. Pierce, D. T., Meier, F., and Zurcher, P. (1975b). Appl. Phys. Lett. 26, 670. Pierce, D. T., Wang, G.-C., and Celotta, R. J. (1979a). Appl. Phys. Lett. 3, 220. Pierce, D. T., Kuyatt, C. E., and Celotta, R. J. (1979b). Rev. Sci. Instrum. SO, 1467. Pierce, D. T., Celotta, R. J., Wang, G.-C., Unertl, W. N., Galejs, A., Kuyatt, C. E., and Mielczarek, S. (1980). Rev. Sci. Instrum. 51, 478. Pierce, J. R. (1954). “Theory and Design of Electron Beams,” 2nd ed., p. 150. Van Nostrand Reinhold, Princeton, New Jersey. Pollard, J. H. (1973). Presented in (Bell, 1973, Appendix C). Prescott, C. Y., Atwood, W. B., Cottrell, R. L. A., DeStaebler, H.,Gamin, E. L., Gonidec, A,, Miller, R. H., Rochester, L. S., Sato, T., Sherden, D. J., Sinclair, C. K., Stein, S., Taylor, R. E., Clendenin, J. E., Hughes, V. W.,Sasao, N., Schuler, K. P., Borghini, M. G., Lubelsmeyer, K., and Jentschke, W. (1978). Phys. Lett. B 77, 347. Raith, W. (1969). At. Phys., Proc. Int. Con$, Ist, 1968, p. 389.
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ADVANCES IN ATOMIC A N D MOL.ECULAR PHYSICS, VOL. 16
THEORY OF ATOMIC PROCESSES I N STRONG RESONANT ELECTROMAGNETIC FIELDS S. SWAIN Deparimeni of Applied Mathematics and Theoreiical Physics The Queen’s University of Belfasi, Belfmt, Northern Ireland
.
. . . . . . 159 . . 159 . . . . . . 161 . . . . 165 . . . . . . . . . 165 . . . . . 168 . . . . . . . 171 . . 171 . .
I. Introduction . . . . . . . . . . . . . . . A. General Introduction . . . . . . . . . . . B. Elementary Discussion of the Basic Phenomena . . . 11. MasterEquations. . . . . . . . . . . . . . A. Derivation of Master Equation . . . . . . B. Master Equation for the Atom Field Problem . . . . . 111. Resonance Fluorescence . . . . . . . . . . A. Introduction . . . . . . . . . . . . . . . . . B. Resonance Fluorescence in Nondegenerate Two-Level Systems. . C. Intensity Fluctuation Spectra . . . . . . , . . . . . . D. Resonance Fluorescence in Multilevel Atoms with Monochromatic Fields . . . . . . . . . . . . . . . . . . IV. The Optical Autler-Townes Effect. . , . . . . . . . . . A. Introduction . . . . . . . . . . . . . B. Three-Level Autler-Townes Theory . . . . . . . . . . C. Experiments on Multilevel Atoms . . . . . . . . . . V. Conclusion , . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . ., . . . .
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I. Introduction A. GENERAL INTRODUCTION
The object of this review is to describe some recent work on the interaction of atomic systems with intense, near-resonant, narrow-band laser fields. We are concerned with multiphoton processes but we may divide multiphoton processes for convenience into two types: 1. Those in which there is net absorption (or emission) of two or more photons by the atom, such as multiphoton ionization or two-photon 159
Copyright 0 1980 by Academic Press, Inc. All rights of reproduction in any form reserved.
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absorption. These phenomena have been reviewed recently by Lambropoulos (1976) in these volumes, and are not our prime concern here. 2. Those in which there are repeated emissions and absorptions of photons by the atom but in which, at the end of the interaction period, the energy of the atom is unchanged or has changed by only a single photon energy. An example is resonance fluorescence where the energy initially residing in the laser field is redistributed by many absorptions and subsequent emissions of photons although, at the end of the interaction period, the atomic energy is unchanged. We concentrate therefore on the topics of resonance fluorescence and the optical Autler-Townes effect. Related phenomena, such as Raman scattering in intense fields, are also treated but only briefly. These effects have been studied for a long time: the first quantum treatment of the scattering of resonance radiation by free atoms was given by Weisskopf and Wigner (1930) in the early days of quantum mechanics. The Autler-Townes effect was reported 25 years later (Autler and Townes, 1955). The development of the laser (particularly the tunable dye laser) made feasible experiments in the optical region with essentially monochromatic fields at saturating intensities. During the last decade there has been a resurgence of interest in these phenomena, presenting as they d o a challenge to experimentalists and theorists alike. I t is fair to say that the basic features are now considered well understood. The importance of these topics in such fields as laser spectroscopy and optical pumping is evident. Because of the repeated interaction with strong fields, conventional perturbation theory is inadequate and a wide variety of methods have been used in the literature. To give a unified treatment of the various phenomena we stick to just one method here, namely the master equation approach. As this method is of considerable interest in its own right (being applicable to an extremely wide range of phenomena), we give brief derivations of the relevant equations in Section 11. An advantage of the master equation approach is the ease with which the various damping processes such as collision broadening and laser linewidth may be incorporated into the equations. We treat the atom/field interaction quantum electrodynamically. Recent reviews that cover some of the material treated here are CohenTannoudji (1977), Feneuille (1977), Stenholm (1978). Bayfield (1979), and the various articles occuring in the ICOMP (1977) and CQO IV (1977) conference proceedings which are referred to explicitly in the text. Here we restrict our attention to essentially isolated atoms and do not consider cooperative effects. See, for example, Lugiato and Bonifacio (1978), Gibbs et al. (1978), and the references quoted there. In addition, we are con-
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cerned mainly with steady state rather than transient effects; no attempt is made to give an extensive bibliography of the latter.
B. ELEMENTARY DISCUSSION OF THE BASICPHENOMENA Consider a two-level atomic system interacting with an intense monochromatic E.M. field of frequency wL close to resonance with the atomic transition. We suppose the atomic levels to have some structure so that the ground state consists of N / degenerate or quasi-degenerate levels I i = 1,2, . . . , N / with energies E,' and the excited state a set of Nu levels, labeled l u , ) , ~= 1,2, . . . , Nu with energies EC. For simplicity we work with a system of units for which h = I , so that energy and frequency have the same dimensions. Assuming the atom to be isolated and at rest, the Hamiltonian for the system in the electric dipole, rotating wave approximation (RWA) (e.g., Loudon, 1973; Stenholm, 1973) is N"
Nl
H = a+awL+ I=
I
c
l~l)E+ ~ ( ~ llu,)EC(u,l ~ /=I
where a + and a are the usual creation and annihilation operators for a laser photon of energy wL and g is the coupling constant, given explicitly by
In (2), 6 is the unit polarization vector of the field, r,, = (/llrluJ) is the dipole matrix element, a is the fine structure constant. and V is the volume of the system. For the moment we have neglected spontaneous emission. n) and I u,, n - l), where n denotes a Fock state of The sets of states the field with n photons present, are nearly degenerate: E,' + n u L - Ei + ( n - l)wL. Thus the eigenvalues W,(n) of the Hamiltonian (1) are given by the roots of
where [ W - El - nuL] is an N , X N , diagonal matrix with elements W E,' - n u L , i = 1,2, . . . , N , ; [ W - E, - ( n - l)wL] is an N u X N u diagonal matrix with elements W - Ei - ( n - l)w,-,j = I . 2, . . . , N u : and d is the
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matrix
gli
g12
g2,
g22
... ...
5”=
(4) gNil
...
gN12
where the gii’s --, are defined in (2). For a nondegenerate two-level system ( N , = Nu = 1) the solution of (3) is easily found:
where a,, = E,, - El - wL is ihe detuning and a,, = 21 g,,lJi is the Rabi frequency (e.g., Allen and Eberly, 1974). I t is an important parameter in the subsequent discussion. At resonance (a,! = 0), W , , , ( n )= nuL + E, 2 Q,,,. Since wL >> Q,,,, the energy spectrum of the interacting system consists of sets of doublets, the spacing between neighboring doublets being wL and the separation between components of the doublet being Q,,. In all of the experiments we are concerned with, the mean photon number ii satisfies E > 1,
Var(n)<< ii2
(6)
so that we can treat n as a constant, n E E , and hence Q,,(ii) = 2 l g U , l 6 = a,,, is also a constant. Thus the doublets may be considered equally spaced as shown in Fig. 1. This splitting of the energy levels was predicted and observed by Autler and Townes (1955) and is referred to as the Autler-Townes splitting, ac Stark effect, or dynamic Stark effect. In the general case, under the conditions (6) the spectrum will consist of sets of multiplets each containing N , + Nu levels and every multiplet being separated from its neighbor by the laser frequency w L :
W,(n)= n u L + 6,,
k
= 1.2,
. . . , Nu + N ,
(7)
where the precise form of 6 , is determined by the solution of the eigenvalue equation (3). The system of atom interacting with applied monochromatic RWA field has been called the “dressed atom” and used with great effect in radio-frequency spectroscopy and optical pumping (e.g., Cohen-Tannoudji, 1968; Haroche, 1971). Let us now consider some ways of investigating experimentally the dressed-atom spectrum.
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FIG. 1. The "dressed" energy level diagram for a two-level atom showing allowed spontaneous transitions.
a. Resonance fluorescence. We add to the Hamiltonian (1) a term representing spontaneous emission and treat this as a perturbation too weak to change the dressed-atom energies but capable of causing transitions between the dressed levels. If one observes the spontaneous emission, the spectrum expected may be deduced from the following simple picture for the nondegenerate case: In Fig. 1 are illustrated the four possible transitions between the four states of adjacent doublets, two of these have frequency w L , one has frequency wL a,,,, and the other has frequency oL - Qu1. Assuming all these transitions have equal weight, we would therefore expect a resonance fluorescence spectrum that has three peaks, a central peak at the laser frequency of twice the height of the symmetrically placed side peaks, which occur at wL 2 aul;that is, the side peaks are
+
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S. Swain
separated from the central peak by the Rabi frequency. Naively one would expect all the peaks to have the same half-width, namely the natural linewidth. These are the predictions of the so-called “one-photon approximation” to resonance fluorescence. This simple picture breaks down because in addition to the transitions shown in Fig. I , there are transitions from the doublet centered at n u L to the doublet centered at ( n - l)wL, and then from the doublet centered at ( n - I)wL to that at ( n - 2)wL, and so on; that is, we have to take into account “cascade transitions” or repeated absorptions and spontaneous emissions. As these transitions involve common levels they are not independent, and there is interference between the processes which causes the spectrum to be modified from that given by the one-photon approximation to the so-called Mollow spectrum (Mollow, 1969) derived in Section 111. By similar arguments applied to the general case, one can show that the spectrum consists of ( N / N u ) lines all centered on the frequency wL and ( N / + Nu)(” + N u - 1) sidebands at frequencies wL 6 , - E j (k Zj). The sidebands are clearly symmetrically placed with respect to the center frequency.
+
+
b. Photon antibunching. The spectrum in resonance fluorescence is determined by the first-order correlation function of the field (Glauber, 1965); it was suggested by Carmichael and Walls (l976a, b) and Cohen-Tannoudji (1977) that Rabi oscillation effects might also appear in the second-order field correlation function. Detailed calculations confirmed this, and in fact “photon antibunching” (e.g., Knight, 1977a; Walls, 1979) and Rabi oscillations have been observed (Dagenais and Mandel, 1978). We discuss this effect in Section 111, C.
Another way in which we could try to detect the doublet structure is to use a weak tunable laser (by weak we mean so weak that it does not significantly perturb the dressed-atom energies) to connect the levels Iuj) to a third set of N p levels 1 ~ ~ ) We . call this weak, tunable laser the “probe laser” and the first intense laser the “saturating laser.’’ As the frequency, upof the probe laser is swept through the resonance frequency of the Iu,) ++-Ip,) transition, we would expect the population of level Ip,) to exhibit maxima at c. Optical Autler-Townes effect.
aP=Ep-tGk;
p = 1 , 2, . . . , Np, k = 1 , 2, . . . , ” + N u
(8)
that is, at the Bohr frequencies of the dressed states with the probe levels. Thus if the population of the probe states can be monitored in some way, we can map out the dressed-atom spectrum. In general we would expect
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+
the output to consist of N , ( N , N u ) lines. This effect is discussed in Section IV. Finally we mention that weaker manifestations of the Autler-Townes splitting are apparent in the interaction of atomic systems with amplitude modulated light (Armstrong and Feneuille, 1975; McClean and Swain, 1976, 1978; Feneuille et al., 1976; Saxena a n d Agarwal, 1979; Kryzhanovsky and Melikyan, 1979). We do not discuss this topic further here.
11. Master Equations A. DERIVATION OF MASTEREQUATION
We give a n outline derivation of the master equation which will be used in all the subsequent sections. For further details one is referred to the original paper by Zwanzig (1964) and the reviews by Haake (1973) and Agarwal (1973, 1974). For brevity we give here only the derivation for a fully quantum-electrodynamic Hamiltonian; the derivation for a semiclassical Hamiltonian is very similar (see, e.g., Agarwal, 1973, 1974). Advantages of the master equation formalism are (a) the interaction with the applied field is treated exactly; it is only the interaction with the vacuum field that is approximated; and (b) it is easy to incorporate phenomenologically into the equations other damping processes such as collisions and laser linewidth broadening. We consider a composite system which can be regarded as a “system of particular interest” (henceforth called simply the system and denoted by S ) interacting with a “system of secondary interest” (henceforth called the reservoir and denoted by R ) . The division into system plus reservoir is to some extent arbitrary: in this paper S is taken to be the atom ( A ) interacting with the applied field ( F ) and R the vacuum electromagnetic field. Thus the Hamiltonian of S may be written
H,
=
HA
+ H , + HA,
(9)
and the Hamiltonian of the composite system ( S + R ) as
H
=
H,
+ H , + Hs,
(10)
HA, being the interaction between the atom and the field and H,, being the interaction between the system ( A + F ) and the reservoir. The equa-
166
S. Swain
tion of motion for the density matrix pS+R of the composite system is d p s + R / d t = - i [H , & + R ]
= - I.C P S + R
(11)
Equation (1 1) defines the Liouvillean operator e; it is often called a tetradic or super operator as it acts in a space of greater dimensions than p. Thus if p is represented by a N X N matrix, e is represented by a N 2 X N matrix. It may be decomposed in a similar way to If:
e = cs + CR + CSR
(12)
In this section we use script to denote superoperators. It is later shown that for the problems which concern us here the properties of interest can be expressed as averages of system operators 2 only, and these may be calculated from p in the usual way: ( i ( t ) > = T ~ ~ + R ( ~ P S + R ( ~=) )~ r s ( S p s ( t ) )
(13)
where Tr, denotes a trace over the X states only and ~ s ( t ) TrR(pS+R(t))
(14)
is the reduced density matrix for the system alone. We can derive from Eq. (1 1) an equation for p s ( t ) alone: this is the master equation. First we define the projection operator
9 = pR(0) TrR
(15)
where pR(r) is a reduced density matrix defined analogously to p s ( r ) : pR(l)
(16)
TrS(p.S+R(f))
and pR(0) is the initial density matrix for the reservoir alone. By operating with 9 on pS+R and tracing over R we obtain p s ( f ) : p s ( f ) = TrR ( 9 p s + R ( t )
1
(17)
It is convenient to introduce the Laplace transform of ~ ~ + ~ ( t ) :
=i
m
PS+R(')
(18)
e-"pS+R(t)dt
and to separate p S + R into two parts: PS+R
= 9PS+R +
= PI + p2
- ')pS+R
(19)
when we find by acting on the Laplace transform of ( I 1) with 9 and (1 - 9)the equations
z p I ( z ) - p,(O)
= -i 9 C p , ( z ) -
i9Cp2(z)
z p 2 ( z ) - p2(0) = - i ( 1 - 9 ) C p , ( z )
-
i ( 1 - 9)Cp2(r)
(20)
ATOMIC PROCESSES IN STRONG RESONANT E. M. FIELDS
167
which have the formal solution zpl(z) - P,(O) =
+ i(1 - 9)t]-'p,(0)
-
iqCpI(z) -
~CPC[Z
-
~ ez +[ i(1
-
CP)C]-'(I - ~ ) e p ~ ( z )
(21)
We now assume that the initial density matrix of the reservoir may be expressed in terms of reservoir states 1R) as PR(O) =
c R
(22)
IR)PR(RI
where H s , has no diagonal matrix elements with I R ) ; and that
(23)
P(0) = PR(O)PS(O)
(These conditions are certainly satisfied for the interacting atom field problem of interest here.) One can then show that p z ( 0 ) = O , so that the second term on the right-hand side of Eq. (21) vanishes. So far the treatment is exact within the stated assumptions; now we make the Born approximation. In the final term on the right-hand side of Eq. (21) we set [z
+ i(l
- T)C]-'-[z
-[z
+ i ( l q e s + !&)]-I + i(1 - T)(CA + C F + -
t,)]-'
(24)
(In the last step we have set CAF = 0; this is not essential if one later takes matrix elements between dressed states. We have made this approximation here so that we can take matrix elements between unperturbed states. The effects of neglecting C,, are very small.) Then, after a little simplification and finally tracing over the reservoir states I R ) we obtain the master equation for p,(z) alone: ZPS(Z)
where i 6 C(z)
= Tr, X(Z)
-
+ iesPs(z) + i 6 f q z ) PS(Z) = 0
P,(O)
(25)
X(z) and ~S,[Z
+ i ( e A + C, + C~)]-'~~,P,(O)
(26)
The first three terms describe exactly the free evolution of the system, that is, the evolution if H s , = 0, and the final term contains all the effects of the reservoir interaction on the dynamics of p s (i.e., in our case the effects of spontaneous emission into the vacuum electromagnetic field). In practice we will wish to evaluate matrix elements of p s ( z ) between states la), Jb), Ic), . . . , say, which are eigenkets of the unperturbed system Hamiltonian H A + H , (HA
+ HF)la)
= Eola>
(27)
S. Swain
168
We now make the Markoff approximation, that is, we set z = - i ( E , Eb) + E , E + O + in 8 e ( z ) . This is an excellent approximation in the long time limit, that is, for t >> OLI . We may then show, with x = pR(0)ps(z), that
(28) Note that it still remains to trace Eq. (28) over the reservoir states to obtain the final term in Eq. (25). Expression (28) is cumbersome but its application to specific problems is straightforward.
B. MASTEREQUATION FOR
THE
ATOMFIELDPROBLEM
1. Monochromatic Fields
We now utilize the master equation (25) to derive some results we need later. The Hamiltonian of the system H = H A + H , H A , is given by Eq. (1). The atom also interacts with the vacuum field which gives rise to spontaneous emission. We treat the vacuum field as the reservoir with the Hamiltonian
+
HR=
2 b:b,,w,, x
and the interaction with the system is
with the g,,;,, E g,,(w,,) given by Eq. (2). Here the b l and b,, are of precisely the same nature as the a and a defined after Eq. (1); the use of 6, instead of a,, is purely a notational convenience to distinguish the applied field mode from all others. The subscript h labels the vacuum mode and includes polarization index j and wave vector k. The initial state of the reservoir is taken to be the vacuum state 10) corresponding to zero occupation of the reservoir modes. Hence +
PR(O) =
(31)
The final term (28) in the master equation (25) can now be straightforwardly evaluated. Further simplification follows by dropping rapidly oscil-
ATOMIC PROCESSES IN STRONG RESONANT E. M. FIELDS
I69
lating terms (valid for nondegenerate, nonequally spaced atomic systems: Aganval, 1973). One finally obtains (2
+ ‘a,w)pab + i [ Hsj
~ ] a b -2SabC yc, a* Pc c ( Z ) a{n,,).(n,)= Pab(0)
(32)
c’
where
y,,,, is the rate of spontaneous decay into level la’) from the atomic state Ic’). In Eq. (32) the simultaneous eigenstates of H A and H , have been written as la) = Ia’)l{n,}),where la’) is an eigenstate of H A and I ( n , } ) an in Eq. (32) ensures that pcc has eigenstate of H,. The &function, the same applied field state as pa,. We have also neglected Lamb shift terms, assuming these to be incorporated already into the atomic energies
E,. . For derivations of the master equation not involving tetradic techniques, see, for example, Louisell (1969) and Cohen-Tannoudji (1977).
2. Laser Linewidth Effects
So far we have treated the laser as monochromatic, whereas in practice both the amplitude and phase of the electric field produced by the laser are subject to random fluctuations (e.g., Haken, 1970: Sargent et a/., 1974). In the following we neglect amplitude fluctuations completely and consider phase diffusion as the only contribution to the laser linewidth. While this simplifies the theory, it should be pointed out that the standard phase diffusion model predicts a Lorentzian line shape, whereas most experimentally determined line shapes appear to be nearer a Gaussian. Burshtein (1965) and Burshtein and Oseledchik (1967) have given general discussions of the effects of various laser-linebroadening mechanisms on atomic transitions. Agarwal ( 1978) using the theory of multiplicative stochastic processes (e.g.- Fox, 1972: see also Schenzle and Brand, 1979: Wodkiewicz, 1979). has shown that when a Hamiltonian possesses a stochastic component
where the random forces pu ( t ) are &correlated, Gaussian random variables
S. Swain
170
of mean zero, =24,
( P a ( f 1 ) P&2)>
6(r, -
(Pa(f)> = 0
12).
(35)
then the density matrix averaged over these fluctuations obeys the same equation of motion as in the absence of fluctuations but with the Liouvillean augmented by the term iS%P =
c AN,[ H,L, [ H p l . 7 P ( 4 ] ]
a.
P
(36)
For the case of the optical Autler-Townes effect, if we assume both lasers are only phase-diffusion broadened, the stochastic Hamiltonian (34) is (Glauber, 1965) HL(l) = Pl(f)a:al
+ P2('>a:a2
(37)
where the p u ( t ) satisfy (35). If we further assume the two lasers are independent (38)
( P d f J P2(f2)) = 0
then the stochastic Liouvillean (36) gives 2
i S eLp=
2I A,(
il:p - 2ilupila
+ phi)
(39)
N=
where hu = a,.' elements.
is the number operator for the Cyth laser. Taking matrix 2
2 a= 1
(i6eLP)ob=
2
- n:)
Pob
(40)
If there is just one laser present, as in resonance fluorescence, then we may set A , = A, A2 = 0. 3. Collisions in Noridegenerate Two-Level Systems When we take collisions into account in our model we have two effects additional to those which occur when radiative damping is the only source of decay: first, inelastic collisions cause transitions from the ground state )1, to the excited state I u ) as well as transitions from l u ) to I[): and second, elastic collisions produce a further dephasing effect. I t is a simple matter to introduce phenomenological collisional damping coefficients into the master equation. They are obtained by adding to the left-hand side of Eq. (32) the term ~ S % P = Qi(IuXuI +
i(Qi
-
I~X~I)(P,,,, - pi/)
+ Q E ) ( I u ) P ~ I I + I[>P/,,(uI)
(41)
ATOMIC PROCESSES IN STRONG RESONANT E. M. FIELDS
171
where Q, and QE are the mean rate of inelastic and elastic collisions, respectively. Note that p,,,, p,,, are still operators in the photon variables. It is convenient to add to the master equation (32) the contributions from the laser linewidth and collision processes, writing the result formally as zp(z)
+ i t ’ p ( z ) = p(0)
(42)
+ iC’p(t) = 0
(43)
in z-space, or in t-space as p(t) where t’= Es
+ 8 C + 8!3, + st,
is time independent.
111. Resonance Fluorescence A. INTRODUCTION
Early calculations were based on perturbative techniques and were restricted to weak ( y >> D) monochromatic fields well below the saturating level. Under these conditions the spontaneous emission is essentially complete in a time (- y -’) much smaller than the time required for significant changes in the atomic population to be effected by the applied field. Weisskopf (1931, 1933) showed that the fluorescence spectrum does not possess the natural linewidth but is monochromatic at the same frequency as the exciting light. Various groups (Wu et a/., 1975: Hartig et a/., 1976; Gibbs and Venkatesan, 1976; Eisenberger et al., 1976) have verified that for weak fields the fluorescence spectrum is a single line of width less than the natural width. At high intensities (D >> y) the atom undergoes many Rabi oscillations in the time available for spontaneous decay and the earlier theories are inapplicable; it is essential to take into account repeated interactions of the atom and applied field (multiphoton processes). The resonant scattering of monochromatic light has been considered theoretically by many authors [Rautian and Sobel’man, 1962, 1963: Apanesevich, 1964; Notkin et a / . , 1967: Newstein, 1968, 1972; Mozorov, 1969; Mollow, 1969, 1970, 1972a, b, 1973, 1975a, b, 1976, 1977; Ter-Mikaelyan and Melikyan, 1970; Sokolovskii, 1971: Stroud, 1971, 1973: Oliver et a / . , 1971; Gush and Gush, 1972; Herrmann et a/., 1973; Baklanov, 1974: Kazantsev, 1974; Carmichael and Walls, 1975, 1976a, b; Smithers and Freedhoff, 1974, 1975; Swain, 1975; Hassan and Bullough, 1975: Renaud er a/., 1976, 1977; Kimble and Mandel, 1975a, b, 1976 (c.f. Ackerhalt, 1978); Polder and Schuurmans, 1976: Sobelewska. 1976; Cohen-Tannoudji, 1975, 1977;
172
S. Swain
Wodkiewicz and Eberly, 1976; Averbukh and Sokolovskii, 1977; Reynaud, 1977; Cohen-Tannoudji and Reynaud, 1977a. b, c; Courtens and Szoke, 1977; Raman. 1978; Kornblith and Eberly, 1978; Krainov, 1978; Knight, 19791. The classic treatment for a nondegenerate two-level atom subjected to only radiative damping was given by Mollow (1969) using a master equation approach together with a Markoff approximation, the quantum regression theorem and a semiclassical model for the atom/field interaction. A statistical factorization of the atom and field variables was assumed. The collisionally damped theory was given by Burshtein (1966) and Newstein (1968). Interest in the phenomena was greatly stimulated when the first experiment, by Schuda et a/. (1974), was reported. This essentially confirmed the three-peaked spectrum predicted by Mollow. A number of theoretical calculations were later made in which the assumptions made in Mollow’s original treatment were variously removed, or his results verified by different methods of calculation (Oliver et a/., 1971; Carmichael and Walls, 1975, 1976a, b; Hassan and Bullough, 1975; Swain, 1975; Mollow, 1975b; Kimble and Mandel, 1975a, b, 1976; Reynaud et a/., 1977; CohenTannoudji, 1975, 1977). Mollow (1975b) found the important result that the use of a c-number applied field was not an approximation but an exact quantum-mechanical method. Thus all the approximations and assumptions made in Mollow’s 1969 treatment were shown to be correct or justified. [We should point out that the form of the spectrum given in Swain (1975) is in fact equivalent to Mollow’s (1969) expression (B. R. Mollow, private communication, 1976).] All of these treatments refer to a two-level atom irradiated by a monochromatic laser subjected to only radiative damping. This model is not merely a mathematical abstraction, because by appropriate optical pumping sodium can be prepared as an effectively two-level system (Abate, 1974). By using an atomic beam to eliminate Doppler and collision broadening and very narrow-band dye lasers, a detailed experimental test of this model is possible. Very careful experiments by Walther (1975), Hartig et at. (1976), Wu et al. (1975), and Grove et a/. (1977) have confirmed the Mollow theory. Brief reviews of the experimental and theoretical situation have been given by Mollow (l978), Walther (1978), Cohen-Tannoudji and Reynaud (1978), and Ezekiel and Wu (1978a). It is important from a basic theoretical point of view as well as a practical point of view to have a theory that takes into account fluctuations in laser output. The theory of resonance fluorescence with finite bandwidth lasers has been discussed by many authors (Aganval, 1976, 1978, 1979; Eberly, 1976; Kimble and Mandel, 1977; Zoller, 1977, 1978, 1979; Zoller and Ehlotzky, 1977, 1978; Avan and Cohen-Tannoudji, 1977;
ATOMIC PROCESSES IN STRONG RESONANT E. M. FIELDS
173
Georges et a/., 1979; Wodkiewicz, 1979: Eberly et a/., 1980) using a variety of methods and models. Most of these papers concentrate on the phasediffusion model for a detailed treatment, but several discuss the effects of other types of broadening mechanisms such as instrumental bandwidth, amplitude fluctuations, and so on. A fundamental difference between the monochromatic and finite bandwidth cases is that the spectrum is no longer symmetric for off-resonant excitation in the latter case. Mandel and Kimble (1978) give a brief review. The effect of collisions on the resonance fluorescence spectrum in strong fields has been considered theoretically by several authors (Burshtein, 1966; Newstein, 1968; Mollow, 1970, 1972b, 1976, 1977: Ballagh and Cooper, 1977; Cooper and Ballagh, 1978; Agarwal, 1978: Nienhuis and Schuller, 1979). The low-field case has been treated near line center by Huber (1969), Omont et a/. (1972, 1973), Shen (1974), Mukamel and Nitzan (1977), Nienhuis and Schuller (1977), Voslamber and Yelnik (1978), and Nienhuis (1978). We do not consider the strong coupling case where the collision integrals are themselves modified by the presence of the applied field (e.g.. Lisitsa and Yakovlenko, 1974, 1975: Lau, 1976: Kroll and Watson, 1976: Yeh and Berman. 1979: Rabin and Ben-Reuven, 1979, and the references quoted there). Variants and generalizations of the original model have been considered by Freedhoff and Smithers (1975) and Freedhoff (1978) who treated resonance fluorescence in systems with permanent dipoles when an rf field is applied with frequency close to the Rabi frequency. Agarwal and Saxena (1978) took account of atomic recoil. Krainov and Kruglikov (1978) discuss multiphoton resonance fluorescence, and Whitely and Stroud (1976), Krainov (1978), and Mavroyannis (1979) discuss resonance fluorescence from a three-level atom. B. RESONANCE FLUORESCENCE I N NONDEGENERATE TWO-LEVEL SYSTEMS 1. General Theory
We use the Hamiltonian ( I ) in conjuction with the master equation (42). For convenience we define the atomic operators (Pauli matrices) (I+
= (u)(ll,
0-
= Il)(ul,
0,
= :(lu>(ul -
l/)
(44)
Now it can be shown, using standard arguments (e.g., Landau and Lifshitz, 1962: Agarwal, 1974; Kimble and Mandel, 1976) that the scattered field at a point x = (r, t ) may be written, for times (E,, - E,)t >> 1 and in the far-field region, E,(x)
=
E,+(x) + E,(x)
(45)
174
S. Swain
with the positive frequency part of the field, E:(x) given by E , + ( x ) = G,+(x) -
(Eu - E d 2
47rr0cZ/r3
[(pxr)xr]o-(r- L, C
(46)
p being the atomic dipole operator and E , ( x ) is the Hermitean conjugate. Equation (46) is nothing more than the retarded solution of the operator Maxwell equations. The first term & z ( x ) is a solution of the homogeneous equation and plays no further role in the subsequent discussion. Thus, in Eq. (46), it is apparent that we have expressed the scattered field in terms of atomic operators only. The experimentally observed field quantities at a fixed space point may be expressed (Glauber, 1965) in terms of the correlation functions
(E<(t)E:(t
+
T))
a g'"(t,
t
+ 7)
Trs+R{&+R(O)u+(:)a-
(i+*)} (47)
( E ; (t)E;
(t
+ T)E: ( t + T)E: ( 1 ) )
= Trs + R { Ps + R(O)U
+
(;).
a g(2)( t, t +
(
+
T)
+ T ) 13- ( f + T ) u - ( L ) }
(48)
+
g ( ' ) ( t ,t + T ) determines the spectrum and g ( 2 ) ( t t, 7) the intensity fluctuation spectrum. We have used Eq. (46) to eliminate the field variables in favor of the atomic operators u + and u - , and t = t - r / c denotes the retarded time. Consider Eq. (47): With a little rearrangement it becomes
+ T ) =Trs{u-TrR(e-'H7[pS+R(t)u+]e'H')}
g(')(t,t
(49)
To evaluate the two time averages in Eq. (49) we need the quantum regression theorem (e.g.. Lax, 1963, 1968; Haken and Weidlich, 1967). Now the formal solution of Eq. (43) is P ( t ) = exp[ - ie'(t
-
to)lp(to)
(50)
According to the quantum regression theorem, the new operators A ( T ) = TrR(e-'HT&O)e'HT),
i(0) = P ~ + ~ ( ~ ) U +
H ( T )= TrR(e-'H'fi(0)e'HT),
fi(0) = u - P ~ + ~ ( ; ) U +
(51)
obey the same equation as p , namely Eq. (43), and have the same solution (50). [It is easy to show this using the methods of Section I1 if we assume the statistical factorization condition of Mollow (1969). p S + R ( t ) = pR(0) . p,(f) for all t . ] Thus using (51) in (49) we obtain g ( ' ) ( t ,t
+ T ) = Tr,[
(I-
A(.)]
Anu:nl(~)
= n
(52)
ATOMIC PROCESSES IN STRONG RESONANT E. M. FIELDS
I75
and similarly for g ( 2 ) we obtain g ‘ 2 ’ ( t , t + T ) = Trs[ u + 6 - H ( T ) ] = n
nnuinu(7)
(53)
In the second stages of (52) and (53) we have used the definitions (44)of u + and u - and we have introduced the Fock states In) for the field. The
validity of the quantum regression theorem in this context has been verified within the rotating wave approximation by Mollow (1975a, b). Because A(T) and II(T)satisfy exactly the same equation as p ( ~ ) namely , Eq. (43), we can evaluate both g(l) and g ( 2 ) by solving for the appropriate matrix elements of this single equation. Consider first the calculation of p,,,,,(r) and pn+ ,,l;nu(r). These are needed later for the calculation of the fluorescence and intensity correlation spectra. Taking matrix elements of Eq. (42), using Eqs. (32), (40), and (41), and making use of approximations of the type Pn, 1,x;n2
I.,v(z)
(54)
Pnx;ny(Z)
(i.e.. changing n by the same integer on borh subscripts leaves the density matrix element unaltered) which are reasonable under the conditions (6), we obtain a set of four coupled equations I
Z
+ K,
+
Z
ih*
-im*
-iL
ih
- im*
im
- Kl
- Ku
- im
KI
i&*
z-iS+l+A 0
0 z
+ iS + 5 + A
where
Y
= Yul
For clarity we have omitted the n subscripts to the density matrix elements. These equations are solved straightforwardly, and from the solution we can determine the steady state inversion, since lim
I--, 00 p n u ; n u ( t )
= 2-0 lim [ z ~ n u ; n u ( z ) I
-
iQ’,({+A)+ K[
S2
K~[S~+({+A)~]
Pn(0) (57) + (5 + A)2] + + A) + K ~ Q: , =4 1 ~ 1 and ~ pn(0)= p,,,,,(O) +
where we have written K = K, P ~ , ; ~ ~ is ( Othe ) initial photon number distribution in the field. Assuming this
S. Swain
176
satisfies the conditions (6) and using C,,pn(0) = 1, we obtain for the probability PU,(co) of the atom being in its excited state after a long period of interaction
where Q2 = 4(x,-12,in agreement with the semiclassical theory (Mollow, 1969) for the case of purely radiative damping:
We may similarly show that the steady state dipole moment amplitude is i&(K,
P/u(m)=
+
[
- ./)(is
+ 5 + A)
( 5 + A)2] + Q2(5 + A)
(60)
Note that since K, - K / = 2y, the collisions do not contribute significantly to Eq. (60); this is because they excite incoherently (cf. Grischowsky, 1976). Now we calculate the spectrum in the steady state ( f + 00) when g ( ' ) ( t .t + 7) is a function of 7 only:
Z(')(Y)
The spectrum is the Fourier transform of Eq. (52) to the Laplace transform Afluzfl/(z):
= 2 Re
c Anu:nl(z
g ( ' ) ( T ) and
=
n
-
iv)
is related from
(62)
We can find A,,u;fl/(z) by taking the appropriate matrix elements of Eq. (42), with p(z) replaced by A(z). As before we obtain a system of four coupled equations when we make approximations of the form (54). The inhomogeneous part involves the initial values A(0). Now as t + 00, A(0) +p(m)(u x I ) and thus
ATOMIC PROCESSES IN STRONG RESONANT E. M. FIELDS
Pnu;nu(a)
where z’ = z
+ io,.
177
J
2. Radiative Damping On&, Monochromatic Source Because of its complexity we do not display the general solution of (64) but consider a few special cases. For radiative damping on(v [Eq. (59)] and monochromatic excitation, A = 0, making use of Eq. (58) and (60) we obtain
-
(z
$ $2’ S’+ y’+
$2
+ iw, + 2y) [ ( z + iw, + y)’ + a’] + i ~ ’ ( +z iw,) (z
+ iw,)f(z + io,)
(65) where f(z) is the cubic introduced by Torrey (1949) in the theory of nuclear magnetic resonance: f(Z) = ( z
+ 27)[ ( z + y)* + 6-21 + Q’(z + y )
(66)
All that remains now is to substitute from (65) into (62) noting that the pole on the imaginary axis in (65) at z = - iw, will give rise to a S function in (62). We obtain (after some algebra) an expression of the form
where explicit forms for the parameters Do, D , Q , , yo, y , are given in Mollow (1969). The first term, representing light scattered monochromat-
I78
S. Swain
ically at the laser frequency, is the coherent part of the spectrum whereas the final three terms comprise the incoherent part. For low powers, Q l I I 0 and the incoherent spectrum is the sum of three Lorentzians all centered on v = oL with widths yo, y + , and y - . However, one of the D’s is negative, which has the consequence that i(’)(v)- wLIP4 as Iv - wLI + cn,instead of Iv - wLI-2 for all D’s positive. At sufficiently low powers the coherent part dominates, producing an essentially monochromatic line. Of more interest is the high intensity case, when Q l # 0. For the special case of very high intensity >> y) and resonance (6 = 0). Q l + Q and we obtain
IY
(a
i‘l’(v)
=
27ry2
S ( v - OL)
+
1
2Y
+
( v - oL)2 y2
This is the Mollow spectrum. Note that the coherently scattered light intensity given by the first term is a small fraction of the total scattered light intensity. The incoherent part of the spectrum consists of a central peak with half-width equal to the natural width and side peaks of 1/3 the height on either side, with a half-width of 1 $ times the natural width, displaced from the central peak by the Rabi frequency Q. The peak heights and widths change when the laser is detuned from resonance, but the spectrum remains symmetric about the incident radiation frequency. This property is lost when collision broadening and finite laser tinewidths are taken into account. The strong field spectrum has been measured by general groups (Schuda et at., 1974; Walther, 1975, 1978: Wu et at., 1975: Hartig et a/., 1976: Grove et at., 1977; Ezekiel and Wu, 1978a. b). The agreement between theory and experiment is very good, as demonstrated by Fig. 2, taken from Grove et at. (1977). Figures 2a-c give the theoretical spectrum for resonant and off-resonant excitation. The coherent contribution is represented by a single vertical line: note that it is very small for resonant excitation but quite large for appreciable detunings. Below each spectrum the experimental data are compared with the theoretical results convoluted with the instrumental linewidth. In spite of the good agreement the experiments also indicated that under certain conditions a slight asymmetry was observable: that is, one side peak was found to be smaller than the other. Various mechanisms have been proposed to account for this. Hartig et al. (1976) have suggested that the use of linearly polarized light excites other hyperfine levels so that the system is no longer two-level: Grove et al.
179
ATOMIC PROCESSES IN STRONG RESONANT E. M. FIELDS Ibl
(a1
lcl
AUA -100
-50
0
50
100
-100
FREQUENCY IMHz)
-100
-50 0 50 FREQUENCY (MHzl
-50
0
50
100
-IM)
Ill
-100
-51
0
50
-53
0
50
100
FREQUtNCY IMHz)
FREQUENCY (MHz)
100
FREQUENCY IMHzI
-100
-53
0
M
loo
FREQUENCY( M H r l
FIG. 2. Theoretical spectra with B = 78 MHz, 2 y = 10 MHz, and detuning (a) 6 = -50 MHz. (b) 6 = 0, (c) S = + 50 HMz. Below are shown the corresponding experimental spectra and convolutions (smooth curve) of theoretical spectra with instrumental linewidth. From Grove er al. (1977).
(1977) suggest that a nonuniform field may be responsible. It is not certain that the various mechanisms are large enough to account for all the observed asymmetries. Renaud et al. (1977) suggest that finite observation times may make a contribution. Agarwal(l976, 1978), Eberly (1976), Avan and Cohen-Tannoudji (1977), and Kimble and Mandel (1977) have taken account of the finite laser bandwidth and find that asymmetries are also introduced from this source. 3. Radiative Damping, Finite Bandwidth Laser
Returning to Eq. (64), again assuming no collisions but allowing this time the laser to be nonmonochromatic, we obtain the exact result (6 = 0)
x Re[
+ 4A) + (z’+ A)[(z’ + y + 4A)(z’ + 3y + A) + f Q 2 ] (z’ + A)[(z’ + y)(z’ + 2y + A)(z’ + y + 4A) + Q 2 ( t+ ’ y + 2A)] 2y2(z’ + y
z’+ i ( w , -
Y)
(69)
180
S. Swain
Expression (69) for the spectrum is still complicated; however, if one takes the strong field limit (St’ >> y’, A’), it reduces to 4Y2(Y + 3 4
a’( y + A)
A ( v - uL)’+ A’
+
3(Y + A)/4
+ ( V - WL
+ 3)’ + 9(y + A)’/4
+ + 2A)’
(v - uL)’ ( y
3(Y + A)/4
+ (V
- uL- St)’
+ 9 ( y + A)’/4
The first term is the coherent scattering contribution, now a Lorentzian of width A instead of a 6 function as in the monochromatic case. The remaining three terms represent the ac Stark spectrum, the first being the central peak and the second two the side peaks. Note that as compared with the monochromatic case the positions of the peaks are unchanged, but now the ratio of the heights R, and the ratio of the widths R , of the side peaks to the central are
R,
= 3(y
+ A)/(y + 2A),
R,
= 3(y
+ A)/2(y + 2A)
(71)
giving the well-known Mollow results for A = 0, but the ratio is being significantly reduced for A 2 y. A number of curves have been given by Kimble and Mandel (1977) showing how the resonance fluorescence spectrum changes with detuning and laser linewidth. There are two obvious features. First the spectrum becomes “washed out” as A increases; and second, if the laser is detuned from resonance the asymmetry increases sharply as A increases from zero for monochromatic excitation, A = 0. A simple physical interpretation of the asymmetry has been given by Knight et al. (1978), who develop a theory based on the Lorentz model and assuming a Lorentzian laser line shape. They interpret the asymmetries as being caused by the transients induced by the frequent random changes in phase of the driving field, and show that in the appropriate limits their results are the same as those of other workers. To conclude the discussion on laser linewidth effects we say a few words about the effects of amplitude fluctuations of the laser light. This is an area in which rapid progress is currently taking place. Here we merely quote the results obtained by Agarwal (1976) and Eberly (1976) for the intense field resonant limit, namely
ATOMIC PROCESSES IN STRONG RESONANT E. M. FIELDS
181
’
where A, is the correlation time of the amplitude fluctuations. Recent discussions have been given by Georges et al. (1979), Agarwal (1979), Zoller (1979), and Wodkiewicz (1979). 4. Collisions
To emphasize the effects we consider the strong collision model (Van Vleck and Weisskopf, 1945; Karplus and Schwinger, 1948). Here it is assumed that the atom suffers random strong collisions which abruptly thermalize its state. The collisions are considered to take place instantaneously, which requires the collision duration 7c to satisfy 7c << I, which is questionable at optical frequencies. The damping parameters are given by Ku = K ( 1 - nT), K, = K n T , 5 =K (73) where nT is the thermal equilibrium value of the excited state as given by Boltzmann statistics and ~ d ist the probability of a collision occurring in the time interval dr. We also assume K >>A, y so that collisions are the principal source of incoherence. The expressions we then find for the fluorescence spectrum differ markedly in the limit of weak driving fields from those found earlier for purely radiative relaxation. In particular, in addition to the coherent component oscillating at the driving frequency wL there is an incoherent component at the resonant frequency E,, - E, = wo. In the limit of strong driving fields, however, the solutions for radiative and collision damping have the same positions for the three components of the spectrum, but different weights and widths. Thus the spectrum for the low-excitation case (a<< K , 6; nT << 1) is
and that for very intense driving fields
g“ I)( Y )
-
(V
- OL)’
+
i K
+
K K2
[a2>> (S2 + K’)]
( V - OL
+ a)*-k K 2
+
is +K
(Y - WL
-
a)2-k
K2
(Newstein, 1968; Mollow, 1970). Note that expression (75) is the same as that which would be obtained in the single-photon approximation. The first term in Eq. (74), the coherent term, arises from the homogeneous part of the solution for the atomic dipole operator as it is driven between collisions; the second term arises from the inhomogeneous part; and the third term, proportional to nT represents the thermal spontaneous emission field contribution.
182
S. Swain
For the remainder of this section we return to the general damping scheme (56). Carlsten and Szoke (1976a, b) and Carlsten et al. (1977) performed a series of experiments in which the collisional relaxation rates exceed the radiative relaxation rates by an order of magnitude. The laser was significantly detuned (161 >> K , 5 ) and the laser power was such that the three components were well separated. Under these conditions the spectrum may be approximated as (Mollow, 1977) ~ “ ’ ( Y ) N A ~ ~ ( Y - o ~ ) + A + ~ ( v - w ~ +A Q - S’ ()Y+- O ~ - Q ’ )
(76)
where we have taken the limit of zero width for the spectral lines. The coefficient, A,, obtained by combining the coherent part with the central line of the incoherent spectrum, is given by
and A , is given by Q2[
A, =
c2Q2+ K , a 2 ( 2 5
-
K,)]
4Q’2(5fi2+ K , S 2 )
(78)
This confirms (following Mollow (1977)l a conjecture by Carlsten and Szoke (1976b) that the strength of the central line is independent of the relaxation mechanism (when the spectral widths are ignored). On the other hand, the weights of the side components do depend upon the particular type of relaxation mechanism operating through the factors S and K,. 5. Time-Dependent Spectra
The results we have derived so far apply when the atom interacts with the field for an essentially infinite time T B y - ’ . I t is of interest to examine the spectrum for finite T ; this has been discussed by Oliver et al. (1971), Herrmann et al. (1973), Kimble and Mandel (1975a, 1976), Carmichael and Walls (1976a, b), Renaud et al. (1977), Knight et al. (1978), and Eberly et al. (1980). The integrated form of the spectrum was used; the output registered by a monochromatic photon detector being taken to be proportional to
j-;4j-;f2
exp[W2 - t J l P ( f l J 2 )
+
(79)
The correlation function g ( ’ ) ( t ,t 7 ) can be evaluaLed in the same manner as in previous subsections except that now p,,(t) and p,”(t) are used
ATOMIC PROCESSES IN STRONG RESONANT E. M. FIELDS
183
instead of p,,(00) and p,,(co). For large T we obtain the expressions discussed earlier in this review; for general T the expression for the scattered spectrum is very complicated and the reader is referred to the original papers. Detailed time-dependent spectra have been calcula:ed by Eberly et al. (1980), taking into account the finite bandwidth Ts of the spectrometer and using a definition of the spectrum that differs from Eq. (79) by having the factor Ts exp{ - Ts[ T - 4 ( I I + t , ) ] } inserted under the integral signs (Eberly and Wodkiewicz, 1977; Stehle, 1979). They also considered the transient spectra when the atom was prepared initially in a linear combination of the ground and excited states. A large number of curves were presented. Courtens and Szoke (1977) have used the dressedatom approach to study the way in which asymmetrical features are introduced into the spectra under pulsed excitation. Raman scattering, two-photon absorption, and Rayleigh scattering were identified with an adiabatic process, and fluorescence with nonadiabaticity.
C. INTENSITY FLUCTUATION SPECTRA
In order to calculate the intensity fluctuation spectrum, Eq. (53), we have to evaluate II,,;,,(t) or equivalently II,,u;nu(z).We find that IInu:,,(z), n,+I./;,+ , , / ( Z ) ? n,+, . / ; n u ( z )and . II,,;,, I./(Z) form a c h e d system of equations which are formally identical to Eq. (55) if p is replaced by II. Since II(0) = ~ l ) p , , , , ( ~ ) ( lthe ~ , initial values are =nu;nu(o)
=
%+
K +I./:nu(O) I./;,+
= ~ n u ; n + l . / (=0 0 )
do)=
P,U,,,(t
)
(80)
The system is easily solved to give n
+ { + A ) ’ + 6’1 + i!J’(z + { + A ) z {(z + ( z + { + A)’ + 6’1 + Q’(z + { + A)} K,[(z
= P,,(t
)
(81)
K)[
under the assumptions ( 6 ) on the photon distribution. Note that because of the formal similarity of the equations for II and p, Eq. (81) also gives j5uu,ll(z),the Laplace transform of the probability of finding the system in the state u at time 7 if it was in state I at time zero, if we set &,(t)-+j5//(0) = 1. For the steady state spectrum we let t + 00 and substitute for p,,(00) the expression (58); we then denote the steady spectrum simply as g(’’(7). In the particularly simple case of exact resonance (6 = 0). Eq. (8 1) is readily
S. Swain
184
inverted to give d2’(T)
[
= g“’(O)]’
I
1-
+ { +A)] {&?’[K , ( { + A) 4- in’] COS(Q’T)} + A) + + Q ’ ] Q ’
exp[ - it(.
[K/({
+ { ~ Q ’ ( K + 5 + A) + f~,[({ + A ) ( { + h - K ) - 2Q2]} sin(Q’7) where 6?” = Q’ g“’(0) = Fuu(@J) = [ K , ( {
I
( 5 4- A - K)’
-
+ A) + +!d’]/[
(82)
(83) K({
+ A) + a’]
(84)
Here [ g“’(O)]’ is defined to be the value of g”’(7) when the two fields are completely uncorrelated, that is, lim
7-m
g(2)(T)
-[
g(’)(0)1’
(85)
The important property of Eq. (82) for the present purposes is that lim 7-+0
8‘2’(T)
[ g( I)(())]’
-
K/(i + A - K) K/({
+ A) + 4 a’
< I
(86)
Note that the right-hand side is less than unity. (In fact it is equal to zero for purely radiative damping, K/ = 0.) Thus we have a n example of photon anribunching; any light field for which g‘2’(T) < [ g“’(O)]’ for some values of T is said to be “antibunched” or “anticorrelated.” Expression (86) indicates that the probability of observing two fluorescent photons a t a given point within a vanishing small interval is less than the probability of observing two uncorrelated photons. This is easily understood; immediately after we have detected a spontaneously emitted photon we know that the atom must be in its ground state. We cannot then expect to observe a second photon until the atom has had time to achieve the excited state again under the influence of the applied field. Thus we would expect g”’(7) to be proportional to the probability for finding a n initially unexcited atom in its upper state u after a time interval T , that is, to F,,,,,,,(T). In fact it can be shown that expression (82) may be written
185
ATOMIC PROCESSES IN STRONG RESONANT E. M. FIELDS
In the present analysis this follows from the comment in parentheses after
Eq. (81). The case of purely radiative damping [condition (59)], which applies, for example, in atomic beam experiments, is of particular interest and has been extensively discussed (Carmichael and Walls, 1976a, b; Kimble and Mandel, 1976, 1977; Cohen-Tannoudji, 1977; Agarwal, 1976, 1979). Then
where now 3' = [ 3' - (A
-
y)']
I /2
(89)
which, in the limits of weak [4Q2<<(A - y)'] and strong [4Q2>>(A fields, respectively, reduces to
g(''(7)
=
[ g " ) ( 0 ) ] 2 ( 1 + [ 2 ~ e - ( ~ +-~(A) '+ Y ) ~ - ~ " ] / ( A
-
7))
-
y)']
(90)
and g ( 2 ) ( 7 )=
[ g(1)(0)1'[
1 - e-(3~+')~/2cos(6/7)]
(91)
In the weak field case, Eq. (90), ~ ( ' ' ( 7 ) increases monotonically from 0 to 7 increases, showing that the antibunching effect is slowly washed out. In the strong field case (Fig. 3) both antibunching and
[g(')(0)l2 as
t+
1.51
1.0
I
€
Time interval T in nsec
FIG. 3. Comparison of the normalized correlation function g'2'(T) derived from measurements with the theoretical curve for C2/(2y) = 4.4 and 0 / ( 2 y ) = 2.2 on resonance. From Dagenais and Mandel (1978).
S. Swain
186
bunching ( g ' 2 ) ( T ) > [ g")(O)l2) are apparent: The probability of observing two photons an interval T after the atom was in its excited state will be larger than that for observing two uncorrelated photons when T 2 n / Q the time taken for an atom to be driven from its excited state to the ground state and back again under the influence of the exciting field. The bunching and antibunching effects are slowly washed out by the random nature of the spontaneous emission processes, as indicated by the asymptotes in Fig. 3. The classic demonstration of second-order correlation effects is the Hanbury-Brown and Twiss experiment (1956a, b) which measured photon bunching. Antibunching for two-photon absorption of coherent light was predicted theoretically by Chandra and Prakash (1970); see also Simaan and Loudon, 1975; Every, 1975) and for second harmonic generation by Stoler (1974). So far the observation of antibunching in these phenomena has not been reported. It is of great interest to observe antibunching since the property cannot be displayed by any classical field. To observe antibunching in resonance fluorescence the atomic beam must be so dilute that there is roughly only one atom present in the section of the atomic beam traversed by the light field at any one time. If many atoms are present, the antibunching is obliterated. Experiments on the observation of antibunching in resonance fluorescence using a dilute atomic beam were performed by Kimble et al. (1977) but the interpretation of these experiments is complicated by the effects of number fluctuations in the atomic beam (Jakeman et a/., 1977: Kimble ef al., 1978; Carmichael et al.. 1978). Later experiments (Dagenais and Mandel, 1978) give a clear, if indirect, demonstration of photon antibunching. The experiments were performed with both resonant and nonresonant excitation, and significant nonclassical features were found. The results were in good quantitative agreement with theory as the experimental points in Fig. 3 indicate. More general situations have recently been treated theoretically. Soboleswka and Sobelewski ( 1978) discuss statistical correlations in the resonance fluorescence from three-level systems. Again, antibunching is predicted. Apanasevich and Kilin (1979) discuss photon bunching and antibunching effects in the time-delayed coincidences of photons from the side components in strong field resonance fluorescence.
-
D. RESONANCE FLUORESCENCE I N MULTILEVEL ATOMSW I T H MONOCHROMATIC FIELDS
In the event that the ground and excited states are not nondegenerate, but possess some structure, we would expect the fluorescence spectrum to
ATOMIC PROCESSES IN STRONG RESONANT E. M. FIELDS
187
be modified. We have in mind particularly magnetic degeneracy; then different magnetic sublevels in the ground and excited states would be connected if the applied field was a-polarized or a-polarized, for example. Thus the dressed-atom energy level structure depends strongly upon the polarization of the exciting laser; likewise the observed spectrum depends upon the state of polarization admitted by the detector. A second effect which operates is that of optical pumping: It is possible for spontaneous emission to transfer eventually all the atomic population into magnetic sublevels of the ground state which are not connected by the applied field to any sublevel of the excited state. In this case there would be no steady state fluorescence spectrum, although there may be a transient one. The theory of resonance fluorescence in multilevel atoms has been considered by Higgins (1975) who took account of the effect of the neighboring F = 2 * F = 2 transition on the fluorescence spectrum of the F = 2 t,F = 3 transition in the sodium D, line, as well as the effects of magnetic degeneracy. Sobelewska (1976) and Kornblith and Eberly (1978) have treated the case where the excited statk is a doublet. Polder and Schuurmans (1976) have generalized Mollow’s (1969) treatment to deal with the case of doubly degenerate ground and excited states. They compute the spectra produced by the three components of the atomic dipole operator when the incident field is a-polarized in, say, the Z direction. Polarization effects have been discussed in detail also by CohenTannoudji and Reynaud (19774 and Kornblith and Eberly (1978). The dressed-atom approach for multilevel atoms, outlined in Section I, B, has been given a more quantitative development by Cohen-Tannoudji and Reynaud (1977a. b, 1978). This is valid for arbitrary numbers of sublevels (say N , and N u . respectively) in the ground and excited states, but in practice numerical methods may be necessary to find the dressed-atom energies if N , + N u is large. In addition to the assumptions made in Section I, B. the strong coupling assumption is made, which requires the difference in energy between any two dressed states to be much larger than the natural width of the excited levels. A master equation is then set up which allows for the spontaneous decay between the dressed states. This equation has a particularly simple form; in fact, for the diagonal elements it has a rate-equation structure. In the first paper they consider the situation where the ground and excited states are quasi-degenerate. From their analysis they were able to draw general conclusions about the fluorescence spectrum, which without going into the details of their calculations we abbreviate to: 1 . The spectrum consists of a central component at w L , the laser frequency, with R( R 1) sidebands symmetrically placed with respect to
+
188
S. Swain
wL at wL 2 Gj,. (Here R = N , + N u is the total number of states in the excited and ground state manifolds, and 6 , = G j - &;/. This property was deduced in Section I, B.) 2. The two components at wL -+ ti.‘J have the same width. 3. The weight of the component at wL Gi? is proportional to the total number of atoms undergoing a w L + 6,, transition in the transit time T. In the steady state, if detailed balance holds, the spectrum is completely symmetric. (Note however that this condition is not always met with in practical experiments.) 4. The central component is a superposition of R components centered on wL but having different widths, one of them corresponding to the coherently scattered component. The total weight is porportional to the total number of atoms undergoing an wL transition in the transit time T .
+
In this summary, polarization effects arising from the Zeeman degeneracy of the ground and excited levels (such as discussed by Polder and Schuurmans, 1976; Cohen-Tannoudji and Reynaud, 1977~)have not been emphasized. Cohen-Tannoudji and Reynaud (1977a, 1978) also discuss the properties of the absorption spectrum when a second weak laser beam crosses the illuminated part of the atomic beam (also at right angles). In this case they find the following general properties of the absorption spectrum: 1. The components appear at the same positions with the same widths as in the fluorescence spectrum except that the central component is missing. 2. The weight of a component at wL Gi, depends on the population difference between the (dressed) states li) and lj). (The central component is associated with the transition Ii,n)+ li,n - 1) which explains why it is missing. Higher order processes may give a finite contribution to the central component, however.) 3. Of the two components at wL 2 ti,,, one is amplifying and the other is absorbing. In the steady state the amplifying component is larger than the absorbing component.
+
For example, applied to a two-level system these rules predict that the absorption spectrum should consist of two peaks, one absorbing and one amplifying, which disappear at resonance since the populations are then equal. The experimental situation on absorption line shapes has been reviewed by Ezekiel and Wu (1978a). The theory of absorption by weak probe fields in the presence of strong saturating fields has been considered also by Rautian and Sobel’man (1962), Mollow (1972b), Haroche and Hartmann (1972), and Sargent and
189
ATOMIC PROCESSES IN STRONG RESONANT E. M. FIELDS
Toschek (1976). Note that the absorption spectrum is determined by the Fourier transform of the correlation function ([ u ( I ) , u ( I 7)]) rather than ( u + (t)a_ ( t 7)). as in Eq. (47) (Mollow, 1972b). Evidence of probe amplification in the rf (Bonch-Bruevich et a/., 1975), millimeter (Senitzky el al., 1963; Senitzky and Cutler, 1964). and optical regions (Wu el a/., 1977) has been found. Bonch-Bruevich and Khodovoi (1968) and BonchBruevich et a/. (1969) have considered absorption in three-level systems with the probe beam connecting the lower two levels and the saturation beam the upper. The theory, taking into account various broadening mechanisms for the saturating laser, has been given by Zusman and Burshtein (1972) and Przhibelskii and Khodovoi (1972). Closely related to these topics is the problem of resonance Raman scattering at high field intensities. It is not our intention to treat this topic in detail but we briefly consider the model treated by Cohen-Tannoudji and Reynaud (1977b). Here we have one excited state Iu) with natural width 2y and two ground states 11,) and [I,) having a n energy difference of S . The laser frequency is w L , the resonant frequency for the l u ) 11,) ~ transition is w o , the detuning is 6 = wo - w L , and the Rabi frequencies for the transitions I , c-) u and 1,- u are 0 , and Q2, respectively. At resonance (aL= wo) for low light intensities (Q, << y ) , we expect the fluorescence spectrum to consist of three lines: at wL (Rayleigh), at wL - S (RamanStokes), and at w L S (Raman anti-Stokes). For intermediate field strengths ( y . 16 I << Q, << S ) the transition u t)I , is saturated, whereas the transition u - I , is weak. The position of the spectral lines may be predicted as described in Section I, B. Thus the strong resonant field splits the degenerate states l u , n ) and II,.n 1) into Autler-Townes doublets as for the two-level problem, whereas the state 112,n) is to a first approximation unaffected by the interaction; allowing spontaneous emission between the dressed states gives rise to a spectrum in which the Rayleigh line is a triplet and the Stokes and anti-Stokes lines are doublets. For large detunings (IS I >> Q,) the separation of the components is I S ( (for both the Rayleigh and Raman lines), whereas at resonance the separation of the components is Q , . By solving the master equation the weight and the widths of the components may be calculated. Similar processes have been investigated by Mollow (1973). At high field strengths ( y , 161, S << a,) we have the same situation as discussed earlier in this subsection of quasi-degenerate excited and ground state sublevels and the rules (l)-(3) for the fluorescence spectrum apply. At these high intensities the Rayleigh and Raman components are intermixed. The transition from the low intensity spectrum to the high intensity spectrum has been investigated by Reynaud (1977).
+
+
+
+
+
190
S. Swain
IV. The Optical Autler-Tomes Effect A. INTRODUCTION
In a well-known experiment on the molecule OCS Autler and Townes (1955) demonstrated that a microwave transition line could be split into two components when one of the two levels involved in the transition was coupled to a third by a strong radio-frequency field. With the development of lasers it became feasible to perform the analog of the experiment in the optical region, and tunable cw lasers have enabled high resolution techniques to be used. Cross-saturated absorption experiments have been performed by Schabert et al. (1975a. b) on neon with the active cell inside the laser cavity. The combination of velocity and radiation power effects and the interaction with standing waves complicate the interpretation of the experiment. An experiment using smaller laser powers with the cell outside the laser cavity has been performed by Cahuzac and Vetter (1976). A two-step optical resonance experiment on a sodium atomic beam using two cw lasers was reported by Picque and Pinard (1976). The use of an atomic beam eliminated the need for velocity averaging in the interpretation of the results, but the complex hyperfine structure of sodium provided further difficulties. Delsart and Keller (1976) have performed an experiment on neon gas which has no hyperfine structure, thus providing a simple atomic model, but requiring Doppler effects to be taken into account. Gray and Stroud (1978) and Ezekiel and Wu (1978a) have made the most direct test of the three-level theory using an atomic beam of sodium prepared as a two-level system as far as the saturating laser is concerned. For weak fields a single peak was observed, whereas for strong fields a doubly peaked structure was obtained. This was symmetric for resonant excitation and the magnitude of the splitting was approximately equal to the Rabi frequency. The results are in good agreement with theory. The case of three-level systems subjected to Doppler broadening has been discussed experimentally and theoretically by Delsart and Keller (1978). Again, reasonably good agreement between theory and experiment is found. B. THREE-LEVEL AUTLER-TOWNES THEORY The theory of three-level systems interacting with monochromatic fields has been discussed by many authors in many different contexts (e.g., Feld and Javan, 1969; Hansch and Toschek, 1970; Feldman and Feld, 1972; Salomaa and Stenholm, 1976; Hermann and Swain, 1977). Particularly
ATOMIC PROCESSES IN STRONG RESONANT E. M. FIELDS
191
relevant here are the works of Mollow (1972a), Feneuille and Schweighofer (1975), Whitley and Stroud (1976). Salomaa (1977), and Bialynicka-Birula and Bialynicka-Birula (1977). Treatments for finite bandwidth lasers have been given by Salomaa (1978), Georges and Lambropoulos (1978, 1979), Agarwal and Narayana (1979), Zoller (1979) and Osman and Swain (1980). In the last reference, transient spectra are also discussed. Here we give the theory for phase-diffusion broadening. An intense laser at frequency wL is near resonance with the atomic transition 1,)I u ) (resonance frequency E,, - E/), and a weak tunable laser at frequency up probes the transition I u ) t)Ip) (resonance frequency Ep - E,,). Using the multilevel master equations in the RWA of Section 11, Eq. (40), and taking account of the finite linewidths A,- and A,, of lasers 1 and 2, respectively, using Eq. (51), we obtain a set of nine equations which are straightforward if tedious to solve. For simplicity we assume that the transitions from I t ) u and u t ) p are dipole allowed, and the transition 1-p is dipole forbidden. (Thus there is no spontaneous decay directly from level J p ) to level I/).) We have also assumed that spontaneous decay is the only decay mechanism in the system (i.e., no collisions, etc.) and that the lasers are sufficiently intense for their mean photon numbers EL, Ep to satisfy EL >> I , iip>> 1. There is assumed to be no cross-correlation between the two lasers (Wong and Eberly, 1977). The solution for the population of the probe level Ip) is complicated, but in most of the experiments of interest we can assume the probe transition Iu) t)I p) to be weak compared with the saturating transition u t)1. Thus we can make the “probe approximation” ( e g , Feneuille and Schweighofer, 1975)
a;,, << a:,
7
y,’
9
YP‘
-
(92)
u and u * I, where QpU, a,,, are the Rabi frequencies of the transitions p respectively. and 2y,, 2yp are the spontaneous decay rates of levels u a n d p , respectively. If further we are only interested in the steady state spectrum, the expression for ppp(w)reduces to
192
S. Swain
a n d p,,(co) is given by Eq. (58) under the conditions of purely radiative decay (59). It is the final factor of Eq. (93) which for fixed saturating laser frequency (a, = constant) exhibits resonant behavior as the probe frequency ,,a is varied. We discuss the following special cases.
I . For very low powers (ni, << y,’, y,‘) the behavior of pPp(0o) as a function of the upper level detuning ,a, for fixed 6,, is still a relatively complicated function of Spu, but essentially the function exhibits a single resonance. 2. For exact resonance of the lower transition, S,, = 0, p,,(co) is given by
where
p,,(.o)
has turning points at
Thus for
Qi, > Q:.,
4,= 0 a n d at
where
the right-hand side of (97) is real, a n d pPp(0o) exhibits a double resonance with a minimum a t ,a, = 0, whereas for < Q: it shows only a single resonance. For the case where the saturating laser linewidth greatly exceeds the natural linewidths and the probe linewidth, the minimum saturating laser power necessary to observe the Autler-Townes splitting is given by
at,
52;
nil
=
3At
(99)
>> rt, r;, one obtains the classical Autler3. For very intense fields, Townes result for the positions of the resonances, = 6, :
[
8 t = I2 - 8
4,
u/
* (G+ Qi/)”21 J
so that a t resonance,
a,
= 0,
(100)
the two resonances are separated by the Rabi
I93
ATOMIC PROCESSES IN STRONG RESONANT E. M. FIELDS
frequency of the saturated transition:
where I , is the intensity of the saturating laser. At resonance, in the vicinity of 6 = 6, or 6 = 6 - , p,,(m) approximately by
is given
showing that the resonances are symmetric of half-width
rp+ tr, = yp + aP+ tY, + :A,
(103)
The state Iu) only contributes : y , to the linewidth because the dressed states of the Autler-Townes doublet each possess only half the character of the state lu) at resonance. Off-resonance the peak at = 6, is generally broader than the peak at 9, = - 6 for 6,, > 0 and vice versa for
4,
~
a, < 0.
The expressions (93)-( 103) generalize the results of Feneuille and Schweighofer (1975) to include the effects of finite laser linewidths. Georges and Lambropoulos (1979) and Zoller (1979) have considered the effects of chaotic fields and find significant changes. As already noted the experiments of Delsart and Keller (1978). Ezekiel and Wu (1978a), and Gray and Stroud (1978) provide confirmation of the three-level theory.
C. EXPERIMENTS ON MULTILEVEL ATOMS
If the experiments are performed on a gas rather than an atomic beam, it is necessary to integrate over the velocity distribution, which of course complicates the resulting expressions. See, for example, Feneuille and Schweighofer (1975) and Delsart and Keller (1978) for a calculation of the monochromatic excitation case. Experiments on a system that can be prepared as a three-level system have been performed using counterpropagating laser beams by Delsart and Keller (1976, 1978) on a gas of "Ne and "Ne, the atoms of which exhibit no hyperfine structure. By the use of linearly polarized lasers with the polarization vectors parallel, the
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neon system behaved like a three-level system. For gaseous atoms at high intensities, after velocity averaging, the positions of the resonances are given by i&= ,,6: where
which agrees with (101) for a,,, = 0, but the displacement of the peaks for from that predicted by (100). They confirmed the dependence of the Autler-Townes splitting on the square root of the saturating laser intensity predicted by (101) and also obtained good agreement between the calculated and observed values of the absolute magnitude of the splitting. When the orientation of the laser polarizations is varied from the parallel configuration, the neon system behaves effectively like two independent three-level systems with different Rabi frequencies. Delsart and Keller observed a change in the separation of the doublets as the angle between the polarizations was changed in accordance with the calculated values of the two Rabi frequencies but were unable to resolve the spectrum into four peaks. The experiment of Picque and Pinard (1976) was performed on a n atomic beam of sodium with the laser beam at right angles so that the atoms could be treated as stationary; but in this case the interpretation of the experiment is complicated by the hyperfine structure and Zeeman degeneracy of the sodium atomic energy levels. Although the AutlerTownes splitting was clearly observed the doublet was found to be asymmetric, the low frequency peak being narrower and less intense than the high frequency peak. It seems natural to associate this asymmetry with the additional hyperfine levels, but a detailed numerical investigation (McClean and Swain, 1977b, 1978; see also Shore, 1978) taking into account all the neighboring hyperfine levels of the 32S,/,, 32P,/2, and 5 2S,,2 states showed that the resulting spectrum was asymmetric but with the low frequency peak the more intense. In any case the spectrum showed additional structure. In this treatment the lasers were assumed monochromatic and the analysis was complicated because the Rabi frequency was of the order of the spontaneous lifetime, so that spontaneous emission played an important role. The steady state spectrum was calculated. When the effect of the Zeeman degeneracy of the atomic levels was taken into account, the calculated spectra rather surprisingly looked much more like those expected from a nondegenerate three-level system for laser intensities not too large. Indeed, the dependence of the magnitude of the splitting on the square root of the laser intensity was recovered, but the low frequency
a,,, # 0 is very different
ATOMIC PROCESSES IN STRONG RESONANT E. M. FIELDS
195
peak remained the most intense. Possibly a more general calculation in which the finite linewidths of the lasers is taken into account and nonstationary spectra found may be necessary. Our calculations indicated that the asymmetry varied strongly with the detuning of the saturating laser and that optical pumping with the lower hyperfine level of the ground state greatly reduced the intensity of the spectra. Bjorkholm and Liao (1977) have also performed an experiment on an atomic beam of sodium, but they used much higher powers, which enables them to argue that the system may be considered as two independent three-level systems. At the high powers used (L?2,,-0.70 GHz), we have Q,,, >> y u , y, and thus to a first approximation spontaneous emission may be ignored so that the dressed-atom approach of Section 11, B is appropriate. The results of this calculation are in good agreement with the predictions of the heuristic two independent three-level systems and with experiment. The ac Stark effect in multiphoton ionization has been observed by Moody and Lambropoulos (1977) and Hogan et af. (1978). The setup was similar to that previously described, except that the saturating and probe lasers were of roughly equal intensities, the probe transition being weak because it was quadrupole and the saturating transition being strong because it was dipole. A photon from either beam had sufficient energy to ionize the atom from the probe state. and this was the mechanism used to monitor the population of this state. The number of ions was measured as a function of the detuning of the probe laser for various strengths of the saturating field. For low saturating powers only a single peak was observed, but with high saturating powers a double-peaked spectrum was obtained, the separation of the peaks being proportional to the square root of the saturating laser intensity as required by the Autler-Townes theory. A detailed theory of these experiments has been given by George and Lambropoulos (1978); the finite laser linewidths being taken into account in a manner equivalent to that of Agarwal’s (1978) treatment. They showed that the asymmetry of the peaks of the resonance curve due to Stark splitting was reversed when the laser bandwidth was larger than the widths of the atomic transitions of the resonant states. The Autler-Townes splitting of photoelectron energy distributions in resonant two-photon ionization was predicted and described by Knight (1977b. 1978) using Heitler-Ma techniques. Effects of the laser lineshape on the photoelectron spectrum have been considered by Armstrong and ONeill (1979). Very recently, Moloney and Faisal (1979) have discussed the possibility of directly observing the Autler-Townes splitting of certain rotational lines in diatomic molecules using a n infrared laser.
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V. Conclusion We have given a unified account, based on the master equation method with the dressed-atom approach for multilevel systems, of the consequences of the splitting of the energy levels of atoms in strong electromagnetic fields. The basic features are considered well understood. In resonance fluorescence for two-level systems and in the optical Autler-Townes effect with three-level systems the agreement between theory and experiment is satisfactory for monochromatic fields with purely radiative damping. Some of the predictions about collision-broadened systems have also been verified, but the predictions concerning laser linewidth effects have yet to be substantially tested experimentally. For multilevel systems also theory and experiment have not been adequately compared. ACKNOWLEDGEMENT C.
It is a pleasure to acknowledge informative discussions with J. H. Eberly, P. L. Knight, and P. Stroud, Jr. on the topics of this review. REFERENCES
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ADVANCES IN ATOMIC A N D MOLECULAR PHYSICS. VOL . 16
SPECTROSCOPY OF LASER-PRODUCED PLASMAS M . H . KEY Science Research Council Rutherford and Appleton Laboratories Chilion, Didcot Oxfordrhire. England
and
R. J . HUTCHEON* Physics Department University of Leirester Leirester. England
1. Introduction . . . . . . . . . . . . . . . . . . . . . . 202 I1. Ionization . . . . . . . . . . . . . . . . . . . . . . . 203 A. The Local Thermodynamic Equilibrium (LTE) Model . . . . . . .204 B. The Coronal Model . . . . . . . . . . . . . . . . . . .204 C . The Collisional-Radiative (CR) Model . . . . . . . . . . . .206 D . Application to Laser-Produced Plasmas . . . . . . . . . . . .207 E. Reduction of Ionization Potential . . . . . . . . . . . . . .208 F . High-Density Effects . . . . . . . . . . . . . . . . . .209 G . Transient Ionization . . . . . . . . . . . . . . . . . 212 . 111. Population Densities of Bound Levels . . . . . . . . . . . . . . 213 A . LTE and the LTE Limit . . . . . . . . . . . . . . . . .214 B . The Coronal Approximation . . . . . . . . . . . . . . . . .214 C . Collisional-Radiative-Level Populations . . . . . . . . . . .215 D . Rate Coefficient Data . . . . . . . . . . . . . . . . . .216 IV . Intensity of Line Radiation . . . . . . . . . . . . . . . . .217 A . Introduction . . . . . . . . . . . . . . . . . . . . . 217 B . Hydrogen-Like Ions . . . . . . . . . . . . . . . . . .218 C . Helium-Like Ion Resonance and Intercombination Lines . . . . . . 219 D . Satellites to Resonance Transitions in One- and Two-Electron Ions . . 220 E. Characteristic X-Ray K Lines . . . . . . . . . . . . . . .223 V . Line Broadening . . . . . . . . . . . . . . . . . . . . . 225 A. Natural Broadening . . . . . . . . . . . . . . . . . . 226 .
*Present address: Nuclear Power Company (Whetstone) Ltd., Cambridge Road, Whetstone, Leicester, England
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Copyright 0 1980 by Academic Press Inc . All rights o f reproduction in any form reserved.
ISBN 0-12-003816-1
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M . H . Key and R. J. Hutcheon
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B. Doppler Broadening . . . . . . . . . . . . . . . . 226 C. Stark Broadening . . . . . . . . . . . . . . . .227 . .234 VI. Continuum Emission . . . . . . . . . . . . . . . . .234 A. Bremsstrahlung . . . . . . . . . . . . . . . B. Recombination Continuum . . . . . . . . . . . . .237 VII. Radiative Transfer . . . . . . . . . . . . . . . . . . . 238 A. The Radiative Transfer Equation . . . . . . . . . . . . . . 239 B. LTE Solutions . . . . . . . . . . . . . . . . . . . .240 C. Collisional-Radiative Solutions. . . . . . . . . . . . .241 D. Radiative Transport with Flow Doppler Shifts. . . . . . . . .245 VIII. Structure and Spectroscopic Characteristics of Laser-Produced Plasmas . .246 A. Plane Targets . . . . . . . . . . . . . . . . . . . . .247 B. Spherical Shell Targets. . . . . . . . . . . . . . .249 IX. Spectroscopic Diagnostics of Laser-Produced Plasmas . . . . . . . ,251 A. Spectroscopy of the Expansion Plume . . . . . . . . . . .251 B. Spectroscopy of the Ablation-Front Plasma . . . . . . . . .258 C. Continuum and K,-Emission Spectroscopy (the Zone of Hot-Electron Preheating) . . . . . . . . . . . .263 .269 D. Implosion-Core Spectroscopy . . . . . . . . . . . . E. X-Ray Shadowgraphy and Absorption Spectroscopy . . . . .272 References . . . . . . . . . . . . . . . . . . . . . .272 Note Added in Proof . . . . . . . . . . . . . . . . . .280
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I. Introduction Study of laser-produced plasmas (LPPs) began about 15 years ago when the development of Q switching of solid-state lasers made megawatt laser power readily obtainable and with it focused irradiance in excess of 10" W cm-2. At such high irradiance solid or gaseous targets are ionized and the resulting plasma absorbs a large fraction of the laser power to form a high-temperature LPP. Present-day high-power lasers can generate up to 20 TW (2 X IOl3 W) focused to lo'* W cm-2 in subnanosecond duration pulses (Gibson and Key, 1980). Much of the motivation for the development of these high-power laser systems originates in the long-term goal of inertially confined fusion (ICF) (see, e.g., Nuckolls et al., 1973; Emmett et al., 1974; Brueckner and Jorna, 1974). A central objective of ICF research is compression of plasma to high densities and pressures. Already pressure of 4OOO Mbar and density of 10 g cm-3 have been reached in laser-driven implosions of deuteriumtritium-filled spherical-shell targets. These plasmas are very small (- 20pm diam.) and their inertially confined lifetime is less than 0.1 nsec. Indeed the general characteristics of LPPs are small size and short lifetime as well as high density, temperature, and pressure. Density prior to expansion ranges from to 10 g cm-3, temperature from kT = 10 eV to
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
203
10 keV, and pressure from to 4 X Id Mbar. Dimensions range from 10 to 1000 pm and laser pulse duration from lo-’’ to lo-’ sec. General introductions to LPPs have been written by Hughes (1975), Bekefi (1976), and Motz (1979). The published proceedings of certain workshops and summer schools (e.g., Schwartz and Hora, 1970, 1972, 1974, 1977; Cairns and Sanderson, 1980) are also useful, as may be brief reviews (e.g., Mulser et al., 1973; Key, 1980a). The physics of LPPs is now a subject of international research on a large scale. We do not seek here to discuss the subject as a whole but rather to consider the spectroscopy of LPPs. The connection between spectroscopy and plasma physics is generally strong, since spectroscopy is a powerful plasma diagnostic technique and plasmas provide novel spectroscopic sources. LPPs are no exception in this respect and their unique characteristics have stimulated much novel work in diagnostics and spectroscopy. Authoritative introductions to general laboratory plasma spectroscopy are available in books by, for example, Griem (1964), Huddlestone and Leonard (1965), Bekefi (1966), Zeldovich and Raizer (1966), and LochteHoltgreven (1968) and in reviews, for example, by Cooper (1966). More specific discussion of laser-produced plasmas is found in short review articles on general laser-produced plasma diagnostics by, for example, Attwood (1978) and of spectroscopic diagnostics in particular by Vinogradov ef al. (1974), Godwin (1976), and Peacock (1978a, b, 1979a,b). In the present review we first compile a theoretical basis of the spectroscopy of laser-produced plasmas in Sections I1 to VII presenting a broad outline with references to detailed sources. We then briefly describe the main physical characteristics of LPPs in Section VIII. In Section IX we review experimental spectroscopic diagnostic studies of these physical characteristics, including discussion of experimental spectroscopic evidence for population inversion on VUV and XUV transitions which is part of the wider field of effort to develop X-ray lasers (see, e.g., a review by Waynant and Elton, 1976). The application of LPPs as sources for the classification of X W spectra of highly ionized ions is an area of active research outside the scope of this review but described in recent reviews by Boiko et af. (1978a) and Fawcett ( 1974).
II. Ionization This section briefly describes the standard models for ionization and discusses how they may be applied to plasmas with parameters typical of
M. H. Key and R. J. Hutcheon
204
LPPs. The effect of optical opacity on ionization is considered in Section VIII. A. THELOCALTHERMODYNAMIC EQUILIBRIUM (LTE) MODEL
In this approximation, which is valid at sufficiently high densities, the distributions of ionization stages are determined solely by balanced collisional processes. Thus the effect of radiative processes is ignored. These distributions are then described by the Saha equation which may be written U z ( T,) ( 2 ~ r n , k T , ) ~ / ~ = 2 U z - ' ( T,) h3 exp(
N,N~ N Z - I
-xz-I
- Axz-' kT,
where N , ( ~ m - is ~ )the electron number density, N Z the density of ions of charge Z, Uz(T) the partition function for an ion of charge Z, xz the ionization potential of an ion of charge Z, and AxZ the reduction of ionization potential. Calculation of partition functions is discussed, for example, by Richter (1968) and reduction of ionization potential in Section 11, E. The LTE model and the criteria for it to be valid have been discussed in detail by Griem (1963), Wilson (1962), and McWhirter (1965, 1967). McWhirter (1965, Eq. 10) gives a necessary (but not sufficient) condition for the LTE model to apply, namely, that the electron density should satisfy N,
> 1.6 X
10l2 T,'/2x(p,q)3
Here T, (K) is the electron temperature and x ( p , q) in electron volts is the largest energy gap in the term scheme of the ion considered. The concept of LTE can also be applied to plasmas where the dominant energy transfer processes are radiative (Richter, 1968) as is often the case in stellar atmospheres. These LTE plasmas, discussed in detail by Thomas (1965) and Page1 (1968), are not considered further here since they are rarely produced in the laboratory.
B. THECORONAL MODEL In the low-density limit the ionization equilibrium is described by the coronal model. This model has been discussed in detail by McWhirter (1965) whose formulation assumes that the ionization distribution is determined by a balance between collisional ionization of an ion from the ground level and radiative recombination to that level, the population
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
205
densities of the other bound levels being negligible. The ionization distribution is then given by N Z + I ( g ) / N Z (g ) = S Z ( g , c ) / a Z ( c 9g)
(3)
where N z ( g) and N z + ' ( g) are the population densities of ions with charge Z, 2 + 1 in their ground levels, and S z ( g , c) and a Z ( c ,g) are the (temperature-dependent) rate coefficients for collisional ionization from and radiative recombination to the ground level of the ion with charge Z. In contrast to the LTE model, the ionization distribution derived from Eq. (3) is independent of density but does depend on the atomic rate coefficients. A general numerical expression for Eq. (3) has been given by McWhirter (1965, Eq. 22). As first pointed out by Burgess (1964) and Burgess and Seaton (1964), Eq. (3) should be corrected for dielectronic recombination. This process can be included formally in Eq. (3) by replacing the term a Z ( c ,g) by a Z ( c ,g ) + az(tot), where af(tot) is the dielectronic rate coefficient summed over all relevant levels. Approximate values for &tot) can be obtained quickly for hydrogenic ions from the tables of Donaldson and Peacock (1976). Their calculational procedure can be extended to other ions if the energy levels of the captured electron can be assumed to be hydrogenic. The relative importance of a z ( c , g) and af(tot) can be assessed by comparing Eq. (18) of McWhirter (1965) [an approximate expression for a Z ( c ,g)] with the values of af(tot) given in Fig. 1 of Donaldson and Peacock (1976). This shows that in the temperature range of interest the dielectronic recombination rate coefficient exceeds a!'(c, g) by an order of magnitude for the recombination of He11 to HeI. As the nuclear charge Z increases dielectronic recombination becomes relatively less important, so that the two coefficients are roughly comparable for Z of 10- 15 and radiative recombination dominates for higher Z. Jordan (1969, 1970), Summers (1974), and Jacobs et al. (1977, 1979) have used the coronal model with corrections for dielectronic recombination to calculate the ionization balance of several ions in the range Z = 6-26 for lowdensity plasmas such as the solar corona. Mosher (1974) described corona model calculations with dielectronic recombination ignored, for plasmas of carbon, aluminum, and the heavy elements copper and tungsten; while House (1964) has similarly computed equilibrium for a wide range of elements from H to Fe. For hydrogenic ions the rate coefficients in Eq. (3) can be scaled as a function of nuclear charge Z (McWhirter and Hearn, 1963). For given normalized temperature T,/ Z * the radiative recombination rate varies as Z and the collisional excitation rate approximately as Z - 3 . Calculation of these rate coefficients shows that for a given normalized
'
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M. H. Key and R. J. Hutcheon
temperature the ratio N z + ' ( g ) / N Z (g) scales approximately as Z - 4 for high Z. Conversely, the normalized temperature required to obtain a given fraction N Z + ' / N Z increases with Z (e.g., McWhirter, 1965, Table 11). The Z-scaling for nonhydrogenic species is similar. Wilson (1962) has shown that the coronal model may be applied if the electron density satisfies the criterion N,
< 1.5 x
10~0(k7-,)4~ - 1'2
(4)
where kT, (ev) is the electron temperature and x (ev) the ionization potential of the ion of charge Z, referred to in Eq. (3). C. THECOLLISIONAL-RADIATIVE (CR) MODEL At intermediate densities where none of the preceding approximations can be used, there is no simple method of calculating the ionization balance. A proper calculation requires that all the collisional and radiative processes involving the bound levels of the ions be considered. This is done in the collisional-radiative (CR) model described by Bates et a/. (1962a, b). These authors consider a plasma containing hydrogenic ions, bare nuclei, and electrons, not necessarily in a steady state and show that for a wide range of plasma parameters the population densities of hydrogenic ions in the ground level N Z ( g ) and of bare nuclei N z + l are described by the equation
d N z ( g ) / d t = - d N Z + l / d t = aCRNZ+'N, S,,Nz(g)N,
(5)
The two coefficients a,, and S,, are the CR recombination and CR ionization coefficients. They may be regarded as net recombination and ionization rates. In a steady-state plasma this equation reduces to
N Z + ' / N Z= SCR/aCR
(6)
This equation is similar in form to the coronal model equation (3). A major difference however is that S,, and a,, are functions of electron density. They are also functions of electron temperature and atomic data but of no other parameters. Values of these coefficients are tabulated by Bates et a/. (1962a, b). There is no contribution to aCRfrom dielectronic recombination since this process, which involves a doubly excited ion, is not a possible mode for the recombination of bare nuclei to hydrogenic ions. Bates et a/. show that the dependence on nuclear charge can be largely removed by introducing the normalized electron temperature T,/ Z z and density N , / Z 7 . For given values of these quantities the normalized CR coefficients a C R / Z and Z3S,, are approximately independent of Z for
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
207
positively charged ions. Collisional-radiative treatment of ionization equilibria for low density (< 1OI6 cmP3)plasmas including nonhydrogenic ions and density-dependent dielectronic recombination rates is described by Summers (1974).
D. APPLICATION TO LASER-PRODUCED PLASMAS In the common case of the ablation-front plasma (see Section VIII) in an LPP generated by a neodymium laser, typical parameters are N , = Id2 e/cm3, kT, = 500 eV with the plasma containing ions with ionization potential -22ooo eV. With the two latter values the condition for the coronal model is N , < 2.1 x 1019 e/cm3 and for LTE N, 2 1.3 x Id5 e/cm3. Thus these LPPs lie between the coronal and LTE regions, and an accurate calculation of the ionization distribution should employ the CR model described in the preceding section (see, e.g., Kilkenny et al., 1980, Fig. 7). Full solutions for ions other than hydrogenic are not available in the literature. The CR model equations cannot reasonably be solved analytically and require a lengthy computer routine which calculates the population densities of all the bound levels of the relevant ions. The calculation of the excited-level populations is not separable from that of the ionization equilibrium in the intermediate density region, as is clear from the discussions of Bates et al. (1962a, b) and McWhirter (1965). Approximate results for the ionization distribution in LPPs have been obtained by several authors. These can be illustrated by writing the ratio of the population densities of consecutive ionization stages [cf. Eq. (6)] in the form, after Salzmann and Krumbein (1978)
In this equation S z - ' is the net ionization rate coefficient such that there are N Z - ' N , S Z - ' ionizations per unit volume and unit time. The symbols a'-', a:- '(tot), and pz- are, respectively, the net radiative recombination, dielectronic recombination, and three-body recombination rate coefficients similarly defined. Specifically the symbol a:- '(tot) has the same meaning as in Section I I , B and there are N 2 N Z p Z - ' three-body recombinations per unit time and volume. Solution of Eq. (7) is equivalent to the solution of the CR model. The coronal model omits the three-body recombination term and replaces S z - ' and az-' with the coefficients for ionization from and recombination to the ground state. The calculations of Peacock and Pease (1969) and Colombant and Tonon (1973), which are explicitly for LPPs, allow for the greater importance of collisional (three-
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M. H . Key and R. J . Hutcheon
body) recombination in these plasmas by including the N , P Z - l term. However, they neglect dielectronic recombination and replace S z - I , a I , and Pz- I by the corresponding ground-level coefficients. A recent calculation for aluminum LPPs (Salzmann and Krumbein, 1978) considers all the terms in Eq. (7) and includes an estimate for the effect of populated excited levels. These authors include throughout their treatment the correction for the reduction of the ionization potential which is discussed in Section II,E. It should be noted that although the equations of Peacock and Pease (1969) and Colombant and Tonon (1973) can be reasonably solved without recourse to a computer, at least for steady-state plasmas, this is not so for the more complex methods of Salzmann and Krumbein or for the similar work of Whitney and Davis (1974) and Landshoff and Perez (1976). The dominant mode of recombination depends on the density. In the ablation plume (see Section VIII) recombination is mainly by direct radiative recombination, or dielectronic recombination; the relative importance of these two processes has been discussed earlier. Near the ablation front three-body recombination tends to dominate. It should be noted that the rate for dielectronic recombination is reduced in LPPs through collisional destruction of the doubly excited intermediate level (Weisheit, 1975; Donaldson, 1975; Salzmann and Krumbein, 1978; Summers, 1974). It should be noted that an unavoidable limitation on the accuracy of even the most elaborate calculation is the availability of reliable rate coefficients for all the relevant processes. The available sources of information are given in Section 111.
'-
E. REDUCTIONOF IONIZATIONPOTENTIAL The models reviewed in the preceding sections for the ionization distribution introduce the ionization potentials x. In a plasma the ionization potential of an ion is reduced by the electrostatic interaction of the ion with nearby charged particles. At lower densities this interaction can be described by the Debye-Huckel approximation (Landau and Lifshitz, 1969, p. 231) which is valid if the average energy of the Coulomb interaction of the ions is small compared with its mean kinetic energy, that is,
(Ze)'/rZz <<$kT
(8)
where rzz is the average distance between the ions of charge Z and is defined approximately by
r2z = ( ~ / ~ T ) ( N) - Iz
(9)
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
The Debye length A, Coulomb field and
209
is a measure of the range of an ion’s screened
The Debye-Huckel model is valid (Griem, 1974, p. 138) for number densities such that
corresponding to an average of more than one sixth of a particle in a Debye sphere. The screening modifies the electronic orbitals of the ion. Rogers et al. (1970) describe a mathematical model of this in which they solve the Schrodinger equation for a hydrogenic ion where the central Coulomb potential V ( r )= - Z e 2 / r is replaced by a screened potential V ( r ) = ( - Z e 2 / r )exp( - r/A,,). These authors calculate the energy levels of all the individual angular momentum ( n l ) sublevels up to n = 9, I = 8, as a where A, is the appropriate Bohr radius a , / Z . A function of X,/A,, (crude) summary of these results is that the energy of each principal quantum shell is raised by Ze2/A,, the I degeneracy is removed, and levels lying above E = - Ze2/A, in the unperturbed atom are no longer allowed states. Thus the number of allowed states are finite and the ionization potential is reduced by Ax, where
Ax
= Ze2/A,
(12)
which result can also be derived by thermodynamic considerations (Ecker and Kroll, 1963). In a real plasma each ion suffers a different instantaneous screening, and values calculated for energy levels and ionization potentials are statistical averages. This discussion has avoided the usual term “depression of the ionization potential” since it may imply (wrongly) that energy levels are lowered.
F. HIGH-DENSITY EFFECTS The standard descriptions of a plasma discussed previously assume that collisions are binary, that interparticle Coulomb interaction energy is small relative to thermal energy, and that the electrons are nondegenerate. The range of plasma parameters which may be created in LPPs is such as to violate these simplifying assumptions in some parts of parameter space as shown in Fig. 1 [adapted from Peacock (1978a, b)]. The conditions A,
210
M. H. Key and R. J. Hutcheon
18
19
20
2
22
23
2A
25
26
27
28
log N, ( c m 3
Fro. 1. Parameter space for laser-produced plasmas (enclosed by the dashed line) showing the region of conventional plasmas (upper left) and where high-density effects modify emission (lower right). Non-Debye plasma (hatched area) is bounded by A, = rzz . Electron degeneracy (cross-hatched area) is bounded by E , = kT,,and pressure ionization occurs for densities such that rzz < a,/Z (vertical line for Z = 2).
-
rzz and k T - ( Z e ) 2 / r z z are shown as well as the condition for electron degeneracy and for pressure ionization (see the following subsection). Reviews of high-density spectroscopic effects have been written by Peacock (1978a, b) and Burgess (1979). I . Reduction of Ionization Potential Theoretical treatments of the reduction of ionization potential for conditions beyond the validity limit of the Debye-Huckel theory [Eq. (1 l)] have been given by Ecker and Kroll (1963), Dekeyser (1965), Stewart and Pyatt ( 1966), Rouse (1 967), and Burgess ( 1979). These high-density models predict that A x scales more like ( N z ) 1 / 3than the ( N z ) 1 / 2of the DebyeHuckel model. A physical reason for this is the dominance at high density of the nearest neighbor ions in determining the potential energy change, so that A x varies as ( Z e ) 2 / r z z . Lee and Hauer (1978) show calculations of A x for a neon X plasma at densities of interest in laser compression research. The reduction of ionization potential enters the Saha equation, Eq. (I), and Burgess (1979) notes that A x might be defined by the value that when inserted in Saha’s equation gives the correct result for the ionization equilibrium in LTE.
2. Degeneracy
-
Electrons being fermions each require a minimum volume of phase space h 3 / 2 . The inclusion of N , electrons in unit volume therefore
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
21 1
implies the occupation of phase space at least up to the Fermi energy EF, where
If E, 2 kT,, the electrons are influenced more by their Fermi energy than their thermal energy and are termed ‘‘degenerate.” This degeneracy criterion is shown in Fig. 1 and includes a substantial region of LPP parameter space. The pressure of Fermi degenerate electrons is determined by their Fermi energy and thus departs radically from the ideal gas thermal pressure. Equations of state in the degenerate regime are discussed by Zeldovich and Raizer (l966), Motz (1979), and Hansen (1980), the latter describing numerical Monte Carlo simulations. If the electron Fermi energy becomes comparable with the binding energy of the ground state of an ion, “pressure ionization” occurs. Another way to view this is in terms of the interparticle spacing being similar to the orbital radius of the bound electron, which for hydrogenic ions implies rzz = a,Z
-
’
(14)
where a, is the Bohr radius. An indication of the pressure ionization regime for hydrogen and hydrogenic ions is shown in Fig. 1. Calculations involving pressure ionization are described by Zink (1968) and Lee and Thorsos (1 978). Fermi degeneracy of ions does not occur in the parameter space of LPPs because of their high mass [see Eq. (13)]. 3. Rates of Collisions and of Radiative Decay The rates of collision processes discussed in Sections 11, B and C are assumed to be due to binary collisions. Burgess (1979) points out that an obvious breakdown of this binary assumption occurs if particle separations are comparable with orbital dimensions of bound electrons, but also notes that this is just the criterion for pressure ionization so that collision rates associated with the bound states are then no longer relevant. If, however, collision processes involve long-range Coulomb interactions, and Burgess cites the case of excitation of ionized emitters, then modification to the simple binary result may be anticipated when Coulomb-induced correlations between particles are on a scale length comparable to the impact parameter for the long-range excitation process. He cites recently observed discrepancies in collision rates as possible evidence of such effects (Burgess et al., 1978).
M.H. Key and R. J. Hutcheon
212
A degenerate plasma evidently has no simple binary collisions, since free electron wave functions always overlap and the density of final states available after a collision is modified. Thus while there is no good theoretical or experimental evidence available there is good reason to expect complications. Some minor effects on radiative rates may also occur in the high-density limit (see Bekefi, 1972; Davis and Jacobs, 1975; Rozsnyai, 1977; Peacock, 1978a, b; Burgess, 1979).
G. TRANSIENT IONIZATION Consider an ionizing or recombining plasma, which is not in a steady state. In the absence of mass motion the rate of change of the population density of an ion species N Z is dNz/dt = NZ-'NeSZ-'
+ N Z + ' N e R Z- N Z N e S Z- N Z N e R Z - '
(15)
where S and R are the net ionization and recombination rate coefficients, is the coefficient for recombination from 2 respectively. The symbol R to Z - 1 and S z is that for ionization from Z to Z + 1. The evolution of the ionization distribution is described by a set of coupled rate equations having the form of Eq. (15).
''
1. The Coronal Model
Solutions of these equations are available in particular cases. For example, in a plasma where the ionization equilibrium is described by the coronal model (Section 11, B) the population density of the highest ionization stage present, say N Z , is described by the equation dNz/dt
= N e N Z - ' S Z - ' ( g , c )-
N e N Z [ a Z - ' ( c g, )
+ ~y,Z-'(fot)]
(16)
where a z - ' ( c , g) and a,Z-I(tot) are the rate coefficients for radiative recombination to the ground level and the total rate coefficient for dielectronic recombination. McWhirter (1965) has solved a similar equation N Z is constant, and for a plasma close to its steady state, where N z - ' his treatment shows that the plasma approaches the steady state with a relaxation time T given by
+
The af-'(tot) term does not apply if the final ionization stage is a bare nucleus, as noted previously. Substituting numerical values into Eq. (17)
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
213
shows (McWhirter, 1965) that the atomic relaxation time for any coronal model plasma is given to within roughly an order of magnitude by Ne7
=
10I2cm-3 sec
( 18)
2. The Collisional- Radiative Model
For plasmas too dense for the coronal model to apply, the terms
+
S z - ' ( g , c ) and a Z - ' ( c ,g) af-'(tot) in Eq. (17) should be replaced by the net rates S z - and R Z - or suitable approximations for them. For a plasma containing bare nuclei and hydrogenic ions, Eq. (1 5 ) can be solved for a wide range of densities by the CR model of Bates et al. (1962a, b) (Section 11, D).
'
'
An algorithm for obtaining a more general solution of the set of equations (15) has been described in detail by Magill (1977, 1978). A similar approach was described previously by Peacock and Pease (1969) who solved (with a computer) the time-dependent ionization balance equations for an expanding spherical iron plasma. Colombant and Tonon ( 1973) also solved the time-dependent CR equations for ionization of Fe, expressing their results in terms of the degree of ionization reached as a function of the density x time product Net. Broadly speaking it is found that the degree of ionization in the quasiisothermal parts of LPP ablation plasma plumes from high-Z targets is less than the steady state equilibrium value corresponding to the temperature, since the (density X time) for an element of plasma passing through this zone is less than that required to reach equilibrium. Conversely, in the adiabatically cooled low density region of the expanding plasma plume the ionization is "frozen in" at levels much greater than the equilibrium value corresponding to the (low) temperature resulting from the rapid expansion. Thus consideration of the transient aspects of ionization is essential in describing the ablation plasma. Dense plasmas produced by laser driven implosions generally reach levels of ionization rather closer to equilibrium.
111. Population Densities of Bound Levels Excited-level populations can be computed in a similar fashion to ionization equilibria using appropriate rate coefficients and thermal equilibrium considerations. Reviews by Griem (1963), Wilson (1962), and McWhirter (1965) are useful here.
M.H. Key and R. J. Hutcheon
214
A. LTE AND
THE
LTE LIMIT
At densities satisfying Eq. ( 2 ) the population densities of all the bound levels are described by the LTE model (Saha-Boltzmann equation). The population density N z ( p ) of a level p of an ion of charge Z is then given by
where u Z ( p ) , w z + ' (g) are statistical weights and xz(p, c ) is the boundlevelp to continuum c ionization energy. At lower densities the Saha-Boltzmann equation can still be used in a restricted sense. This is because as the principal quantum number of a bound level increases, the probability of radiative decay decreases and that of collisional transitions increases. Thus there is some level above which population transfers are dominated by (balanced) collisional processes, and the bound-level population densities, being collisionally tied to the free electrons, are described by Eq. (19). The level above which this applies can be found by regarding Eq. (2) as an equality and solving for xz(p,q), which then indicates the highest energy gap above this level. The principal quantum number of this LTE limit can be estimated from the hydrogenic relation xz(p,q) = ( Z + ~ ) ~l / xn 2 ~- [I/(n
+ 112]
(20)
where xH is the ionization potential of hydrogen.
B. THECORONAL APPROXIMATION In the coronal limit excited-level populations may be treated as due to the balance between collisional excitation and resonance radiative decay. Thus,
The collisional excitation rate coefficient X ( g, p) for dipole-allowed transitions may be approximated in terms of the absorption oscillator strength f( p ) as g 9
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
215
McWhirter (1965, p. 212), where x( g, p) is in electron volts, g is a mean Gaunt factor whose value is 1, and Te is in degrees Kelvin. This model is applicable where the radiative decay rate of a state dominates over collisional decay, that is, 2-
Cq A (P,4)>>Ne C4 X(P,4)
(23)
McWhirter (1965) gives a criterion for validity of Eq. (23) for levels with principal quantum number n < 6, namely
N,
< 5.6 X 108(Z+ 1)6T:’2
exp
(
1 162(ge+ 1)*
)
(24)
LPPs are generally too dense for validity of Eq. (24), but a coronal description may be useful for n < 3. Direct application of Eq. (23) may be used as a test. C. COLLISIONAL-RADIATIVE-LEVEL POPULATIONS At intermediate densities it is again necessary to allow for many collisional and radiative processes. McWhirter and Hearn (1963) use the CR model to calculate the population densities of bound levels of hydrogen atoms or hydrogenic ions in a plasma containing such ions, electrons, and bare nuclei. These authors show that in a transient plasma the population densities of the excited bound levels of hydrogenic ions come into equilibrium effectively instantaneously with the relatively slowly varying groundlevel population. The excited-level population densities may then be expressed in terms of the ground-level and bare-nuclei population densities and two coefficients which are functions of electron temperature, density, and atomic coefficients. The population density N ( p ) of a principal quantum levelp (over all sublevels) is given by
where ro(p) and r , ( p ) are the two coefficients, which are tabulated by McWhirter and Hearn. The term N , ( p ) is the Saha-Boltzmann population of level p . Departure from LTE is then readily assessed for hydrogenic ion levels. The solutions can be scaled with Z for any hydrogenic ion as described earlier and give, for example, a simple method for determining the possibility of population inversions in supercooled LPP expansion plumes (see Section IX).
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M. H. Key and R. J. Hutcheon
The CR model can in principle be adapted to allow for further atomic processes and nonhydrogenic ions. However, as pointed out by McWhirter ( 1965), these other ions may contain metastable levels whose relaxation time (for the population densities to equilibrate to suddenly changed ambient plasma parameters) are comparable to that of the ground level. In this case population densities of other bound levels, as predicted by the CR model, must be expressed in terms of both the ground-level and metastable-level population densities, as well as the other parameters listed earlier. Even for hydrogenic ions a limitation of the CR model as formulated by McWhirter and Hearn (1963) is that it does not consider the effect of doubly excited levels in the helium-like ion (Burgess and Summers, 1969). There have been relatively few CR calculations for plasmas with parameters characteristic of LPPs. However, the work of Davis and Whitney (1976), Landshoff and Perez (1976), and Salzmann and Krumbein (1978) presents rather detailed numerical calculations for aluminum LPPs. If it is desired to calculate population densities in the intermediate density region without resource to lengthy (computer) calculations, approximate values can be obtained by considering only the most important lower bound levels and only the dominant radiative and collisional processes. Gabriel and Jordan (1972) have reviewed calculational models with broadly these limitations for helium-, lithium-, and beryllium-like ions. However, much of their discussion concerns low-density astrophysical plasmas where the coronal model may be applied. Laser-produced plasmas have densities many orders of magnitude higher, and hence collisional processes that are negligible at the densities considered by Gabriel and Jordan may be dominant in LPPs. For helium-like ions these authors do extend their calculations to LPP densities. Researchers at the Lebedev Institute, Moscow, have published a series of papers describing similar calculations for hydrogen- and helium-like ions for plasma parameters typical of LPPs. These treat transitions from doubly excited levels, which are important in LPPs, and are discussed in Section IV. DATA D. RATECOEFFICIENT Calculations of bound-level population densities require atomic data for the radiative and collisional processes. For hydrogen-like ions the transition probabilities can be calculated exactly. Values for many transitions, including most of those relevant for LPP calculations have been given by Wiese et al. (1966). For other ions radiative transition probabilities can be obtained from the Coulomb approximation of Bates and Damgaard (1949), or from computer programs for atomic structure calculations (Bromage, 1978).
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
217
Cross sections and rate coefficients for collisional processes cannot be calculated exactly. The (quantum-mechanical) theoretical methods for calculating them have been reviewed by Seaton ( 1975) for hydrogen-like (and other) ions. Gabriel and Jordan (1972) have referenced much of the atomic data published up to February 1971 for helium-, lithium-, and beryllium-like ions. For optically allowed transitions the cross section for collisional excitation can be obtained from the oscillator strength and Eq. (22) (see Moseiwitsch and Smith, 1968). Papers discussing ionization equilibrium, excited state populations, and radiated intensity (Section IV) also contain many references to sources of rate coefficients. See, for example, Summers (1974), Whitney and Davis (1974), Weisheit (1979, Vainshtein (1979, Davis and Whitney (1976), Landshoff and Perez (1976), Boiko et af. (1977a), Jacobs et af. (1977, 1979), Shevelko et al. (1977), Burgess et af. (1977), Davis et af. (1978), Salzmann and Krumbein ( 1978), and Vainshtein and Safronova (1978).
IV. Intensity of Line Radiation A. INTRODUCTION The number of photons emitted per unit time and volume in a transition from level p to q is
The absorption coefficient for the line radiation may be written
wheref(q, p) is the absorption oscillator strength, and P ( v ) is the normalized line-shape function (see Section v). The line-center value P(v,) is approximately l / A v , where Av is the linewidth. If the optical opacity JK(v,,)dfis small (see Section VII), absorption can be neglected; and this section discusses radiated intensity under such optically thin conditions. The excited-level population N ( p) in Eq. (26) determines the intensity. Approximate solutions for N ( p ) are given in Section I11 in the coronal limit [Eq. (21)], in the LTE limit [Eq. (19)], and in the general CR case [Eq. (25)]. The calculation of intensity in the optically thin coronal or LTE limits is straightforward but attention must be paid to the validity of the models and to the possibility of optical opacity.
218
M . H . Key and R. J. Hutcheon
Calculations of resonance line emission using CR models are complex, and some cases specific to LPPs have been described in the literature, for example, by Whitney and Davis (1974) for Al XI1 and Al XI11 emission, by Davis and Whitney (1976) for Al XI emission, and by Landshoff and Perez (1976) and Salzmann and Krumbein (1978) for AlXII and AlXIII also. Other authors have used intensity calculations in interpreting diagnostic measurements. These are discussed in Section IX. The most detailed treatments of optically thin line intensity have been made in the context of modeling density and temperature-dependent line ratios which potentially give simple methods of diagnosis of T, and N,. In general terms, if two excited levels have very different radiative decay probabilities and/or collisional excitation rates, their population ratio in the low-density coronal limit [Eq. (21)] will be significantly different from that in the high-density LTE limit. In the intermediate range their population ratio will be density dependent, and with appropriate choice of transitions, a wide range of density can be diagnosed. The intensities of Hand He-like ion resonance lines, their fine structure, and forbidden or dielectronic satellites have been extensively analyzed; and this work is reviewed in the following section. Much of it has come in a series of papers from the Lebedev Institute, Moscow, and is described in a detailed review by Boiko et al. (1979b).
B. HYDROGEN-LIKE IONS Vinogradov et al. (1977a) and Boiko et al. (1979b) have shown that the intensity ratio of the two components of the Lyman (Y (lSl,2-2Pl,2,3,2) doublet, hereafter called the “doublet ratio,” may be used as a density diagnostic in LPPs. The two components are separated by about 5 mA for all hydrogenic ions of interest (Garcia and Mack, 1965). In an optically thin plasma between the coronal and LTE regions the doublet ratio departs from the statistical value of 1/2. This is because (Hutcheon and McWhirter, 1973) the ratio of the collisional population processes 2S,,,2P,!, and 2Sl,,-2P3,, differs from 1/2, and in the intermediate density region these processes are significant and unbalanced. Vinogradov et al. (1977a) have calculated the doublet ratio in an optically thin plasma by solving the steady state balance equations. Beigman et al. (1976) have pointed out that the strong electric microfields in a dense plasma causes mixing between the 2P,,, and metastable 2S,,, levels (see Section VII) and hence greatly increases the probability of radiative decay of the 2S,,, level. The enhanced transition probability can be obtained from Eq. (10) of Weisheit et al. (1976). This affects the doublet ratio because the induced 2S,,,- IS,,, decay is spectroscopically indistinguishable from the 2P,,,-
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
219
lSi,2 line. Beigman et a/. also note that the Lyman (Y line is often optically thick for highly ionized ions in LPPs and that this too affects the doublet ratio. The value of the doublet ratio determines the mean wavelength of the unresolved doublet. Thus if the unresolved doublet is used as a wavelength standard, it is necessary to allow for the doublet ratio departing from the statistical value [as was recognized by Hutcheon et al. (1980)l.
C. HELIUM-LIKE ION RESONANCE AND INTERCOMBINATION LINES The intensity ratios of the resonance ( Is2 'So-ls2p 'PI), intercombination ( Is2 1So-ls2p 'Pi), and forbidden ( Is2 'S0-ls2s 3Sl)lines in helium-like ions (see, e.g., Fig. 2) depend on electron density over a wide range of electron density. Gabriel and Jordan (1972) have discussed the theory of these line ratios in detail. Although these authors consider all significant collisional and radiative processes between the ground (1 s2 IS,,) and the various n = 2 levels, they do not consider the higher (n 2 3) levels, in particular, neither cascade from these levels nor radiative or collisional (three-body) recombination from the hydrogenic ions. Vinogradov et a/. ( 1975) describe similar calculations, developed from earlier calculations of Aglitzkii et a/. (l974), of the resonance/intercombination line ratio but performed for densities typical of LPPs. These authors include interactions with levels with n = 3 and 4 and also recombination by three-body and radiative recombination. Vinogradov et a/. show that the resonance/intercombination line ratio for ions N a X to KXVIII are density sensitive in ranges in the region 10'8-ld3 e/cm3, depending on the ion, as shown in Fig. 2. (They do not consider the forbidden line which is suppressed at Sixm Sxv
Mgxi
',1 I,
Arxn XVm
16 8
4
2 1 10''
10'~
1020
lo2'
lo2'
an
Ne
FIQ.2. The ratio I R / I s as a function of electron density N, for He-like resonance and intercombination line intensities showing calculated behavior for ions from Na X to K XVIII and experimental measurements from the ablation plasma emission of LPPs. (After Boiko et al., 1979b.)
220
M. H . Key and R. J. Hutcheon
these densities by collisional depopulation of the upper ls2s 'S1 level.) Boiko et al. (1979b) point out that it is necessary to correct the observed intensity of the intercombination line for blending with satellite lines. Some broadly similar calculations have been reported by Weisheit et al. (1976). These authors consider the effect of opacity on the resonance line intensity (see Section VII) and show that radical departures from optically thin line ratios occur for typical LPP parameters. They criticize earlier Russian calculations (Aglitzky et al., 1974) for an oversimplified allowance for opacity and for neglecting the fact that the plasma microfields enhance the decay rate of the forbidden ls2s 'S,-ls2 IS,, through mixing of the upper level with that of the ls2p 'Pl-ls2 IS, allowed line. (This is analogous with the induced decay of the hydrogenic 2s 1Sl,2-ls 1S1,2 transitions discussed previously.) This effect is important because this line blends with the intercombination line. Weisheit et al. gwe estimates for the relative intensities of the two components of the blend. Boiko et al. (1977b) have described a similar theoretical analysis for the ratio of the intercombination (1s' IS0-ls3p 3P,) and resonance (1s' IS,ls3p 'PI) lines from the n = 3 levels.
D. SATELLITES TO RESONANCE TRANSITIONS IN ONE-AND TWO-ELECTRON IONS In the spectra of LPPs the resonance lines of highly ionized hydrogenic and helium-like lines are accompanied by satellite lines from doubly excited levels of ions from the next-lower ionization stage. These satellites are interesting since their intensities may be interpreted to determine the electron temperature and density of the LPP. Several recent papers have discussed this, notably again a series of papers from the Lebedev Institute, Moscow. The theoretical models for the line intensities are discussed here by using as a starting point the calculations of Gabriel (1972) for the satellites to helium-like ion transitions. Gabriel points out that the doubly excited levels whose decay results in the satellites are populated by (a) dielectronic recombination and (b) inner-shell excitation of the lithium-like ion. The rate of photon emission of a satellite line formed by dielectronic recombination is
In this equation n(v,) is in photons per unit volume and unit time: NH' is the population density of the helium-like ions, a, the Bohr radius in neutral
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
22 1
hydrogen, d s ) , w(g) statistical weights of the doubly excited level and'the helium-like ground level, and A r , A a the transition probabilities for decay of the doubly excited level by radiation and autoionization. The summation C A , is over all possible radiative transitions from the satellite level. xs is the energy of the satellite level above the helium-llke ground state. The derivation of Eq. (28) assumes that the density is low enough that radiative decay and autoionization are the only significant depopulation processes for the satellite level. In an LPP, however, collisional depopulation may also be significant, as discussed later. The intensity I , of the helium-like resonance line is, on the coronal model,
I , = NH"N,( 1
+ a)X
(29)
where X is the rate coefficient for collisional excitation, and a a correction for satellites not spectroscopically resolvable from the resonance line. The intensity ratio I s / I R may therefore be written from Eqs. (28) and (29),
Is/IR= (1
+ a)-'F,(T,) + F 2 ( S )
(30)
where F,(T,) is a function of temperature and F,(S) a function of the satellite line. Thus in the coronal model approximation of Gabriel the ratio Is/ZR is independent of the population densities of the helium-like ions and electrons but does depend on the temperature and so may be used to determine it. The parallel argument for the satellites due to collisional excitation of the next lower ion gives
I;/IR=(l + a ) - ' ( N L ' / N H e ) P F ; ( S ) where N L ' is the population density of the lithium-like ions, and /3 and F 2 ( S )are functions of the satellite lines. Thus the ratio I ; / I , gives the ionization temperature T, through the ratio NL1/NHe. If the plasma is not in a steady-state ionization equilibrium, T, # T,. The ratios I s / I R and I ; / Z , increase rapidly with Z, as can be seen from the tables of Gabriel (1972) or experimental spectra (e.g., Boiko et af., 1978a). Gabriel's original calculations have been refined by Bhalla et al. (1975). The following sections describe particular applications of this general approach to hydrogen- and helium-like ions. 1. Hydrogen-Like Ions
Bolko et al. (1977~)and Vainshtein and Safronova (1978) have calculated accurate theoretical wavelengths for all Lyman a satellites of the type
222
M.H. KeyandR. J. Hutcheon
FIG. 3. The calculated relative intensities of the Ne IX 2121' dielectronic sate1lite)ransitions as functions of electron density. Lorentzian line profiles, with full widths 10 mA (left) and 40 mA (right), have.been added to simulate instrumental and Stark broadening. The neon L, ljne is at 12.132 A, and the density-sensitive dielectronic satellite line is indicated ( 8 ) at 12.32 A. The electron temperature is 400 eV. (After Seely, 1979.)
1~21-2121'and 1~31-2131'for the ions MgXI to SXVI inclusive and have used a model similar to that of Gabriel (1972) to calculate the temperaturedependent resonance to satellite line intensity ratio. Boiko et al. (1979b) note that discrepancies between experimental and theoretical intensities occurred for two groups of satellite transitions: (A) ls2p 3P0,,,,-2p2 3P,z and (B) ls2s 'SI-2s2p 3P0,,,2.They point out that theoretical calculations of Vinogradov et al. (1977b) resolve this discrepancy by allowing for collisional transitions between doubly excited levels. Vinogradov et al. (1977b) show that the intensity ratio of the two groups of satellite lines (A) and (B) depends on the electron density and derive an expression for its value. Advantages of this diagnostic technique include the fact that all the lines involved are satellite lines and so have small optical depths and that inner-shell satellite transitions by their nature do not occur for hydrogenlike ions. Recently, Seely (1979) has made accurate calculations of these density-sensitive satellite line ratios for NeX Lyman a (Fig. 3) obtaining results somewhat different from those of Boiko et al., 1979b.
2. Helium-Like Ions The intensities of the satellites of the ls2 'So-ls2p 'PI resonance line have been studied theoretically by the coronal-model treatment of Gabriel (1972) and Bhalla et af. (1975) as discussed earlier. Boiko et al. (1978b) have published similar calculations for the satellites to the 1s' 'S0-ls3p 'PI line for ions in the range MgXI to KXVIII but note some disagreement
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
223
between theory and experimental results. Boiko et al. (1978~)have also published similar calculations for satellites, due to doubly excited beryllium-like ions to the helium-like ls2 'S,-ls2p 'PI resonance line with fairly good agreement between theory and experiment. Weisheit et al. (1976) and Boiko et al. (1979b) both review the line-ratio based spectroscopic diagnostic methods discussed earlier. In our own experience of analyzing LPP spectra we find that in the majority of cases it is not possible to neglect modifications to resonance line intensity due to opacity, except far from the target surface in the expansion plume or with deliberately low fractional content of the atom of interest in the plasma. Where opacity is important calculation of the corrections is generally too difficult to give useful diagnostic results. This consideration is therefore a serious limitation in the usefulness of resonanceto-satellite intensity ratio data. 3. Other Line Ratios
A simpler form of line intensity diagnostic is the ratio of intensities of lines from successive ionization stages. Optically thin H- and He-like X-ray resonance transitions from high quantum number levels in LTE with the next-higher ion stage can give a simple measurement of the population ratio of the hydrogenic and bare nucleus ion stages in LPPs (Valeo et al., 1976; Key et al., 1979a; Kilkenny et al., 1980). With an appropriate treatment of the ionization equilibrium such data give a temperature estimate. A similar approach using VUV 2s22p"-2s2p"+I transitions in ions of charge 20-30 has been described by Boiko et al. (1970), for measurements of kT, in the range 100-1OOO eV. Analogous methods can also be used for the low-temperature, low-density region of the expansion plume based on visible/UV transitions between high quantum number levels of highly ionized ions (e.g., Boland et al., 1968).
E. CHARACTERISTIC X-RAYK LINES Penetration of energetic electrons and photons into solid material ahead of the thermal ablation front in LPPs may result in K-shell ionization and thus in characteristic X-ray line emission. Photoionization of K-shell electrons has a cross section uK(hv)which for the simple case of a hydrogenic ion rises abruptly at the threshold energy hv = Z 2 E , = E, with the functional form for hv > E,, uK(hv)= (8 x lo-'') Z - 2 ( h v / E , ) - 3
cm2
(32)
M. H. Key and R. J . Hutcheon
224
For neutral atoms the behavior is similar and is given in several tabulations such as the Handbook of Spectroscopy (1974, p. 28). The cross section for ionization of electrons in outer shells (L, M, . . . ) is always less than for the K shell for hv > E,, so that creation of K vacancies is the dominant photon absorption process. Electrons of energy E > EK also cause K-shell ionization with a cross section uK( Ee) = 7.9
X
10-’O(0.85
+ 0.0047 Z ) ( E K . Ee)-’ In(E,/EK)
(33)
where EK and E, are in keV (Brown and Gilfrich, 1971). Here the cross section does not rise to an abrupt peak at E, = E K but increases to a maximum for E, 2 E, . The relative magnitude umaX(hv)/umax( E,) can be estimated from Eqs. (32) and (33) using EK Z’E, and is approximately 6 X lO-’Z’. In contrast to photons, electrons do not lose energy predominantly by K-shell ionization. Coulomb collisions with weakly bound electrons are more important as described by the Bethe Bloch analysis (see, e.g., Dyson, 1973; Berger and Seltzer, 1964; Handbook of Spectroscopy, 1974, p. 248). The energy loss rate aE/ax is of the form
-
-
aE/ax = const(NZ/E,) ln(2Ee/x)
(34)
for density N of atoms of average ionization/excitation energy x. Detailed numerical analysis of the scattering and absorption of electrons in solids has been made by Spencer (1959). Their penetration range scales roughly as E:. Electrons thus deposit only a small fraction of their energy in K-shell ionization. The ratio R ( E ) of this energy to the total deposited energy tends to a constant value lo-’ for energies above 15 keV. Decay of K-shell vacancies occurs either by radiative fluorescence (emission of K lines) or by nonradiative Auger decay (primarily K + LL). The ratio of the radiative decay rate A to the Auger decay rate r gives the fluorescent efficiency c - A / ( A + ,)?I and since A scales as Z4 and r is large and approximately Z-independent the fluorescent efficiency becomes low for low-Z elements. Data on fluorescent efficiencies are available, for example, in the Handbook of Spectroscopy (1974, p. 219). K, fluorescence radiation is reabsorbed through photoionization of L-shell electrons, and there is thus an escape depth for the radiation which is the reciprocal of the absorption coefficient. The latter is typically several times smaller than the absorption coefficient for hv > E K and is tabulated, for example, in the Handbook of Spectroscopy (1974). The escape depth varies from 12 to 26 pm for elements from Na to Fe, for example.
-
-
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
225
The spectroscopic structure of the K fluorescence radiation includes the basic K,, K,, . . . transitions arising from the filling of the K vacancy by decay of electrons from the L, M, . . . shells. In the case of the K, transition there is a spin relativity doublet structure K,,K,, arising from the transitions 1 2Sl,2-22Pl,2,3,2. This doublet splitting which scales as Z4 is tabulated in the Handbook of Spectroscopy. The doublet separation just exceeds the natural linewidth due to the rapid Auger and radiative decay rates of the K vacancies (see, e.g., Dyson, 1973, Fig. 3.28). The K, transition is at a significantly lower energy than the corresponding resonance transition of a helium-like ion due to the screening effect of the atomic electrons. This screening is reduced by multiple ionization. Thus the K, lines have weak satellite features arising from the finite probability of multiple ionization (K and L vacancy production) in a single ionizing collision (Parratt, 1936). More pronounced satellite structure in the case of LPPs is created through the high levels of thermal energy deposition by fast electrons which may lead to ionization of the atom. The shift in K, transition energy as a function of the degree of ionization has been calculated by House (1968). The main feature of the results is that measurable shifts occur when ionization reaches the L shell. A series of K-type lines is thus obtained in the spectral region between normal K, and the ls2 'S0-ls2p IP, resonance line of the helium-like ion (see, e.g., Boiko et al., 1979a, Fig. 7). The ratio of K, yield to thick target X-ray bremsstrahlung for electrons incident on solid targets has been studied (Dyson, 1973) and found to reach values of the order of unity for electron energies well above the K-ionization threshold.
V. Line Broadening There has recently been a significant stimulus to theoretical study of line broadening in plasmas arising from new experimental data on line shapes, particularly in resonance series of highly ionized hydrogenic and heliumlike ions in dense LPPs. Prior to this there was already an extensive body of established theoretical work which had been applied mainly to the study of line shapes of neutral or singly ionized emitters in low-density plasmas. Space does not permit a full review of this work here, and the reader is referred to other books and reviews (e.g., Baranger, 1962; Griem, 1964, 1974; Wiese, 1965; Cooper, 1966; Traving, 1968; Burgess, 1972). A brief summary of the essential elements of the theory is given in the following section.
226
M. H. Key and R. J. Hutcheon
A. NATURALBROADENING A transition between levels 1 and 2 with decay rates for each level y I and y2 has a natural Lorentzian line shape
1 P(Aw) = r
Y/2 Ao2 + ( ~ / 2 ) ~
(35)
where Aw is the angular frequency difference from line center and y = y , + y 2 . Natural line broadening is negligible relative to other line broadening for transitions of interest here with the exception of K, lines, where y is large due to Auger decay (see Section IV, E).
B. DOPPLER BROADENING Doppler shift of emitted frequency due to the thermal motion of the emitting ions with temperature Ti gives a Gaussian line shape
with Awo = ( ~ / c ) ( 2 k T , / m ~ ) ' / ~
(37)
If the mean free path for velocity changing collisions is less than the emitted wave length, normal Doppler broadening is suppressed. Dicke (1953) and Sobelman (1957) have discussed this Doppler narrowing. Burgess (1979) estimates that this occurs for ion density N Z , where N Z = (kTi/Z2e2)1/2(?rhlnA)-'
A
(38)
and In A is the Coulomb logarithm. Thus for h = 10 and kTi = 300 eV, Doppler narrowing occurs for N Z > Idocm-'. Clearly therefore it ought to be considered in evaluating particularly the core line shape in resonance transitions with central components unshifted by the Stark effect in LPP spectra. The separability of Doppler and Stark broadening permits them to be calculated separately and combined by convolution. However, Burgess (1979) notes that since the spectrum of electric field fluctuations seen by an ion, and thus its Stark broadening, has some dependence on the ion velocity this separability can also be queried.
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
221
C. STARKBROADENING Perturbation of bound states by an electric field (the Stark effect) causes splitting and shifting of energy levels and changes in oscillator strength (particularly increased oscillator strength for transitions forbidden in the absence of perturbation). In a plasma, emitting ions experience an electric microfield due to the plasma electrons and ions. The field varies statistically for different individual emitters and fluctuates in time. The net result is an ensemble averaged line shape with an overall width related to the average strength of the perturbation, and a detailed shape whose calculation is the object of line-broadening theory. I. Elementay Theoy A useful introduction to the orders of magnitude is to evaluate the static electric field due to an ion situated at a distance rzz from the emitter
Eo = Z,e/r$,
(39)
where Z , is the charge on the perturber ion. A measure of rzz is given by
$miz= ( N Z )- I and a more useful expression obtained from (39) and (40) with the same parameter dependence but a different numerical constant is the average field for statistically independent ions given by Griem (1974, p. 22) in the form 213
Eh = 8 Z , e ( N Z )
(4')
The Stark effect is a strong first-order perturbation in the special case of hydrogenic energy levels. It is a second-order effect, quadratic in Eo for isolated nonhydrogenic levels which are therefore less sensitive to Stark broadening. The terms split symmetrically in hydrogenic ions. The shift of the Kth component from the center is
Au,=
3h . KEo 2meeZe
where Z , is the charge on the emitter nucleus and K is an integer. The intensity pattern of the Stark components follows from quantum-
228
M. ff. Key and R. J. Hutcheon
mechanical perturbation theory (see, e.g., Underhill and Waddell, 1959). A weighted average
giving a measure of the value appropriate to estimating the half-width of the line. k is discussed by, for example, Underhill and Waddell (1959) and Edmonds et al. (1967). Griem (1974, p. 8) estimates the full width of hydrogenic lines between quantum levels p and q using
which with (41) and (42) gives
Equation (45) is suitable for rough estimates of the linewidth of hydrogenic lines with no unshifted center component, that is, L,, L,, . . . and H,, H,, . . . . Full calculation of the line shape is a more complex matter and must include both the effect of high frequency electric fields due to electron impacts and low frequency fields due to ions. 2. Impact and Quasi-Static Broadening These two aspects of the broadening may be treated as separate “impact” and “quasi-static’’ effects combined by simple convolution. The impact approximation for electrons can be developed classically or quantum mechanically (see, e.g., Traving, 1968; Griem, 1974). It can be shown by Fourier analysis that for a classical collision of duration 7 the impact approximation holds, where rXpp/vp
(46)
and pp is the perturber impact parameter, up its velocity, and Ao the shift from line center. Thus the maximum shift from line center for which the impact effect is operative is obtained for the minimum value of pp. Because the most frequent impact perturbations are by relatively distant Coulomb encounters with pp -A,, where A, is the Debye length, it follows that since up is typically the thermal velocity, electron impact broadening is effective for Ao
< ope
(47)
where opeis the electron plasma frequency. By the same argument ion impact broadening is effective only for Aa Q api<
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
229
Conversely the quasi-static approximation holds for slowly changing electric fields where the time scale for fluctuation 7
> Aa-’
(48)
Thus quasi-static broadening may dominate most of the line shape except for the region of the line core for lines that are strongly broadened, for example, hydrogenic transitions especially those from levels of large principal quantum number [see Eq. (45)]. The probability distribution for the field P(E) for static uncorrelated ions is the well-known Holtzmark function W( p), where p = E / E o and the function W ( p ) is of a unique form scalable to any density and ion charge Z , through the scaling of E, [see Eqs. (39) and (40)]. A similar unique function S(a) gives the line shape, where a is a dimensionless scalable Aa.These functions have been tabulated for hydrogenic lines (e.g., Underhill and Waddell, 1959). 3. Coulomb Eflects The validity range of the scaling is limited by Coulomb interaction correlations between the charged particles. Two such effects occur. Debye shielding of the perturbing ion field by electrons reduces the field at an emitter and is significant if rzz/h, > 0.1. This ratio a is given by
and it follows that for a dense, cool laser compressed plasma with, say, kTe = 100 eV, Z , = Z , = 10, and N, = Id3that a = 0.13 (see Table 11). Thus Debye shielding effects will be important but are not included in the Holtzmark field distribution for which a = 0. Correlations also occur through the emitter charge Z , having a Coulomb interaction with a perturber charge 2, which is > kTe. This criterion can be expressed in the form NZ
>[
7 x 106(kT,/1 eV) ZeZp
I’
and for the above Z , Z , and kTe gives N Z > 3.3 X Idoagain well within the range for typical LPPs. Detailed computation of the Debye screened quasi-static field distribution P ( E ) have been made by Hooper (1968a,b) and Mihalas (1978, p.
M.H. Key and R. J. Hutcheon
230 0.6
0.5 a z O.O(HOLTZMARK
0.4 0.3
0.2 0.1
0
0
1.0
2.0
30
.O
P Fm. 4. The electric microfield distribution function W(8 ) at a neutral point for several values of a. (After Hooper, 1968a.)
292), and Fig. 4 from Hooper’s work illustrates the narrowing of W( p ) for increasing a. The problem of a charged emitter with Coulomb interactions has also been included in more recent work (OBrien and Hooper, 1972), and P ( E ) has been evaluated for a limited range of conditions for T,# Ti and a multicomponent plasma (Tighe and Hooper, 1977). Given these more sophisticated quasi-static fields, a corresponding quasi-static line profile follows by simple summation over the Stark components of a line as in the Holtzmark model (see, e.g., Underhill and Waddell, 1959). However, such profiles are always inaccurate near the line center because they neglect the electron impact broadening. Vinogradov et al. (1974) have discussed the range of plasma parameters for which quasi-static broadening is dominant and relatively simple density diagnosis is possible. 4. The Standard Theory
Several authors have described full computations of hydrogenic ion line shapes for plasma parameters and ions typical of LPPs using what may be termed the “standard” method. This treats electron impact broadening by the quantum-mechanical methods developed by Baranger and Griem (see Griem, 1974) and uses quasi-static microfields computed following Hooper’s method. Examples of such calculations are those by l c h a r d s (1974), Kepple and Griem (1976), and Tighe and Hooper (1978), the latter including multicomponent plasma and T, # T i .
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
23 1
5. Recent Developments-The Ion Dynamic Model
Even these rather elaborate calculations have been shown to be inaccurate in predicting broadening in the line core. Grutzmacher and Wende (1977) measured optically thin La linewidths in a low-density plasma that were -2 times wider than predictions by the “standard” methods described previously. It was subsequently found possible to apply further developments of the theory to explain these observations (Lee, 1978a; Seidel, 1977; Griem, 1978). The improved formulation of the line-broadening problem due to Dufty (1968, 1970) and Lee (1973a, b) is particularly interesting. In this work the full Fourier spectrum of microfield fluctuations due to electrons and ions is calculated in a quantum-mechanical random phase approximation which is closely related to the derivation of the plasma density fluctuation spectrum in light-scattering theory. This method includes the highfrequency microfield of the ions which has been. termed an “ion dynamic” contribution. The quasi-static approximation is still used to treat the major broadening effect with the high-frequency effect being treated as a perturbation. Thus the line profile is expressed in the form I ( w ) = w 4 1 P ( E ) J ( w E )dE
with P ( E ) being a Hooper-type quasi-static field and J ( w E ) the Fourier spectrum of the microfield minus the quasi-static term (see, e.g., Lee, 1978b). The principal effect of the ion dynamics is to give further broadening of the line center and the formulation includes collective effects due to plasma and ion wave resonances. In addition to the improved modeling of the microfield, the effect of a high-Z emitter on the surrounding plasmas is treated giving a red shift to the line scaling as Z , - 1. [Griem (1974) has reviewed this type of “polarization shift” and Burgess (1979) has discussed the theoretical uncertainties associated with it.] The action of the charge Z , - 1 on the emitter dipole is also included and causes asymmetry of the line wings. Examples of these effects are given by Lee (1979a) for hydrogenic ion line profiles, some of which (see, e.g., Fig. 5a) are for plasma parameters and transitions typical of LPPs. The line shapes for high-Z helium-like ion transitions at high densities have been analyzed using the “standard” theoretical approach with Richards’ code (Key et a[., 1979a) and also using the Dufty-Lee method (Lee et a[., 1979). Examples of line profiles of ArXVII for a plasma electron density of 5 x Id3cm-3 are shown in Fig. 5b. The main feature of these profiles is the appearance of “forbidden” components, labeled on the profiles in Fig. 5b. These occur because of the mixing of states by the perturbation. In the limit where the level splitting due to the perturbation
1000
100
10
1
FIG.5. Computed line profiles using the "standard" theory (solid lines) and the Dufty-Lee formulation (broken lines). (a) Si XIV La (left) and L (right) for T, = 5.2 x lo6 K and N, = 3 X Id3~ m - Abscissas ~ . in angstroms. (After Lee, 1979a.) (b) Ar XVII 1 'S-2'P, 2's (right) and 1 fS-3'P, 3'S, 3 'D (left) for T, = 8 x 106 K and N, = 5 X i d 3 ~ m - Abscissas ~ . in angstroms. (After Lee et al., 1979.)
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
233
is greater than the separation of the unperturbed levels, the line shape resembles the corresponding hydrogenic line. The fine structure splitting of Lyman lines may be significant for high-Z emitters due to the Z 4 scaling of the spin relativity splitting and has been included in line profile calculations by Lee (1979b). The effect of a nonthermal electron velocity distribution on collective resonance features of line profiles have been discussed by Lee (1979~). 6. Plasma Satellites
Lee’s work cited previously includes plasma satellites. This subject is considered in general terms by Vinogradov et al. (1 974), Bekefi (1976), and Peacock (1978) and in theoretical detail by Courtens and Szoke (1977) and Baranger and Mozer (1961). The process is essentially the ac Stark effect due to the oscillating electric field of plasma waves and (see Fig. 6) induces satellites to transitions which may have a significant intensity relative to other features, particularly in the case of forbidden lines.
Induced transitions
n (wi, 2 wP a
b
FIG. 6. Schematic energy-level diagram of satellites to a forbidden line induced by an
oscillatory field at the plasma frequency up:a, allowed line; b, forbidden line. (After Bekefi, 1976.)
7. The Inglis- Teller Limit
A further result of line broadening is the merging of broadened spectral lines at the Inglis-Teller limit (Inglis and Teller, 1939). The highest resolvable principal quantum numberp, for ions of charge Z , perturbed by singly charged ions is [Cooper, 1966, Eq. (16.10); or Griem, 1964, Section 5-71 7.5 log p, = 23.26 - log N ,
+ 4.5 log Z
(52)
This may be used as an approximate indicator of electron number density with due regard to the question of whether this effect or the reduction of ionization potential (Section 11, E) is dominant.
M. H . Key and R. J. Hutcheon
234
In concluding this section we note that a recent appraisal of the status of line-broadening theory in high-density plasmas by Burgess ( 1979) gives a good account of the limitations of current approaches.
VI. Continuum Emission A. BREMSSTRAHLUNG
The continuum spectrum radiated by an electron flux interacting through binary collisions with ions of charge Z is given in the classical approximation (see, e.g., Zeldovich and Raizer, 1966, Vol. I, p. 248; or Dyson, 1973, p. 32) by
dZ(v) = NZun(u)P(v)dv e r g ~ m - ~ s e c Hi-' -' (53) for N Z ions ~ r n and - ~ n(u)udu electrons cmP2sec-' in the velocity range
+
u to u du. P ( v ) denotes the spectral variation, and in the classical approximation P( v) is constant and given using standard cgs symbols by
P(v)= 32a2Z2e6
(54)
3 6 m,2c3u2
Quantum-mechanical calculations (e.g., Karzas and Latter, 1961) modify the classical spectrum P ( v ) in a manner conveniently described by a correction factor, g(hv, E, - hv), where E, is the electron kinetic energy and g is referred to as a Gaunt factor. Dyson (1973, p. 32) has reviewed comparisons between experimental measurements and theory of I(v) for monoenergetic electron beams and thin targets. Thermal bremsstrahlung continuum from plasmas has been discussed by, for example, Zeldovich and Raizer (1966, Vol. I, p. 258), Richter (1968), and Griem (1964, p. 105). If plasma electrons have an energy distribution n,(E,), then from Eq. (53) the plasma emission spectrum Z&v) is I'(v) = A N z l m n(Ee)EJI2dE, ergcmP3sec-' Hz-' E, = hu
with
-)
A = z2( 321r2
3(6)'/'
e6 = 7.1 x m2/*c3
10-46
(55)
235
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
If n ( E e ) is Maxwellian with temperature T,, the emission spectrum IP(v, T,) can be evaluated by integration and, including integration over the Gaunt factor correction, this gives
Ip(vT,) = BNZNeg(hv,T,)T;’/’exp(
e r g ~ m sec-’ - ~ Hz-’
- hv/kT,)
(56)
where B = A ( 2 / k ’ / 2 7 r ’ / 2 )= 6.8 x low3’
and g(hv, T,) is an integrated Gaunt factor (Karzas and Latter, 1961). The inverse process of bremsstrahlung is collisional absorption which may be significant in LPPs for low photon energy and high density, causing saturation of emission at the blackbody limit and also being an important mechanism in absorbing laser radiation. The absorption coefficient K ( vT,) follows from Kirchhoff‘s law (Richter, 1968) (57)
€(ITe) = K(vTe)B(vTe)
where c(vTe) is the emission per unit solid angle and B(vTe) is the Planck function. Taking c from Eq. (56) and using 2hv3 1 B(VTe) = c2 [exp(hv/kT) - 1 3
the inverse bremsstrahlung absorption coefficient is
[
( $)I
K(vT,) = Cg(hv, Te)NZNeZ2T;’/2v-3 1 - exp -
cm-’
(58)
with e6 = 3.7 x lo8 k I 12md/2hc
Where the bremsstrahlung emission spectrum Ip(v) in a homogeneous plasma is known and the non-Maxwellian form of n(Ee) is unknown, it may be possible to follow a deconvolution procedure to deduce n(Ee)from I p ( v ) as discussed, for example, by Key (1975) and Brueckner (1977). A particular case of relevance to LPPs arises where the distribution function can be approximated as having two temperatures due to a majority of “cold” electrons characterized by a temperature T, and a minority of
236
M. H. Key and R. J. Hutcheon
“hot” electrons with a temperature TH >> T,. In this case the spectrum has the form exp( - hv/kT,) at low photon energies hv < kTH and exp( - hv/ kT,) at high photon energies hv > kT,. The logarithmic slopes of the exponentials may therefore be used to determine TH and T, (see Fig. 13). Such analysis is complicated, however, by the fact that the mean free path of the electrons in the “hot” Maxwellian may exceed the dimensions of the plasma and may be sufficient to penetrate significantly into soliddensity cold matter ahead of the thermal conduction front. The bremsstrahlung of the hot electrons is modified under these conditions, since the radiation is produced by interactions at varying energy as the electrons are slowed by interaction with the solid matter (as discussed in Section VIII). Such “thick-target” bremsstrahlung has been analyzed in connection with X-ray tubes. The spectrum due to a monoenergetic beam shows a linear decrease of intensity up to the maximum photon energy hv, E,, which is reasonably represented by the theoretical expression of Dyson (1973, p. 39) for the emission per electron
-
’
I ( Y, E,) = D Z ( E, - hv) erg Hz-
(59)
with
D x (4n/96)(e2/c3m,) = 7.6 X lopz4 The spectrum emitted by any specified polyenergetic electron flux follows by integration as before. In the case of a three-dimensional Maxwellian source where the electron flux is n(u)udu, the integration results in a spectrum with the same form, exp( - hv/kT,), as in Eq. (56). Thus the logarithmic slope of the spectrum still gives a valid temperature measurement. Shay et a/. (1978) suggest, on the other hand, that the slope of the observed bremsstrahlung spectrum is not given by exp( - h v / k T , ) but do not elaborate on their comment which seems to arise from their numerical simulation. The total bremsstrahlung energy in the hot component of the spectrum may be related to the total energy carried into the target by hot electrons as discussed by Eidmann et a/. (1976) and Brueckner (1977), using arguments similar to the thick-target bremsstrahlung model just discussed. The angular distribution of bremsstrahlung for a directed electron source varies with the nature of the collision producing it (see, e.g., Dyson, 1973, p. 11; Compton and Allison, 1935, p. 80). At the high-energy limit (E, = hv) collisions are “head on” with the deceleration vector along the direction of motion. The radiation pattern is of the dipole type with a forward bias due to relativity effects. Emission with h v / E , < 1 arises in deflecting collisions and the radiation pattern is more isotropic. Thick targets cause scattering of a directed beam and thus increase the isotropy
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
231
of radiation for h v / E , < 1, but at h v / E , = 1 the dipole pattern persists. A homogeneous plasma has no anisotropy in its bremsstrahlung. The polarization of bremsstrahlung varies with the angular pattern. For hv/ E, = 1 radiation is polarized in the dipole fashion, whereas polarization decreases as angular isotropy increases as discussed previously (see, e.g., Dyson, 1973, p. 47).
B. RECOMBINATION CONTINUUM In LPPs of moderately high-Z elements with ionic ionization potentials
xz > kT,, the dominant source of continuum at photon energies hv > xz is often radiative recombination, described by A'+ + e + E , e A ( Z - ' ) + +hv
(60)
+
with hv = E, x z . Useful basic discussions of this subject are given, for example, by Griem (1964, p. 105) and kchter (1968). A convenient expression for the combined recombination and bremsstrahlung continuum emission of a homogeneous plasma is given by Mandelshtam et al. (1966) and can be expressed as z(vT,) = FZ2N,Nf(Te)-'/2exp( -
e)
where F = B / Z 2 [see Eq. (56)], Nf is the total number density of ions of average charge 2, and NZ the number density of ions of charge labeled Z. xH is the ionization potential of hydrogen, xZ-'(n,c) is the ionization potential of shell n of an ion charge Z - 1, is the number of available states for recombination into shell n. The dominant effect is recombination into the ground state ( n = 1) because of the exp[Xz-l(n, c ) / k T , ] dependence. Bremsstrahlung (1 in the bracket) is weak compared to recombination radiation when xz-l(n, c ) >>kT, that is, in a plasma of high-Z ions. The spectral variation exp( - h v / k T ) is the same for both bremsstrahlung and recombination continuum except that the latter terminates at photon energy hv < xZ-'(n, c) (see, e.g., Fig. 12). The semiclassical result of Eq. (61) can also be
238
M. H. Key and R. J. Hutcheon
corrected in accord with quantum-mechanical results using an integrated Gaunt factor g(v, n, Z ) (see, e.g., Karzas and Latter, 1961; Griem, 1964, p. 105; Richter, 1968). The cross section for radiative recombination is approximately for hydrogenic ions (Zeldovich and Raker, 1966, Vol. I, p. 236)
The fall of a(Ee) with electron energy for E e > > x z is quadratic, and thus hot electrons with kTH>> kT, may have negligible probability of recombining in the ablation plasma. They may then lose energy by penetrating into the solid target, as discussed in Section VIII. Since the cool solid-density matter has no ions of xz comparable to kT,, bremsstrahlung typically dominates over recombination [Eq. (6 l)] in the hard X-ray continuum radiation due to hot electrons. A typical solid-target LPP X-ray continuum spectrum (see Fig. 13) therefore has a soft X-ray component of high intensity and high slope corresponding to the ablation plasma “cold electron” temperature and due to recombination, together with a low slope, hard X-ray continuum of lower intensity due to thick target bremsstrahlung of the hot electrons in cold solid material, indicating a temperature TH >> T, . The inverse process of recombination radiation is photoionization absorption. This follows from the Kirchhoff-Planck relation, the Saha equation, and Eq. (61), and for absorption by ground-state ions of charge Z - l is
(63) where -=
1.3 x 1040
Z2 [see Eq. (56)]. In the limit of high density, LPPs may become optically thick to their own recombination radiation (Section VII).
VII. Radiative Transfer It is generally found in studying LPPs that where the intensity of spectral lines is sufficient for spectroscopy, particularly with space or time
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
239
resolution as well as high-spectroscopic resolution, the emergent spectral line intensities and shapes are significantly modified by absorption in the plasma. The theoretical treatment of these effects has an extensive background which is discussed in general terms by, for example, Chandresekar (1960), Sobolev (1963), Zeldovich and Raizer (1966, Vol. I.), Richter (1968), Zwicker (1965), Hummer and Rybicki (1971), and Mihalas (1978). A. THERADIATIVE TRANSFER EQUATION
The emission coefficient C(Y) and absorption coefficient K(v) of a plasma for line and continuum radiation were discussed in Sections IV-VI. In a plasma of finite size the optical opacity ~ ~ (isv )
where K ( Y X ) is the space- and frequency-dependent absorption coefficient and the integration is along the line of observation. The absorption is in general due to a mixture of processes. The main contributions are from free-free, bound-free, and bound-bound transitions, [Eqs ( 5 8 ) , (63), and (27), respectively], and a net absorption coefficient due to these processes is obtained by addition. If T << 1 the intensity emerging from the plasma has its optically thin value
where reabsorption is not negligible, then in any element of path dx ~ Z ( V X= ) [~
( Y X)
K(vx)Z(VX)] dx
(66)
so that with the introduction of the source function S(v),where S(v)= W / K ( 4
(67)
it follows that 1 dZ(4 K(vx) dx
- S ( v x ) - I ( ux)
Equation (68) is the equation of radiative transfer and has the integral form ~ ( v=) z 0 ( u ) exp( - 70) + J T 0 ~ ( v 7 exp( ) - T) d7 0
.
* *
(69)
using the optical depth T as a variable. In general the solution of the transfer equation presents major problems
240
M.H. Key and R. J. Hutcheon
because of the coupling of the opacity to the intensity through the effect of reabsorption on energy-level populations.
B. LTE SOLUTIONS A limiting case where no such coupling exists is the LTE approximation. The definition of LTE requires level populations to be collision dominated, that is, independent of the radiation field. The LTE source function is, from Kirchhoff‘s law,
S(vx) = B[vT(x)]
where T ( x ) is the local “distribution temperature” characterizing, via the Boltzmann factor, the population density in the upper and lower levels of the radiative transition at frequency Y, and B(vT) is the Planck function. For a homogeneous plasma with T ( x )= T it follows from Eqs. (68) and (70) that the emergent intensity is Z(V) =
B ( ~ T )1{- exp[ - T , ( Y ) ] }
(71)
where T, is the optical opacity along the line of sight. The homogeneous slab solution [Eq. (71)] predicts a saturation of the intensity of a spectral line at the blackbody limit. When the opacity is large there is opacity broadening of the line which has a flat region in the center of its profile over the frequency band where the opacity is > 1. In LPP studies where accurate intrinsic line profiles have been computed and compared with experiment, they have commonly been corrected for opacity broadening using Eq. (71) (see Section IX). If the plasma is inhomogeneous but in LTE, then it is a straightforward matter to integrate Eq. (68) numerically along any line of sight. Numerical integration of Eq. (68) has been used to compute emergent line profiles in computer simulations of laser-driven implosions by Yaakobi et al. (1978). In their LILAC code the implosion hydrodynamics are solved on a onedimensional Lagrangian mesh. From the plasma parameters the local spectral emission and absorption coefficients and line shape are obtained in the LTE approximation, using Hooper’s line-shape data (Section VI). The transfer equation is then integrated along chordal lines of sight through each successive radial zone to produce radially varying simulated neon X resonance series spectral line shapes. A spatially integrated line shape is then obtained by simple summation. This enables the effect of, for example, reabsorption in cold dense material around the compression core to be included in the calculation producing self-reversal of line intensity at the line center.
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
24 1
An analytic treatment of the effect of reabsorption on line profiles in a plasma where the source function (and therefore the temperature) decreases toward the periphery has been given by Bartels (1953) and is discussed by Zwicker (1965). This approach gives a straightforward appreciation of the way in which with increasing opacity an emission line becomes self-reversed and ultimately an absorption line in underlying continuum.
C. COLLISIONAL-RADIATIVE SOLUTIONS I . General Observations It is typical of LPPs that the intense resonance spectra of principal interest originate from levels whose populations are strongly influenced by radiative decay and therefore also by reabsorption. Bound states lying below the LTE limit (Section IV) have population density less than the LTE value because of their high rates of radiative decay. Reabsorption of resonance radiation opposes this decay process and leads to population densities closer to LTE values. The solution to the radiative transfer problem can be viewed as a self-consistent result where integration of the transfer equation [Eq. (68)] over a self-consistent spatial distribution of plasma parameters and excitedlevel populations gives the radiation field in any direction; and the rate of photoexcitation at any point in the plasma by radiation integrated over frequency and direction is such that, when coupled with the collisional and spontaneous radiative decay processes in the plasma, it produces the selfconsistent level population densities at that point. In this view of the problem there is an implicit assumption that there is no connection between the frequency and direction of an absorbed photon and that which may subsequently be reemitted from the excited level. The emission is assumed isotropic and with the line shape appropriate to the plasma parameters. This is often referred to as the assumption of complete redistribution of frequency. A more detailed picture of radiative transfer introduces the idea of scattering. The spontaneous emission from the plasma is computed from its excited-level populations in the absence of photoexcitation. The frequency and angular distributions of the emergent photon flux are computed by considering the subsequent history of the emitted photons. They propagate until absorbed by excitation of an atom. This is treated as a scattering process in which the photon may then be either reemitted or destroyed by a collision-induced change of state of the atom. The reemission may be with complete redistribution of frequency or may allow for
242
M. H . Key and R. J. Hutcheon
correlation between the incident and reemitted photon characteristics, which is termed “partial redistribution.” The photon, if not destroyed by a collision, eventually escapes from the plasma and in the limit of high line center opacity does so by frequency diffusion, that is, by being reradiated in the line wing at a frequency with a low probability of reabsorption. In LPPs we are interested in levels that are broadened by Doppler and pressure broadening. In this case the assumption of complete redistribution is accurate in the line center and in the line wings with some enhancement of the probability of reradiation close to the frequency of the absorbed photon in the intermediate region (Mihalas, 1978, Fig. 13.4). It is found that despite the greater rigor of the partial redistribution scattering picture, for practical purposes the assumption of complete redistribution is satisfactory for Doppler- and pressure-broadened lines (Mihalas, 1978, p. 430) and therefore the problems may be treated either as one of level populations or of scattering with equal validity. 2. The Escape-Factor Approximation A useful approximate method giving the change in excited-level population and of emergent radiation intensity is based on calculation, for simple line-profile shapes (Gaussian or Lorentzian), of an “escape factor” g ( T o ) from a plasma of homogeneous level densities and specified geometry (plane slab, cylinder, or sphere). The escape factor g is the volume direction and frequency-averaged probability of escape without reabsorption for a photon emitted in the transition of interest. It may be expressed as a function of the line-center opacity T~ along a characteristic dimension of the plasma. McWhirter (1965) and Zwicker (1965) have given good introductions to work of this type, while Irons (1979) has reviewed the concept, associated literature, and analytic expressions for g ( T o ) in detail. For a two-level “atom” the modification due to opacity of the resonance line of excited-level populations in the coronal model (Section IV) is obtained simply by reducing the radiative decay rate A ( p , g) in Eq. (21) to an effective rate g A ( p , g). This gives an increased upper-level population by the factor g - ’ which, with the reduced probability of direct escape of emitted photons g, results in a rate of escape of photons that is unchanged from the optically thin value while the line shape is opacity broadened as before. In the intermediate CR regime the population ratio for a two-level atom is
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
243
where X denotes a collisional rate coefficient. The rate of escape of photons is reduced by a factor g " W g) + N e X ( P * 811
[ gA (P.8 ) + NeX(P9 s>l
(73)
The previous statements are valid when the excited-state population is small relative to the ground-state population and the latter is not significantly changed by photoexcitation. In fact opacity of the resonance transitions can sufficiently reduce the recombination rate due to CR cascading to change significantly the ionization balance in CR equilibrium. This is shown in the work of Bates et al. (1962a, b) where CR equilibria are calculated assuming trapping ( g = 0) of resonance line radiation. Weisheit et a/. (1976) have used an escape factor g(7J # 0 in CR calculations of ionization balance and radiated intensity in LPPs. 3. Self-consistent Solution of the Radiative Tramfer Equation
The escape factor approach is useful as a simple approximation and may give results for the rate of photon emission and for average population ratios in an optically thick transition which are close to those obtained by full solution of the radiation transfer problem. Such full solutions using numerical methods and assuming redistribution have been described by Hearn (1963), Hummer (1963), and Cupermann et al. (1963) for an idealized two-level atom with Doppler line broadening and homogeneous slab geometry. They show the spatial variation of the radiation field and level population ratio, the rate of escape of photons, and the emitted line shape as a function of direction. Being idealized they do not relate directly to LPPs but do give a useful qualitative appreciation of, for example, the change from intense isotropic radiation to weaker outward directed radiation in moving from the center of the boundary of the plasma. The effect of the collisional quenching probability for excited states
is made clear in the dependence of photon escape rate on opacity. A drop in excited-level to ground-level density ratios toward the plasma periphery results from the fall in radiation energy density, and an associated selfreversal of the emission line profile [see Fig. 4 of Hearn (1963)l also emerges from the calculations. Comparison of the detailed results with the escape factor estimates for the rate of photon emission given by Hearn (1963) illustrates the usefulness of the latter over a wide range of opacity and quenching probability.
244
M. H . Key and R. J. Hutcheon
Numerical solutions while accurate are lengthy and alternative analytic approximations have been developed for these ideal two-level atom problems by, for example, Averett and Hummer (1965), Wilson (1972), and Kunasz and Hummer (1974a, b). The details of the mathematical methods both numerical and analytic are beyond the scope of this review but are discussed at length by, for example, Mihalas (1978) and Richtmeyer and Morton (1967) as well as in the other general references cited in the introduction to this section. To date there has been relatively little work specific to LPPs involving proper solution of the radiative transfer problem. One of the earliest attempts was by Whitney and Davis (1974). They developed a numerical model of radiation from a laser-produced aluminum plasma. Quite detailed time-dependent CR atomic physics was incorporated as well as treatment of Al XI1 and Al XI11 resonance line radiation without opacity corrections. In an improved model, Doppler line shapes and a simplified form of frequency diffusion were used to describe the escape of photons from the plasma. No detailed emerging line shape was therefore obtained, though total radiated intensity was estimated reasonably well, and a sophisticated two-dimensional hydrodynamic model of the formation of an LPP was included (Colombant et al., 1976). Further detail in the modeling of the transport of line radiation was developed using a combination of a diffusion approximation in the opaque center of the plasma with ray tracing near the periphery (Apruzese et al., 1976). The numerical method was due to Hummer et al. (1973). Although those calculations produced line shapes for optically thick Al XI1 and Al XI11 transitions based on 10 frequency elements per line, they were not useful for detailed comparison with experimental line profiles since they were derived from Doppler profiles. Useful line intensities were obtained, however. The effect of photoexcitation on CR ionization dynamics in a carbon plasma with homogeneous sphere or spherical shell geometry and LPP relevant density has been studied in a further development of the model (Davis et al., 1978), but still the line-shape calculation has been based on an intrinsic Doppler profile. Landshoff and Perez (1976) have also calculated the total power radiated from an aluminum LPP using Doppler line shapes and an averaged transport method. The only work to date in which full line-shape calculations have been coupled with a CR atomic physics model and radiative transport is due to Skupsky (1978). He has used a simplified CR model for Ne XI, Ne X, and Ne IX in which only the Ne X 1s-2p transition is not in LTE. The radiative transport solution was a simple adaptation of the LTE model of Yaakobi et al. (1978) described earlier with integration along chordal lines of sight through each spherical shell giving in effect n directions of solution in the nth radial shell. Hooper’s line
245
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
/';,m;latL
Lei
Spect,'urr'
'
'
'
'
'
'
'
'
Hornogenews Sphere
Fie. 7. Simulated Ne X emission spectra at 0" from a spherical laser-compressedplasma obtained with CR transport modeling and full line-shape calculations.The broken line is for a radial distribution of p and T, (inset) and the solid line for a homogeneous sphere of the same average parameters. (R. W. Lee, private communication, 1979.)
profiles (Section VI) for Ne X La were used to compute the emergent line shape on each line of sight and by summation the net line shape for a homogeneous spherical source and for a source of uniform density with a radial temperature gradient. In recent unpublished work, Lee has developed a numerical treatment of radiative transfer using his line-profile calculations (Section VI) coupled to a CR model of the plasma with solutions based on the methods of Averett and Hummer (1965). He has compared, for example, line profiles in the Lyman series of NeX in two cases (see Fig. 7), first using density and temperature profiles from a hydrodynamic model of compressed neon in a laser-imploded microballoon and second from a uniform plasma of the same average parameters. Changes in conclusions that a full analysis might give relative to simplified methods are thus seen to be small in this case. Bailey (1977) has described a similar model which uses, however, only Doppler and Lorentz line profiles.
D.
RADIATIVE
TRANSPORT WITH FLOWDOPPLER SHIFTS
The expansion of the plasma plume from an LPP on a plane solid target at distances larger than the focal-spot radius has both axial and radial components of velocity (see Fig. 8). If the plasma is viewed transverse to
246
M,H. Key and R. J. Hutcheon
the laser axis, its radial velocity component is along the line of sight and varies from zero at the center of the plasma to f u , at its near and far boundaries, respectively. Since the plasma has cooled by adiabatic expansion and its initial internal energy has been largely converted to flow kinetic energy, the width of the spectral line is due to Doppler and Stark broadening and can be smaller than the flow Doppler shift Av where A u / v = f v,/c
(75)
Optically thin emission line profiles under those conditions exhibit doublepeaked structures. With optical opacity the observer sees a blue-shifted self-reversal due to the velocity component directed toward him of the nearer outer zone of cooler plasma. Computation of emission line profiles from expanding plasmas is a familiar problem in astrophysics (Sobolev, 1960; Kunasz and Hummer, 1974a,b; Mihalas, 1978). In the LPP context it is relatively new and the first computation of emergent line shapes for optically thick lines is due to Irons (1975) who integrated the transfer equation (68) for Doppler and Lorentzian line shapes with a specified source function variation (i.e., not a self-consistent coupled solution and therefore equivalent to an LTE solution). His results covered a wide parameter range illustrating the possible types of asymmetric self-reversal and double-peaked optically thin features. Similar computations were described more recently by Tondello et al. (1977) for the Lyman spectrum of Be. They used better line profiles scaling Griem’s results for hydrogen lines (Griem, 1974) and used a specification of the spatial variation of the source function (population ratio) which was based on experimental data. Thus they integrated the transfer equation without a self-consistent solution for the CR atomic physics. Their computed line profiles being based on fairly good intrinsic line shapes could be compared with experiment. The converse problem of a red-shifted NeX La self-reversal in an imploding plasma was computed by Skupsky (1978) using his non-LTE radiative transfer model described in Section VII, C, 3 and by Yaakobi et al. ( I 979) using an LTE model.
VIII. Structure and Spectroscopic Characteristicsof Laser-produced Plasmas It is convenient to divide LPPs into their two main categories of plane targets and spherical shells, though the interaction of the laser radiation with the surface of a solid spherical shell is closely analogous to that with the surface of a solid-plane target.
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
247
A. PLANETARGETS 1. The Expansion Plume
Referring to Fig. 8a the overall structure of an LPP is shown alongside a typical X-ray image. The expansion plume (1) is the region where the LPP energy is converted from thermal energy to kinetic energy by adiabatic expansion in vacuum. Density and temperature decrease in accord with the expanding volume at increasing distance from the surface. There is a quasi-stationary hydrodynamic flow at a velocity several times lo7 cm sec- during the laser pulse. When the pulse ends the luminous plasma detaches itself from the target and moves down the expansion plume as a source of expanding radial and axial dimensions but of decreasing brightness. The rapidity of the expansion is such that recombination rates are too slow to bring the ionization into equilibrium with the decreasing temperature and highly ionized ions persist in a “supercooled” condition. This frequently leads to population inversion among the first few excited states of the ions. Visible and near-W spectroscopy is most useful at the low-density extreme at distances from 0.5 to 2 cm from the target, for temperature from a few eV to a few tens of eV, and electron density in the region 1OI6 to 10l8 cmP3. Farther away the plasma emission becomes too weak to detect. Closer to the target opacity limits the usefulness of the spectra. Transitions between high quantum number states of supercooled highly charged ions are a feature of the visible spectra. VUV spectroscopy in the grazing incidence region is suited to the region of higher density and temperature from a fraction of a millimeter to several millimeters from the target, with electron density from 10l8up to lo2’ ~ m - and ~ , temperature from 20 to 200 eV. Stark and opacity broadening dominate line shapes at the high-density limit. The Doppler effect is dominant at low densities. An unusual feature of the spectra is strong motional Doppler structure arising from the highly supersonic flow. Optically thin emission lines have double peaks, while self-reversal features are blue shifted.
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2. The Ablation Plasma The zone numbered (2) in Fig. 8a is the region of high temperature plasma between the thermal conduction front and the onset of significant adiabatic expansion and cooling. The temperature may be as high as lo00 eV at high irradiance (10” Wcm-2) down to about 100 eV at low irradiance (10” W cmP2).This is the region of greatest radiative emission
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FIG.8. (a) A typical plane target LPP. The schematic diagram shows (1) the expansion plume, (2) the ablation-front plasma, and (3) the zone of hot-electron preheating. T h e photograph is an image in I-keV X rays of an LPP produced by a 5-nsec 20-J Nd laser pulse focused to a 50-pm spot on a solid Cu target. (b) A typical exploding pusher LPP. The schematic diagram shows (I) the ablation-front plasma, (2) the imploded glass, and (3) the compressed gas. The photograph is a 2.5-keV X-ray image and the target was a 122-pm g cm-3 D, + T, gas fill. It diameter 1.0-pm wall thickness glass microballoon with 4.2 X was imploded by six-beam irradiation with a 2-TW 50-psec Nd laser pulse. (From Thorsos et al., 1979, by permission.)
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since the electron density is also high ranging from several times greater than to somewhat less than the critical density (102’ cm-3 for wavelength X = 1 pm). Most of the emission occurs in the soft X-ray region at photon energies of the order of kT,. Continuum optical opacity is high at lower photon energies. Grazing-incidence VUV spectra show strong continua with low line to continuum contrast ratio and often self-reversed lines or absorption lines in the continuum. Soft X-ray crystal dispersed spectra have a large information content with Doppler and Stark broadened lines of good contrast relative to recombination and bremsstrahlung continua. Intercombination and dielectronic satellites are prominent in the spectra. The emission intensity is very large. Up to 200/0 of the laser power can be converted to soft X-ray power. Very highly ionized ion X-ray emission spectra from high-Z elements are also generated in this region. The ablation plasma region is where much of the important physics of the laser-plasma interaction occurs (see, e.g., Key, 1980a). 3. The Zone of Hot-Electron Preheating
At low irradiance ( I X 2 < IOI4 Wcm-* pm2) laser light is absorbed by inverse bremsstrahlung absorption [Eq. (SS)], but at higher irradiance the dominant absorption process is collisionless excitation of plasma waves whose damping gives energy to a small fraction of the electrons in the critical density region. These electrons have been termed hot electrons since their energy spectrum is approximately Maxwellian with kT, >> kT,, where TH and T, are the hot and cold electron temperatures, the latter being the temperature of the majority of ablation plasma electrons. The irradiance is quoted above in units of ZA2, where X is the laser wavelength since this parameter determines T , (see, e.g., Forslund et al., 1977). to 10’’ The hot electrons which for high irradiance (ZA2 from W cm-* pm2) carry most of the absorbed energy have kT, from 10 to 40 keV and thus a range A, scaling as ( I c T ~from ) ~ several microns to several tens of microns in solid-density matter. These electrons do not therefore deposit energy in the ablation plasma which they pass through essentially without collisions. Instead they carry energy ahead of the ablation front to preheat the solid target. The spectroscopic effects of this preheating are mainly in the form of emission from zone (3) in Fig. 8a of thick-target, hard X-ray bremsstrahlung and of K, radiation.
B. SPHERICAL SHELLTARGETS These targets have been imploded in two different modes, namely, exploding pusher implosions where the hot-electron preheat pressure P,
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drives the process and ablative implosions where the ablation pressure Pa is dominant. 1. Exploding Pusher Implosions The term exploding pusher has been applied to implosions in which a thin-walled microballoon ( A r 1 pm, r 50 pm) is irradiated at I A 2 from 10’’to lo” W cm-2 pm2. Hot electrons deposit energy over a scale depth A, > Ar. The laser pulse rise time and duration are short enough ( < 100 psec) to give a high value of hot electron preheat pressure P , in the shell before it can relax by explosive expansion. The explosion of the wall sends approximately half the mass inward and half outward. The imploding half compresses gas filling the shell. This gas has already been preheated by the hot electrons and by shock waves. The implosion stagnates when the pressure in the gas is approximately equal to P, the initial pressure in the solid wall. Electron temperature in the gas on stagnation of the implosion ranges from kT, 0.2 to 1 keV. The ion temperature is higher because of shock heating and the absence of thermal conduction cooling and ranges from 0.5 to 10 keV. The final temperature is proportional to the ratio E / M between the laser pulse energy E and the shell mass M , and this ratio needs to exceed 0.3 J ng- to produce a typical compression core X-ray source. Electron number density in the implosion core is in the range ld2 to ld3cmP3,and it has been found that significant increase beyond Id3is not possible with this mode of implosion because increased laser power increases the temperature rather than the density (see Note added in Proof). Fuller discussion of the hydrodynamics of this type of implosion is found in Rosen et a/. (1979), Ahlborn and Key (1979), and Goldman et al. (1 979). The nature of the spectroscopic source is illustrated by Fig. 8b. The novel feature is the emission from the 10- to 50-pm diameter compression core. This is due to the compressed gas (3) but also to a small fraction of the shell material which either surrounds or is partly mixed with the gas and is at a similar density and temperature (2). As the E / M of the implosion increases the emission from this compression core becomes progressively more dominant. The high density of the implosion makes only X-ray spectroscopy of interest, since the plasma is optically thick to continuum emission for softer radiation. The X-ray line spectra show unusually large Stark broadening due to the high density which has exceeded Id3 e/cm3. Strong opacity broadening of resonance lines, strong continuum intensity relative to line intensity, and strong intercombination and dielectronic satellite lines are features of the spectra.
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The ablation plasma (1) is generated at the surface of the solid shell before the shell explodes and thus appears as a static shell of emission. Time-resolved studies show it to be emitted 100 psec before the implosion-core emission (Section IX, C). It is similar to the ablation plasma on a plane solid target. The expansion plume has not been studied much for microballoon targets. This is partly because the more rapid spherical compression causes a much sharper falloff with radius in intensity of spectral lines in space-resolved spectra, but also because interest has focused on the compression-core plasma.
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2. Ablative& Driven Implosions Increased wall thickness Ar and/or reduced irradiance IA2 can make Ar > A,. This stops hot-electron preheating and leads to the inward acceleration of an unheated part of the shell wall by the ablation pressure acting on the shell surface. This mode of implosion with longer pulse (- I nsec) irradiation during the whole implosion process makes it possible to produce higher compressed-core density scaling up with the magnitude of the ablation pressure. A fuller discussion of this mode of implosion is given by Key (1980b). I t produces a compression core that is dense but too cool to show in X-ray emission, though this limitation should be removed as more powerful compressions are attained. As spectroscopic sources, they have weak emission from an ultradense but cool implosion core (kT,-50 to 500 eV and N , from ld3to Id5 ~ m - ~The ) . features of the core have therefore been studied in absorption of externally generated X rays (see Section IX, E). The ablation plasma is similar to that on a plane target. The zone of emission is different from the exploding pusher, however, since the continuous acceleration creates an inward-moving shell of emission from the ablation plasma which therefore appears in a static image not as a ring but as a more uniform emission source.
IX. Spectroscopic Diagnostics of Laser-Produced Plasmas
A. SPECTROSCOPY OF THE EXPANSION PLUME
Study of the expansion plume of LPPs (see Fig. 8a) has involved: 1. Stigmatic visible/UV and normal incidence VUV spectroscopy;
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2. grazing incidence VUV and XUV spectroscopy; and 3. X-ray spectroscopy with Bragg reflecting crystals.
Roughly speaking, the density and temperature of the plasma region under study has increased from ( I ) to (3), so these subdivisions are used in the discussion that follows.
I . Stigmatic Visible/ UV and Normal Incidence VUV Spectroscopy Boland et al. (1968) reported the first detailed study of the expansion plume of an LPP. The plasma was generated from a plane polyethyelene target with a 17-nsec ruby laser pulse focused to 3 X 10" W cm-2. Transitions in C V I to C I were identified in the space-resolved spectra, with transitions between high quantum numbers ( e g , CVI 6-7 at 3434 A) being features of the highly ionized ion spectra. Time- and space-resolved measurements were also made with a stigmatic monochromator. Line emission was thus recorded photoelectrically, and delayed pulses of emission due to the transit of the luminous plasma at various distances from the target surface were observed. The results gave the velocities of the various ion stages, which increased from 0.7 to 3.3 x lo7 for C I to C VI. The electron temperature was deduced from the Boltzmann relationship between level populations in LTE and their associated line intensities. Ion densities were obtained from the absolute intensity of the lines. The results indicated k T e - 5 eV and total ion density N f - I O l 7 cmP3 in the region 3-5 mm from the target surface. Aglitskii et al. (1970; see also Basov et al., 1973) also studied spectra from a carbon target. In this case a 1-GW Nd laser was used and stigmatic spectra were obtained with normal incidence VUV and quartz instruments. Electron number densities from 1017to 10l8cm-3 in the range 0.7-1.8 mm from the target surface were deduced from the Stark width of the C I V 3434A line. A Doppler shifted self-reversal in the CIV line at 1550 was also observed and used to estimate the 2 X lo7 cm sec- expansion velocity of the cool boundary of the plasma. Further development of this line of research was reported by Irons et al. (1972) who obtained visible/UV photographic spectra and multiple-shot time-resolved photoelectric spectra for the same target conditions as in their previous work (Boland et al., 1968) described earlier. With suitably chosen C I to C V I transitions they built up a detailed picture of the velocity and flow pattern of the various ions. They produced line-shape measurements showing double peaks due to flow Doppler effects and deduced that ions of different charge predominated in the flow in zones of a conical shell-like geometry in which slower moving lower-charged ions
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were concentrated in the outer zones. Irons (1973) also analyzed the Stark broadening of several high quantum number visible/W lines studied in the above work and found electron densities between 2 X 10l6and 2 X 10I8 in the range 5-1 mm from the target. Similar methods have been used to study the expansion phase of LPPs in gas breakdown, for example, by Ahmad et al. (1969), who also used an electron optical streak camera to time resolve the He I 2 'S,-3 'P line and observed a strong Doppler-shifted self-reversal due to the expanding cool boundary of the plasma. 2. Space-Resolved Grazing-Incidence VU V Spectroscopy Incorporation of a slit perpendicular to the spectrograph slit in a grazing-incidence spectrometer gives one-dimensional imaging which produces VUV spectra with spatial resolution (see, e.g., Fig. 1 of Galanti and Peacock, 1975). Under optimum conditions 50- to 1Wpm spatial resolution has been achieved. In early work Burgess et al. (1967) and Boland et al. (1968) obtained 0.1-0.5-mm spatial resolution in grazing-incidence VUV spectra. They showed how the slope of C VI and C V recombination continua (Section VI, R) could be used to measure kT, which decreased from 100 to 10 eV in the range 0.1-2 mm from the target surface, and how Stark widths of Li-like 0 VI and Na-like K IX could give density estimates in the range 6-12 X IdocmP3.They also discussed the effects of transient ionization and of spectral line opacity. Their data showed that the relationship between density and temperature in the expansion plume was consistent with adiabatic expansion. Irons and Peacock (1974) made a detailed study of the absolute intensity of space-resolved C VI and C V resonance line spectra and the associated continuum produced by 3 X 10" W cm-2 irradiation. They determined the spatial distribution of plasma parameters in the plasnia plume in the range kT, = 9-40 eV, N , = 2 x 10"-4 X 10l8 ~ r n and - ~ the associated recombination rates which were compared with theory. Furthermore they were able to compute the populations of the bound states and to note the location of the LTE limit. Below the LTE limit they found inversions of population between levels of quantum number 2-5 around 3 mm from the target where N, loi8and kT, 20 eV. It had been appreciated for some time that fast adiabatic expansion and cooling, producing a depleted ground-state population and depleted nonLTE population for n = 2 (with LTE for n 2 4 ) , could cause population inversion. Dewhurst et al. (1976) set out to optimize the process by irradiating a 5-pm diam. fiber with a O.I-J, 140-psec Nd laser pulse focused to a 40-pm diam. spot. They recorded multiple-shot space-resolved C V 5
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M.H. Key and R. J. Hutcheon
and C VI resonance spectra and deduced population inversion between n = 3 and n = 2 from the L,/L, intensity ratio, finding an inversion density of 3 X loi4cm-3 when the plasma had expanded to N,-3 X
~ m - ~ . Key et al. (1979b) developed a new method to study such transient population inversion by coupling space-resolved grazing-incidence dispersion of the C V and C V I resonance spectra to subnanosecond time resolution with a VUV streak camera. Their results were for 2-5, 150-psec pulse irradiation of solid C targets. Measurements were made in a single shot and showed the pulsed emission spectrum as the luminous plasma moved past the observation plane. Their analysis of the data emphasized the importance of opacity of the La transition and the use of ratios of LTE C V to C VI line intensities to estimate ionization ratios and thus opacity. Inversion of population between n = 3 and n = 2 was found for N, = 8 X 10” and kT, = 35 eV, the initial plasma parameters having been N, = Id’ and kT, = 600 eV. Dixon et al. (1978) have described study of the C VI Lyman and Balmer spectrum in the expansion plume of a carbon-target plasma mixing with a background gas at electron density loi4 cm-3 and kT,- 1 eV. They found enhanced inversions of population for C VI 3 +2 and 4 +3 transitions due to charge exchange with neutral carbon atoms. Kononov et a/. (1976) studied Li-like Al XI space-resolved VUV resonance emission spectra in the expansion plume of an LPP and found inversions of n = 5 and n = 4 levels relative to the n = 3 level analogous to those just described. They estimated that the inversion occurred for N, l d o cm-3 with inversion number density 10’’ cm-3 for the ,154-A 3d-4f line of 4x1. Zherikhin et a/. (1977) observed anomalous line intensities in the range 58-78 in the resonance VUV spectrum of Na-like CI VII in the expansion plume of an LPP, though with some doubts as to whether the effects were due to population inversion or to optical opacity. Similarly the earlier reports of anomalous intensity of an A1 V resonance line at 117 A (Jaegle et al., 1971) may also be evidence of population inversion but could be due to optical opacity. Very interesting calculations (Vainshtein et a/., 1978) and experimental data (Illyukhin et al., 1977) have described the population of 3p and 3s levels in excited neon-like Ca XI in the expansion plume of an LPP. Slitless VUV spectroscopy was used by Illyukhin et al. to produce evidence for a directional “laser” beam on a circa 600-A 3p-3s transition. The mechanism was attributed to collisional 2p -+3p excitation, laser action from 3p to 3s, and radiative terminal state depopulation from 3s to 2p. A necessary condition for this is non-LTE population of the n = 3 levels and thus N,-3 X loi9cm-3 and kT,- 150 eV.
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lon
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3 FIG. 9. Time-averaged density and temperature profiles along the normal of a polyethylene target surface for 7 x W cm-2 4-5 nsec irradiation with a Nd glass laser. Experimental data are: (I) N, from C VI continuum intensity, (2) N, from the Inglis-Teller limit, (3) N, from the non-LTE C VI LB/L, intensity ratio, (4) N, from Stark profiles of C VI Ha,(5) T, from C VI continuum slope, (6) T,from the La blackbody peak intensity. (After Galanti er al., 1975.)
In work more closely related to study of the ablation plasma (Section IV, B), Galanti et af. (1975) and Galanti and Peacock (1975) recorded space-resolved C V and CVI resonance spectra (see Fig. 9) as the laser irradiance was varied from 10” to 7 x 1OI2 Wcm-2. A wide range of spectroscopic methods was used to obtain spatial profiles of electron density and temperature (see Fig. 9). Stark-broadened profiles of Lyman and Balmer series were computed by the full “standard” methods (Section VI) and compared with experiment. Opacity corrections by the simple homogeneous slab LTE method [Section VII, Eq. (71)] was applied to the C V I L, line profile which was found to have a line-center opacity of 55 and self-reversal due to cool boundary plasma. Comparison of the spectroscopically deduced ionization balance with CR models suggested that ionization was transient. Tondello et af. (1977) have used very similar experimental methods to record the BeIV resonance spectrum produced by 10-nsec, I-GW ruby laser irradiation. Their results show very pronounced self-reversal of the La and Lp lines near the target surface, strong Stark and opacity broadening, and Doppler shifts of the self-reversed features. They describe a detailed analysis of the spectra using comparisons with a model of the expanding plasma with a treatment of radiative transfer discussed in Section VII.
3. Space-Resolved X - Ray Spectroscopy Observations analogous to and overlapping with those made by spaceresolved grazing-incidence spectroscopy have been obtained with a Bragg
M. H. Key and R.J. Hutcheon
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reflecting X-ray crystal spectrometer having a slit to give spatial resolution in a fashion similar to that described previously [see, e.g., Fig. 2 of Kilkenny et af. (1980)]. The advantage of the approach is that shorter wavelengths enable study of hotter denser parts of, the plasma and also reduce diffraction, allowing about 10 times better spatial resolution (5- 10 pm) than in the V W work. The first results in this field came from Boiko et af. (1975) who used convex mica crystal spectrographs with 65-pm space-resolving slits to study plasmas formed by Nd laser pulses focused to 5 x lOI4 W cm-2 on a wide variety of targets in the range Mg-V. They investigated the structure of the plasma plume, finding like Irons et af. (1972) that emission from more highly ionized ions came from smaller inner regions of the plasma. Their measurements were of intensity ratios between helium-like resonance, intercombination, and dielectronic satellite lines (Section IV). These revealed the now familiar fact that continuum and satellite lines are confined very close to the target surface, whereas resonance and intercombination lines are observed in the expansion plume up to distances of millimeter order with the intercombination line intensity increasing relative to the resonance line with distance (see, e.g., Fig. 10). The quantitative conclusions of Boiko et af. (1975) are open to doubt for plasma parameters close to the target surface because of their neglect of resonance line opacity. They found N , up to Id' cm-' and T, up to 800 eV and measured axial profiles of these variables in the range 0.1-0.8 mm. Boiko et af. (1979b) have discussed these earlier data and presented a detailed review of extensive recent work from the Lebedev Institute. Density measurements based on He-like resonance to intercombination line ratio data (see, e.g., Fig. 2) are compared with data from H-like resonance to dielectronic satellite line ratios, from ratios of densitysensitive dielectronic satellites, from La-doublet intensity ratios (see Section IV), and from Stark broadened linewidths of Lyman series lines near the Inglis-Teller limit (see Section V). They conclude that certain discrepancies in the deduced plasma parameters can be explained by the differing spatial distributions of ions and the spatial and time averaging implicit in the data, but it should be emphasized that they do not analyze in detail the effect of resonance line opacity on the data (see Section IV). Yaakobi and Nee (1976) improved spatial resolution in X-ray spectros15 pm and recently reported study of the expansion copy of LPPs to plume from a plane Al target (Bhagevatula and Yaakobi, 1978). They found evidence of inverted populations between n = 2 and n = 3 and 4 in helium-like Al XI1 resonance spectra 400 pm from the target surface where N , - 102' cm-3 and k T - 100 eV, suggesting gain of 10 cm-I on the 4'F-3 'D 129-A transition. The suggested mechanism was analogous to that for C V I inversion discussed above. They argued that opacity of the
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FIG.10. A space-resolved X-ray spectrum from a plane solid target of SiO with 10-pm spatial resolution perpendicular to the target surface. Densitometer traces at 0 and 160 pm from the surface show the one- and two-electron resonance spectra of Si. The intensity ratio of the ls2-ls4p and Is2-ls3p lines 160 pm from the surface indicates a 4 : 1 population inversion ratio, and the intercombination line intensity exceeds that of the ls2-ls2p resonance line. (After Lunney, 1979.)
resonance line did not affect the interpretation of their data citing numerical modeling, and they also suggested that enhanced cooling by a cold Mg plate close to the target was a central part of the mechanism. Key et af. (1977) reported further improvements of space-resolved X-ray spectroscopy to 7-ym resolution using minimum instrument size to optimize spectral brightness. Figure 10 (Lunney, 1979) shows a space-resolved
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spectrum from a SiO target irradiated with 10 J in 100 psec focused to 100 pm. It is clearly seen that the resonance and intercombination lines extend far from the target surface, while dielectronic satellite and continuum emission falls off much more rapidly. Measurements of the population ratio N (Is4p)/N (ls3p) at 160 pm from the target surface show inversion (similar to that reported by Bhagevatula and Yaakobi) with the ratio -4. In this case there was no special cooling device, suggesting the inversion is a feature of the adiabatic expansion alone. The intercombination line far from the target is brighter than the resonance line, which is not consistent with steady-state CR calculations (Section IV and Fig. 2). This anomaly has been discussed in detail by Boiko et al. (1979a) who studied Nd and CO, LPPs by space-resolved X-ray spectroscopy. They calculated resonance to intercombination line ratios I,/ I, for nonequilibrium “overheated” and ‘‘supercooled’’ plasma in which the mechanism populating the excited states differs from the steady-state CR situation discussed earlier. For the latter case the excited states are populated by collisional excitation in the coronal regime and I , / l , is close to unity. In the supercooled case they are populated by recombination at rates proportional to their degeneracies, leading to a 3 : 1 triplet : singlet ratio and thus l , / l R 3. Their results obtained with the CO, laser show an interesting series of screened satellites extending between the He-like resonance line and the K, line (See Section IV, C). Recently the first time-resolved X-ray spectra of resonance, satellite and intercombination lines were obtained by Key et al. (1980a; see also Rutherford Laboratory, 1979, pp. 4-57) using an X-ray streak camera coupled to a crystal spectrometer. The time development of the lines closely mirrors the spatial structure in Fig. 10, reflecting the relationship between time and space through the motion of the luminous region of the plasma down the expansion plume.
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B. SPECTROSCOPY OF THE ABLATION-FRONT PLASMA Since the ablation front is the region of high temperature and it has a high density, it is the most intense emission region in an LPP and predominates in time- and space-integrated spectra. Interest has centered on the temperature and density in this region as a function of laser irradiation and target parameters and particularly on the effect of thermal conduction in maintaining a temperature comparable to that at the critical density in regions of supracritical density in the steep density gradient formed at the target surface. Useful information on the ablation plasma is obtained only in XUV or X-ray spectroscopy because of opacity of the plasma at lower photon
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energies. Early work used filtered X-ray continuum measurements (discussed in Section IX,C) and grazing incidence XUV spectroscopy (see, e.g., Fawcett er al., 1966; Basov et al., 1967; Burgess et al., 1967; Peacock and Pease, 1969; Seka et al., 1971). Work by Galanti et a/. (1975) and Galanti and Peacock (1975) referred to previously gives a good discussion of the ablation plasma characteristics at low irradiance (10" to 7 X 10l2 W cm-2) for low-Z (carbon) targets and 1.06-pm Nd laser wavelength, showing that kT, varies from 100 to 350 eV . of deducing ablation plasma and N , from Id' to 3 X Id' ~ m - A~ method N , up to 2 x Id' cm-3 from the Stark-broadened wings of opacitybroadened C VI resonance lines is discussed by Smith and Peacock (1978). The latter work emphasizes the opacity problem which is mitigated by observations at higher frequency with crystal diffraction X-ray spectroscopy. When sufficiently intense X-ray emission was obtained with higher laser power, Mead et al. (1972) initiated such studies. The approach was rapidly developed especially for the study of highly ionized ion resonance spectra (Boiko et al., 1978a), but also for diagnostic measurements in the ablation plasma. Aglitskii et al. (1974) described helium-like ion resonance and satellite spectra in which density and temperature in the ablation plasma were measured from intercombination and satellite intensity relative to resonance line intensity (see, e g , Fig. 2), but Weisheit et al. (1976) reported similar observations and emphasized the limitations of the interpretation proposed by Aglitskii et al. (see Section IV). Boiko et al. (1974) observed a plasma satellite line to the forbidden ls2s 'So-ls2 'So transition in MgXI (see Section IV). The implied value of N , was 1.8 x Id3cm-2 in a plasma formed by 5 x lOI4 W cm-2 irradiation in a conical depression in a solid target. (No sobsequent work has repeated this type of observation, however.) An alternative approach to diagnostic interpretation of X-ray resonance line spectra was used by Colombant et al. (1976) and Landshoff and Perez (1976) when they compared absolute and relative AlXII and XI11 line intensities with a computer model of the hydrodynamics and radiative transfer in the plasma (see Section VII). This approach has not been widely used since, owing to its complexity, effort has been directed rather to space- and time-resolved plasma parameter measurements. The information content of X-ray spectra was greatly enhanced by the introduction of spatial resolution by Boiko et al. (1975). This work and its extensive further developments reviewed by Boiko et al. (1979b) was based on line intensity ratios, which are in the optically thin regime in the expansion plume, but are severely complicated by opacity problems in the ablation plasma and are therefore not as well suited as other approaches to diagnosis of the ablation plasma.
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1
0
X-RAY PINHOLE PHOTOGRAPH
Ib
A l n 1s'-ls2p
b
-ii
2b
ib
AE lev1
N o X I Lp
'1 N a X 1s2-1s3p
NoXlL,
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ABLATION
NoX l s 2 - 1 s 2 p
PLASMA'
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SiXJZls2-ls2p
I A l X U I La
( 2 n d ORDER)(Znd ORDER)
I
I
FIG. I I. (a) A theoretical fit (for N, = 7. x l d 2 and kT, = 300 ev) to an experimental Ne X L, line profile from a 8.6-bar neon-filled microballoon. Broken line, optically thin; solid line, line center opacity T = 0.5; points, experimental profile. (After Yaakobi el at., 1979, Fig. IS.) (b) An experimental 7-pm space-resolved X-ray spectrum from the implosion shown in the X-ray pinhole image. The target was an aluminum-coated glass microballoon of 70 pm in diameter, 1.1-pm wall thickness filled with 2.5-bar neon, irradiated with a 16-5, 100-psec Nd laser pulse. Densitometry of the central spatial zone of the spectrum shows the core spectrum and of the outer zones the ablation plasma spectrum. (After Lunney, 1979.)
26 1
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
Yaakobi and Nee (1976) refined the space-resolved spectroscopy tech16-pm spatial resolution in X-ray spectra from an niques to give “exploding pusher” glass microballoon irradiated at -2 X lOI5 W cm-2 with a Nd laser. They applied analyses in addition to those of resonance to intercombination and satellite intensity ratios. These included Stark widths of OVIII resonance lines near the Inglis-Teller limit for density using a simple quasi-static broadening model and temperature measurement from Si XIV/Si XI11 line intensity ratios assuming coronal ionization. Their conclusions are somewhat at variance with current results (discussed later) in giving higher temperature (kTe-900 ev) and lower density ( N e l@’ cm-*) with 30-pm density plateau at the critical density. Key et a/. (1977, 1978a, 1979a,c), Lunney (1979), and Kilkenny et a/. ( 1980) developed more sensitive space-resolving X-ray spectrographs with higher (7 pm) spatial resolution and absolute sensitivity calibration. The more intense spectra thus obtained (see, e.g., Figs. 10 and 11) made possible several kinds of spectroscopic analyses. A detailed study was made of the ablation plasma formed by 100-psec, 2 X loi5 Wcm-2 Nd laser irradiation of Al-coated microballoons. Density was deduced primarily from Stark broadened H- and He-like line-widths by methods described in more detail in Section IX, D with absolute continuum intensity used to corroborate density values. Temperature was obtained primarily from the slope of the recombination continuum (see, e.g., Fig. 12) and corroborated with data on the ionization ratio obtained from line ratios (Section IV, D, 3). Values of kTe 600 eV and N , Id2cm-3 were thus obtained (see Table 11) and the fraction of absorbed energy coupled to the ablation process estimated (Rutherford Laboratory, 1979, pp. 4- 1 1). Aglitski et a/. (1977) studied X-ray spectra from plasmas produced by 250-psec Nd laser pulses emphasizing transient ionization effects in Ti, Fe, and Cr targets. Spectra produced by CO, laser irradiation (A = 10.6 pm) of Al targets have been studied by Enright et a/. (1977) for 2 x 1014Wcm-2 irradiance and by Courtraud et a/. (1977) for 4 x 10l2 W cm-2 irradiance, but in general there has been less work with CO, lasers because the lower critical density in the plasma leads to less intense spectra. Measurement of the penetration rate of the ablation front in solid targets has given important information revealing inhibition of thermal conduction. X-Ray spectra from layered targets first used by Seka et a/. (1971) show the arrival of the ablation front at an underlying layer when its resonance lines appear in the spectrum. This technique has been used by Young et a/. (1977) and Yaakobi and Bristow (1977) to record ablation rates with Nd laser irradiance on plane targets of 0.4-1 x lOI5 Wcm-2 and by Key et a/. (1978a) (see also Rutherford Laboratory, 1979, pp. 4-1 1)
-
-
-
M . H. Key and R. J. Hutcheon
262
20
I
hV kQV
FIG. 12. X-ray recombination continuum of SiXIII and SiXIV recorded by spaceresolved X-ray spectroscopy from the imploded glass in an exploding pusher target (see Table 11). (After Kilkenny el aL, 1980.)
for spherical targets irradiated by 2 x IOl5 W cm-2. The above work gave evidence of inhibited thermal transport. Work by Zigler et al. (1977) at 2 x IOl3 W cm-2 suggested an unusually large ablation rate as did that of Mizui et a/. (1977), but it is difficult to reconcile these results with the other observations. Ablation rates for 10.6-pm irradiation were measured from X-ray spectra by Mitchell and Godwin (1977) and for 0.53-pm irradiation by Kilfor a kenny et al. (1979). The rate increases for shorter wavelengths as given value of I X 2 . Time-resolved measurements of ablation rate on layered targets and of transient Stark widths of lines in the ablation plasma have been obtained recently by X-ray streak photography of crystal dispersed X-ray spectra (Key et al., 1980a; see also Rutherford Laboratory, 1979, pp. 4-57; Kilkenny et al., 1979). The absolute intensity of integrated X W emission from LPPs has been discussed by Peacock (1976) and measured by, for example, Rosen et al. (1979), Shay et a/. (1978), Eidmann et al. (l976), Nagel et al. ( 1974), and Mallozzi et al. (1974). Conversion of up to 35% of the absorbed laser energy to XUV emission was noted by Rosen et al, for Au targets.
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
263
C. CONTINUUM AND K,-EMISSION SPECTROSCOPY (THEZONEOF HOT-ELECTRON PREHEATING) 1. Space- and Time-Averaged X-Ray Continuum Emission We are concerned here mainly with observations of the spectroscopic effects of hot-electron preheating of solid-density material, that is, with effects associated with the hot-electron temperature TH. However, since these observations also yielded the early results for the ablation plasma temperature T,, some discussion of T, is included. Recording of soft X-rays transmitted through absorbing filters and determination of the continuum slope gave the first data on the ablation plasma temperature T, (see, e.g., Basov et al, 1969; Bobin et al., 1969; Puell, 1970; Donaldson et al., 1973). These measurements recorded soft X-ray continuum at photon energies above the highest recombination edge but only up to several kiloelectron volts. As more powerful lasers became available and higher irradiance was achieved, scintillator/photomultiplier detectors were used to record X rays of photon energy in the range 10-100 keV. It was observed that the simple exponential decay of the continuum spectrum measured for soft X rays did not continue out into the hard X-ray continuum. The hard X-ray “tail” was much more intense than expected (Basov et al., 1971; Mead et al., 1972; Shearer et al., 1972; Buchl et al., 1972). Since the spectrum was not a simple exponential, deduction of its form from intensities transmitted by various absorbers was complicated (see Section VI, A). Iterative numerical methods have been commonly used to fit trial spectra to the experimental data, for example, Slivinski et al. ( 1975) and Johnson ( 1974). A more direct approach was therefore adopted by Kephart et al. (1974) who dispersed the spectrum by crystal diffraction and measured absolute spectral intensity in the range 3-50 keV for 10l6 and lOI4 Wcm-* irradiance with 1.06- and 10.6-pm wavelength lasers, respectively. Ripin et al. (1975) also measured absolute spectral intensity from 1 to 300 keV using the filter method with 10l6 Wcm-’ irradiance at 1.06 pm. Recording of X-ray continuum spectra using absorbing filters is now used widely to give information on the electron temperature in LPPs. It has been found in many systematic surveys that the spectrum can generally be subdivided into a soft X-ray component at photon energy below 10 keV with a n exponential slope of temperature T, characterizing the ablation-front plasma electrons. This emission is mainly recombination radiation originating in the ablation-front plasma. T, is < 1 keV even for the highest irradiances studied (- lo” W cm-2). The hard X-ray spectrum
264
M . H . Key and R. J . Hutcheon
hv(keV1
FIG. 13. X-ray continua recorded with multichannel absorbing filter system. (a) Tungsten glass targets; dot-dash line and black squares, 5.7 X IOl4 W cm-2; dashed line and open squares, 4.4 X IOI4 W cm-2; solid line and open triangles, 7.2 X lo” W cm-2. (b) Parylene (CH) targets; dashed line and open circles, 5.3 X lot4W cm-2; dotted line and black squares, 4.4 X loi4W cm-2; double-dot-dash line and black triangles, 1.4 x IOl4 W cm-2. (After Shay er al., 1978.)
between 10 and 100 keV has a slope corresponding to the hot-electron temperature TH. The emission is thick-target bremsstrahlung in the solid material ahead of the ablation front. Figure 13 from Shay et al. (1978) illustrates these remarks. Work with 1.06-pm lasers has established the irradiance dependence of TH. Manes et al. (1977) reported kT, ranging from 4 to 30 keV in the irradiance range 10’4-1017WcmP2,and Estabrook and h e r (1978) have discussed the theoretical interpretation of these data. Shay et al. (1978) studied the change in the hard X-ray spectrum for irradiances between lOI4 and 1015 W cm-2 when the target atomic number was vaned from low Z (parylene) to high Z (tungsten glass). Their data show a 100 times greater hard X-ray intensity from the high-2 target (see Fig. 13). This is associated with about a factor two times greater TH . Rosen et al. (1979) have made a detailed study of interaction with Au targets for irradiances between 3 X 1014 and 3 X 1015 Wcm-2, finding kTH varying from 11 to 30 keV. Their work supports the conclusion that TH increases substantially with 2 as does the earlier work of Manes et al. (1977). Pulse duration also affects electron temperature in the limit of pulses shorter than 100 psec for whch the hydrodynamic response of the plasma
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
265
flow has insufficient time to approach a steady condition. Cold-electron temperature measurements for such pulses are described by Salzmann (1973) and by Donaldson et al. (1980). The latter give data for 35-psec pulses at irradiances between 2 x 10" and 2 x IOl4 W cm-, which show about three times lower temperature T, than for longer pulses in the hydrodynamic steady state. The magnitude of TH is not, however, reduced from its long pulse limit as seen in results reported by Luther-Davies W ern-,. This might (1977) for 25-psec pulse irradiation at loi4to 3 x be expected since TH depends on the instantaneous irradiance at the critical density more than on hydrodynamic response. Investigation of X-ray continua from CO, LPPs (A = 10.6 pm) began later than the 1.06-pm work because of the later development of highpower CO, lasers. It was more difficult to achieve short-pulse high-power operation, and thus focused irradiances were initially low. Martineau et al. (1974), Fabre et al. (1975), and Hall and Negm (1976) have discussed results at < lo1, W cm-, which show qualitatively similar features to the 1.06-pm data, that is, cold- and hot-electron temperatures with kTH reaching -2 keV at 5 X 10" W c m p 2 and kT,- 100 eV. With large electron beam sustained CO, laser discharges high irradiances were generated, notably in the CO, laser program at the Los Alamos Laboratory. Giovanielli (1976) has compiled THdata from CO, laser experiments in the range 1011-10i5W cm-, including high irradiance results from his own laboratory and from Manes et al. (1976). The X-ray spectra from the CO, laser experiments show higher TH for a given irradiance when compared with results compiled from Nd laser experiments. However, by plotting all TH values as a function of the parameter (Ih') Giovanielli (1976) found a single-valued relationship between THand (Ih'). A practically important consequence of this scaling is that it suggests that minimum hot-electron preheating range is obtained with minimum laser wavelength. First results of investigations with h = 0.53 pm seem to confirm the I X 2 scaling. Amiranov et al. (1979) found kTH- 3.5 keV at lOI5 Wcrn-,, and this is in fair agreement with the 1.06- and 10.6-pm data. In similar work (Rutherford Laboratory, 1979; Kilkenny et al., 1979) both short (70 psec) and long (0.5-1.5 nsec) pulses have been used at intensities up to W cm-, again giving results consistent with the I X 2 scaling.
2. Absolute Energy Deposition by Hot Electrons As noted in Section VI,A the absolute intensity of hard X-ray bremsstrahlung is a measure of the total thermal energy carried into a solid target by electrons of lunetic energy in excess of the X-ray photon energy at which the spectral intensity is measured. One of the first such estimates
266
M . H . Key and R. J . Hutcheon
was by Eidmann et al. (1976) who found that 5 J of energy was deposited by hot electrons from 20 J incident at 3 X IOl4 W cmP2 and X = 1.06 pm. Brueckner (1977) has considered the problem in more detail and has found that the deposition of energy by hot electrons amounts to typically 25% of the absorbed energy in experiments with intensities between I O l 5 and loi6 W ern-,.
3. Polarization and Isotropy of X-Ray Bremsstrahlung The extent to which hot-electron generation gives a directional flux of electrons may be investigated by examination of the isotropy of intensity and polarization of hard X-ray bremsstrahlung (see Section VI, A). Eidmann ( 1975) has considered the theoretical anisotropies for a directional electron beam with a non-Maxwellian velocity distribution. Eidmann et al. (1976) found no intensity anisotropy in 2.2- 16.5-keV X-ray emission from a solid D, target irradiated at 3 X loi4Wcrn-, by a Nd glass laser. Young (1974) also found isotropy of X-ray intensity for photon energies between 15 and 30 keV with targets of various atomic number irradiated at up to lot6 Wcrn-, with a Nd laser. Significant anisotropy of intensity of 2-4-keV X rays with orientation relative to the laser polarization vector was reported by Krokhin et al. (1975) for plane solid targets irradiated at 2 X loi4 W cm-, and h = 1.06 pm, but no corroboration of this result has been reported subsequently. Godwin et al. (1972) examined polarization in hard X-ray continuum emission with inconclusive results. 4. Time-Resolved Continuum Spectra
Recently there have been developments in instrumentation that have yielded time-resolved X-ray continuum spectra. X-Ray vacuum photodiodes with subnanosecond response time have been applied to recording soft X-ray emission from plasmas produced by nanosecond duration laser pulses (Key et al., 1974b; Kornblum and Slivinski, 1978; Tirsell et al. 1978). Faster time resolution has been obtained with X-ray streak cameras using filters selecting X rays of varying energy on different spatial sections of the streak record. Streak camera measurements of soft X-ray continuum emission have been reported by Key et al. (1976) and by Attwood et al. ( 1976). Time-resolved hard X-ray spectral information has been obtained by Lee and Rosen (1979).
5. Space-Resolved Continuum Emission X-Ray images recorded through absorbing filters can give spectral information from multiple images similar to that above but with spatial
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
267
resolution. The first such observations were made with pinhole cameras (Donaldson et af.,1973) and the technique has since been widely applied. The qualitative picture of the plasma obtained this way is very useful (see, e g , Fig. 8), revealing features such as the structure of the ablationplasma plume on plane targets (Eidmann et af., 1976). Evidence of selffocusing of the laser beam in the plasma in the form of bright spots in X-ray images was found by Key et al. (1974a) and Manes et af. (1977). The successful production of a high-temperature implosion core in exploding pusher implosions was first revealed in X-ray images by Campbell et af. (1975). Seward et af. (1976) obtained improved imaging with X-ray microscopes. High-quality 5-pm resolution images showing the symmetry of exploding pusher implosions under six-beam irradiation have been reported (see Fig. 8b) by Thorsos el af. (1979). Coded aperture zone plate imaging has high sensitivity and has been used for hard X-ray (10 keV) and particle-emission imaging of LPPs by Ceglio and Smith (1978). Time-resolved images have been obtained with an X-ray streak camera. In this way the temporal development of the ablation plasma plume on plane targets was studied by Key et af. (1976) and the implosion dynamics in exploding pusher targets was recorded by Attwood et af. (1977) and Key et af. (1979d). Quantitative interpretation has been used for measuring spatial profiles of temperature variation from continuum intensity ratios in images obtained via different absorbing filters (Zakharenkov et af., 1975; Eidmann et af.,1976) and for measuring electron density profiles from absolute X-ray intensity in soft X-ray images (Key et af.,1974b). An alternative has been to compare details of the experimental image with computer simulated images from models of the plasma hydrodynamics. This has been done to derive evidence for thermal transport inhibition in the interaction of laser radiation with solid-plane targets by Haas et af. (1977), Mead et af. (1976), Shay et af. (1978), and Rosen et al. (1979). A similar approach has been adopted comparing both streaked and timeintegrated exploding pusher X-ray images with computer simulations, notably in the work of Campbell et af. (1975), Manes and Storm (1977), Attwood (1978), and Goldman et al. (1979). 6. K, Emission If the target material is of suitable atomic number, K, line emission may be generated by long-range hot electrons, and it may be observed in crystal dispersed X-ray spectra. An early study of K, lines from Si in experiments with Al-coated SiO, targets is that from Mitchell and Godwin (1976) whose experiments involved 5 x 10” W cm-, irradiation with a CO, laser. The depth of Al
268
M . H . Key and R. J. Hutcheon
needed to suppress Si K, radiation was used to estimate the average range of the hot electrons and thus the 18-keV average energy. Zigler et al. (1977) used 3 x lof3W cm-, Nd laser irradiation of polymer-coated Al targets and studied the behavior of Al K, radiation, which they attributed to K-shell ionization by hot electrons. Mitchell and Godwin (1977) presented further detailed results from their study of K, radiation from CO, laser irradiated targets estimating both the range and temperature of the hot electrons but concluded that quantitative measurement of hot-electron energy deposition was not possible from their data. A quantitative study of hot-electron energy spectra and the preheating due to the hot electrons was made by Hares et al. (1979; see also Rutherford Laboratory, 1979, pp. 4- 13): 100-psec, 1.06-pm laser pulses were focused to 1015 W cm-2 on layered targets consisting of 0.1-pm Al, 1.0-pm SiO, 3-pm KCl, a variable thickness (2.5-50 pm) of polymer, and 2.5-pm CaF,. K, emission was recorded from K, C1, and Ca (shown in Fig. 14) together with Al and Si line and recombination continuum emission from the ablation plasma. The ablation depth was sufficient to penetrate through the Al layer and partly through the SiO layer. The intensity of the spectra was determined absolutely from instrument calibrations, and the variation in the K, yield from the rear CaF, layer with thickness of the polymer layer was recorded. The contribution of photoionization to the K, intensity was calculated from the measured recombination continuum intensity and from the known photoionization cross sections, as well as being checked experimentally. The conclusion was that it could be significant if the K-shell ionization energy was low enough to fall near the peak of the recombination continuum (seen in Fig. 14). Thus the experiments described earlier where Si or Al K, emission was observed probably had significant photoionization contributions to the K, yield. A second relevant process is the shift of K, wavelength with progressive ionization of the atom (Section IV). With sufficient hot-electron energy deposition the K, yield saturates through shifting of the emission wavelength outside the neutral K, linewidth. Hares et al. (1979) showed that their observed K, yield data could be interpreted to yield the total energy deposition from hot electrons as a function of depth in the target, and the energy spectrum or temperature of the hot electrons. Table I gives the energy deposition results. I t follows from these data that from 20 J incident 2.2 J went into fast electron energy deposition, and this amounts to about 35% of the absorbed energy, in broad agreement with the analyses based on hard X-ray continuum discussed in Section IX,C,2. The fact that the deposition is measured as a function of depth allows the deposition density and thus the pressure due to hot-electron preheating to be evaluated. In a
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
269
FIG. 14. X-ray spectrum of a layered target with KCI and CaF, fluors emitting characteristic K lines. The thermal spectrum of Si and Al from the ablation plasma is also seen. (After Hares ef al., 1979.)
TABLE I K, MEASUREMENTS OF HOT-ELECTRON ENERGY DEPOSITION IN A LAYERED TARGEP
Fluor layer
Depth in target (i.e., mylar layer thickness pm)
Deposited energy density in fluor layer (J cm-’)
KCl CaF, CaF, CaF,
Front fluor 2.5 pm 12.5 pm 25 pm
1.4x 107 L O X 107 5.1 X lo6 1.7 X lo6
‘After Hares ef 01. (1979).
recent experiment Bond et al. (1980) and Kilkenny et al. (1979) used the same technique to show that hot-electron preheating can be significantly suppressed by the resistive inhibition of the hot-electron current in a layer of low-density gold.
D. IMPLOSION-CORE SPECTROSCOPY The first reliable spectroscopic data on density and temperature in exploding pusher implosion cores was obtained using X-ray spectroscopy
270
M . H . Key and R. J. Hutcheon TABLE I1 PLASMA
PARAMETERS DEDUCED FROM SPACE-&SOLVEDX-RAY SPECTRA OF EXPLODING PUSHERIMPLOSION^
Parameter
Ablation plasma
Imploded glass
Compressed gas
Element Continuum temperature (ev) Line ratio temperature (ev) Source sue ( crm) N J X 10-*~cm-’) N H (gxx cm-’)
A1
Si
Ne
6502 100
430 2 40
300 f 50
500k 50
600 5 20
310 f 30
45 2 5
22 2 2
1 f 0.3 4 2 1 0.15
1224 0.29
28 2 2 XI221 2.8 2 0.5 1424 0.23
a*‘zzlh
‘After Kilkenny et
I5 2;
a/. (1980).
in simultaneous experiments by Key et al. (1977) and Yaakobi et al. (1977). Eight-bar neon-filled glass microballoons were imploded with Nd laser pulses of 50-100 psec duration at an intensity 2 X 10’’ Wcm-’. Heavily Stark-broadened resonance spectra of Ne X and Ne IX were recorded in space-integrated spectra by Yaakobi et al. and in 7-pm space-resolved spectra by Key et al. The Ne emission was from the implosion core as was established from the 15- to 20-pm spatial spread of the emission in space-resolved spectra (e.g., Fig. 11) and also more recently by timeresolved X-ray spectroscopy (Key et d.,1980a; Rutherford Laboratory, 1979, pp. 5-6), the latter showing 100 psec duration N e X La emission with 100 psec delay (the implosion time) relative to emission from the microballoon ablation plasma. Temperature and density in the implosion core were deduced from the spectra (see, e.g., Table 11). Line-profile calculations with opacity corrections based on Eq. (71) were fitted to experimental data. Profiles of transitions from n = 3 and above were typically not much modified by opacity, whereas the transitions from n = 2 were strongly opacity broadened. Yaakobi et al. fitted Hooper’s full standard model calculations for Ne X to La, L,, and L, lines (see Fig. 11). The opacity broadening of La gave a measure of J , N H ( g ) d l , and, with estimates of the ionization fraction, a measure of the line density pr. The optically thin L, line and near-optically thin L, lines gave a measure of N , (see also Yaakobi et al., 1978, 1979). Key et al. (1977, 1978a, 1979a) used Richards’ full standard
-
-
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
27 1
calculations for N e X and also for NeIX to fit their data, and later the more refined Dufty-Lee calculations for Ne IX, Ne X, Al XII, Al XIII, Si XIII, Si XIV, Ar XVII, and Ar XVIII (Key et al., 1979~;Kilkenny et a/., 1980). The latter paper discussed the spectroscopic analysis in detail. These analyses all led to similar conclusions for the density of the compressed gas, which was found to be Ne-00.5-1 x Id3cmP3. The high sensitivity and absolute intensity calibration of the miniature spectrographs developed by Key et al. enabled space-resolved measurements of other features of the implosion-core spectra, notably the SiXIII and SiXIV spectra of imploded glass surrounding and possibly mixed with the compressed gas, and the slopes and absolute intensity of continua. The latter (see Fig. 12 and Table 11) gave the most direct temperature diagnostic as well as a check on the density. A determination of the ionization fractions from optically thin LTE lines (see Section IV) gave an alternative T, estimate via the CR ionization model (see Section 11). The compressed gas temperature was typically kTe 350 eV. The parameters of the compressed glass could be separated from those of the ablation plasma in space-resolved spectra (see, e.g., Fig. I I ) , and analysis of the Si spectra (Table 11) showed similar density and temperature to the compressed gas. Only a small fraction of the glass shell was included in the emitting region with the remainder forming a nonluminous cooler blanket causing self-reversal of some Si XI11 lines (Kilkenny et al., 1980). Extension of these methods has led to observation of spectra from Ne at up to 56-bar initial pressure (Yaakobi et a/., 1978, 1979; Skupsky, 1978), from neon-doped D, and (D, + T,)-filled targets (Auerbach el a/., 1979; Key et a/., 1978a), from ArXVII in Ar-filled targets (Key et al., 1979~; Kilkenny et a/., 1980), * and from CO, laser-driven implosion of Ne-doped (D, + T,)-filled targets (Mitchell et a/., 1979). The mixing of outer layers of the microballoons with the implosion core has been studied by Key et al. (1978a, 1979c, 1980a) from Al X-ray lines in the implosion-core spectrum of glass microballoons with a surface coating of Al. New work on the interpretation of implosion-core spectra includes density determination from continuum edge shifts (Lee and Hauer, 1978) and from satellite line intensities (Seely, 1979, and Fig. 3). Better radiative transport modeling is being developed (Section VII; Yaakobi et al., 1978; Skupsky, 1978; Kilkenny et a/., 1980; and Fig. 7) but has not made any change in the essential conclusions about implosion-core parameters, though it should lead to more detailed knowledge on parameter distributions as opposed to average values.
-
'see Note Added in Proof, p. 280.
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M . H . Key and R. J . Hutcheon
E.
SHADOWGRAFWY AND &SORPTION SPECTROSCOPY
X-RAY
A new and promising area involves the spectroscopic absorption characteristics of LPPs. Key et al. (1978b) used a 100-psec pulse of soft X rays from a secondary LPP as a background source for producing pulsed shadowgraph images of an exploding pusher-type implosion. The implosion time of the glass microballoon target was - 5 0 0 psec due to its dense (87 bar) neon gas fill, and there was therefore negligible X-ray emission from the implosion core. The spectrally averaged continuum opacity [due to photoionization Eq.'(63)] was measured by densitometry of the shadowgraph images and hence the 50 times volumetric compression of the neon to 4 g cm-3 was determined. This approach is particularly advantageous for the study of ablative implosions, and Key et al. (1979~)described a study of the implosion of polymer-coated glass microballoons irradiated with two beams at 4 X IOl3 W cm-2 with a 2-nsec laser pulse. A shadowgraph image was projected onto the slit of an X-ray streak camera, and the resulting streaked shadowgraph &splayed the implosion in 1.5 nsec to a high-opacity core. Analysis of the results gave the ablation pressure and data on an apparent breakup of the shell which could be attributed to hydrodynamic instabilities. Lewis et al. (1980) refined the technique using six-beam irradiation at IOl4 W cmP2 of an argon-filled polymer shell with an X-ray microscope producing a shadowgraph image at the streak camera slit. Computer simulations indicated the possibility of direct determination of the gas density from continuum opacity at peak compression due to the X-ray absorption of a low-Z polymer shell with a high-Z gas fill. Spectrally resolved absorption studies are at an even earlier stage though Jaegle et al. (1974) have used an auxiliary LPP as a VUV source to study absorption (also negative absorption) in VUV lines in the Al IV resonance spectrum of an LPP.
-
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280
M . H . Key and R. J. Hutcheon
Weisheit, J. C. (1975). J. Phys. B 8, 2556. Weisheit, J. C., Tarter, C. B., Scofield, J. H., and Richards, L. M. (1976). J. Quant. Spectrosc. & Radial. Transfer 16, 659. Whitney, K. G., and Davis, J. (1974). J . Appl. Phys. 45, 5294. Wiese, W. L. (1965). In “Plasma Diagnostic Techniques” (R. H. Huddlestone and S. L. Leonard, eds.), p. 265. Academic Press, New York. Wiese, W. L., Smith, M. W., and Glennon, B. M. (1966). Natl. Stand. Re$ Data. Ser., Natl. Bur. Stand. 1. Wilson, R. (1962). J. Quanr. Spectrosc. & Radial. Transfer 2, 477. Wilson, R. H. (1972). Report MRC-R-I 1. Analytic treatment of the line radiation from dense plasmas. Mission Research COT. (unpublished). Yaakobi, B., and Bristow, T. C. (1977). Phys. Rev. Lett. 38, 350. Yaakobi, B., and Nee, A. (1976). Phys. Rev. Lett. 36, 1077. Yaakobi, B., Steel, D., Thorsos, E., Hauer, A., and Perry, B. (1977). Phys. Rev. Lett. 39, 1526. Yaakobi, B., and 14 co-authors (1978). Report No. 74. Laboratory for Laser Energetics, University of Rochester, Rochester, New York (unpublished). Yaakobi, B., et al. (1979). Phys. Rev. A 19, 1247. Young, F. C. (1974). Phys. Rev. Lett. 33, 767. Young, F. C., Whitlock, R. R., Decoste, R., Ripin, B. M., Nagel, D. J., Stamper, J. A., McMahon, J. M., and Bodner, S. E. (1977). Appl, Phys. Lett. 30,45. Zakharenkov, Yu. A., Zorev, N. N., Krokhin, 0. N., Mikhailov, Yu. A., Rupasov, A. A., Skliskov, G. V., and Shikanov, A. S. (1975). JETP Len. (Engl. Transl.) 21,259. Zeldovich, Ya. B., and Raizer, Yu. P. (1966). “Physics of Shock Waves and High Temperature Hydrodynamic Phenomena,” Vols. 1 and 2. Academic Press, New York. Z h e k i n , A. N., Koshelev, K. N., Kryukov, P. G., Letokhov, V. S., and Chekhalin, S. V. (1977). JETP Lett. (Engl. Transl.) 25, 300. Zigler, A., Zmora, H., and Schwob, J. L. (1977). Phys. Lett. A 63, 275. Zink, J. W. (1968). Phys. Rev. 176, 279. Zwicker, H. (1965). I n “Plasma Diagnostic Techniques” (R. H. Huddlestone and S. L. Leonard, eds.), p. 214. Academic Press, New York.
NOTE ADDED IN PROOF A recent publication (Yaakobi et al., 1980) reports measurements of Ar XVII and Ar XVIII resonance line profiles emitted from laser-imploded Ar-filled microballoons. The targets were similar to exploding pusher targets but with somewhat thicker walls (up to 5 pm) obtained by coating glass microballoons with plastic. Short pulse (50 ps) irradiation at high energy (100 J) gave an implosion of the exploding pusher type, but the effect of radiation cooling of the Ar together with the effect of the thicker microballoon wall resulted in electron-number densities up to 1.5 x l d 4 e/cm’ in the compressed Ar.
ADVANCES IN ATOMIC A N D M O L E C U L A R PHYSICS, VOL. 16
RELATIVISTIC EFFECTS IN A TOMIC COLLISIONS THEOR Y B. L. MOISEIWITSCH Department of Applied Mathematics and Theoretical Physics The Queen’s Universily of Belfast Belfast, Northern Ireland
I. Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11. Excitation and Ionization
A. Meller Theory for High Energy Incident Particles
. . . . . . .
B. Impact Parameter Treatment . . . . . . . . . C. Scattering of Electrons and Positrons by Free Electrons D. Kolbenstvedt Theory of K Shell Ionization. . . . . E. Inner Shell Ionization Theory Using Meller Interaction F. Relativistic Effects at Low Impact Energies. . . . . 111. Electron Capture . . . . . . . . . . . . . . . A. Impact Parameter Formulation . . . . . . . . . B. Wave Formulation . . . . . . . . . . . . . C. Classical Double Scattering . . . . . . . . . . D. Radiative Capture . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . .
. . . .
. . . .
. . . .
. . . .
.281 .282 .282 .286 .289 .291
.294 .304 .307 ,307 ,309 ,312 .315
.316
I. Introduction Because the experimental data on ionization of atoms by charged particles have become more precise and detailed at low impact energies, and also because the experimental work has been extended to very high impact energies in recent years, it has been found necessary to generalize the nonrelativistic theories of atomic scattering to allow for the effects of relativity. This has been accomplished by (1) Using relativistic atomic electron wave functions (2) Using relativistic continuum electron wave functions (3) Taking account of the high velocity of relative motion of the colliding particles by using relativistic kmematics and mechanics, as well as relativistic transformation theory 28 1
Copyright 0 1980 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-12-003816-1
282
B. L. Moiseiwitsch
In Section I1 of the present article we shall discuss excitation and ionization, the subject of a first relativistic theoretical study by M ~ l l e rand by Bethe as long ago as 1932, and in Section 111 we shall examine electron capture, both collisional and radiative, by incident ions.
11. Excitation and Ionization A. MBLLER THEORY FOR HIGHENERGY INCIDENT PARTICLES
The relativistic analysis introduced by Mdler (1932) is based on a first-order time-dependent perturbation theory. He discussed the scattering of electrons by hydrogenic atoms, but the theory can be readily extended to include protons by considering the general case of incident fermions. The incident fermion, which we shall denote by subscript 1, having mass m,,momentum pI = hk,, and spin component s, along the axis of quantization in the rest frame, may be represented at high energies of impact by the Dirac relativistic plane wave function = Nlaklsl
&,s,
exp i(kl
* rl
- @It)
(1)
where the angular frequency w , is given in terms of the energy E l of the fermion by w, =
E l / h = c ( m ; c 2 + p:)
1/2
/h
Here N , is a normalization factor, r, the position vector of the fermion referred to the nucleus of the target atom, and u ~ ,a~free , particle spinor having components
[ 1,
0,
klz/K,,
( h x
+ iky)/K,]
for spin s, = t and components
[o.
1 7
( h x
- ikly)/KI,
-kIz/Kl]
for s, = - i , where K , = (mlc2 + E , ) / h c , the z axis being taken as the axis of quantization. Denoting the initial and final states of the incident fermion by k;s; and k{s{, respectively, its charge and current densities are given by P =
zle@k{s{@k',s{ t
j = Z,ec~~~s,,a"'~k~,;
(3) (4)
RELATIVISTIC EFFECTS IN ATOMIC COLLISIONS THEORY
283
where Z , e is the charge of the incident particle, a is the vector composed of Dirac matrices, and denotes the adjoint. Then the scalar and vector potentials at the point r2 and time t produced by the incident charged particle are given by - r2l dr,
V'"(r270 = J[P(rl)l/lr, A ( v ~ + ~=)
f J[j(rl)1/~rl
-
r21drI
(5) (6)
where [p(r,)] and [i(rl)] are the charge and current densities at the retarded time t - (r, - r21/c. The perturbation causing the atomic electron, denoted by subscript 2, to undergo a change of state is
-e
[
v"' (r2, t )
-
-
a(') A(')( r2 ' t )]
(7)
so that the matrix element characterizing the atomic transition is; +j s i is accordingly - eJ+j:(r2, t ) [ v(I)(r2,t ) - a(') A ( ' )(r29 )]+ls:(r2. t)dr2
(8)
where +i3i and denote the initial and final relativistic wave functions of the atomic electron. This matrix element may be written (k{ j I U I kii) = N(N;Ju(t+~(r2)(/(rl, r2)af+i(r2)e'q''1drldr,
(9)
where the M d e r interaction U ( r l , r2) is given by
+,(r2) and +,(r2) being the time-independent Dirac relativistic atomic wave functions. Here we have dropped symbols describing spin, and set q = kf - k{ so that Aq is the momentum change of the incident fermion, and denoted the energy loss of the incident particle by AE, = El - E { . Now integrating over the position vector rl of the incident particle we find
where
284
B. L. Moiseiwitsch
and qo = AE,/hc. The J, term arises from the Coulomb interaction - Z ,e 2 / ( r , - r21 between the incident fermion and the atomic electron, while Fii comes from the current-current interaction - Z,e2a(I) a ( 2 ) / l r l - r21. The quantity qi in the denominators of J, and F, arises as a consequence of making allowance for retardation and appears in the Msller interaction U(r,, r2) as the factor exp(i AE,Ir, - r21/hc). It follows from (11) that the differential cross section for the scattering of the incident fermion into the element of solid angle da is
where qij(q) = N{Nisa{t#/(r2)(l - a(’) * a(2))u~#i(r2)eiq*rzdr2 (15)
This form for the differential cross section was obtained by Bethe (1932), although it is based on the original work of Msller (1931, 1932). We may rewrite the differential cross section as
where a = e2/hc is the fine structure constant. Since the deflection of the incident fermion is very slight and a change of spin of the incident fermion is very unlikely to occur, we may set (E{E;)1’2N{N;a{ta;
N
c h k ; P - l S,;,,
(E~E;)”2N{N;a(ta(’)~a; N chk‘, Ssrsr where
P = u / c and u is the speed of the incident fermion. Hence
taking the z axis along the direction of motion of the incident particle. We have used here the original Msller treatment to derive the differential cross section (17) in order to show in detail how the relativistic corrections arise. An alternative treatment leading to the same formula can be based on the Lorentz transformation of the Coulomb potential due to the incident charged particle, which is the approach we shall employ to discuss capture at relativistic energies in Section 111.
RELATIVISTIC EFFECTS IN ATOMIC COLLISIONS THEORY
285
Mdler (1932) chose the atomic wave functions to have the approximate form
where x,(r) is the Schrodinger nonrelativistic wave function for the state i with nonrelativistic eigenenergy E, and aXiis a spinor shown by Darwin (1928) to take the semirelativistic forms
where K; = (2mc2 + E , ) / A c . Then the differential cross section may be expressed as
with
and where we have averaged over the initial spin states s; = i , summed over the final spin states s{ = 4 , - of the atomic electron. The nonrelativistic formula for du,,(q) may be obtained by neglecting Pe,; compared with c,, and dropping the retardation term 4;. For quo << Z , , where a, = h 2 / m e 2 is the Bohr radius and Z , is the effective charge of the atomic nucleus at the orbit of the active electron in
+
286
B. L. Moiseiwitsch
its initial state i , the Bethe (1930) approximation may be used giving
k&I)
(24)
+ iP~&(q)I2-s21('s)~12
where
I
(25)
(rJU= %*(r)~(r)& rdr -
and s = (qx,qy, q, - Pq0). Since q, N q 0 / P it follows that s2 N
q 2 - q; - y -24;
(26)
where y = (1 - P 2 ) - ' I 2 ,and so
The excitation cross section for the transition i + j is given by 0'I . .=
~q"'"duv( q) Qmin
where qmin= k ( - k( and qmax= k ( + k { . The upper limit of the q integration may be replaced by 2k; N _ 00, while the lower limit is determined by conservation of energy and to good approximation is given by qmin= 40/ P. Following Bethe and assuming that the integrand duq/dq in (28) becomes small for q2 > 2mlAE21/h we obtain the excitation cross section 4nz;a2 cJ..N ~
'I
P2
In
2mc2P2 -
~
cg
ln(1
-
p')
-
1
P2
(29)
where Cg is a constant having a value of the order of the excitation energy AE, = E/ - E,. This is essentially the asymptotic formula obtained by Mnller (1932) and by Bethe (1932). I t differs from the nonrelativistic asymptotic formula derived by Bethe (1930) because of the presence of the terms ln(1 - P 2 ) P2, which causes the excitation cross section aii to pass through a minimum at a certain energy as can be seen from Figs. 6 and 7. The ln(1 - P 2 ) term arises from the inclusion of retardation in the interaction (7).
+
B. IMPACTPARAMETER TREATMENT The method of impact parameters was first used by Williams (1933) to treat distant collisions between atoms and incident charged particles at relativistic impact energies. In this approach the incident particle having
RELATIVISTIC EFFECTS IN ATOMIC COLLISIONS THEORY
287
charge Z , e is assumed to move along a straight line path with constant velocity v. We let S , and Sh denote frames of reference with origins at the nucleus A of the target atom and at the incident particle B, respectively. Then S , is the rest frame of the target atom and Sgis the rest frame of the incident particle. We shall suppose that the incident particle B reaches the point of closest approach C to the target at time t = t’ = 0 and that the distance of C from A, the impact parameter p, is considerably greater than the dimension of the atom. Choosing the z and x axes parallel and perpendicular, respectively, to the path of the incident particle B, and t h e y axis perpendicular to the plane of motion, we see from Fig. 1 that at the nucleus A of the target atom, the electric field strength 6 of the incident particle B referred to the rest frame S , of the target atom has components
and &,, = 0, using the Lorentz transformation. We may neglect the presence of the magnetic field since the velocity of the electron in the atom is taken to be small compared with v . It follows that the perturbation acting on the target atom is composed of two terms, V,,(z,t) = ezE, and V , ( x , t) = ex&,. Hence the probability of
C
Vt
B
V
FIG. 1. Collision between an incident charged particle B, moving in a straight line path with velocity v, and an atomic system with nucleus A. The point of closest approach is C and the impact parameter is p.
288
B. L. Moiseiwitsch
a transition i +j caused by these is given by Pv = PI,+ P , , where
AE2 being the energy of excitation. Now carrying out the integrations over the time t we find
where KO and K , are modified Bessel functions of the second kind. To derive the cross section uv for the i -+ j excitation of the target atom taking account of distant collisions only, we now integrate over impact parameters p greater than a certain least value po chosen to be somewhat larger than the dimension of the atom. Using the asymptotic formulas for Ko(x) and K , ( x ) , where x is large, we obtain
uii' = 2nJWP* ( p ) p d p PO
and since, after summing over the azimuthal quantum numbers of the initial and final states of the target atom, we have l(x)v12 = I ( Z ) ~ [ ~ it follows that
which is in accordance with the asymptotic formula (29) derived by Msller (1932) and Bethe (1932).
289
RELATIVISTIC EFFECTS IN ATOMIC COLLISIONS THEORY
C. SCAITERING OF ELECTRONS AND POSITRONS BY FREEELECTRONS Let us denote the two free leptons involved in the collision by 1, 2 and take a frame of reference S* with origin at the center of mass. The two leptons have the same energy E* and equal but opposite momenta. Then if i , j denote the initial and final states, the differential cross section may be written (Mdler, 1932) in the form da = ( e / h ~ ) ~ E *S~)(dS2*
(39)
where dS2* is an element of solid angle located about the direction of Scattering in the center of mass system, and
2
the summation being over all spin values of the two leptons in the initial and final states. The first term of S corresponds to direct scattering, while the second term arises from exchange scattering. If u* is the initial speed of each lepton referred to the frame of reference S* and y* = (1 - u * ~ / c ~ ) - ’we / ~ have , E* = y*mc2. Then, integrating over the azimuthal angle +*, we obtain
+
da = ( 2 ~ e ~ m ~ / h ~S)sin y *8*~dB*
(41)
where 8* is the angle of scattering of each lepton referred to S*. Also the denominator of the first term of (40) is (2m~*y*/A)~sin~+ 8*, while the denominator of the second term of (40) is (2mu*y*/h)’ cos’ 8* where u* = y u / ( y + I), u being the relative velocity of the two leptons, and y = (1 - u2/c2)-’l2 so that y*2 = ( y + 1)/2. Performing the summation over spin values in (40) for the case of collisions between two free electrons
+
B. L. Moiseiwitsch
290
then yields the result obtained by M ~ l l e r(1932) du=4n(
-)mu2 &J
2y+l
sin4t?*
- sint?*dt?*
Y2
3 sin2@*
(Y -
4
The case of the scattering of positrons by electrons has been investigated by Bhabha (1936). He showed that the effect of exchange must also be taken into account in the analysis, since the positron may be regarded as an unoccupied electron state of negative energy. Thus the original electron may jump into this unoccupied state while another electron in the sea of negative energy states simultaneouslyjumps into a state of positive energy. This may be considered as an annihilation of the original electron and positron together with a simultaneous creation of a new electron-positron pair. For positron-electron collisions the denominator of the second term of (a), which corresponds to exchange scattering, is therefore - (2E*/hc)’ since kf*= ki* and Ef*= - Ei* = E*. After carrying out a summation over spin values, Bhabha (1936) obtained the differential cross section formula
du+ =
.( 2 I2 mv2
sint?*dt?*
+ a (y - 1)2(1 + cos 8*)2]
I
(43)
An alternative treatment of Mdler and Bhahba scattering can be found in the book by Jauch and Rohrlich (1976).
RELATIVISTIC EFFECTS IN ATOMIC COLLISIONS THEORY
29 1
D. KOLBENSTVEDT THEORY OF K SHELLIONIZATION A simple theory of K shell ionization by relativistic electrons was developed by Kolbenstvedt (1967, 1975). For small impact parameters p < po, that is, close collisions, Kolbenstvedt treated the K shell atomic electrons as free and used the Maller theory outlined in Section II,C. Introducing the ratio w = sin2(8*/2) of the energy transfer t ( y - 1) mc2(1 - cos8*) to the kinetic energy of the incident electron (y - l)mc2, we may write the differential cross section (42) in the form
2.rr(e2/ mc2I2 da =
P 2 ( Y - 1)
1
x [ -w2+
1 (1 - w)2
+(
+) yz .1 2y-1
-
w(1-1 w)
(44)
If I , is the ionization energy of a K shell electron, the least value of w is wmin= I K / ( y - l)mc2. Since there are two K shell electrons, the ionization cross section for close collisions is
the upper limit being f since the scattered electron is regarded as the faster one after the collision. Neglecting terms of order wkin, Kolbenstvedt (1967) found that
+-2 yy2- 1
y-1
lnT;7;;;;])
For distant collisions with impact parameters p > po, Kolbenstvedt used a method introduced by Williams (1935) in which ionization is regarded as arising from photoejection. The photons come from a pulse of electromagnetic radiation produced by the incident electron and propagating along the z direction. The frequency spectrum of the photons is given by the energy
B. L. Moiseiwitsch
292
per unit area per unit frequency interval, where &, (w) and &,(a) are the Fourier transforms of the electric field strength components whose relativistic forms are (31) and (30), respectively. On carrying out the Fourier integrations we get
and then integrating over impact parameters p 1975) that do
dw
> po, it is found (Jackson,
P dP
for low angular frequencies w. If we set c = ho, the photon energy, and let N ( c ) denote the number of photons per unit energy interval, we obtain 2 a A’(€)- -In
€0
-
p4
’
where c0 = 1.123y p h c p ; exp( - 4 p2).Taking po to be the K shell radius ao/Z,, where Z2e is the nuclear charge of the target atom, the K shell ionization cross section for impact parameters p > po is given by
where U(€)
=
cot ( 2 ) exp(1 4ne-’””
128 a3z;
-
-
-
In) 60
is the photoionization cross section, uo = (81~/3)(e’/rnc~)~ is the Thomson scattering cross section, I , = z,2e2/2a0, and n = [ I K / ( € - IK)]”2. Since exp(-4n cot-’n)/(l - e - 2 n n )N e-4( 1 + 4/3n2) it follows that
neglecting terms of order wkin.
RELATIVISTIC EFFECTS IN ATOMIC COLLISIONS THEORY
293
The total K shell ionization cross section is then gwen by
cm2) and give the ionization If we express the cross section in barns energy I, and the lunetic energy E of the incident electron in units of the electron rest energy mc2, we have
x [ In
1.19E(E IK
+ 2)
-
+ 2) ( E + 1)2
E(E
1
In his calculations Kolbenstvedt (1967) used experimental values Z F P for the ionization potential of the K shell. In subsequent publications Middleman et a/. (1970) and Kolbenstvedt (1975) used a modified cross section formula involving a screening number 2IrP/( a2J2 which gave better accordance with the experimental data at very high electron impact energies although overestimating the lower energy cross sections. The Kolbenstvedt method can also be applied to the K shell ionization of atoms by incident positrons e + . For distant collisions we may again use a,(p > pa) given by (53) or (56) since the magnitudes of the components of the electric field strength produced by a positron are the same as those for an electron. However for close collisions the Bhabha cross section (43) must be used rather than the M ~ l l e rexpression (42) and this yields (Tawara, 1978)
(57)
+
where a = 2 - ( E + 2)-’ and b = 3 E ( E + 2)-’ E ( E + 2)-3 - 4 E 2 ( E + 2)-2 - 3 E 3 ( E + 2)-3 neglecting terms of order ( I K / E ) ’ and taking the upper limit of the w integration in (45) as unity. This produces a smaller K shell ionization cross section for close collisions than the formula (55)
294
B. L. Moiseiwitsch
corresponding to electron scattering. It is found by Tawara (1978) that the ratio (u; - uK+)/ui of the difference between the cross sections u i and u l for e- and e + scattering to uc increases with increasing atomic number Z , of the target atom and attains a maximum in the incident lepton kinetic energy region 0.5-0.7 MeV, independent of Z,, ranging from 0.15% for Ne to about 20% for U.
E. INNERSHELL IONIZATION THEORY USING M0LLER INTERACTION The first analysis of K shell ionization of atoms by electron impact using the Meller interaction (10) was carried out by Arthurs and Moiseiwitsch ( 1958). Subsequently more detailed analyses were performed by DavidoviC and Moiseiwitsch (1975), by DavidoviC et al. (1978) who also investigated the case of proton impact, and by Moiseiwitsch and Norrington (1979) who considered the case of L, shell ionization by electrons and protons. These calculations all used the Darwin approximate relativistic wave functions (18) with (19) and (20) to describe the atomic electron before and after ejection has taken place. The inner shell electron is assumed to move in a Coulomb potential having an effective charge Z,e where Z , = Z , 0.3 for the K shell and Z , = Z , - 4.15 for the L shell. In the initial state of the system, denoted by zero, the bound atomic electron has nonrelativistic energy E, = - ( Z , / n ) * e 2 / 2 a , where the principal quantum number n = 1 and 2 for the K and L shells, respectively, while in the final state the ejected electron has kinetic energy Ek = c(m2c2+ h2k2)'I2- mc2 where k is the wave number of the free electron. The double differential cross section with respect to q and k for the ionization of a full inner shell by an incident fermion having charge Z , e is given by
where
5 = Z , / a , and J is a complicated function of k , q, 5. and the angle h between q and ky. For small Z , a and small ka,, the normalization factors
RELATIVISTIC EFFECTS IN ATOMIC COLLISIONSTHEORY
295
which arise in (1 8) are given by
where the initial and final states are denoted by a zero and a prime, respectively. Also, assuming no change of spin, it is found that
+ [( ~ ~ , a / 2 ) s i n X ][({/n)’+ ~(
q2 - k2I2+ (2k{/r1)~)7,
where T~ = ({/n)’ + k2 + 3q2 and T~ = ({/n)’ + k 2 for the K shell, with more involved expressions for T~ and T~ in the case of the L, shell. The formula (61) is very nearly the same as that obtained earlier by Arthurs and Moiseiwitsch (1958) but differs from it in a significant respect, namely that ({/n)’ + k2 - q2 is replaced by .({/n)’ + k2 in the first term of expression (61) for J. In the nonrelativistic limit 1/c +O, (58) becomes
-- ak aq
16aZ:a2
AT^ (62)
P2
which agrees with the formulas obtained by Burhop (1940). For ionizing collisions involving a spin-flip of the K shell electron it is found that J
=(
P ~ , a / 2 ) ~+ ( {k2 ~
+ 3q2)q4sin2h
(63)
) ~ been neglected in the formulas (60) for All terms of order ( Z , C ~have the normalization factors N:, N ; and in the expressions (61) and (63) for J . They are simplifications of the formulas obtained by Davidovic et a/. (1978) and Moiseiwitsch and Norrington (1979). Similar simple formulas have also been derived by Anholt (1979). The double differential cross section a2uK/ak aq has been calculated by Davidovik et al. (1978) for the K shell ionization of 47Ag and 7 9 A by ~ incident protons having 1.836, 3.672, and 4.880 GeV impact energies for collisions without and with spin-flip of the atomic electron. I t is found that the differential cross section has nearly the same values for high energy incident electrons and protons possessing equal velocities. Norrington (1 978) has evaluated the double differential cross section
296
B. L. Moiseiwitsch
where the scattering angle 8 is related to the momentum change h q by q2 = (kD2 + (k;)’ - 2kyk; cos 8, for the case of electrons impinging on 47Ag and 7 9 A ~His . cross sections for incident electrons having 1 and 2 MeV impact energies are displayed in Figs. 2 and 3 for collisions without and with spin-flip of the atomic electron, respectively. has a sharp peak at small 8 In the case of collisions without spin-flip, and low values of the wave number k of the ejected electron whose position is determined by retardation, the prominence of the peak becoming greater with increasing electron energy. For high values of k the peak at small B becomes less pronounced and another much broader peak appears at larger values of 8. When spin-flip occurs the sharp peak in is a less significant feature except at very high impact energies and the broad peak occurring at large values of 8 dominates the double differential cross section. The calculations of DavidoviC et al. (1978) and Norrington (1978) are in good accordance with the differential cross sections, summed over all spin
0 (Degrees)
-
~ Fro. 2. Differential cross sections uko for the K shell ionization of 47Ag and 7 9 Aby
electrons with 1 and 2 MeV energy for values of the wave number k the ejected electron without spin-flip.
10,30,50,70,90a;
of
RELATIVISTIC EFFECTS IN ATOMIC COLLISIONS THEORY
b
i
47 a0 I H H El~clronr
,631
'10)
'
o;
' Id.0 ' Ii.0
297
19 A" 1MeV Elrctmnr
,
, 5.0
,
,
10.0
, 15.0
0 (Degrees)
FIO. 3. Differential cross sections uke for the K shell ionization of "Ag and 7 9 A by ~ electrons with 1 and 2 MeV energy for values of the wave number k = 10,30,50,70,90a~ of the ejected electron with spin-flip.
'
values of the ejected electron, evaluated by Das (1972) and Das and Konar (1974) for the case of electron impact using a somewhat different formulation of the relativistic theory. The cross section for producing a free electron with energy Ek is given by da
&:L=
z
where qmin2: qo/p with qo = (Ek - Eo)/Ac and qmax2: 2ky2: 00. The total cross section for inner shell ionization can then be written
where k,,, corresponds to the event in which the incident fermion transfers all of its kinetic energy to the atomic electron and to quite good approximation may be taken as infinite.
B. L. Moiseiwitsch
298
This analysis ignores the effect of the outer electrons in the target atom. Assuming that they produce the difference between the theoretical ionization energy ( Z , / n ) , rydberg and the experimental value J e x p , their effect may be taken into account by including a contribution d2)to the ionization cross section arising from states lying in the negative energy range IcxP - ( Z e / n ) 2to zero. The total inner shell ionization cross section is then given by = u(l)
+
(67) Moiseiwitsch and Norrington (1979) have found that d2)is proportionately much larger for L, shell ionization than for K shell ionization. Nevertheless for K shell ionization the contribution of uk2) becomes of the important for small values of the atomic number and is nearly WO total for Z, = 25. For inner shell ionizing collisions without change of spin of the atomic electron, it is found that the total cross sections for electrons and protons having equal high velocities of incidence are quite close together (Tawara, 1976; DavidoviC et al., 1978), although as the atomic number increases their ratio alters significantly from unity with the electron cross section lying below the proton cross section because k,,,, although large, is much less for incident electrons than for protons. A comparison between the K shell ionization cross section calculated by DavidoviC et al. (1978) for 3.672 GeV protons and 2 MeV electrons having the same incident velocity given by p = 0.979 is made in Table I for a selection of values of the atomic number Z , . TABLE I
K SHELLIONIZATIONCROSS SECTION aK AS A FUNCTION OF ATOMIC Z2"sb NUMBER
Atomic number
Without spin-flip
With spin-flip
22
Proton
Electron
Proton
Electron
30 40 50 60 70 80 90
181 85.5 47.0 28.3 18.1 12.0 8.16
179 83.7 45.4 26.9 16.9 10.9 7.17
2.26 1.98 1.77 1.60 1.45 1.32 1.20
1.90 1.61 1.40 1.22 1.07 0.94 0.82
"For 3.672 GeV protons and 2 MeV electrons. bFrom Davidovii er ul. (1978).
RELATIVISTIC EFFECTS IN ATOMIC COLLISIONS THEORY
299
The total cross section for inner shell ionization involving a spin-flip of the atomic electron is negligible for small atomic numbers, but for the K shell it has been shown by DavidoviC et al. (1978) that it becomes appreciable for Z, > 40 and amounts to about IWOof the total K shell ionization cross section for Z, = 80. Also the ratio of the spin-flip cross sections for electron and proton impact with equal incident velocities differs considerably from unity for large atomic numbers, as can be seen from Table I. The investigations just described have involved the use of Darwin approximate relativistic wave functions for the active atomic electron. The relativistic effect arising from the atomic electron is negligible for small atomic numbers but becomes important for large 2,. In Fig. 4 we have displayed the momentum distributions of K shell electrons in Au (Z, = 79) calculated using nonrelativistic and Dirac relativistic wave functions.
FIG.4. Momentum distributions 19(q)l2of the K shell electrons of gold plotted as functions of uKq using nonrelativistic (NR) and Dirac relativistic (R) hydrogenic wave functions. The minimum momentum transfer qOmin IE,,I/Ao for K shell ionization comesponding to zero-energy ejected electrons and 0.25,0.5, 1,2,5, 10 keV incident electrons are indicated by arrows.
-
300
B. L. Moiseiwirsch
Clearly the greatest difference between the relativistic and nonrelativistic distributions occurs for large electron momenta. Now the main contribution to the ionization cross section comes from electron momenta close to the minimum momentum transfer Aq,, N (Ek- E J / u which decreases with increasing incident particle velocity. These are indicated in Fig. 4 for various electron energies taking the wave number of the ejected electron k to be zero, from which it can be inferred that for electrons with energy greater than about 10 keV (or incident protons with energy more than 20 MeV) the role of relativity in the atomic wave function is not very important for ionizing collisions. Thus the use of Darwin approximate relativistic atomic wave functions is justified at high energies of impact. Recently the K and L shell ionization of atoms by relativistic electrons has been investigated by Scofield (1978) using solutions of the Dirac equation for a Hartree-Slater potential to describe the atomic electrons. P. H. Norrington (private communication, 1979) has made a comparison between the inner shell ionization cross sections evaluated by Scofield (1978) and those calculated by DavidoviC et al. (1978) and Moiseiwitsch and Norrington (1979). He finds that for K shell ionization the agreement is very good for 28Ni for electron energies ,>0.1 MeV and for 47Agabove about 0.2 MeV, whereas for 7 9 A the ~ accordance is very satisfactory for electron energies 2 0 . 4 MeV. However for L, shell ionization by electron impact the agreement is poor for "Ni probably owing to the inadequacy of the hydrogenic orbitals used by Moisewitsch and Norrington (1979). The agreement improves, however, with increasing atomic number because the hydrogenic orbitals are better for heavy atoms, and the accordance also improves with increasing energy in the same way as for the K shell case. Calculations of the L, shell ionization cross sections by incident electrons for atomic numbers from Z , = 56 to 83 have also been carried out by Ndefru and Malik (1980). They used bound and continuum state solutions of the Dirac equation for the Coulomb potential - Z , e 2 / r to describe the atomic electrons. The Pauli principle was satisfied by employing symmetrized products of atomic orbitals. Ndefru and Malik (1980) obtained cross sections differing by not more than about 1oo/o from those calculated by Scofield (1978) for 1-MeV incident electrons and also not greatly different from the L, shell ionization cross sections calculated by Moiseiwitsch and Norrington ( 1979) using Darwin approximate relativistic atomic wave functions. The agreement between relativistic theory and experimental data on inner shell ionization is, on the whole, quite good both for incident electrons and for protons. Let us first examine electron impact ionization of the K shell. In Fig. 5 we show uK for 2, 50, and 500 MeV electron energies as a function of the
RELATIVISTIC EFFECTS IN ATOMIC COLLISIONS THEORY 1
I
I
I
1
I
30 1
I
Atomic number 22
FIG. 5. Total cross sections oK for K shell ionization of atoms as functions of atomic number 2, for 2, 50,500 MeV incident electrons. Relativistic theoretical cross sections: -, Scofield (1978); - - - - -, Davidovic er al. ( 1978). Experimental data: 0 , Li-Scholz el al. (1 973); 0, Hoffmann er al. (1978); +, Middleman er al. (1970).
atomic number Z , of the target atom. The accordance between the theoretical curves and the experimental data of Li-Scholz et al. (1973) for 2 MeV, Hoffmann et al. (1978) for 50 MeV, and Middleman et al. (1970) for 500 MeV electrons is very good. The theoretical cross sections for 2-MeV incident electrons calculated by Scofield (1978) using Dirac atomic wave functions and by Davidovii: et af. (1978) using Darwin semirelativistic atomic wave functions are almost indistinguishable in Fig. 5 except for very high Z , . The dependence of uK for a given impact energy on the atomic number Z , has been found empirically to be of the form AZ,", where a 2: 2.6. However a much better fit to the theoretical K shell ionization cross sections can be obtained by using the expression AZ,-2.6
-
BZ,-2.4
-~ ~ ~ - 2 . 8
302
B. L. Moiseiwitsch
Electron energy (MeV1
FIG. 6. Total cross section aK for the K shell ionization of *'Ni by incident electrons: Experimental data: A, Pockman er 01. (1947); A, Dangerfield and Spicer (1975); 0,Hoffman er al. (1978); +, Middleman er al. (1970) for 29Cu.
-, Scofield (1978).
2 4020-
I
303
RELATIVISTIC EFFECTS IN ATOMIC COLLISIONS THEORY
0.1
I
10
10
lo3
Electron energy (MeV)
FIG. 8.
Total cross section uK for the K shell ionization of 79Au by incident electrons:
-, Scofield (1978). Experimental data: 0, Motz and Placious (1964); A,Dangerfieid and Spicer (1975); 0, Hoffmann et al. (1978); +, Middleman er al. (1970).
Further, in Figs. 6, 7, and 8 we compare the experimental data with ~ a wide spread relativistic calculations of uK for 28Ni,47Ag, and 7 9 Aover of electron impact energies ranging up to 10oO MeV. The agreement is quite satisfactory. An experimental investigation of L, shell ionization by electrons having 1.04, 1.39, and 1.76 MeV impact energies has been carried out by Park et al. (1975). Here again the accordance with relativistic calculations (Moiseiwitsch and Norrington, 1979; Ndefru and Malik, 1980) is fairly satisfac~ tory. Also experimental investigations of L shell ionization of 7 9 A by electron impact have been made by Middleman ef al. (1970), Schlenk et al. (1977), and most recently by Hoffman et al. (1979) at 20,40, and 60 MeV. Their experimental data for uL are in satisfactory agreement with the relativistic calculations of Scofield (1978). However G e m et al. (1979) find that the theoretical values of Scofield (1978) lie below the uL, points, and lie above the uL, and uL, points, deduced from the experimental data using La and Lp X-ray production cross sections, even though theory and experiment are in accordance for the whole L shell. We now turn, finally, to proton impact ionization of the K shell. Anholt et a/. (1976) have carried out an experimental study of uK for protons
B. L. Moiseiwitsch
304
lo3
4.88 GeV
t -
1
30
40
50 60 70 Atomic Number 22
80
90
FIG.9. Total cross section uK for the K shell ionization of atoms by 4.88-GeV incident protons as a function of atomic number 2,. Relativistic theoretical cross section: -, Davidovii er al. (1978) and Anholt (1979). Experimental data: d o l t er 01. (1976).
+,
having a 4.88-GeV energy incident on atoms with a wide range of atomic numbers. It can be seen from Fig. 9 that the agreement with the relativistic calculations of DavidoviC et at. (1978) and Anholt (1979) using Darwin approximate relativistic atomic wave functions is quite satisfactory.
F. RELATIVISTIC EFFECTS AT Low IMPACT ENERGIES We have noted in the previous section that as the incident particle velocity is decreased, the effects arising from the relativistic nature of the atomic inner shell electrons become increasingly significant for large atomic number Z , > 50. The first use of relativistic atomic wave functions in the calculation of ionization cross sections was made by Jamnik and ZupanEiE (1957). More recently Amundsen et al. (1976), in an investigation of K shell ionization by ions, have discussed the role of relativity in the atomic wave functions employing the following simple approach. As long as the energy of the incident particle is not so great that relativistic effects arising from its translational velocity relative to the
RELATIVISTIC EFFECTS IN ATOMIC COLLISIONS THEORY
305
target atom are important, the K shell ionization cross section may be written
where c k . , s ( q )is the matrix element (22) between the initial Is state of the K shell electron and the final state corresponding to an ejected electron with wave number k. Amundsen et al. (1976) assumed that the dominant contribution to uK comes from q=qrn,,, and from an ejected electron having kinetic energy Ek = 0 corresponding to wave number k = 0, and zero angular momentum. This implies that uK is proportional to I@(q:ln)12, where &,,, = (E0(/Avand @(q)is the momentum distribution given by @( q ) = 4 -
'I
rnRk= o( r ) Rls(r ) sin( qr)r dr
(69)
0
R l s ( r )and Rks0(r) being the radial functions of the initial and final states, respectively. Thus
where R and NR denote the use of relativistic and nonrelativistic hydrogenic orbitals. Denoting the radius of the K shell by aK = ao/Z, and using nonrelativistic orbitals we have
since qlin is sufficiently large that the important region of integration in (71) is small r where R k z 0 ( r ) is effectively constant. On the other hand, using relativistic orbitals we have
= qK1-21(qLin)-'irne-r/~K sin(q:inr)r2S-I where s = ( 1 - U ~ Z ~ )the I / factors ~, r'-
I
dr
(72)
arising from the relativistic forms
of Rls(r) and R k = o ( r ) . On carrying out the integrations in (71) and (72) we obtain the result found by Amundsen et al. (1976)
306
B. L. Moiseiwitsch
Amundsen et al. (1976) have also performed the integrations involved in (69) and (70), without making the approximations (7 I) and (72), and obtained
neglecting terms of higher order in (&,,,)-'. The function f(s) is well approximated by s2 although s has proved a more convenient choice. A similar but less accurate, though simpler, formula for U,"/U,"" had been derived previously by Bang and Hansteen (1959) using the semiclassical approxima tion. The semiclassical approximation together with relativistic electron wave functions has been employed extensively to study inner shell ionization by several investigators including Bang and Hansteen (1959) using a straight line path for the incident charged particle, Amundsen and Kocbach (1975) and Amundsen (1976, 1977a) using a straight line path tangential to the classical Kepler orbit at the point of closest approach, and Amundsen (1977b, 1978) using an hyperbolic trajectory. Satisfactory agreement has been obtained with the experimental data when relativistic electron wave functions are used, as can be seen from Fig. 10 for the case of K shell ionization of 7 9 A ~by protons with energies ranging up to 20 MeV (Amundsen, 1976). The semiclassical tangential approximation has also been used by Amundsen (1977a) to investigate the L, , b,L, shell ionization of 7 9 A although ~ the accordance with the experimental data is less certain in these cases.
1
e
12
16
Proton energy [MeV)
Fro. 10. Total cross section oK for the K shell ionization of "Au by low energy protons. Relativistic theoretical cross section, -; nonrelativistic theoretical cross section, -----, Amundsen (1976). Experimental data: 0,A.
RELATIVISTIC EFFECTS IN ATOMIC COLLISIONS THEORY
307
111. Electron Capture The first investigation of nonradiative electron capture, taking account of relativistic effects, was carried out by Mittleman (1964) using the Oppenheimer-Brinkman-Kramers (OBK) approximation for the process H+ + H ( l s ) + H ( l s ) + H +
(75)
He found that the capture cross section falls off as E - l when the kinetic energy E of the incident proton is exceedingly high. This contrasts greatly with the rate of decay E P 6 given by the OBK approximation neglecting the effects of relativistic kinematics and mechanics, or E -''/* given by the nonrelativistic second Born approximation. More recently Shakeshaft (1979) and Moiseiwitsch and Stockman (1979a. b, 1980) have carried out a new study of the problem of electron capture at relativistic impact energies. Although the details of their capture cross section formulas differ from those derived by Mittleman (1964), they obtained the same slow E - I falloff at ultrahigh impact energies. Radiative capture was first discussed by Oppenheimer (1928). However its great importance at high energies of impact was not appreciated until much later (Briggs and Dettman, 1974). A. IMPACT PARAMETER FORMULATION
We consider a nucleus B with charge Z,e moving with high constant velocity v along a straight line path relative to a target hydrogenic atomic system having nucleus A with charge 2,e. Our analysis is concerned with the capture of the electron attached to A by the incident nucleus B. If S , and SBare frames of reference with origins located at nuclei A and B, respectively, and t and t' denote time measured relative to these frames, we choose t = t' = 0 at the instant when B is at the point C of closest approach to A. The distance of C from A is the impact parameter p as depicted in Fig. 1. We let rA and rk be the position vectors of the electron e referred to S , and Sg , respectively, and express the time-dependent electronic wave function referred to S, in the form \k = a(r)$p(r,)exp
( - h Eat' 1 + b(r)S$B(fB)exp(- f E i f )
(76)
where $p(r,), %B(rg),and EL, E$ are the normalized Dirac wave functions and eigenenergies corresponding to the states i and j of the hydrogenic ions with nuclei A and B, respectively. Here S is an operator which
B. L. Moiseiwitsch
308
transforms the Dirac wave function $jB(fB)referred to S g to a wave function referred to S,, and is given by (Berestetskii et al., 1971)
.=(
T) +( y+l
y-1
(77)
where, as before, y = ( 1 - u ~ / c ' ) - ' /and ~ the components of a are the Dirac matrices. Taking the direction of the velocity v along the z axis we have EjBt'
=
Y E i t - hkz,
(78)
where k is the linear momentum and yEL is the energy of the electron attached to nucleus B but referred to S,. In the nonrelativistic limit, (76) approaches the form used by Bates (1958) in his analysis of electron capture using translational factors. The interaction prior to the capture event is represented by the Coulomb potential VB(r;)= - Z,e2/rL between the electron and the nucleus B referred to SB. The Coulomb interaction potential between the two nuclei also occurs in the first Born approximation, although, to avoid difficulties arising from nonorthogonality of the initial and final atomic wave functions +p(r,) and $B(rb), it should be removed by using a procedure analogous to that introduced by Bates (1958) but generalized to allow for relativity. The additional terms that arise fall off sufficiently quickly with increasing impact energy to be small where relativistic effects become significant except for high nuclear charges. If these terms are dropped, we obtain
I
x . ~ B ( r g ) + L V B ( r g ) ~ ~ (exp( r A )- ikz,) dr, taking the initial conditions a( - co) = 1, b( - 00) = 0 corresponding to the situation in which the electron is initially attached to nucleus A, and assuming that b ( t )<< a ( t ) for all time. Expression (79) is the relativistic form of the OBK formula for the capture amplitude in the impact parameter treatment. It should be noted that it involves an operator L which transforms the potential VB(rb)referred to Sg to a potential referred to s,. The cross section for capture is then given by
RELATIVISTIC EFFECTS IN ATOMIC COLLISIONSTHEORY
309
where the dependence of the amplitude b on the impact parameter p is explicitly indicated.
B. WAVEFORMULATION If we denote the wave vector of the incident nucleus B by k, and that of the hydrogenic atomic system composed of B together with the captured electron by k,, it can be shown using the principle of conservation of energy that k , - k 2 = (EL - y E L ) / h c ( y Z - 1)1’2
(81)
neglecting terms of order m / M , where M , is the mass of nucleus B. It follows that l / h ( y E g - &)t-
kz,+(k, -k,).peq-r,+q’.r’,
(82)
d
where p = AC, q = k, - k,, (q + q’) - p = 0, q * 8 = (EL - y E A ) / h c ( y2 q‘ * 8
=
(EA - yEg)/hc(yZ -
1y2 1)II2
Then it can be shown that we may express the relativistic OBK scattering amplitude for capture in the form
The capture cross section is then given by
where qmlnN Ik, - k,l. For simplicity we shall now suppose that the nuclei A and B are protons so that Z, = 2, = 1 and M A = M , = M , , the mass of a proton. Also we shall suppose that i a n d j are both the ground state of a hydrogen atom so that EA = E; = rnc2(1 - a2)’/’, where a is the fine structure constant. Thus the process we are concerned with now is (75), the case originally considered by Mittleman (1964). Then, making use of (81), we find that
3 10
B. L. Moiseiwitsch
qminN [ / a o where 6 = (1 - a2)’/2a- ‘X - ‘ I 2 and X = 1 + 2Mpc2E- I , denoting the kinetic energy of the incident proton B by E = ( y - 1)M,c2. There are two cases to consider: (1) the electron spin does not change; and (2) the electron spin changes. The capture cross sections for these two cases can be written, respectively, as
(86) where s = (1 - a2)I/’ and I = aoq. To the lowest order in the fine structure constant, Moiseiwitsch and Stockman (1979a, 1980) have shown that
P(I) 2:
4[ 1 - ( 2 X ) - ’ ] I 2
(1 + 1 2 f
Ta2
+-
13
and
Q ( 4=
-212 (1 + 12)’
For E 5 10 MeV, nonrelativistic atomic wave functions may be used, which corresponds to taking only the first term of P ( l ) in (87) without the factor ( 2 X ) - I . Then the OBK capture cross section becomes
and we obtain readily
up2:128 M,c2 ra,2a(I 2 5
E
x5
[ 1 + (2a2Mpc2/E)]’
(90)
which approaches the nonrelativistic OBK formula
in the limit where E/M,c2 << 1. For E > 10 MeV, formula (90) begins to become increasingly inaccurate
31 1
RELATIVISTIC EFFECTS IN ATOMIC COLLISIONS THEORY
because the relativistic character of the atomic wave functions cannot be ignored. Making use of (87) and (88) and carrying out an expansion in powers of a, Moiseiwitsch and Stockman (1979a, 1980) find to a good approximation that for the case without spin-flip the capture cross section takes the form
[1 - (2x)-']xs
128
E
[ 1 + (21y*MPC2/E)]
5
9
- 10
- I6
- ia 3
2 Logo
FIG. 11.
[Proton inpoct energy E
4
(MeV)]
Electron capture cross sections uc for H + +H(ls) collisions. , relativistic
OBK cross section uil)using relativistic atomic wave functions without spin-flip; - - - -, relativistic OBK cross section)':u using relativistic atomic wave functions with spin-flip; - - _ , relativistic OBK cross section uil) given by (90) using nonrelativistic atomic wave Functions; -----,nonrelativistic OBK cross section u,"" given by (91); . . . . ., relativistic classical double-encounter cross section 1979a, b).
given by (107) (Moiseiwitsch and Stockman,
3 12
B. L. Moiseiwitsch
and for the spin-flip case the capture cross section is given by
The most significant aspect of these capture cross sections is that they both fall off as E - at ultrahigh energies E 2 105 MeV as originally found by Mittleman (1964) although with different coefficients. In the ultrahigh energy limit E -+ 00, we have X + 1 so that a:l)/aL2)-+ 4,neglecting terms of order a. Numerical evaluations of the integrals over I involved in the full analytirespectively, are found to cal expressions (85) and (86) for a:') and agree very well with the approximate formulas (92) and (93). The results of these calculations are displayed in Fig. 11. Shakeshaft (1979) has also carried out numerical integrations for the case without spin flip. Very good accordance is found with the calculations of Moiseiwitsch and Stockman (1979a, 1980). The formulas (92) and (93) for the capture cross sections have been generalized to arbitrary nuclear charges by Moiseiwitsch and Stockman (1980) for small values of a Z , and aZ,.
'
C. CLASSICAL DOUBLE SCATTERING By treating the capture problem using classical mechanics involving two successive binary collisions, Thomas (1927) showed that the capture cross section decays as u - l l or E - I 1 / * at high, but nonrelativistic, impact energies. Subsequently it was found by Drisco (1955) that the nonrelativistic second Born approximation to the capture cross section exhibits the same decay with increasing energy of impact. Recently Shakeshaft (1979) and Moiseiwitsch and Stockman (1979b) have used the Thomas model and relativistic classical mechanics to investigate electron capture. The differential cross section for the first binary encounter between the incident proton B moving with velocity v and a stationary electron attached to a proton A forming the target hydrogen atom, illustrated in Fig. 12, is given by q=2m(6)sin6d6
(94)
where 6 is the angle of scattering of the electron referred to the frame Sb, in which the incident proton B is at rest, and
a(6) =
( -")
2myu2
2
+
cosec4 6
(95)
313
RELATIVISTIC EFFECTS IN ATOMIC COLLISIONS THEORY
\
\ \
\
\
I
FIG. 12. Double encounter between an incident nucleus B moving with velocity v and an atomic system composed of a nucleus A and an electron e.
is the cross section for the scattering of an electron by the Coulomb field of the proton B, in which relativity is taken into account. If v’ is the velocity of the electron after the first encounter, referred to the frame S, in which the target hydrogen atom is at rest, it can be shown using relativistic classical mechanics that
(
;: )
coss= V c22 1 - -
where y’ = (1 - u ’ ~ / c ~ ) - ’ /neglecting ~, terms of order m / M , so that 0’
Yf3
sinSdS = - 7 do‘ (97) v2 Y which enables q to be evaluated. We next consider the second binary encounter between the electron and the proton A of the target atom. The probability that the electron is scattered through a polar angle 8” into an element of solid angle dS2 is given by
where r is the distance of the initial position of the electron from the proton A and
B. L. Moiseiwitsch
3 14
For capture to occur we need the final velocity V” of the twice scattered electron referred to S, to satisfy v” 2: v so that v” N v’ 2: v and 8 ” 2: 8’, where 8’ is the angle between v‘ and v given by
(
;!1
cos8’= 7 1 - W c2
Also for capture to take place, lv” - vI referred to S, must be less than u, where m u 2 / 2 = e 2 / r ’ . Hence
where D N y - 4 is the Jacobian of the transformation from S, to S, of the relative velocity between the twice scattered electron and the proton B. Thus we obtain
~ ( 8 ’ ’j )n(2e2/rnr’)3/2 y - 4
P=
4 ~ r dv” ~ v ~ ~ ~
( 102)
Now it can be shown that, after the second binary encounter, the distance r’ of the electron from B, referred to S;, is equal to r. Hence setting r = r’ = ao, the Bohr radius of the orbit of a 1s electron, and noting that do’’ = dv‘ we obtain for the capture cross section uclass c
=
-
P4 ( E / M , c ~ ) ” /+ ~ ([ E ~/M,c~)]~/~
where E = (y - 1)MPc2 is the kinetic energy of the incident proton B referred to S,. In the ultrahigh energy limit
which is a much faster rate of decay with increasing E than the E - I decay exhibited by the relativistic OBK capture cross section. At nonrelativistic incident particle velocities
which is the same as the result obtained by Thomas (1927). If spin effects are allowed for in the Coulomb scattering cross sections
RELATIVISTIC EFFECTS IN ATOMIC COLLISIONS THEORY
315
(95) and (99), an extra factor
arises in the capture cross section (103). However this does not alter the limiting formula (104). With the inclusion of the factor (106) arising from spin, the relativistic classical capture cross section may be written class 0,
- 4fi 3
’(Mg2) 3
I
Ix
- 3/2
[+(2+x+X2)]
-
(107)
+
where X = 1 2M,c2E - I . It can be seen from Fig. 11 that the classical double encounter cross section (107) is much smaller than the relativistic OBK capture cross section given by (85). However, it is important to note that the nonrelativistic classical capture cross section (105) is a factor 37r/fi smaller than that obtained using the nonrelativistic second Born approximation, and thus we should expect that the relativistic capture cross section (103) should be too small by the same factor. This indeed seems to be the case. Using an “updated Thomas model” to take account of quantum mechanical effects, Shakeshaft (1979) obtained a cross section which is in accordance with the term of the relativistic second Born approximation corresponding to classical double encounters and is a factor 37r/fi times (103). He also found that for the case of a nucleus of charge Z,e incident on a target hydrogenic ion having nuclear charge Z,e, the cross section has an additional factor 2 ( Z , Z , ) 5 / ( Z , + Z,).
D. RADIATIVECAPTURE It has been pointed out by Briggs and Dettman (1974, 1977) that radiative capture dominates the capture cross section at sufficiently high impact energies. Since the energy of binding of an electron of the target atom is small, the radiative capture cross section uRc for incident protons can be derived from the cross section up, for the photoionization process H
+ h v - + H ++ e
by using the principle of detailed balancing. This gives
( 108)
B. L. Moiseiwitsch
316
where p = ymu is the momentum of the electron and hv ~ the energy of the photon. Thus we have the relation ‘RC
-[
- ‘)/(Y
( -yl)mc2 is
+ ‘)lUPI
(110)
where the photoionization cross section has the form (Sauter, 1931; Heitler, 1954): = 4na,y(y* - 1)3/2(y
-I)-~
(111)
The radiative capture cross section has been calculated by Raisbeck and Yiou (1971) using these formulas and compared with their experimental data on capture by high energy protons from Al, Ni, and Ta. The accordance between theory and experiment is moderately successful. In the nonrelativistic limit upI= (2’/3)~a&r~(c/o)’
(112)
which yields uRC = (
1 6 a /3)7a;a8( M ~ c ~ / E ) ~ ’ ~
(113)
This formula has been given by Berestetskii el al. (1971) and Briggs and Dettman (1974) who note that it dominates the OBK capture cross section at proton impact energies above 13 MeV. At ultrahigh energies
-’
which falls off as E in the same way as the relativistic OBK capture cross section. However, the ratio of (92) to ( 1 15) is 32a4/5 N 1.8 X lop8in the high energy limit so that uRC>> 6,.
REFERENCES Amundsen, P. A. (1976). J . Phys. B 9,971 and 2163. Amundsen, P. A. (1977a). J . Phys. B 10, 1097. Amundsen, P.A. (1977b). J. Phys. B 10, 2177.
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Amundsen, P. A. (1978). J . Phys. B 11, 3197. Amundsen, P. A., and Kocbach, L. (1975). J. Phys. B 8, L122. Amundsen, P. A., Kocbach, L., and Hansteen, J. M. (1976). J. Phys. B 9, L203. Anholt, R. (1979). Phys. Rev. 19, 1004. Anholt, R., Nagamiya, S., Rasmussen, J. O., Bowman, H., Ioannou-Yannou, J. G., and Rauscher, E. (1976). Phys. Rev. A 14, 2103. Arthurs, A. M., and Moiseiwitsch, B. L. (1958). Proc. R. Soc. London, Ser. A 247, 550. Bang, J., and Hansteen, J. M. (1959). Mat. Fys. Medd. 31, No. 13. Bates, D. R. (1958). Proc. R. SOC.London, Ser. A 247, 294. Berestetskii, V. B., Lifshitz, E. M., and Pitaevskii, L. P. (1971). “Relativistic Quantum Theory.” Pergamon, Oxford. Bethe, H. A. (1930). Ann. Phys. (Leiprig) [5] 5, 325. Bethe, H. A. (1932). Z. Phys. 76, 293. Bhabha, H. J. (1936). Proc. R. SOC.London, Ser. A 154, 195. Briggs, J. S., and Dettman, K. (1974). Phys. Rev. Lett. 33, 1123. Briggs, J. S., and Dettman, K. (1977). J . Phys. B 10, 1 113. Burhop, E. H. S. (1940). Proc. Cambridge Philos. Soc. 36, 43. Dangerfield, G. R., and Spicer, B. M. (1975). J . Phys. B 8, 1744. Darwin, C. G. (1928). Proc. R. SOC.London, Ser. A 118, 654. Das, J. N. (1972). Nuovo Cimenio E 12, 197. Das, J. N., and Konar, A. N. (1974). Nuovo Cimenfo A 21, 289. Davidovic, D. M., and Moiseiwitsch, B. L. (1975). J. Phys. B 8, 947. Davidovii, D. M., Moiseiwitsch, B. L., and Norrington, P. H. (1978). J. Phys. B 11, 847. Drisco, R. M. (1955). Ph.D. Thesis, Carnegie Institute of Technology, Pittsburg. Genz, H., Hoffman, D. H. H., Low, W., and Richter, A. (1979). Phys. Left. A 73, 313. Heitler, W. (1954). “Quantum Theory of Radiation.” Oxford Univ. Press (Clarendon), London and New York. Hoffmann, D. H. H., Genz, H., Low, W., and Richter, A. (1978). Phys. Lett A 65, 304. Hoffman, D. H. H., Brendel, C., Genz, H., Low, W., Muller, S., and Richter, A. (1979). Z. Phys. A293, 187. Jackson, J. D. (1975). “Classical Electrodynamics.” Wiley, New York. Jamnik, D.. and ZupanEiE, C. (1957). Mar. Fys. Medd. 31, No. 2. Jauch, J . M., and Rohrlich, F. (1976). “The Theory of Photons and Electrons.” SpringerVerlag, Berlin and New York. Kolbenstvedt, H. (1967). J . Appl. Phys. 38, 4785. Kolbenstvedt, H. (1975). J . Appl. Phys. 46, 2771. Li-Scholz, A., Colle, R., Preiss, I. L., and Scholz, W. (1973). Phys. Rev. A 7, 1957. Middleman, L. M., Ford, R. L., and Hofstadter, R. (1970). Phys. Rev. A 2, 1429. Mittleman, M. H. (1964). Proc. Phys. Soc., London 84, 453. Moiseiwitsch, B. L., and Norrington, P. H. (1979). J. Phys. E 12, L283. Moiseiwitsch, B. L., and Stockman, S. G. (1979a). J . Phys. B 12, L591. Moiseiwitsch, B. L., and Stockman, S. G. (1979b). J . Phys. B 12, L695. Moiseiwitsch, B. L., and Stockman, S. G. (1980). J . Phys. B. 13, 2975. Msller, C. (1931). Z. Phys. 70, 786. Msller, C. (1932). Ann. Phys. (Leiprig) [5] 14, 531. Motz, J. W., and Placious, R. C. (1964). Phys. Rev. 136, A662. Ndefru, J. T., and Malik, F. B. (1980). J . Phys. B. 13, 2117. Norrington, P. H. (1978). Ph.D. Thesis, Queen’s University, Belfast. Oppenheimer, J. R. (1928). Phys. Rev. 31, 349. Park, Y. K., Smith, M. T., and Scholz, W. (1975). Phys. Rev. A 12, 1358. Pockman, L. T., Webster, D. L., Kirkpatrick, P., and Harworth, K. (1947). Phys. Rev. 71, 330.
318
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Raisbeck, G., and Yiou, F. (1971). Phys. Rev. A 4, 1858. Rester, D. H., and Dance, W. E. (1966). Phys. Rev. 152, I . Sauter, F. (1931). Ann. Phys. (Leipzig) [ 5 ] 9, 217 and 454. Schlenk, B., Berenyi, D., Ricz, S., Valek, A., and Hock, G. (1977). J. Phys. E 10, 1303. Scofield, J. H. (1978). Phys. Rev. A 18, 963. Shakeshaft, R. (1979). Phys. Rev. A 20, 779. Tawara, H. (1976). Phys. Lerr. A 59, 199. Tawara, H. (1978). Proc. Int. Con& Phys. Elecrron. At. Collisions, Invited Pap. Prog. Rep., 10th 1977 p. 311. Thomas, L. H. (1927). Proc. R. Soc. London, Ser. A 114, 561. Williams, E. J. (1933). Proc. R. Soc. London, Ser. A 139, 163. Williams, E. J. (1935). Mat. Fys. Medd. 13, No. 4.
\I
ADVANCES IN ATOMIC AND MOLECULAR PHYSICS. VOL. 16
PARITY NONCONSERVATION I N ATOMS: STATUS OF THEORY A N D EXPERIMENT E . N . FORTSON and L. WILETS Department of Physics University of Washington Seattle. Washington I . Introduction . . . . . . . . . . . . . . I1. The Neutral Current Interaction in Atoms . . . . I11. Observable Effects . . . . . . . . . . . . A . El-MI Interference . . . . . . . . . . B . Stark-PNC Interference . . . . . . . . . IV . Atomic Calculations . . . . . . . . . . . . A . Independent Particle Approximations . . . . B . Further Complications and Corrections . . . . V. Optical Rotation Experiments: Bismuth . . . . . A . Introduction . . . . . . . . . . . . . B. Status of Atomic PNC Calculations for Bismuth . C . General Experimental Features . . . . . . . D . BismuthOptical Rotation at 8757 A . . . . . E. Bismuth Optical Rotationat 6477A . . . . . F . Summary of Results and Discussion . . . . . VI . Stark Interference Experiments: Cesium and Thallium A . Overview . . . . . . . . . . . . . . B . Calculations . . . . . . . . . . . . . C . Experiments . . . . . . . . . . . . . VII . Atomic Hydrogen Experiments . . . . . . . . VIII . Conclusions . . . . . . . . . . . . . . References . . . . . . . . . . . . . . .
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I. Introduction In the two-plus decades since the 1956 Parity Revolution. there have been profound theoretical and experimental developments in weak interactions which have culminated in the establishment of the unified theory of weak and electromagnetic interactions proposed by Weinberg ( 1967) and 319
.
.
Copyright 0 1980 by Academic Press Inc All rights of reproduction in any form reserved. ISBN 0-12-003816-1
320
E. N . Fortson and L. Wilets
Salam (1968). In this theory, the weak interaction between any fermion pairs is mediated by the exchange of massive vector bosons, W' and p . A variety of renormalizable gauge theories had been proposed by the 1970s but the Weinberg-Salam theory was the most attractive: It introduced only one free parameter (Ow, the mixing angle between the bare neutral vector meson and the bare photon). The charged bosons W' are involved in P-decay processes where, for example, a neutrino is converted into an electron while a neutron is converted into a proton (n + p + e- + E), In the earlier language of current-current interactions, the process involves the charged (or more properly charge-changing) currents of the leptons and hadrons. These can also contribute to neutrino-electron scattering. An important prediction of the Weinberg-Salam version is the existence of neutral weak currents mediated by the neutral vector boson 2.Neutral currents were observed in neutrino-nucleon scattering in 1973 (Hasert, 1973; Benvenuti et al., 1974; Barish et al., 1974). At least one key question remained: Is the neutral weak current parity nonconserving as required by Weinberg and Salam? The electron-nucleus system, where there can be interference between weak and electromagnetic interactions, provided the possibility of such a test. Before any experiments or popular theories demanded neutral currents, Zel'dovich (1959) speculated that if neutral currents were to exist, they could lead to detectable parity nonconservation (PNC) effects in atoms. The interaction of the atomic electrons with the nucleons could admix atomic states of opposite parity and lead to a handedness of emitted photons. Zel'dovich estimated the size of the optical rotation expected in a gas of hydrogen atoms, but concluded that this particular effect would be far too small to observe. Later, Michel (1965) analyzed possible experiments involving excited states of hydrogen. Parity mixing would be enhanced by the near degeneracy of states of the same major quantum number and opposite parity, but the experiments still appeared to be very difficult. Poppe (1970) and Bradley and Wall (1962) were able to set experimental upper limits to PNC circular polarization in magnetic dipole transitions in atomic lead and molecular oxygen, respectively, but neither experiment was sensitive enough that such effects could be observed. Several possible new experiments to detect neutral currents in atoms were under consideration when Bouchiat and Bouchiat (l974a) pointed out the considerable advantages of looking to heavy atoms. They showed that neutral current effects should increase rapidly with 2, and that heavy atoms such as TI and Bi should exhibit effects more than six orders of magnitude greater than ground state hydrogen. This provided immediate
PARITY NONCONSERVATION IN ATOMS
32 I
impetus to world-wide experimental efforts in heavy atoms. Later, new experimental approaches to atomic hydrogen, involving the n = 2 (metastable) states, were considered, and several experiments to measure the neutral current effects there were begun. Now, after six years of intensive effort, results from a number of experiments give evidence of PNC in heavy atoms. Over the same period, the complex atomic calculations have been refined considerably. At this point, most atomic experiments agree in sign and approximate magnitude with current atomic calculations using the Weinberg-Salam theory. However, there are contradictory experimental results and it is still too early to draw a definitive conclusion. In the meantime, high-energy electron-proton and electron-deuteron scattering experiments by the SLAC-Yale group (Prescott, 1978) have shown PNC in agreement with Weinberg-Salam. For the future, more work is needed to clarify both the experimental and atomic theoretical situations in heavy atoms. We also await results from the important atomic hydrogen experiments, which are free from atomic theory uncertainties. In addition, there has been a variety of proposals (Bouchiat and Bouchiat, 1974b; Bernabeu et al., 1974; Feinberg and Chen, 1974; Missimer and Simons, 1979) to study PNC in muonic atoms. Although the atomic physics is simple for muonic atoms, there are complications in the nuclear calculations involving heavy nuclei. This would, however, provide an important test of lepton universality.
11. The Neutral Current Interaction in Atoms The mass of the Zo is - 8 5 GeV corresponding to a reduced Compton wavelength of 0.002 fm. For low and intermediate energy processes, therefore, it is adequate to replace the intermediate boson propagator by a contact function, and the model reduces to the current-current interaction form. The PNC part for nucleon (N) and electron (e) interactions is given by
where GF is the universal Fermi coupling constant, GF = 89.6 eV fm3 = 2.22 X
a . u.
E. N . Fortson and L. Wilets
322
In the Weinberg-Salam theory, uo = -sin2 B, U,
a.
= ) ( 1 - 2 sin2
ow),
=0
(2)
a , = $ ( 1 - 4 sin2 B,)g,
where g , N 1.25. The two terms in Eq. (1) are, respectively, vector-axial vector (VA) and AV coupling for the nucleons and electrons. Alternatively, one frequently uses the vector and axial vector coupling coefficients for neutrons and protons,
c,,= uo + u , = + ( I cIn = u 0- U = I- L
- 4sin28,),
c,, = a, + a , = ) ( I - 4sin2B,)g, ~ ~ , = a , - a , =- ) ( I -4sin2B,)g,
2 ’
(3) where the second form in each case is that given by the Weinberg-Salam theory. The subscripts “ I ” and “2” on the C’s refer to nucleon V and A coupling, respectively. Nonrelativistically, yo-
1,
w5-0,
(4)
y5-(u-p)/mc
For nonrelativistic nucleons in the atomic nucleus, we can neglect the y 5 term. In first-quantized notation, we have
where N stands for n or p, and the sum extends over all electrons and nucleons in the atom. Recall = ++yo and y = yoa. Because the electron wave functions are large compared with nuclear dimensions, the C , , terms are additive (coherent) for the sum over nucleons. The C,, terms depend upon the nuclear spin and have a net contribution only from unpaired nucleons. Thus the C , , terms normally dominate in heavy nuclei and provide an interaction
where pN is the nuclear density, normalized to IpN dr = I , and
Q = 2( ZC,,
+ NC,,)
= Z ( 1 - 4 sin2 0,) -
N N 0.08 Z
-
N
(7)
Z and N being the proton and neutron numbers of the nucleus. The second expression for Q in (7) is for the Weinberg-Salam theory and the third expression uses the best current value sin’ 8,210.23 t 0.01 (Abbott and Barnett, 1979).
PARITY NONCONSERVATION IN ATOMS
323
There are some cases in heavy atoms where the C,, terms, although much smaller than the C,, terms, might be measurable (Novikov et al., 1977). The PNC neutral current interaction of the electrons with each other produces effects similar to the C,, terms, but far smaller in heavy atoms (Bouchiat and Bouchiat, 1974b). In principle, the hydrogen experiments can be made equally sensitive to C,, and C2p.Note, however, that in Weinberg-Salam theory these quantities are small (0.04 and 0.05 for sin2@, = 0.23) and would vanish for sin2@, = 1/4; in contrast C,, = -0.5 (C," = - C,,). Combinations of all four coefficients enter in the deuteron experiment and, taken together with hydrogen, each can be extracted. Thus more information is provided by the H and D experiments than the heavy atom experiments and the atomic calculations are free of uncertainties, but the experimental problems are of at least comparable difficulty. Although it is essential to treat the electrons relativistically in heavy atoms, there is some conceptual value is considering the nonrelativistic form of the interaction, and this is also a good approximation for light and intermediate atoms. We further approximate the nuclear density to be a delta-function at the origin. Using the nonrelativistic limits (4), we obtain
+ iu,
- C2,(u, *pe
* (re
x p,)]
+ h.c.
(8)
The operator combinations that appear in (8) are pseudoscalars and hence the interaction mixes states of opposite parity. In fact, the only nonvanishing matrix elements of (8) are between s and p stafes. Relativistically, even including finite nuclear size, the only important admixing is between sIl2 and pl/, states. Other states have negligible amplitude over the nuclear volume. The strong dependence of PNC on atomic number 2, as first pointed out by Bouchiat and Bouchiat (1974a,b), can be seen by considering qualitatively the matrix elements of (8). Fermi and Segre (1933) have given an approximate expression for the probability density of an s-electron at the origin: 1$~(0)12
-
zzo'
= -( 1 -
an *'a:
2)
Z
an*3
(9)
where 2, 1 is the outer charge seen by the electron, n* = n - u is the effective principal quantum, and (I is the quantum defect. (The formula
324
E. N . Fortson and L. Wilets
was originally derived for alkali metals, but the qualitative behavior is more general.) Similarly Ip+,f, a ~ ' + ~ / n * ~
( 10)
or
Bouchiat and Bouchiat (1974a) have given a more detailed treatment based on a modification of the Ferrn-SegrC formula. The Q of Eq. (6) contributes a factor proportional to N, which increases somewhat more rapidly than Z . Although n for the valence electron increases as Z 'I3, n* increases much more slowly. In Cs ( Z = 55), for example, n = 6 but n* N 2 (and du/dn is very small). Relativistically, there is further enhancement of nearly an order of magnitude for the heaviest elements. These considerations are the bases for the statement that PNC is enhanced in heavy atoms by more than six orders of magnitude over the ground state of hydrogen. In perturbation theory, an energy denominator enters and near degeneracies can give further enhancement in special cases (e.g., n = 2 hydrogen).
111. Observable Effects In the absence of H,,,, the states of the atom have definite parity. Electric dipole (El) transitions, which normally dominate atomic spectroscopy, are forbidden between states of like parity. However, H,,, mixes s and p states and causes, in some cases, a small El transition amplitude to appear and to interfere with transitions (MI, E2) normally present but weak between states of like panty, giving rise to an observable PNC effect. For instance, when H,,, couples an El amplitude into an MI transition, the transition rate acquires a dependence on the sense of circular polarization of incident radiation. The optical rotation experiments utilize this effect. An applied static electric field Es can create a Stark El amplitude between two states of the same parity in the unperturbed atom. Interference between Stark and PNC induced El transitions also produces observable PNC effects, and forms the basis for many of the experiments now being carried out.
PARITY NONCONSERVATION IN ATOMS
325
A. El -M I INTERFERENCE A simple illustration of El-MI interference is provided by the radiation from a helical antenna that is small compared to the radiation wavelength. Charge moving back and forth through the antenna produces oscillating El and MI moments parallel to the helical axis, proportional respectively to the charge displacement along the axis and to the current about the axis, and therefore having a relative phase difference of m / 2 . The radiated electric and magnetic field components in the plane of the axis thus also differ in phase by n / 2 , which means the radiation has partial circular polarization. Conversely, there is a circular polarization dependence associated with absorption of radiation by the antenna. The introduction of a parity-nonconserving term in the electron Hamiltonian u a p introduces a handedness or screw sense to the atom. Classically the electronic orbit becomes helical with the axis of the helix aligned with respect to u.Note that the screw sense of a helix is invariant with respect to rotation by m about an axis normal to the helix axis. In the atom, time reversibility of H,,, guarantees that the phase difference between El and MI amplitudes is ? m / 2 , just as in the simple antenna just mentioned. This is because the El and MI operators have opposite signs under time reversal.
-
1. E l and MI Matrix Elements
We now write down expressionsAforthe transition matrix elements. In optical experiments, the operators E 1 and H,,, are both small, and both may be treated in first-order perturbation theory. Let In) and En denote exact wave functions and energy eigenvalues in the absence of the above perturbations, and let InpNc) denote the wave function when H,,, is included. There is no first-order energy shift due to H,,,. Then
where E 1 = - e C , r , . The two terms represent, respectively, perturbation of the final and initial states by HPNc. We introduce a reduced matrix element G P N C ( j ) defined by (fPNC
I B1I iPNC)
= FPNC(fi)'mrm,
(13)
E. N. Fortson and L. Wilets
326
where the orientational dependence given by a,,,*
= C,(2 2 9 ) 6 ( m , , m 2 1) + Coi6(mf,mi)
is the same as for a vector operator (Messiah, 1968) because H,, is (pseudo-) scalar, The Clebsch-Gordon coefficients C , and Co depend upon the initial and final total angular momenta and 2 components mi and The M1 matrix element is (fPNC I ” I iPNC) m(jlA?l I i ) = - ( e h / 2 m , c ) ( f I L + 2 S I i ) = G X ( f i ) ~ 3 ~ , , ~(14)
and we have introduced the reduced matrix element a($). We will often suppress the indicesf, i in both F,,, and a. We chpose the phases of*thestates to make the reduced matrix elements of ( m I E 1 I m‘) and ( m I M 1 1 m ’) real. (One can readily show that matrix elements of r, S, and L can have a common phase for any given spatial component.) The matrix elements of H,,, will then be pure imaginary, as follows from either Eq. ( 5 ) or Eq. (4, and holds true for any PNC interaction that obeys T invariance (Bouchiat and Bouchiat, 1974b). 2. Circular Dichroism and Optical Rotation As already noted, the interference between G P N c and 9 2 leads to circular polarization effects, which we now derive rigorously. A circularly polarized electromagnetic wave traveling in the + i direction has electric and magnetic field vectors,
Ell = E , / 6 ( 2 - i @ ) , B, = B , , / f i ( i q 2 +$) (15) where q = ? 1 gives the two states of circular polarization, the positive sign denoting right-hand circularly polarized light by the standard optical convention, which corresponds to negative helicity. The transition amplitude is proportional to
A,(fi)
= (fPNC
I(
*
E, +
A
’ Bq)
I iPNC)
The degree of circular polarization in emission is (again suppressing the indices f,i )
PARITY NONCONSERVATION IN ATOMS
327
where we have used Eqs. (13), (14), and (15), and in the last expression we have assumed GPNC<<9R, which will always be the case in what follows. Similarly, in the case of absorption, the absorptivity K will change between right and left circularly polarized light by a fractional amount 26:
28 = ( K + - K -
)/K
=4 h(&pNc/%)
(17)
where 6 is called the circular dichroism, and K x K + x K - . An important case is the rotation of plane polarized light. Such optical rotation arises from the difference in the real part of the refractive index n between right and left circularly polarized light. From dispersion relations we know, to a very good approximation, that ( n + - n - ) / ( n - 1) = ( K + - K - ) / K (18) Because of the differing phase velocities c / n + , one can easily show that plane polarized light of wavelength h has its polarization plane rotated in a distance I by an angle (19) where we have used the optical convention in which the rotation is considered positive when it appears clockwise looking into the source. Rotation of the form shown in Eq. (19) is the observable quantity measured in four experiments (at Seattle, Oxford, Novosibirsk, and Moscow) discussed later. B. STARK-PNCINTERFERENCE When an external static electric field Es is applied to the atom, a Stark El amplitude appears with a form similar to the PNC amplitude in Eq. (1 2),
+ (fl
Es ‘ r I n>(n I r I i> Ei - En
1
(20)
where the Stark perturbed states are given a subscript S. For analyzing the geometry of experiments, it is helpful to use the vector Ci defined after Eq. (13):
E. N . Fortson and L. Wilets
328
The Stark-EpNc form of interference is utilized in experiments with heavy atoms at Berkeley and Paris, and in many experiments with hydrogen. We defer to a later section the discussion of hydrogen. Here we present a qualitative sketch of the scheme used at Paris and Berkeley. The basic idea, originally pointed out by Bouchiat and Bouchiat (1979, is that an electronic polarization (i.e., a nonzero expectation value of the electronic angular momentum (J)) in the excited state of the atom is induced by absorption of a circularly polarized photon directed perpendicular to an applied static electric field. For definiteness, let the static field be in the x direction, Es = Es$ and let the incident photon have its momentum k in they direction, so that its circularly polarized electric vector is given [cf. Eq. (12)] by E = E ( 2 - iqa)/fi. We look for electronic polarization in the z direction given by
( P , Ie=
c
1 B11 is>
m,l(fs
mrm,
<J,'> = I(fs
mrmi
I
+ (fPNC
I 's> + ( f P N C I
1 "1 'PNC)lz I 'PNC)I2
(22)
The sums over m,, mi, Fi give in general a nonvanishing result proportional to q. Thus, a measurement of J,' provides the pseudoscalar J' * qk x & which reveals the PNC interaction in the atom. More details of the calculation of Eq. (22) are given in Section VI.
IV. Atomic Calculations We turn now to the methodology used in calculating the PNC-induced El matrix elements introduced in Eq. (12). An earlier review is to be found in Wilets (1978). The operators l?1 and H p N c have very different spatial dependence. The integrand for the matrix elements of E l i is large over most of the atom (and depends on details of the atomic wave function). HpNc= C,pN(ri) y ' ( i ) is concentrated at the nucleus. The electronic wave functions near the nucleus can be solved for very precisely, except for their normalizations, which do depend upon the whole atomic wave function; Only s,/* and electronic functions are connected by HpNc,whereas E 1 can connect any functions for which AI = ? 1. A. INDEPENDENT PARTICLE APPROXIMATIONS The first elementary approach is to approximate the exact wave function In) by a single determinant of independent particle model wave functions
329
PARITY NONCONSERVATION IN ATOMS
(IPM). These wave functions can be derived from parametric or HartreeFock (HF) potentials. This is quite a good approximation for the alkali metals, such as Cs, which contain one valence electro? outside of closed shells. Since the electromagnetic operators, L?l or M1, and HPNc are single-particle operators, only one (valence) electron will be involved in the transition. Let us denote by In) and E,, the single-electron states and energies. Then (fPNC
I
''
I iPNC)
The sum over intermediate states is unrestricted; it includes not only unoccupied states, but also occupied states. The latter correspond to two-particle/one-hole excitations, as shown in Fig. 1. The sums in (23) can be executed utilizing a technique pioneered by Sternheimer (1954). Let
Then 11) and (FJsatisfy the inhomogeneous equations ( h - Ei)lI)
= - HpNCli),
(Fl(h - ~ f = ) -(flH,Nc
(25)
where h is the parity-conserving IPM Hamiltonian. Both 11) and (FI are of opposite parity to li) and (fl. Since there are no eigenstates of that parity with eigenvalue or q , there are no homogeneous solutions to Eq. (25). The required solutions can be constructed, for example, by adding to any particular solution which is regular at the origin a multiple of a solution to
PNC
(b)
El
(d)
FIG. 1. Lowest order (in the residual interaction) diagrams contributing to one-electron approximation.
&pNc
in the
E. N . Fortson and L. Wilets
330
the homogeneous equation, also regular at the origin, such that the resultant is regular at infinity. The E l matrix element is then given by <jpNc I
~ i p N c ) =(FIi 1 l i ) + ( j l g 1 1 1 )
(26)
Because of the symmetry between k1 and HPNcin Eq. (3), this procedure can also be carried out by the interchange HpNct)I?1, E , t)E , .
B. FURTHER COMPLICATIONS AND CORRECTIONS Anything beyond the IPM can be classified as configuration mixing, correlations, polarization, shielding, and so on. Some of these terms have more specialized connotations than others, and some of the terms imply certain methods of calculation. 1. Intermediate Coupling
Except for the case of a single valence electron outside of closed shells, it is necessary to take a linear combination of (Slater) determinants in order to construct an atomic wave function of good J and M . If the various determinants all have the same occupation of n and I , we call this coupling within a configuation; the two most familiar schemes are L-S and j - j coupling. Spectroscopic notation usually specifies a state as though it were L-S coupled. Because relativistic effects are important in heavy atoms, it is more convenient to use a j - j coupling representation. Since actual atoms are intermediate between L-S a n d j - j coupling, one must take a linear combination of j - j coupled states. Bismuth is a case in point. The low-lying configurations are 6p3 coupled to 4S3/2,2D3/2,and 'DsI2. The j of each electron can be either 1 / 2 or 3/2, and these states can be constructed from various combinations of and p3/2 electrons. [Note that the state J = 5 / 2 , M = 5 / 2 is unique, p3/2 ( m = 3/2) p3/2 ( m = 1/2) p,/* ( m = 1/2), and is represented by a single determinant.] The intermediate coupling coefficients for the J = 3/2 states have been determined by Landman and Lurio (1962) from an analysis of hyperfine splitting. The calculations of the PNC effects are not very sensitive to the uncertainties in these coefficients. 2. What Is h,? The single-particle Hamiltonian in Eqs. (23) and (25) is required to execute the sum over intermediate states or to solve the inhomogeneous differential equations. If a Hartree-Fock potential is used, the potential is
PARITY NONCONSERVATION IN ATOMS
FIG. 2. Diagrams summed by Martensson er a / . (1980) in order to treat H,,, consistently in HF.
33 I
self-
determined for the occupied ground state orbitals and is, in principle, completely undetermined for excited orbitals. This point has been emphasized by H. P. Kelly (private communication, 1979). The problem can be circumvented by solving the HF equation in the presence of H,,,, which is a one-electron operator (Sandars, 1977). Mlirtensson et al. (1980) have performed such a calculation for Bi transitions. This consists of replacing the PNC vertex by the dressed vertex obtained from the integral equation depicted in Fig. 2. For the 4S,/2 -+'D,/, transition they obtain for -R the value 15.1 x compared with 9.6 x lo-' using a parity conserving HF. (The value through first order only in this correction is 12.9 X lo-*.) 3. Nearby Configuration Mixing Multiparticle-hole excitations can be admixed to either of the transition states through the residual two-body electric interaction. If the mixing is small, it will have little effect on the matrix elements of H,,,, but, as first pointed out by R. D. Cowan, S. Meshkov, and S . P. Rosen (private communication, 1977), a small d-state admixture could ppsibly have a large effect on the El transjtion amplitude, since ( p 1 E 1 I d ) can be considerably larger than ( p I E 1 1 s). The point was investigated in considerable detail by Henley et al. (1977) for Bi, who found that inclusion of all nearby configurations produced a correction of less than 4%. A simple argument shows that the correction is of orde;
where V is the residual interaction, A E is the energy from the unperturbed transition state p to either of the admixed intermediate states s or d. Note that A E is not the splitting between the s and d intermediate states. They could be degenerate and not affect the argument! 4. Shielding
Harris et al. (1978) have called attention to and performed calculations on a particularly important kind of configuration mixing which leads to a
E. N . Fortson and L. Wilets
332
DIRECT
EXCHANGE
J
I
FIG. 3. A few of the perturbation diagrams summed in TDHF/RPA. Note that exchange diagrams begin to proliferate in second order since we have included, on that side, mixed direct-exchange terms.
reduction of El matrix elements. It is shielding of the radiation field by the core electrons. Although the effect is contained in the initial and final state wave function correlations, it can also be calculated in time-dependent Hartree-Fock (TDHF) or the random phase approximation (RPA) in a way that is specific to the physical effect. The diagrams which are summed are shown in Fig. 3. The effect is known to be important in cases involving penetrating orbits or easily polarizable cores. The electromagnetic field is treated classically; the electric dipole component is described by
@,(r, t ) =
- 2E,r cos 8 cos at
(28)
In the following eE, is set equal to 1. The HF wave function is given by
*(rl,
. . . , rz, t ) = (Z!)-”*det[J/,(r,,t)] (29) $,,(r, t ) = e-%‘[+,,+
+,
U+e-Iu/
+ U-e+iuf
1
Here and E, are eigenfunctions and eigenvalues of the spherical HF Hamiltonian h,. The u’ are first order in E,, and satisfy the inhomoge-
333
PARITY NONCONSERVATION IN ATOMS
neous equation
The term C[ ] is the dipole part of the potential due to the response (polarization) of all of the other electrons in the atom. The exchange operator P., comes from antisymmetry; it is the Fock term. It is instructive to consider first the zero-frequency limit, w = 0. This is actually not a bad approximation for low-lying states of Bi, since the transition frequency in that case is low compared with all other characteristic electron frequencies. This corresponds to solving the wave equation for the Hamiltonian 3c = H - eE,rcosO
to first order in E,. The static H F approximation is then obtained from (30) by setting w = 0 and dropping the superscript 2 ; 2eE0 cos or 3 1 . The resultant dipole component of the field is the sum of @, plus the selfconsistent contributions of the atomic electrons. It has the limiting forms - cP/cos
8 = cr’,
r +0 ;
-@/cosO
= r - (a/r2),
r+m
(31)
The small r limit obtains because a self-consistent solution cannot produce a net force on the atom and, in particular, a net force on the nucleus. The large r limit exhibits the applied field modified by the atomic dipole field; a is the atomic polarizability. Shielding calculations to date (in this context) have neglected the exchange term in the self-consistent potential [the P, in Eq. (30)l. This has led to some ambiguity in the treatment of the self-field of the electrons. If the P, is merely dropped (as is usually the case), then the self-field is included. The true Hartree treatment would explicitly exclude the self-field term. In practice, the difference in the two approaches appears to be smaller than the polarization field of a single electron, due to selfcompensation. [See, however, Mdrtensson er al. (1980).] A more serious problem with the T D H F method is that the effect in question is dependent on the magnetic substate M , which is disturbing since the exact result is not (except for geometry through Clebsch-Gordon coefficients). A prescription for “spherical averaging” is usually invoked.
334
E. N . Fortson and L. Wilets
FIG. 4. The net dipole electric potential in the presence of an external static electric field, as calculated by E. N. Fortson and R. Katz (private communication, 1978) and Hams er al. (1978). The radius is in atomic units. Asymptotically, the difference between the unshielded and shielded potential is given by a / ? , where a is the dipolarizability. Superimposed on the figure is the unnormalized curve of the radial transitive integrand rR,,R,, where R, is the parity mixed s-state function. The shielding correction consists of replacing r by the shielded potential function.
There is also the ambiguity as to whether to use initial or final state or (probably best) intermediate state configurations, and the results do depend upon the choice. Finally, published calculations have been performed only with single determinants, not coupled determinants (and hence not intermediate coupling). An interpretation of the variations among different results is that they give a measure of the uncertainties in the calculations. In Fig. 4 are shown static calculations for Bi: "HF" by Harris et al. (1978), and "HF" by E. N. Fortson and R. Katz (private communication, 1978). The difference between r and -cP/cos8 is the shielding. Also imposed on the figure is the integrand that enters into the calculation of (fPNCIEllipN,-), namely, rR6pRs, where R, is the radial PNC function which satisfies Eq. (25), with arbitrary normalization. The true integrand is obtained by the replacement r + - cP/cos 8. In the case of Bi, the integrand lies well into the atomic interior; the largest single compo, ~ state. In this region the shielding is great and nent of R , is the 6 ~ ,hole the reduction factor from the IPM calculations is 1/2 (Harris et a/., 1978) to 2/3 (E. N. Fortson and R. Katz, private communication, 1978). This is also consistent with a nonrelativistic calculation by S. Fraga (private communication, 1979).
PARITY NONCONSERVATION IN ATOMS
5. The Dipole Transition Operator The dipole transition operator can be written in the length form El=D,= -eCri i
or the equivalent velocity form
61 = D, = i e C v , / w i
where w is the transition frequency (q - q)/h and vi is the velocity operator, p/ me nonrelativistically or ca relativistically. If one has exact many-body wave functions for the two states, then the matrix elements calculated with either form should be exactly the same. Because approximations must be made, the two operators frequently yield very different results. One can argue that either: (1) a discrepancy in the results is a measure of the quality of the wave functions and the reliability of either number (Carter and Kelly, 1979); or (2) that one form is inherently better because it is less sensitive to the accuracy of the wave function. There is a long history on this subject. We believe that the weight of empirical and experimental evidence favors the length form. As one argument, consider the Hartree-Fock approximation. Since it is an independent particle model, the velocity operator is
where h is the single-particle, nonlocul, H F Hamiltonian. The resultant v is not p/m, . If the right-hand side of Eq. (34) were used in the calculation of E 1, one would obtain the same result as with D,. On the one hand, this demonstrates the inconsistency of H F with the exact result, but it also demonstrates that H F is internally consistent with gauge invariance [p + p + ( e / c ) A ]if the length form is used [equivalent to the right-hand side of (34)). This is aiso the conclusion of Sandars (1980) and, including relativistic field theoretical corrections, of Hiller et al. (1980b). Sandars (1980) offers the observation that if one begins with a local potential, then in each order of perturbation theory, the length and velocity forms give the same result for E 1. The local and H F results are not very different from each other if the length form is used. Carter and Kelly (1979) find that they can be very different if the velocity form is used, and also the length and velocity forms can yield very different results in HF. We conclude: Stick to the length form.
336
E. N . Fortson and L. Wilets
6. Many- Body Perturbation Expansion
In compact notation, all perturbation diagrams (for one-valence electrons) through second order in the residual electrostatic interaction are If shown in Fig. 5. The one-body potential U is assumed to include If,,,. one wishes to display H,,, explicitly as a perturbation, then an IfpNc vertex must be placed on each propagator, but only one per diagram since first order in H,,, is sufficient. Only topologically different types of diagrams are shown. Various sequencing of vertices and “time” orderings are not displayed. The classification ( n - j ) is according to the order n ‘in the residual interaction. The zeroth-order diagram (0) encompasses the set of four diagrams in Fig. 5 discussed previously. Diagrams (1-1) and (2-1) are the first two terms in the infinite series which is summed by the RPA of TDHF shielding equations. If the I/-potential is taken to be some HF potential, then the “box” vertex vanishes everywhere it appears. Note, however, that the HF potential can be self-consistent for the initial (i) or final ( f ) occupancy, but not both. Furthermore, it is most inconvenient to use a self-consistent potential for open shells, since one loses spherical symmetry. Thus the box vertex is the correction to lack of self-consistency. It is usually a small correction, although diagrams formed from individual “pieces” of the box vertex may be large. Diagram (2-2) is a core polarization diagram where the valence electron polarizes the core (in contradistinction to polarization by the external photon). (2-1), and (2-2), all other diagrams to Except for diagrams (0), (1-l), this order represent corrections to the electron propagator. The objective of the diagrammatic approach is this: RPA or TDHF have ambiguities, especially in the presence of open shells. One can sum infinite subsets of diagrams by these techniques and then correct these sums (for omitted diagrams) in each order by evaluating the contributions exactly. This corrects for the type of (usually) local potential and the problem of intermediate coupling. In the case of Bi, Sandars (1980) has evaluated all diagrams through first order. Whereas we show two in first order, there are eight when the dots and boxes are displayed explicitly. This was a formidable task. The largest contributions came from the “direct” parts of (1-1) which were greater in magnitude and opposite in sign to the zeroth order term. The sums of the remaining terms were roughly 10% of (1-l), the actual value depending on the transition. The sum of the direct RPA/TDHF diagrams, namely, (0), (1- l), (2- l), . . . was such as to yield a reduction in the zeroth order value O the discussion of Bi in the next section). by about ~ W(see
331
PARITY NONCONSERVATION IN ATOMS
f
(2-3)
fL
(0)
i
(1-2) (2-8) FIG. 5. All perturbation diagrams through second order in the residual electrostatic interaction for one-valence electrons. A very compact notation is used. Signs and weights are not indicated; all “time” orderings must be included. In the label (n-j), n refers to the order.
7. Work in Progress or Projected
In addition to refinements and checking of calculations already published, various improvements and novel approaches have been proposed, some of which are currently in progress. We mention a few here.
E. N . Fortson and L. Wilets
338
Grant et al. (1980) are working on a multiconfigurational relativistic Hartree-Fock (MCRHF) in Bi, where 6p3, 6p27s, and 6s-'6p4 configurations are handled simultaneously. The configurations are admixed by HPNc.Preliminary results have been presented by Sandars (1980). Hiller et al. (1978) have formulated a new technique for evaluating matrix elements of a contact interaction with the nucleus, proportional to C i y s ( i ) 6 ( r , ) ,such as is required for H,,,. (They also considered contact potentials between electron pairs.) By use of identities, they can express such matrix elements in terms of an integral over the entire wave function rather than the value at r, = 0. That is, the expression is less sensitive to specific details of the wave function. Tests on helium with Hylleraas-type wave functions indicated that their technique gives a significant improvement over straightforward evaluation using at r; = 0. Hiller et al. (1980a, b) have also formulated the many-body problem for atomic PNC transitions in a manner that is consistent with relativistic field theory. They have made specific tests on helium. Wilets et al. (1980) have proposed a variational method to evaluate electron-electron correlation structure in atoms, with the intent of including shielding and higher order corrections. The method has been tested on helium and proved to be practicable. It can be extended relativistically.
+
V. Optical Rotation Experiments: Bismuth A. INTRODUCTION Experiments to look for PNC optical rotation began in 1974 at Seattle (Soreide and Fortson, 1975; Soreide et al., 1976), Oxford (Sandars, 1975), and Novosibirsk (Barkov and Zolotorev, 1978a; Barkov et al., 1979; see also Khriplovich, 1974). All three groups picked atomic bismuth because it has MI absorption lines that can be reached by available tunable lasers. A fourth experiment, also using Bi, began more recently at Moscow. The bismuth energy levels and the transitions studied by the different groups are shown in Fig. 6. The Seattle experiment uses the J = 3/2+ J = 3/2 line at 8757 The other experiments use the J = 3 / 2 + 5 = 5/2 line at 6474 A, although the Oxford group has begun working recently at the 8757 A line also. All experiments take advantage of the chraracteristic change in +PNC with wavelength to separate it from other rotations in the apparatus. Equation (19) tells us that GPNc follows the dispersion curve determined by n - 1 and has a particularly striking asymmetry about each absorption
A.
PARITY NONCONSERVATION IN ATOMS
339
5 8 790
B i l l Limit 2 3 6 P ( P2)7S
49 457
2
44 8 6 5
4 +
6p2(3P1)7s
+ p3/2 '312
6 P3 62
15 4 3 8
2
I I 419
2
5./ 2_ -
0 Bi I
6 P3
2 +
2 +
FIG.6. Energy levels of 83Bi.
line. The maximum change in angle about an absorption line is determined by the line shape and by / K O , the number of absorption lengths at line center. For an isolated Lorentz-shaped line, the change between dispersion peaks is readily found to be &PNC
=
/KoR
which serves as a useful estimate even when the line shape is not Lorentzian. Here we have defined
R
=W & P N C / W
(35)
As we will see, calculations give I R 1% lo-'.
It is the ratio R that is quoted in comparisons between theory and experiment.
B. STATUS OF ATOMICPNC CALCULATIONS FOR BISMUTH The calculation of 311. depends only on the intermediate coupling coefficients and not on details of the radial wave functions [see Eq. (17)]. The coupling coefficients have been chosen by most authors to be those
E. N. Fortson and L.. Wilets
340
obtained by Landman and Lurio (1962) from an analysis of hyperfine structure. The M1 matrix element is known more reliably than the El. The most systematic calculations of G,,, have been performed by Sandars and collaborators (Sandars, 1980) and by Miirtensson et al. (1980). The first group begins with a parametric IPM potential and calculates first the lowest-order diagrams of Eq. (23). Beyond this they calculate shielding corrections in TDHF/RPA, without exchange, and all first-order corrections. Miirtensson et al. also includes exchange terms in the shielding corrections and treat H,,, self-consistently, as discussed in Section IV, B, 2; they use a spherically averaged Hartree-Fock potential. The resultant values for R are: (Sandars, 1980) (MBrtensson 1980)
x 10-8,
R(3/2+3/2):
- 11
R(3/2+5/2):
- 13 X
x - lox -8
(36)
An estimate of the reliability of these calculations is difficult, but we attempt here to give some perspective. Several other researchers (Novikov et al., 1976; Henley and Wilets, 1976; Henley et al., 1977; Carter and Kelly, 1979) have also evaluated the lowest order, one-electron terms using various IPM potential?; they obtain agreement to within about 1oo/o when the length form for El is used. In contradistinction, Carter and Kelly (1979), using Hartree-Fock wave functions, found that the velocity form for k1 yielded significantly smaller results than the length form. For the ratio D,/ D , they found 0.82 for 3/2+3/2,
0.14 for 3/2+5/2
(37)
For the reasons discussed in Section IV, B, 5, we favor the length form. The shielding correction was found by Sanders et al. (Harris et al., 1978; Sandars, 1980) to be about 50% of the lowest order result, and of a sign to reduce the matrix element. The total correction factor of shielding and all first-order terms (relative to the lowest order calculation) was found to be 0.67 for 3/2+3/2, 0.56 for 3/2--+5/2 (38) Some of the uncertainty or arbitrariness in their parametric potential is “forgiven” by calculating all first-order terms. However, further corrections or uncertainties include: (1) Exchange in the shielding TDHF/RPA calculations. (2) The parametric potential is also used in the shielding calculations, and this may not be optimal. (3) The handling of open shells and intermediate coupling in the shielding calculations.
PARITY NONCONSERVATION IN ATOMS
34 1
(4) Special higher order diagrams, such as the second-order electron-core polarization diagram, Fig. 5 (2-2).
An interesting and significant semiempirical approach to the problem was taken by Novikov et al. (1976) before shielding corrections were considered. They calculated El matrix elements in Au, Hg, and Tl using a parametric potential. They found that the calculated 6 p -m matrix elements agreed well with experiment, with the exception that the 6p-+6s element was too large by a factor of 1.6 in Tl. Although the optical 6p -+6s transition is not observed in Bi, it is the dominant (56%) matrix element in the sum, Eq. (14). On this basis, they reduced this calculated PNC-induced El matrix element in Bi by the factor 1/1.6. This semiempirical approach probably accounts for some shielding and perhaps other higher-order effects. It should be noted that shielding is frequency dependent. The optical 6p +6s transition is higher in frequency than the intraconfiguration transitions used in testing PNC in Bi. Thus, the semiempirical approach tends to underestimate the shielding. Furthermore, Bi is more polarizable than TI due to two extra valence electrons. There are other differences in the Sandars and Novikov calculations, and the latter end up with the TABLE I CALCULATIONS OF R
=
Im(GpN,/9R)
FOR
fyBi,,,“
3/2+ 3/2 Independent particle models Parametric potential (Novikov et al., 1976) “Relativistic” Hartree-Fock (Henley and Wilets, 1976; Henley ef al., 1977) Parametric potential (Brimicombe er al., 1976; Hams et al., 1978) Dirac- Hartree-Fock (Carter and Kelly, 1979) Plus shielding TDHF/RPA (Harris e l al., 1978) Plus shielding and first-order Semiempirical (Novikov er al., 1976) Perturbation theory (Sandars, 1980) (Mihemson, 1980)
3/2 + 5/2
- 17
- 23
- 18
24
- 17
-
- 16
- 22
- 9
-
11
13
-
17
- 11 - 8
-
13
-
10
-
Results of the various authors have been renormalized to sin2 0 , = 0.23.
E. N. Fortson and L. Wilets
342
slightly larger in magnitude values: R ( 3 / 2 + 3 / 2 ) = - 13
R ( 3 / 2 + 5 / 2 ) = - 17 X
X
(391
Theoretical researchers are reluctant to assign errors to their calculations. Our judgment is to adopt the mean of the Sandars (1980) and MBrtensson (1980) values as quoted in Eq. (36), with the difference giving a measure of the uncertainty, -30%. A summary of various calculations on Bi is given in Table I for the purpose of comparison.
C. GENERAL EXPERIMENTAL FEATURES From the discussion before and after Eq. (35) it follows that AcpPNC lo-’ rad for a single absorption length at line center. All experiments rad. are thus designed to resolve rotations smaller than ‘c.
I . Measurement of Small Rotations
The small rotations involved can be measured by placing a column of Bi vapor between two nearly crossed polarizers. A slight rotation of the plane of polarization of light between the two polarizers will produce a large fractional change in the light intensity I transmitted by the second polarizer. Let rp be the angle by which the plane of polarization differs from the plane of minimum transmission through the second polarizer. Then Z varies as sin’ cp. For cp << I , as is always the case in these experiments, I = lo(@: + rp2)
(40)
where I, is the incident light and rp, is a constant (called the “extinction angle”) dependent upon the polarizer and light beam quality. A rotation 6rp produces its largest fractional intensity change when cp is set equal to rp,, in which case 6I/Z = a+/+,. If rpi = lo-’, a typical value of extinction for good calcite polarizers, then a rotation acp = 3 X lop7rad will produce aZ/ZIt is this enhancement in the fractional effect that makes the rotation experiments feasible. 2. Angle Resolution Shot noise in the rotation experiments is very small. The rms fluctuation in angle Arps due to shot noise in the detected photons is
Arps 2: (210t)-”2
(41)
PARITY NONCONSERVATION IN ATOMS
343
where t is the observation time. The lasers used in these experiments produce photon fluxes of typically I , > 10l6 photons/sec. Thus, in 1 sec, A+s < lo-* rad. In practice, the noise in the measured angles has turned out to be larger than the shot noise limit, due to a combination of mechanical and optical instabilities, and detector noise in some cases. Typically, lo-* rad can be resolved in less than 5 min.
3. Faraday Rotation Faraday rotation of plane polarized light occurs in a medium when a magnetic field is present. The rotation is proportional to the component of the magnetic field along the direction of light propagation. The contribution from each hyperfine component of the transition is conveniently divided into two parts, one which is symmetric about the line center of the hyperfine component and another which is antisymmetric. We give here a brief semiquantitative discussion. The symmetric portion arises because the magnetic field causes a Zeeman splitting of each hyperfine component. Because of the Zeeman splitting and the selection rules governing circularly polarized light, the index of refraction curve associated with right circularly polarized light is shifted in frequency with respect to the index of refraction curve associated with left circularly polarized light. Thus, at a given frequency, the two circular components of plane polarized light have a difference in index of refraction n - n - , which, as in Eq. (19), leads to a symmetric Faraday rotation, +
piB I dn, C#+(s) 'v - hh dv
(42)
where p, is of the order of a Bohr magneton, and v = c/A. From this approximation, we see that magnetic field strengths of G can produce rotation comparable to the expected size of +PNC. The antisymmetric portion of the Faraday effect arises from the fact that the magnetic field mixes pairs of hyperfine states of a given J level proportionally to M , the magnetic quantum number of the pair. The state mixing results in a different transition amplitude for different AmF values, which causes the refractive index curve to differ in size (but not center frequency) for the two circular polarizations. As in Eq. (42) we find the antisymmetric rotation
E. N . Fortson and L. Wilets
344
where Au, is of the order of the splitting among the hyperfine energy levels. The total Faraday rotation is +F =
c [+&) + +;(a)]
(44)
i
In a practical case, the precise formulas for the effects (42) and (43) can be complicated, and the calculations, although straightforward, are very tedious. The results for the Bi transitions of interest are presented later. (See Figs. 7 and 12.) D. BISMUTHOPTICALROTATION AT 8757 A We will illustrate the optical rotation method by a somewhat detailed discussion of the most recent Seattle experiment. A review of the other optical rotation experiments will then be given. The bismuth absorption line J = 3 / 2 + J = 3/2 at 8757 A is both magnetic dipole and electric quadrupole allowed. The intensity of the E2 moment is about lop2that of the M1. The nuclear spin is 9/2. There are nine prominent MI hyperfine structure (hfs) components to the line, not all of which are Doppler-resolved. Figure 7 shows the expected patterns for absorption, Faraday rotation per gauss, and PNC optical rotation as a function of wavelength, where the vertical scale has been set for an optical 65
9
33 4 3
1.00
+Ik X
su
a
i 0.50
K
I-
-
c
1.40-
>-J L F
(b)
Za ?2
0
-
ASSUMED R = I O - ~
Pi3
22
a o at a m' k2
0
-
-1.66
0
3
6
9
12
15
18
21
24
27
30
GHz
FIG.7. Theoretical curves showing the hyperfine structure for the 8757-A line in atomic bismuth. The optical depth corresponds to two absorption lengths at the peak of the ( 6 , 6 ) hfs component. Shown are (a) absorption, @) PNC optical rotation, and (c) Faraday rotation.
PARITY NONCONSERVATION IN ATOMS
345
depth of two absorption lengths at the center of the strongest hfs component. The curve shown for @PNC has been computed for R chosen illustratively to be lo-'. The tunable light source in the most recent Seattle experiment is a gallium aluminum-arsenide semiconductor laser diode. The laser is operated by applying a forward bias to the diode junction. Light is emitted by injected minority carriers undergoing stimulated recombination at the junction. The laser intensity is a linear function of the injection current and can be remarkably stable, limited only by the current regulation of the power supply. Both Mitsubishi TJS lasers (Namizake, 1975) and Hitachi CSP lasers (Aiki et al., 1977) produce stable, highly monochromatic single-mode radiation, in a beam of excellent optical quality. Maximum intensities ranging from 5-10 mW are available, with 85-95% of the power on a single mode. The rest of the power mostly is scattered on other modes separated by at least 4 cm-' from the main mode. The wavelength of a given mode is a function of both diode temperature and injection current. Typical coefficients are 1 cm-'/"C and 0.03 cm-'/mA. In this experiment, the wavelength is swept by varying the injection current. The laser temperature is adjusted to position the sweep to cover a desired portion of the bismuth absorption, and the temperature is regulated at that setting to "C. Figure 8 shows the bismuth absorption pattern taken with a laser diode and plotted together with the theoretical profile. The detailed agreement is apparent. The absence of molecular bands and other background is clear. A schematic outline of the latest version of the Seattle experiment is shown in Fig. 9. The light leaving the diode laser is focused to a 3-mm
T
5 0.8
2 0.6 I-
+
5 0.4
v)
z a 0.2 LZ I-
FIG. 8. Comparison between the theoretical absorption curve and experimental data points. The optical depth is the free parameter in the fit. The data was taken with a laser diode at 8157 A.
E. N. Fortson and L. Wilets
346
R
FIG. 9. Plan of the Seattle bismuth experiment at 8757 A.
diameter beam and passes in order through a polarizer, a water cell Faraday rotator, the heated bismuth cell, a second polarizer, and then into a PIN silicon detector. Light reflected from the front surface of the second polarizer is further split into two beams, one detected for a reference signal, the other sent into a four-quadrant detector which measures beam movement. Calcite prism polarizers of both the nicol and Glan-Thompson variety have been used. All windows are wedged and AR-coated to avoid etaloning effects. The Faraday cell produces a modulation angle +,, which varies sinusoidally with an amplitude of about rad, and a frequency of 1 kHz. The total angle is = +,, +s, where rpS includes all rotations besides the modulation angle. Equation (40) shows there should be a I-kHzharmonic in the detected signal, given by 2Z+,+,, which serves to measure &. This harmonic is measured by a phase-sensitive detector (PSD) after variations in I, are divided out using a reference beam signal as shown in Fig. 9. The PSD output, together with other signals for use in the data analysis, is fed into a PDP-81 computer for storage as a function of laser wavelength. The time average of GS is held to rad by feedback to an auxiliary coil on the Faraday cell. This control coupled with the division by the reference signal makes the PSD output highly insensitive to changes in light intensity at the second polarizer, and in particular rejects the Bi absorption pattern. A change in I, of 500/0 affects the angle measurement by lo-* rad.
+
+
PARITY NONCONSERVATION IN ATOMS
347
The bismuth cell is an alumina tube with cooled quartz windows. About 1 m of the tube is heated by the oven, which is regulated at temperatures
usually between 1250 and 1400 K. He gas at 30 T o n pressure confines the bismuth vapor to the heated part of the tube. For some of the measurements a movable tube with open ends was located inside the main tube in order to move a column of Bi vapor in and out of the optical path without moving the windows and disturbing the optical alignment. A solenoid wound around the oven provides a uniform magnetic field to control the Faraday rotation +F associated with the bismuth absorption line. Two concentric cylinders of permalloy magnetic shielding surround the solenoid to exclude external fields. Early laser diode data (Fortson, 1978a,b) taken prior to completing the system for controlling and analyzing spurious effects were not considered reliable. To obtain the rotation signal +s as a function of wavelength, the laser wavelength is swept over a selected portion of the Bi absorption pattern at a 1-Hz rate. A triangular injection current modulation is used, which reverses the sweep direction every half period. On alternate periods the solenoid is adjusted to minimize +F and the oven heater current (ac) is switched off to avoid an observed spurious effect due to ac GF (which can lead to the appearance of d+,/dX if the laser wavelength has any synchronous ac modulation). These sweeps are added together in the computer and constitute the PNC data. The reference beam signal is also recorded in the computer to give the absorption pattern. During the alternate periods when the oven current is on, the solenoid current is turned up to give a large Faraday pattern, and the sweeps are recorded separately in the computer. During all sweeps, a number of potential sources of systematic error are also monitored and stored in the computer. As one example, the output of the beam movement detector is stored as a function of laser wavelength to check on possible light bending due to the bismuth index of refraction. After 10oO sweeps, all data stored in the computer are put on tape, and the procedure is repeated. The PNC data in each 1000-sweep curve is fit with the theoretical +PNC curve. The size of GPNC found, taken together with the measured +F in a known magnetic field, yields the value of the desired quantity R independently of the optical depth and of the rotation angle sensitivity. An independent calibration uses the measured absorption and the rotation angle sensitivity. Thus far, some 300 data curves have been accumulated in this manner in about 60 hr of running time. Three different laser diodes were used, the bismuth density was varied over a range of a factor of 8, and the polarizers were reoriented many times. The PNC curves have been analyzed for various possible spurious features including Faraday and absorption-
348
E. N . Fortson and L. Wilets
related effects. The latter two do not contribute to PNC above the 1x level. An important part of the analysis of the data consists of a search for correlations between the measured values of R and of other variables, such as residual +F, beam movement signal, polarizer configuration, etc. During the early stages of taking data, a number of possible systematic errors were uncovered this way, and the apparatus was modified to eliminate them. For example, a correlation was observed between R and the overall slope of angle versus wavelength in the PNC data, which turned out to come from an asymmetry about each absorption line caused by the finite linewidth (50 MHz) of the laser diode then being used. A new laser diode having a far smaller linewidth eliminated this spurious contribution to +PNC*
Of somewhat special interest was the beam movement signal, to which was fit a GpNc-shaped curve. No correlations were found between this fit and the R value for the regular rotation curves. This result seems to rule out a possible spurious effect from spatial variations of optical depth in the bismuth tube. A serious problem has been an oscillatory dependence of the polarization angle with wavelength, apparently an interference phenomenon generated within the polarizers. In principle, this background can be measured and subtracted. In practice, much care is needed in the alignment and positioning of the polarizers to reduce the background below lo-’ rad. Since the summer of 1978 when the data analysis system was completed, four separate sets of curves have been taken at Seattle. The average values of R for each set agree within 2 I X lop8with each other. A sum of 50 of the curves is displayed in Fig. 10, together with the fitted +PNC curve. Because of the strong absorption used for these curves, data near the centers of absorption lines were omitted. The average of all laser diode data that has been analyzed for systematic errors yields the value R = -9.8 k 2.4X (Hollister et al., 1980), where most of the quoted error comes from systematic uncertainties. The statistical deviation is only 20.2 x Although these data agree with the very first result from Seattle [published in Baird et al. (1976)], R = -8 ? 3 x in which there were acknowledged possible systematic errors, they disagree, with the later independent determination (Lewis et a/., 1977), R = -0.7 2 3.2 x The former experiment used data within the absorption profile, the latter used data only from the wings of the line. Both of these earlier results came from experiments that used an optical parametric oscillator laser which did not resolve the hyperfine components of the line. The quoted errors were purely statistical, and no analysis of systematic errors as thorough as in the present Seattle experiment was feasible then. In particu-
PARITY NONCONSERVATION IN ATOMS
349
FIG. 10. Display of Seattle PNC data points accumulated from 50 sweeps as described in text, compared with the best fit of a theoretical $ J curve. ~ The ~ region ~ between lines 1 and 4 of Fig. 7 is covered. High bismuth density was used for this data, giving eight absorption ~ ~ of strong absorption is lengths at the center of line 2. Altered sensitivity to c $ in~ regions shown on the theoretical curve.
lar, the possibility of a spurious effect from dispersive beam movement, apparently ruled out in the present Seattle experiment, was noted at the time of the earlier experiments (Fortson, 1977). They were operated at higher oven temperatures than now, which might amplify beam movement. In view of the recent work at Oxford on beam movement to be discussed in Section V, D, 1, there might well have been such problems. At Seattle, the present result is regarded as far more reliable than the 1976/1977 measurements. Not only is it now free of at least those systematics to which the earlier experiments were known to be vulnerable, but as stated previously, the present result is a synthesis of four mutually consistent and considerably independent measurements stretching over a period of two years. We mention here also the Oxford experiment at 8757 A (Baird, 1979). They use a dye laser system which sweeps over all the hyperfine components of the line. Their experiment appears very close to producing significant results.
E. BISMUTH OPTICAL ROTATION AT 6477 A The bismuth absorption line at 6477 A is J = 3 / 2 + J = 5/2, and is composed of 18 hyperfine components of which 12 are mixed MI and E2, and 6 are pure E2. The E2 intensity is about 0.2 that of the MI. This line is
350
E. N . Fortson and L. Wilets
I
2
3 45 I
I ,
6 789 1 I.
101112I3 141516 I .
I
I I
1718 I
FIG. 11. Theoretical atomic bismuth (a) absorption lines, (b) Faraday curve, and (c) PNC optical rotation curve, all at 6477 A. Both MI and E2 components contribute to (b), but only MI contributes to (c). The field in (b) is opposite that in Fig. 7c. The abscissa is reversed from Figs. 7 and 12. The observed absorption taken at Novosibirsk, shown in (a), displays the complicated pattern of absorption due to Bi, molecules in this wavelength region (Barkov and Zolotorev, 1978a).
about 1/6 the strength of the 8757 A line. This fact plus the use in all experiments at 6477 of a shorter column of Bi vapor (< 30 cm) means that rather higher oven temperatures (- 1500 K) are required for sufficient Bi density. The location and relative intensity of the hyperfine components is shown in Fig. 11 together with the expected Faraday and PNC rotation as a function of wavelength. Also shown is an absorption spectrum in this region taken at Novosibirsk. In addition to the atomic absorption there is an allowed absorption band in this part of the spectrum due to Bi,
A
PARITY NONCONSERVATION IN ATOMS
35 I
molecules, which explains the complicated absorption pattern actually observed. Fortunately, the Faraday rotation of the molecules is much smaller than that of the atoms because the molecular states producing this band have zero electronic angular momentum. One consequence of the molecular absorption is to limit the usable optical depth to about one absorption length on the strongest M 1 hfs component. 1. The Oxford Experiment
A
The tunable light source for the 6477 line at Oxford is a SpectraPhysics model 580 A dye laser that produces several milliwatts of singlemode radiation with a frequency stability of a few megahertz. The laser cavity has longitudinal modes spaced about 400 MHz apart, and the laser is made to operate on a single one of these modes by using etalons inside the cavity. Fine tuning of the laser wavelength is accomplished by changing the length of the cavity while maintaining etalon adjustment for optimum power. Figure 12 shows the Faraday and expected PNC rotation patterns for the largest ( F = 6 + F = 7) hyperfine component at 6477 A. The Oxford group has concentrated much of their attention on this line. In the figure is shown a comparison between the theoretical and experimental Faraday curves. The agreement is very good, and impressively verifies the interference pattern on the low frequency wing due to the pure E2 transition F = 5 + F = 7.
0.6-
. . . EXPERIMENT
-CALCULATED - - - PNC SHAPE 04-
. I GHz
0.2-
/' /
,
/
t
\
\ \
FREOUENCY
FIO. 12. Theoretical PNC and Faraday curves for the (6,7) hyperfine component at 6476
A,where much of the Oxford experimental data has been taken. The experimental Faraday points taken at Oxford show excellent agreement with the theory (Baird er al., 1977).
352
E. N . Fortson and L. Wilets
In the figure one sees there are two points of zero Faraday rotation where the PNC rotation has practically its maximum change A + p N C m R . The plan of the Oxford experiment is to determine A+pNC by measuring the change in angle as the laser wavelength is switched between these two points, thereby eliminating the Faraday rotation. The wavelength is switched by shifting the intracavity etalon tuning so as to move three cavity modes which have the same spacing as the separation between the two points on the bismuth line. This simple method of switching the laser wavelength causes the least disturbance to the geometry of the laser beam and the smallest systematic effect on the rotation signal. The overall optical layout is similar to the Seattle experiment already discussed. The bismuth vapor cell and a Faraday cell for angle modulation are placed between crossed calcite polarizers. The laser beam passes through these components and is detected by a silicon diode and compared with a reference beam reflected in front of the second polarizer. The Oxford experiment has a movable double-oven arrangement which allows rotations to be measured with and without a column of Bi vapor between the polarizers, but without disturbing any optical components while moving the oven. The change in At$ between bismuth-in and bismuth-out constitutes the signal. Data have been taken also by changing the Bi oven temperature rather than moving the oven as a means of changing the Bi optical depth. The Oxford experiment is operated by an on-line computer system. This system is programmed to make a number of systematic checks. We will discuss here mainly the most recent Oxford data using the on-line computer. They have uncovered what appears to be a systematic effect associated with the orientation of the second polarizer. Using the Glan-Thompson polarizer, and tilting the normal to its front face by about 7 t 0 from the light beam axis, they obtained different values of A+ with a given orientation and with the polarizer rotated 180" about the beam axis. They attribute the difference to a possible light beam movement on the second polarizer due to bending of the light by Bi density inhomogeneities in the vapor cell. There would be a wavelength dependence of the bending because the refractive index of the vapor changes sharply with wavelength. The possible appearance of such an effect in the Seattle experiment was discussed in Section V,C. A theory of this effect worked out at Oxford predicts it would reverse sign when the polarizer tilt is reversed, and would go to zero when the polarizer face is normal to the light beam. The Oxford group has followed up this work with a set of measurements using a Glan air polarizer with its face set normal to the beam, and periodically flipping the polarizer by 180". The arrangement required a modification of the detection scheme to eliminate the necessity of reflecting a reference beam from the front face of the second polarizer. The result
PARITY NONCONSERVATION IN ATOMS
353
of this set of runs is a measured value of R roughly midway between the values found from the two tilted polarizer measurements-a result consistent with their theory of beam movement effects. The results from these recent measurements at Oxford yield a prelimifor the 6477 nary value (discussed by Baird, 1980) R = - 10.7 ? 1.5 X A line, using the (6 -+7) hyperfine component. A separate measurement at two points in the hyperfine structure which should give zero is consistent with zero. The errors quoted are statistical and are one standard deviation. This result disagrees with an earlier measurement by the same group This earlier (Baird et a]., 1977) which yielded R = +2.7 ? 4.7 x measurement did not have the benefit of the computer control and came prior to the discovery of the polarizer systematic effects. Nevertheless, it was a careful measurement and the difference with the present data is not readily understood. At Oxford, they have appreciably more confidence in their present data, but until further experiments have been completed they are not prepared to quote their new number as a final result.
2. The Novosibirsk Experiment The tunable light source at Novosibirsk is also a dye laser. They use a Spectra-Physics Model 375 that does not have an intracavity etalon for mode selection and tuning. They have developed instead an interesting alternative in which a tilted glass plate inside the laser cavity acts as a Michelson-type interferometer and selects one of the laser cavity modes. By changing the tilt of the interferometer, the laser output is tuned in discrete hops, as successive cavity modes (spaced about 400 MHz apart) are selected by the interferometer. The absorption pattern at 6476 A displayed earlier in Fig. 1 1 shows the discrete tuning jumps. Figure 13 shows the overall layout of the Novosibirsk experiment. It shows many features in common with other optical rotation experiments, but there are notable differences as well. A heat-pipe oven is used to produce the Bi vapor. Helium is maintained at a pressure near 20 Torr. The oven is heated until, near 1500 K, the Bi pressure just balances the He pressure. The movement of Bi vapor outward from the center of the oven expels the helium gas, which then provides a 20 Torr barrier at either end of the oven, where the bismuth condenses and returns to the center of the oven by wicking action along the tube walls. The bismuth vapor pressure will tend to regulate at the applied helium pressure. The prism polarizer and analyzer are used as the entrance and exit windows of the bismuth cell. There is no additional glass window between the polarizers to introduce spurious rotations into the system. This arrange-
3 54
E. N. Fortson and.!I Wilets BISMUTH
FIG. 13. The plan of the bismuth experiment at Novosibirsk (Barkov and Zolotorev, 1978s).
ment prevents the use of a Faraday rotation cell for rapid angle modulation. Instead the polarizers themselves are rotated to settings of between 2X and rad on either side of extinction. The procedure for taking PNC data is to scan the laser at a 1 KHz rate back and forth across a selected MI hyperfine component. The absorption pattern from the reference photomultiplier (PMl in Fig. 13) is fed back through a PSD to the laser interferometer and is used to keep the wavelength scan between two points of equal absorption. Because of the complicated Bi, absorption pattern overlying the M1 and E2 lines, such a scan may not be symmetric about the desired M1 line. The first harmonic at 1 kHz in the output of the rotation signal from PM2 should reveal the presence of some part of A+PNC. Intensity fluctuations are reduced both by subtraction of the reference signal and by periodic rotation of the polarizers to reverse their angle of offset from extinction. The use of magnetic shielding around the oven is designed to reduce external fields below the level where c#+ would seriously affect the measurements. The 50 Hz magnetic fields due to the oven heater current are reversed in phase at regular intervals with the aim of subtracting off an observed systematic effect due to these fields. In separate measurements, an axial magnetic field is produced by a current through coils inside the magnetic shield, and GF studied carefully. These measurements are used to calibrate the bismuth optical depth in terms of the measured He pressure. The Faraday rotation also serves to locate the M1 lines relative to the selected sweep ranges of the laser. PNC data has been taken on a number of M1 hyperfine components, and also on certain test lines (Bi, or E2 components) that should not show a PNC effect. The result of the most recent round of measurements at Novosibirsk is R = -20.6 2 3.2 X lo-' (Barkov et al., 1979), where the error is purely statistical. This result agrees with the earlier value R = -18 ? 5 X lo-' (Barkov and Zolotorev, 1978a,b), taken on basically the
355
PARITY NONCONSERVATION IN ATOMS
same apparatus, although with some changes having been effected, such as a revised scheme for rotating the analyzing polarizer. 3. The Moscow Experiment
Another experiment to measure optical rotation in bismuth has been underway for the past two years at the Lebedev Institute in Moscow. This experiment also operates on the 6476 A line and uses a dye laser light source. A schematic diagram of the experiment is shown in Fig. 14. A brief description of the apparatus and experimental procedure has appeared recently (Saskyan et al., 1979). Additional information and later results will appear shortly (Bogdanov et al., 1980). The most recent cumulative result of the Moscow experiment is R = -2.3 ? 1.2 X lo-' (I. I. Sobel'man, private communication, 1980).
ROTATION CONTROL
LOGIC
*-
FIG. 14. The plan of the bismuth experiment at Moscow: (1) optical fiber to transmit pure laser mode; (2) absorption reference photodiode; (3) and (6) crystal polarizer and analyzer; (4) Faraday cell; (5) oven with bismuth vapor; (7) and (8) rotation reference photodiode and PM; (9) and (10) analog dividers (Bogdanov er al., 1980).
F. SUMMARY OF RESULTS AND DISCUSSION We group together in Table I1 the most recent bismuth experimental results from each of the four groups. As discussed earlier, quoted results for
356
E. N. Fortson and L. Wilets TABLE I1 RESULTSOF RECENT“ MEASUREMENTS OF F”C OFTICAL ROTATIONIN ATOMIC AND COMPARISONS WITH THE MOSTRECENT ATOMIC CALCULATIONS~ BISMUTH, USING WEINBERG-SALAM THEORY
8757 (3/2+3/2) Seattle (Hollister et al., 1980) 6476 (3/2+5/2) Oxford 111 (Baird, 1980) Novosibirsk I1 (Barkov el al., 1979) Moscow (Bogdanov el at., 1980)
-
9;
I .o
-9.8 f 2.4
- 11; (- 10.7 f 1.5)e
(0.9)‘
- 20.6 t 3.2
1.8
- 2.3 t 1.2
0.2
‘Earlier and, in some cases, contradictory results are discussed in the text. ’Sandars (1980) and Mirtensson et al. (1980). ‘Unpublished preliminary result.
some groups have changed significantly with time. Those listed in the table are believed by each group to be its most reliable. For comparison with theory, the values of R in Table I1 are the means from Eq. (36). The ratio r = Rexp/Rtheor is shown in Table I1 for each result. A straight average gives ( r ) = 1.0. The spread in the four values is too large for the apparent agreement to be meaningful. We would prefer to pick the best experiments, but we see no way of doing that objectively. It appears from Table I1 that there is an effect of the same sign and same order of magnitude as the prediction of the Weinberg-Salam theory. In addition, the Seattle and Oxford results are now in reasonable agreement with each other and with atomic theory. However, we prefer to wait in the hope of better agreement among all the experimental and theoretical groups before attempting quantitative conclusions. Thus we hope all groups will continue their experiments, making measurements under as wide a variety of conditions as possible, with changed polarizers and optics, over a range of bismuth pressures and possibly buffer gas pressures, and using different hyperfine components and test lines. Optical rotation experiments with other heavy elements besides bismuth are underway (Fortson, 1978a) at Seattle. The thallium J = 1/2 + J = 3/2 and the lead J = 0 -+ J = 1 transitions, each near 1.28 pm wavelength, are both under study using cw GaInAsP diode lasers (Hsieh, 1976) suitable for
PARITY NONCONSERVATION IN ATOMS
357
that wavelength. Calculations of the expected effects for these lines already exist (Henley and Wilets, 1976; Novikov et al., 1976), although further work similar to the recent improvements in the bismuth calculations will be needed. Within the next year a reasonably clear picture of the situation with optical rotation experiments will probably emerge. There should be experimental results from TI and Pb, a result from the Oxford measurement at the Bi 8757 A line, and new results from improved versions of all the experiments thus far discussed.
VI. Stark Interference Experiments: Cesium and Thallium A. OVERVIEW
In their original paper, Bouchiat and Bouchiat (1974a) presented a very clever experimental approach which would utilize highly forbidden M 1 optical transitions in heavy elements such as Cs and Tl. Since then such experiments have been undertaken at Paris using Cs and at Berkeley using T1. The optical energy levels of these two elements are shown in Fig. 15. The transitions used are 62Sl/2+72Sl/2in Cs and 62Pl/,-+72P,/2in TI. In the nonrelativistic approximation, these are each forbidden to M 1 because the radial quantum number changes. Spin-orbit coupling and other relativistic
o o0o ~0l o o o o=F; , ; ;$p26
6's
21.2 GHz I F=O
55cs
8 IT1
FIG.15. Relevant cesium and thallium energy levels.
358
E. N . Fortson and L. Wilets
effects permit these transitions to take place, but at an amplitude of order a2 (eh/m,c).
The original concept was to look for a circular polarization dependence (circular dichroism) in the M 1 transitions through El -M 1 interference, with a strong enhancement of the fractional effect in Eq. (16) because of the exceedingly tiny M1 amplitude. The idea was to monitor the fluorescence from the upper state when a vapor of the atoms was illuminated with circularly polarized light at the M 1 transition wavelength. Changes in the intensity of the fluorescence accompanying changes in the sense of circular polarization of the incident light would measure the PNC effect of Eq. in Tl might be in Cs and (16). Fractional effects as large as expected. In practice, because the M1 transitions are so weak, other sources of background light have turned out to be too large compared to the desired fluorescence for the experiments to be feasible in the original concept. However, the transitions can be observed readily by Stark mixing using a static electric field E s , and thus there is the possibility of a PNC-Stark interference of the form of Eq. (22). The fractional PNC effect is reduced in proportion to the size of E s . In practice, the electric fields required to raise the signals above background lead to expected PNC fractional effects for Cs and TI. of order
B. CALCULATIONS I. ThaIlium
Thallium has a relatively simple electronic structure. In first approximation, the atom may be treated as having only one active electron, 6p,/Z in the ground state. In Bi, core polarization is dominated by five electrons: 6s26p3.Thus one might expect a comparable but somewhat smaller effect for the three electrons in Tl: 6s26p or 6s27p. Indeed, Novikov et al. (1976) noted a factor of 1/ 1.6 reduction in the 6s +6p transition which we now identify as due in part, at least, to shielding. The PNC effect on the transition of interest, 62Pi/2+ 72P,/2 involves contributions from both levels, and it happens that the PNC admixture is considerably larger in the upper state, due to the proximity of the even parity 72S,,, and 82S,/2states. But polarization effects in the 7P state are considerably smaller than in the 6P state because its wave function extends further out. Refer to Fig. 4 for Bi to see how the effect falls off with radius. The effect is probably less than IWO(Sandars, 1980). Corrections to the Stark amplitude may be significant (Commins, 1981). The transition of interest, 62P,/2-72Pi,2, is MI-forbidden. The MI
PARITY NONCONSERVATION IN ATOMS
359
transition can go when relativistic effects and configuration mixing are considered, but as mentioned previously, the MI transition amplitude is so weak that it has not been feasible to look for interference between the PNC El amplitude and the MI amplitude. Instead, as suggested by Bouchiat and Bouchiat (l974a, b, 1975), Commins and collaborators looked for interference between the Stark-induced El amplitude and the PNC El amplitude (Conti et al., 1979) and also used the Stark interference with the weak MI amplitude to measure the M1 transition amplitude (Chu et a/., 1977). The calculations on TI by Neuffer and Commins (1977) begin by solving the one-body Dirac equation in the parametric potential
which is a “modified Tietz potential.” The parameters y and 77 were chosen to yield agreement (0.1%) with the observed 62P1/2 and 72P,/2levels. Other low-lying states are then obtained to within 2%. Tests of the one-electron model were made by comparison with experimental and other theoretical fine structure, hyperfine structure, and allowed El transition rates. N o serious discrepancies were found. Comparison with hfs is of interest because it tests the wave function near the origin. Small discrepancies there were attributed to admixtures of configurations which do not affect PNC calculations. The general agreement (52m)with observed El transitions lends confidence to the oneelectron model and the smallness of core polarization effects. The most difficult and uncertain part of the theoretical calculations is the MI transition rate. Using the one-electron central field (OECF) Dirac wave functions, it was calculated to have the very small value %o,,,
= - 1.757 x 10-’pB
Higher order configuration mixing due to electrostatic, spin-orbit, and two-body Breit interactions were considered. The net result of these calculations yielded
G ~ R .= -(3.2
-t-
I ) x 10-spL,
(47)
Hyperfine mixing was also considered. The PNC El amplitude was evaluated within the one-electron approximation by the techniques described in Section IV, A. The result is ImG,,,
=
-0.79
X
IO-’’a.u.
(48)
the mean of two different calculations which are compared in Table 111.
E. N. Fortson and L. Wilets
360
TABLE 111 CALCULATIONS OF h(EpN,-) FOR CESIUM AND T H A L L I U ~ Cs: 6 ' S , / 2 +7% Modified Fermi-Segrk (Bouchiat and Bouchiat, 1974b, 1975) Parametric potential (Loving and Sandars, 1975)
- 12 x 10-10 - 15 X
TI:62Pl/,+72P1/2
- 0.76 X
Semiempirical (Neuffer and Commins, 1977) Parametric potential (Sushkov ef al., 1976)
- 0.82 X
"In atomic units. Results of the various authors have been renormalized to sin28, 0.23.
=
The Stark mixing, Eqs. (20) and (21), was also calculated by techniques similar to those described in Section IV, A. There are some computational differences between the Stark and PNC calculations, however. H,,, is a pseudoscalar, and hence does not mix different j. H s = eE, r is a vector and does mix j with j ? 1. Both S and D states are admixed with the P states. Let 8 be the angle between the light beam plane of polarization and E s . Then the Stark-induced El transition amplitude of Eq. (21) can be represented by the 2 X 2 matrix
-
(72P,12m; (Stark)[ 1162P,/2mJ(Stark)) mi, mJ
-
t - L2
where
and
I
- _I
acos8 - ip sin 9
-$sin8 a cos 8
2
2
(49)
TABLE IV DIPOLE TRANSITION AMPLITUDES( M 1 ) + ( E 1PNC) + ( E 1Slart) FOR 62P,/2(F, mF)-+72P,/2(F', mF.) TRANSITIONS~,~
12P,,,
F' 0
mF'
F
0
1
mF
0
0
6'P,/,
0
a'cos 0
(;/fix% sin e - /?'sin 0 +~,Nccose a'cos e - 3~.ntCose +EPNCsinO
-
cos e
+ 6 PNC sin 0 ( - i / h)(%sin
+ /?'sin B
+ E,,, ( i / h x s~ ine
e
cos e)
a'cos B
- B'sin 0 + E,,, cos e) 0
+
+ E,,, ( ; / f i x % sine - B'sin 0
+ E PNC COS e) "a' = e2Eoa; /?'= e2E,b. bNeuffer and Commins (1977).
(-i/hx%sine
+
e cos e)
a'cos f3 9u. cos 0 - &,, sin 0
E. N . Fortson and L. Wilets
362
with R7P,nS= (72P,,2 1 r I ~ I ’ S , / ~ )etc., , and the remaining notation selfevident. Neuffer and Commins evaluated the sums both by explicitly evaluating contributions of nearby levels and by using the Sternheimer (1954) method to execute the complete sums. Further Clebsch-Gordan algebra (but no further integrals) is required to include hyperfine structure. The general dipole transition matrix is the sum of MI ElPNC Elstarkcontributions (see Table IV), where mF and mF‘ are taken along the direction x k*, k^ being the direction of the light beam propagation. Interference between the PNC and Stark mixing amplitudes can be observed by shining circularly polarized light on the sample and detecting the polarization of the 72P,/2state given by Eq. (22). If we add the M1 contributions as well to the right-hand side of Eq. (22) we obtain an expression true for large electric fields (E,, >> 1 V/cm):
+
+
Pz 5 P, * k* x Lsx (Em + q E p N C ) / E s
(52)
where GS is a measure of the Stark-induced amplitude given in terms of a and /3 above. For certain transitions:
Pz ( F = O-+ F = 0 ) = 0 Pz(F= wheref= 311. ES.
o+
F = 1) =
2f
t(P-fI2- i(P+fI2 * - P t ( P +fI2 + t ( P -fI2 + f 2
(53)
+ vFpNCand again the final approximation is valid for large
2. Cesium From a theoretical point of view, Cs is especially attractive. As an alkali metal, it contains one valence electron outside of a “noble gas” core which is probably quite rigid against polarization. As we have often noted, it was the Bouchiats (1974a, b, 1975) who first proposed looking for PNC in heavy elements, and Cs was their first choice. The transition involved is the 62S,/,+72S,/2; the spectrum is shown in Fig. 15. The M1 transition is forbidden: It can proceed only by spin flip [the u term in Eq. (IS)], but the radial wave functions are orthogonal. In order to calculate a nonvanishing transition amplitude, higher order effects must be invoked. These include retardation, relativistic corrections, core polarization, and so on. The Bouchiats estimated the reduced matrix element 3n. to lie between and a.u., and as discussed later, it has since been
PARITY NONCONSERVATION IN ATOMS
363
measured. However, as in the case of T1, the search for PNC-induced El interference with M 1 has been abandoned for PNC-Stark interference. The calculations of G,,, in Cs are straightforward and should be reliable. Results employing a Tietz potential (Neuffer and Commins, 1977) and a Norcross potential (Pignon and Bouchiat, 1980) are displayed in Table 111. The Stark amplitude is more sensitive to the IPM potential and shielding. The relevant quantity, ImG,,,/ES, is 1.3 X lop4 and 1.9 X lop4,respectively in these cases. C. EXPERIMENTS 1. The Paris Experiment The transition of interest is 6Sl/,+7Sl/, in atomic Cs, shown in Fig. 15, which occurs at 5393 a very good wavelength for stable highly monochromatic dye laser operation. The experi$ent *seeks to measure the electronic polarization component P, = P, Es x k defined by Eq. (22) that is induced in the excited state by absorption of circularly polarized light in the presence of a static electric field &. An expression for P, can be obtained for this Cs transition just as obtained in Section VI,A, 1 for T1, and the result is of the same form as in Eq. (52). The electronic polarization may be measured by the circular polarization it causes in the 7S+6P radiation emitted along the z direction. The degree of circular polarization is proportional to P, and will be reduced by increasing E s . Although reducing the fractional effect in this way may make the experiment more vulnerable to systematic effects, it has very little influence on the effect of pure shot noise. The counting rate increases as E;, and the fractional shot noise on this rate varies as E;' precisely as does P,. The size of Es here plays a similar role to that of the offset angle in the optical rotation experiments. In Cs, the Stark amplitude is about the same size as the MI when Es-2.6 V/cm. In practice, several hundred volts/cm is required to raise the 7s +6P fluorescence rate above background, which dilutes the fractional size of the effect expected iq Cs to P, < In the Paris experiment (Bouchiat et al., 1977), Cs atoms at a pressure of about lo-' Torr are excited by a single-mode cw laser beam tuned to the 6s -+7s transition frequency. An external dc electric field Es is applied to the Cs vapor cell. Inside the cell are placed two mirrors which reflect the laser beam repeatedly through the Cs vapor, increasing by > 100 the fluorescence signal but preserving the circular polarization of the laser light. The fluorescence is detected along the isx k* direction through a circular analyzer consisting of a quarter-wave plate and plane polarizing prism. The circular polarization of the incident laser beam is created by a similar
A,
-
364
E. N. Fortson and L. Wilets
arrangement. The incident polarization and the circular analyzing power are both modulated, at frequencies wi and wf respectively, by shifting the relative angle of polarizer and quarter-wave plate. The desired signal proportional to P, thus appears at both wf wiand wf- wifrequencies in the fluorescent light detected after the circular analyzer. To calibrate the detection sensitivity, including the circular analyzing power, a magnetic field is applied parallel to E,. This field produces an additional electronic polarization component along the detection direction which is readily calculable. This polarization is also independent of Es and is thus readily distinguished from P, given by Eq. (22). According to Eq. (52) one can measure %/&, by measuring P, averaged over 17 = f I, that is, i(P,+ + P,-) where k refers to 17 = f 1. One can measure&/,&,, by observing the difference P,' - P,- . The contribution is enhanced by multiple reflections of of 3R. is canceled and that of, , ,& the incident light (Bouchiat and Pottier, 1976b). An accurate measurement of %/&, has been carried out already at Paris (Bouchiat and Pottier, 1976a). The result falls within the calculated , has reached the limits (see Section VI,B,2). The measurement of , ,& stage where the statistical uncertainty is comparable to the predicted size of the PNC effect. Thus, an important result from the Paris experiment may be imminent (Bouchiat and Pottier, 1980).
+
2. The Berkeley Experiment As mentioned already, this experiment is similar in concept to the Paris experiment with cesium. The 6Pl/,-7P,,, transition in atomic T1 is used, which falls in the UV at 2927 and can be reached by frequency doubling the visible output of a pulsed tunable dye laser. The smaller incident average intensity available from such a light source is compensated by the larger GpNC amplitude in TI compared with Cs because of the larger 2. As with Cs, the PNC-Stark interference is measured by driving the transition with circularly polarized light in the presence of a static electric field, and observing electronic polarization P, as given by Eq. (52) that is produced in the excited state. At Berkeley a new method to measure P, has been exploited. The 7P,,, atoms are pumped to the 8S,,, state by a 2.18-pm (IR) circularly polarized laser beam directed along k,, x E,, and the intensity Z+,- of the 8S,/,6P,/, fluorescence is monitored as the 2.18-pm circukr polarization qIIR = f 1 is changed. Using the customary alignment of axes, ,k and Es are parallel to x andy, respectively, and P, is to be measured. The asymmetry
PARITY NONCONSERVATION IN ATOMS
365
is the measured quantity, and is the sum of asymmetries bM and GOPNC due to M I and El,,,, respectively. Note that, by Eq. (52), AM is odd under Es reversal, while APNC is odd under both Es and quv reversal. Thus, care must be taken that reversing quv does not cause any changes that would change the contribution of MI to P,. The dilution factor 0.7 in Eq. (54) may be calibrated by directing the IR beam along x and measuring the large and known value of P, proportional to quv. Alternatively, the ratio &pNc/9R may be determined independently of calibration by measuring AM and APNC at the same time. The use of laser pumping to measure P, offers some advantages over the alternative of measuring the circular polarization of the fluorescence from the 7P,/, state. The 7P,/2-7S,/, fluorescence is at a wavelength of poor detection sensitivity and high blackbody background, whereas cascade fluorescence from 7S,/, suffers from cascade depolarization. Further, the need to measure circular polarization imposes a limit on the usable solid angle of detection. Finally, laser pumping allows the fluorescence to be displaced to a quieter part of the spectrum. The Berkeley apparatus is shown schematically in Fig. 16. L1 is a flashlamp pulsed tunable dye laser operating at 5854 with pulse width of 0.5 p sec and rep rate of 19 sec-I, and an average output power of 0.13 W. Doubling in an ADA crystal produces the 2927-i UV beam which is then
A
FIG.16. The plan of the thallium experiment at Berkeley (Conti et al., 1979).
366
E. N . Fortson and L. Wilets
circularly polarized and directed into the T1 cell containing TI vapor at T = 1050 K and density loi5atoms/cm3. Tantalum electrodes inside the cell generate Es which is set at 300 volts/cm to boost the fluorescence above background. After the main cell, the UV beam enters a second TI vapor cell where the fluorescence is used to set L1 to the desired hyperfine component of the line. A second dye laser L2 is pumped synchronously with L1 and used to drive a Chromatix CMX4/IR optical parametric oscillator laser tuned to 2.18 pm. The IR output is circularly polarized and directed through interaction region 1 of the main cell and then reflected back with opposite J, through a similar region 2. The fluorescence signal (II- Z2)/(Zl + Z,) is proportional to P, while strongly rejecting intensity variations. Let A be the observed part of A I 2 that is odd under both Es and q,, reversal. Under quv reversal, A should have an even part AM that measures 3R. and an odd part APNC that measures G,,, . Data has been taken on the two hyperfine components (F-+F') O + O and O+ 1 separated by about 2 GHz. Calculation of P, shows that the O+ 1 transition has PNC and MI contributions &? and whereas for the 0 -+ 0 transition, = GN" = 0. The procedure for taking data was to switch the UV laser wavelength from one hyperfine transition to the other approximately every 20 min. were found to vary systematically The observed values of Gopf"' and GNc over periods of hours, but to be correlated such that A":, = Gopf""appeared to have only a random statistical variation. The result of over 200 h of data is a value ALNC = - 169 & 74 X lo-' when normalized such that IAZl = 55,000 x lo-'. The experimental result is quoted in terms of a circular dichroism 6 defined in Eq. (17) and given here by
-
L\oprc,
a
GNc
6/2 = ALNc/ 1.1747
(55)
where the factor 1.17 corrects an estimated 8% reflection from the rear window of the main cell, which should diminish but not A'$"". The experimental result is
aexp= +(5.2 f.2.4) x
(56)
which can be compared with the theoretical value (Neuffer and Commins, 1977; Sushkov et al., 1976)
atheo= +(2.2
0.9) x
10-3
(57)
for sin' 8, = 0.23. The calculation leading to Eq. (57) has been outlined in comes from estimated uncertainSection VI, B, 1. The uncertainty in
PARITY NONCONSERVATION IN ATOMS
367
ties in (15%) (Chu et al., 1977) and in G,,, (25%) (Neuffer and Commins, 1977). Although quoting the result in terms of 6 is convenient, it should be borne in mind that the actually observed fractional effect ApNc is much smaller than 6. At this stage, it seems reasonable to take this experimental result as suggestive of a possible effect of the same order of magnitude as that predicted by the Weinberg-Salam theory. The statistical uncertainty quoted in Eq. (56) is the same size as the expected effect. A thorough analysis of possible systematic effects has not yet been published.
-
VII. Atomic Hydrogen Experiments Experiments with hydrogen and deuterium, as noted in Section I, B, offer valuable opportunities. The atomic theory is totally reliable. All four neutral current coupling constants of Eq. (3) can in principle be determined (Cahn and Kane, 1977). Of fundamental interest, if the experiments achieve sufficient accuracy, is the possibility of checking higher order predictions (Marciano and Sanda, 1978) of the Weinberg-Salam theory in deuterium. These predictions involve exchange of two bosons and are analogous to the radiative corrections of quantum electrodynamics-and go to the heart of any gauge theory. The most recent interest in hydrogen experiments started with Lewis and Williams (1975). Attention at many laboratories began to focus on radiofrequency transitions among sublevels of the 2S,,, state. Several experiments of this general type are underway or are beginning (Dunford et al., 1978; Adelberger et al., 1978; Hinds and Hughes, 1977; V. Telegdi, private communication, 1978). Although some of the experiments are well advanced, none is at the stage of taking data. In Fig. 17 we show the energies of the 2S,,, and 2P,,: magnetic sublevels in hydrogen as a function of external magnetic field. H,,, connects opposite parity levels of the same angular momentum. The P,e crossing near 575 G emphasizes the PNC mixing of Po and e,. The only term in Eq. (8) which couples these two states is the C,, term. Thus, a measurement near this crossing would determine C,, in hydrogen, or C,, + C,, in deuterium. The size of the matrix element in hydrogen is h-'(
Po I H,,, I e,)
=
-0.026C2,
(in Hz)
(58)
The p , f crossing near 1150 G is sensitive to both C , , and C2, in hydrogen.
E. N . Fortson and L. Wilets
368
100
500
Magnetic neld (Gauss)
PIO.17. Hydrogen 2S,,, and 2P,,, energy levels as a function of magnetic field, showing one possible Stark-PNC interference scheme.
For a complete tabulation of all the relevant matrix elements see Dunford et al. (1978). The Weinberg-Salam theory predicts C,, C,, = 0, as seen in Eq. (3). Thus, the first-order effect practically vanishes at the P, e crossing in deuterium. The second-order Weinberg-Salam prediction for C,, C,, does not vanish, but is of order 0.02 (Marciano and Sanda, 1978). A measurement to such accuracy is a formidable task. If later-generation deuterium experiments can achieve this precision, they will be checking loop diagrams in the Weinberg-Salam theory with momentum transfers 100 GeV/c, unless nuclear PNC effects in the deuteron, now under study (E. M. Henley, private communication, 1980), are large enough to mask the effect. Alternatively, one can search for the nonstandard isoscalar axial currents proposed by Wolfenstein (1979). Most experiments in hydrogen are designed to observe Stark-PNC interference. One possible scheme at the p, e crossing is shown in Fig. 17. A radio-frequency field tuned to the a. + Po resonance has components E, parallel to the magnetic field and Ex. A static electric field Es along y induces a Stark amplitude between a. and Po. This interferes with the PNC-induced amplitude between the same two sublevels. Equations ( 12), (13), (20), and (21) may be used if Es is small. Introducing
+
+
-
R, = E,.? .(eol&l lao),
R, =E,P.(e+ IB1 lao)
vy= -eEs(PoI
y I e+ )
(59)
PARITY NONCONSERVATION IN ATOMS
369
which give the dominant terms in (12) and (20), respectively, we find that the PNC and Stark amplitudes are
eo) given in Eq. (58), and where Vp,, is the matrix element ( polHpNCl where we have added to the energy denominators the decay rate y of the 2p, ,2 levels heretofore omitted because the energy separations were much larger (y/27r = 100 MHz). VpNC is intrinsically imaginary as discussed at the end of Section 111, A, 2, while R, is real and 5 is imaginary from the form of 6 in Eq. (21). The transition rate a. -+ Po contains an interference term proportional to the real part of the product of the two right-hand sides in Eq. (60), and is in general nonvanishing. The handedness of the components R,, Vy,R, reveals the PNC effect. Reversing the sign of any of the components reverses the sign of the observed interference. Clearly increasing R, increases the fractional size of the small PNC contributions. In practice, however, an upper limit is set by the decay of a. via the eo state that is induced by R,. This latter effect varies with the energy denominators in the same way as the PNC mixing, so that the fractional size of PNC interference actually obtainable is rather independent of the magnetic field in Fig. 17. A measurement at zero magnetic field in principle can be of comparable sensitivity as at the level crossings. However, most experiments are working initially at the p, e crossing, and utilize either a, + Po, ao+ Po,or p- + Po transitions. We illustrate the method of carrying out hydrogen experiments with a brief description of the experiment at the University of Washington (Adelberger et al., 1978). The experimental geometry is shown in Fig. 18. 500-eV protons from a duoplasmatron, converted by charge exchange in Cs vapor into a beam of H(2S), enter a 570-G solenoid. The beam passes through a transverse, static electric field to quench the p levels. The resulting beam of atoms in the a+ and a. states then enters two successive cavities oscillating coherently near 1608 MHz. The first cavity contains static and rf fields ,to drive the Stark amplitude of Eq. (60). The second cavity about 50 cm long, contains static and rf fields along i.The rf field drives the PNC amplitude. The static field along i drives a useful probe transition for adjusting the relative phases of the cavities. The beam then passes through a cavity containing a perpendicular rf field oscillating at 2143 MHz which depopulates the a + and a. levels by a-f mixing. The remaining p states are detected by passing the beam through a static
E. N . Fortson and L. Wilets
370
D - STATE DIFFERENTIAL PUMPING TUBE
STATIC FIELD
DIFFUSION
E L ECTRODES
1500 I / s ( l i p ) ION PUMP DUOPLASMAT I ON ION SOURCE
0 25 50cm M 0 10 20in
FIG. 18. The apparatus of the atomic hydrogen experiment at the University of Washington.
perpendicular electric field. The Lyman a radiation from this P-e mixing is the signal. Metastable beam intensities of 3 X lOI3 particles/sec at the detector have been obtained. Background counting rates with the beam turned on but the Stark and PNC cavities turned off are < lo7 particles/ sec. The expected fractional PNC interference is 10-6C,p when the Stark transition yields a counting rate at the detector of about 3 x lo7 particles/ sec. About 3 hr of integration time would be required to resolve C2p.=1. Possibilities of developing slower metastable beams of comparable intensity, perhaps at thermal energies, are being explored (R. Deslattes, private communication, 1979), which would improve the sensitivity greatly. Otherwise the major problem in the experiments will be finding and eliminating systematic effects from stray electric fields, from motional electric fields due to a component of the atoms' velocity perpendicular to the magnetic field, and from other causes. [For recent discussion of hydrogen experiments, see Williams (1979).]
-
VIII. Conclusions We have reviewed diverse experimental and theoretical programs engaged in the study of parity nonconservation in atoms. The existence of parity nonconservation in atoms is now reasonably well established by experiment.
PARITY NONCONSERVATION IN ATOMS
37 I
Four different laboratories have performed PNC optical rotation experiments on bismuth. Although there is strong evidence for an effect of the sign and order of magnitude expected from the Weinberg-Salam theory, the various experiments are not consistent among themselves. We judge the atomic calculations to be reliable to about 30%. Experiments based on interference between PNC and Stark El transitions have been performed on thallium and cesium. Here the atomic calculations are judged to be more reliable. Both experiments are quoted by their authors as having an error u equal to the expected (Weinberg-Salam) effect. In TI, an effect 2 . 5 ~is found; in Cs, the effect is 5 u. We look forward in the near future to further improvements in the calculations and the experiments. The weight of evidence thus far favors at least order of magnitude agreement with the Weinberg-Salam theory, but until consistency is found among most of the experiments, a quantitative conclusion cannot be drawn. Experiments are underway with atomic hydrogen. Here the experiments are possibly even more difficult than the heavy atom experiments, but the atomic calculations are free of uncertainty to the required accuracy. REFERENCES Abbott, L. F., and Barnett, R. M. (1979). Phys. Rev. D 19,3230. Adelberger, E. G., Trainor, T. A,, and Fortson, E. N. (1978). Bull. Am. Phys. SOC.[2] 23, 546. Aiki, A. er al. ( I 977). Appl. Phys. Lerr. 30,649. Baird, P. E. G. (1979). Ar. Phys., Proc. Inr. Con$. 6rh, 1978 p. 653. Baird, P. E. G. (1980). I n “International Workshop on Neutral Current Interaction in Atoms, 1979” (W. L. Williams, ed.), p. 77. Baird, P. E. G. et at. (1976). Nature (London) 264, 528. Baird, P. E. G., Brimicombe, M., Hunt, R., Roberts, G., Sandars, P. G. H., and Stacey, D. (1977). Phys. Rev. Lerr. 39, 798. Barish, S. J. e f al. (1974). Phys. Rev. Lett. 33,448. Barkov, L. M., and Zolotorev, M. S . (1978a). Pis’ma Zh. Eksp. Teor. Fiz. 27,379; JETP L e r r . (Engl. Transl.) 27, 357 (1978). Barkov, L. M., and Zolotorev, M. S. (1978b). Pis’ma Zh. Eksp. Teor. Fir. 23, 544. Barkov, L. M., Khriplovich, I. B., and Zolotorev, M. S. (1979). Comments Ar. Mol. Phys. 8, 79. Benvenuti, A. er al. (1974). Phys. Rev. Lerr. 32,800. Bernabeu, J., Ericson, T. E. O., and Jarlskog, J. (1974). Phys. Lerr. 505, 467. Bogdanov, Yu. V., Sobel’man I. I., Sorokin, V. N., and Struk, 1. I. (1980). (to be published). Bouchiat, C. C., Bouchiat, M. A., and Pottier, L. (1977). Ar. Phys., Proc. Inr. Con$. 5th, 1976 p. 1. Bouchiat, M. A., and Bouchiat, C. C. (1974a). Phys. Lerr. E 48, 1 1 1. Bouchiat, M. A,, and Bouchiat, C. C. (1974b). J . Phys. (Paris) 35, 899. Bouchiat, M. A., and Bouchiat, C. C. (1975). J. Phys. (Paris), 36, 493. Bouchiat, M. A,, and Pottier, L. (1976a). J . Phys. Len. 37,L79. Bouchiat, M. A., and Pottier, L. (1976b). Phys. L e r r . 62B,327. Bouchiat, M. A,, and Pottier, L. (1980). I n “International Workshop on Neutral Current Interactions in Atoms, 1979” (W. L. Williams, ed.), p. 122.
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Bradley, L. C., 111, and Wall, N. S . (1962). Nuovo Cimento 25,48. Brimicombe, M., Loving, D., and Sandars, P. G. H. (1976). J. Phys. B 9, L1. Cahn, R. N., and Kane, G. L. (1977). Phys. Lett. B 71, 348. Carter, S. L., and Kelly, H. P. (1979). Phys. Rev. Lett. 42, 966. Chu, S.,Conti, R., and Commins, E. D. (1977). Phys. Lett. A 60, 96. Commins, E. D. (1981). A t . Phys. Proc. Int. Con& 7th, 1980 (to be published). Conti, R.,Bucksbaum, P., Chu, S., Commins, E., and Hunter, L. (1979). Phys. Rev. Lett. 42, 343. Dunford, R. W., Lewis, R. R., and Williams, W. L. (1978). Phys. Rev. A 18,2421. Feinberg, G., and Chen, M. Y. (1974). Phys. Rev. D 10, 190 and 3789. Fermi, E., and Segre, E. (1933). Z. Phys. 82,729. Fortson, E. N. (1977). At. Phys., Proc. Int. Con&, Slh, 1976 p. 23. Fortson, E. N. (1978a). In “Neutrinos-78” (E. Fowler, ed.), p. 417. Purdue. Fortson, E. N. (1978b). I n “Proceedings of the SLAC Summer Institute on Particle Physics” (M. C. Zipf, ed.), p. 305. Grant, I. P., Rose, S. J., and Sandars, P. G. H. (1980). To be published (referred to in Sandars, 1980). Hams, M. J., Loving, C. E., and Sandars, P. G. H. (1978). J. Phys. B 11, L749 Hasert, F. J. et al. (1973). Phys. Lett. B 46, 138. Henley, E. M., and Wilets, L. (1976). Phys. Rev. A 14, 1411. Henley, E. M., Klapisch, M., and Wilets, L. (1977). Phys. Rev. Lett. 39,994. Hiller, J., Sucher, J., and Feinberg, G. (1978). Phys. Rev. A 18,2399. Hiller, J., Sucher, J., Bhatia, A. K., and Feinberg, C. (1980a). Phys. Rev. A 21, 1082. Hiller, J., Sucher, J., Feinberg, G., and Lynn, B. (1980b). Ann. Phys. (N. Y.) 127, 149. Hinds, E. A., and Hughes, V. W. (1977). Phys. Lett. B 67,486. Hollister, J. H., Apperson, G. A., Lewis, L. L., Vold, T. M., Emmons, T. P., and Fortson, E. N. (1980). Phys. Rev. Lett. (to be published). Hsieh, J. J. (1976). Appl. Phys. Lett. 38, 283. Khriplovich, I. B. (1974). Pis’ma Zh. Eksp. Fir. 20, 686; JETP Lett. (Engl. Trunsl.) 315 (1974). Landman, D. E., and Lurio, A. (1962). Phys. Rev. 127, 1220. Lewis, L. L., Hollister, J., Soreide, D., Lindahl, E., and Fortson, E. N. (1977). Phys. Rev. Lett. 39, 795. Lewis, R. R.,and Williams, W. L. (1975). Phys. Lett. 59,70. Loving, C. E., and Sandars, P. G. H. (1975). J. Phys. B p. L336. Mirtensson, A. M., Henley, E. M., and Wilets, L. (1980). (Tobe published.) Marciano, M. J., and Sanda, A. I. (1978). Phys. Rev. D 17, 1313. Messiah, A. (1968). “Quantum Mechanics,” Vol. 11, p. 569ff. Wiley, New York. Michel, F. C. (1965). Phys. Rev. 138,B408. Missimer, J., and Simons, L. (1979). Nucl. Phys. A 316,413. Namizake, H. (1975). IEEE J. Quantum Electron. 11, 427. Neuffer, D.V., and Commins, E. D. (1977). Phys. Rev. A 16,844. Novikov, V. N., Sushkov, 0. P., and Khriplovich, I. B. (1976). Zh. Eksp. Teor. Fir. 71, 1665; J . Exp. Theor. Phys. (Engl. Transl.) 44,872 (1976). Novikov, V. N., Sushkov, 0. P., Flambaum, V. V., and Khriplovich, 1. B. (1977). J . Exp. Theor. Phys. (Engl. Transl.) 46, 420. Poppe, R. (1970). Physica (Ufrecht)50, 48. Prescott, C. Y. et al. (1978). Phys. Lett. B 77,347. Salam, A. (1968). Elem. Part. Theory, Proc. Nobel Symp., 8 4 1968 p. 367. Sandars, P. G. H. (1975). At. Phys., Proc. Inr. Conj, 41h, 1974 p. 71. Sandars, P. G. H. (1977). J. Phys. B10, 2983.
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Sandars, P. G. H. (1980). Phys. Scr. 21, 284. Saskyan, D. V., Sobel’man, 1. I., and Yukov, E. A. (1979). Pis’ma Zh. Eksp. Theor. Fit. [JETP Lett.-Engl. Trawl. 19, 258 (1979)l. Soreide, D. C., and Fortson, E. N. (1975). Bull. Am. Phys. Soc. [2] 20, 491. Soreide, D. C., Roberts, D. E., Lewis, L. L., Apperson, G. R., and Fortson, E. N. (1976). Phys. Rev. Lett. 36,352. Sternheimer, R. M. (1954). Phys. Rev. %, 951. Sushkov, 0. P., Flambaum, V. V., and Khriplovich, I. B. (1976). JETP Lett. (Engl. Trans/.)24, 461. Weinberg, S. (1967). Phys. Rev. Lett. 19, 1264. Wilets, L. (1978). I n “Neutrino-78” (E. Fowler, ed.), p. 437. Purdue. Wilets, L., Henley, E. M., and Mirtensson, A. M. (1980). J. Phys. E 13, 2335. Williams, W. L. (ed.) (1979). “International Workshop on Neutral Current Interactions in Atoms, 1979,” pp. 182-312. Wolfenstein, L. (1979). Phys. Rev. D 19, 3450. Zel’dovich, Ya. B. (1959). Zh. Eksp. Teor. Fiz. 36, 964.
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master equations in, 165-171 resonance fluorescence and, 171-190 Atomic spectra, Bates-Damgaard wave functions and, 65 Atomic unit, defined, 58 Autler-Townes effect, 160 optical. 164,170. 190-196 Autler-Townes splitting, 162. 165. 193 in resonant two-photon ionization, 195 Zeeman degeneracy and, 194 Autler-Townes theory, three-level. 190-
A
Ablation-front plasma, 207,247-249 spectroscopy of, 258-262 Absolute energy, hot-electron disposition of, 265-266 Absorption coefficient. 239 Alkali atoms. as "good" model atoms. 57 Alkali-rare gas interactions, 58 Alkali-rare gas molecules. production of.
76-77
I93
Alkali-rare gas potentials, equilibrium data for, 80 Antibunching. 184 for two-photon absorption of coherent light, 186 Atom(s) neutral current interaction in. 321-324 panty nonconservation in. 319-371 Atomic calculations, in panty nonconservation, 328-338 Atomic collisions theory. see also Collisional decay excitation and ionization in. 282-306 impact parameter treatment in. 286-288 relativistic effects in. 281-316 Atomic Hartree-Fock theory, 1-52.see also Hartree-Fock theory extended frozen core approximations in.
B Bates-Damgaard wave functions, 65 Bayless model. 91. 100 Beryllium calculated and observed ionization energies of. 45 electric dipole oscillator strengths for,
46-48
Beryllium states. total energies of. 30 Bessel functions of second kind, 288 Binary collisions. ionization and. 21 1-212, see also Collision processes Bismuth. PNC calculations for, 339-342 Bismuth optical rotation, 338-356 at 6477 A. 349-357 at 8757 A. 344-349 Moscow experiments in. 354 Novosibirsk experiments in. 353-355 Oxford experiments in. 351-353 Seattle experiments in. 344-349 Born approximation. 167 Boron calculated and observed ionization energies for. 39 electric dipole oscillator strengths for, 40
23- 34 improved frozen core approximations in.
34-49 and properties o f frozen core approximations, 16-23 relativistic energy level corrections and.
so- 52
Atomic hydrogen experiments. PNC and. 367-37I Atomic processes in strong resonant electromagnetic fields. lS9- 196 hasic phenomena in. 161-165
375
376
INDEX
Boron states. orbital and total energies of, 18- I9 Bosons. in parity theory. 320 Bound levels. population densities and. 213-217 Breit equation, 23 Bremsstrahlung absolute intensity of. 265 in continuum emission. 234-237 Maxwellian source of, 236 recombinant radiation and, 237 X-ray. 265-266 Brillouin's theorem. H F wave functions and. 14-16
C
CA. see Coulomb approximation Calcite prism polarizers. 346 Carbon. electric dipole oscillator strengths for, 32 Carbon states. ionization energies for. 31 Central field spin orbitals. 4-6 Cesium in Fano effect source, 116- 117 in GaAs photoemission activation. 147 Stark interference experiments and. 360- 363 Chemi-ionization. of optically oriented metastable helium, 107-1 12, 153 Circular dichroism. in PNC. 327 Collisional decay. vs. radioactive decay. 215 Collisional-radiative atomic physics, resonance line radiation and. 244 Collisional excitation rate coefficient, 214215 Collisional-radiative level populations. 215- 216 Collisional-radiative model of ionization. 206- 207 population densities and, 215-216 in transient ionization, 213 Collision broadening, in resonance fluorescence. 178 Collision processes ionization and. 21 1-212 in resonance fluorescence. I8 1 - 182
Configuration mixing, in PNC, 331-332 Continuum emission. 234-238 bremsstrahlung in, 234-237 recombination continuum and, 237-238 space-time averaged. 263-265 space-resolved, 266-267 time-resolved spectra in. 266 Corecore interaction experimental data for. 79 model potential and. 65 Coronal approximation, collisional excitation rate coefficient and. 214-215 Coronal model of ionization, 204-206 in transient ionization, 212-213 Coulomb approximation, frozen core procedure and, 49-50 Coulomb interaction. long-range. 21 I Coulomb interaction correlations, in Stark broadening, 229-230 Coulomb interaction energy. interparticle, 209 CR codel. see Collisional-radiative model of ionization Cutoff radius. model potential and. 65-66
D
Darwin approximate relativistic wave functions, 299 Debye-Huckel approximation. 208 Debye length. 209. 228 Debye sphere. 209 Dielectronic recombination. 208 Differential scattering processes. 68, see also Scattering Dirac equation. 128 Dirac wave functions one-electron central field. 359 relativistic. 299 Doppler broadening. 226 Doppler shifts. radiative transport with. 245-246 Double scattering. in electron capture. 3 12-315 Doublet ratio. 218 Dressed atom, defined, 162 Dyadic operator, 166
377
INDEX
E El-MI interference, in parity nonconservation, 325- 327 El matrix element, 325-326 Ecole Polytechnique, 135 Effective Hamiltonian, H F methods and, 4 Electromagnetic fields atomic processes in. 159- 196 basic phenomena in. 161- 165 Electron(s) free, 289-290 scattering of, see Scattering Electron beam energy, FWHM and. 107 Electron capture classical double scattering in, 312-315 impact parameter formulation in. 307309 OBK approximation in, 307-311 radiative capture in, 315-316 relativistic effects in, 307-316 wave formulation in, 309-312 Electron diffraction, see Spin-polarized low-energy electron diffraction Electron Fermi energy, 21 1 Electronic model potentials electronic interaction and, 69 electron scattering and. 72-76 experimental sources for, 70-91 experimental vs. theoretical results with. 91-96 general behavior and actual forms of. 62-67 interatomic potentials and, 58-70 parameters of. 69 parameter sources and, 66-67 polarizabilities and, 71-72 pseudopotential theory for. 62 spectroscopy and. 76-78 Electron polarization, extracted beam current and, l l l see also Polarization Electron scattering. model potentials and. 72-76. see also Scattering Electron spin concept. 102 Emission coefficient. 239 Emittance, defined. 105- 106 Energy levels. relativistic corrections to. SO- 52 Energy-selective spin polarization. 123
Escape-factor approximations, 242-243 ETH, see Swiss Federal Institute of Technology Europium sulfide-tungsten. field emission for, 120-127, 153 Excitation in atomic collisions theory, 282-306 impact parameter treatment in, 2 8 6 2 8 7 Mjdller theory in, 282-286 Excited state calculations, upper bound in, 14 Excited state functions, orthogonality of, 13-14 Excited states, valence orbitals for, 37-41 Expansion plume spectroscopy, in laserproduced plasmas, 251-258 Exploding pusher implosions, 250, 270, 280 Extended frozen core approximation. 2334 derivation of energy expression in, 2426 EFC valence radial equation and, 26-28 Extended frozen core calculations, results of, 29-34 Extended frozen core theory, 3 Extended frozen core valence radial equation, 26-28 Extended frozen core wave functions, 27 orthogonality of, 28-29 Extended Hartree-Fock energy expression. derivation of, 24-26 Extended Hartree-Fock procedure, complex equations and. 23 Extended Hartree-Fock theory, 3, see also Hartree-Fock theory Extreme ultraviolet transitions, population inversion on, 203
F Fano effect apparatus used in. 118 e-Cs spin-exchange collisions and, 119 polarized electrons and. 153 source in, 116- 119 Faraday cell, 346 Faraday cup system, 143
378
INDEX
Faraday rotation, in optical rotation experiments. 343-344 FC approximations, see Frozen core approximations Fermi-Segre formula. 324 Ferromagnetic europium sulfide on tungsten, field emission from. 120-127, 153, Flow Doppler shifts, radiative transport with, 245-246 Flowing afterglow polarized electron source, 109 Fluorescence, resonance, see Resonance fluorescence Four-electron parent ion, average energy of, 13 Free electrons, Scattering of electrons and positrons by, 289-290 Frozen core approximations, see also Extended frozen core approximations defined. 2-4 gound state problem in. 41-43 improved, 34- 49 ionization potentials in, 16- 17 model potential in. 46 multiconfiguration frozen cores and, 3436 orthogonality of frozen core functions and, 17-18 polarized. 43-48 properties of. 16-23 SOC calculations in. 47-49 valence orbitals from excited states in. 37-41 results in, 18-23 Frozen core theory, 3-4 FWHM (full width at half-intensity maximum), 107
G
Gallium arsenide. depolarization in, 138139 Gallium arsenide crystal, cleaning of, 145147 Gallium arsenide photocathode, in UHV source chamber, 144
Gallium arsenide photoemission. 134- 152 activation of. 147- 148 electron optics in, 148- 150 incident radiation in, 143- 144 negative electron affinity in, 135- 143 positive electron affinity in, 137- 143, 1 50
spin orientation during photoexcitation in. 135- 136 Gallium arsenide polarized electron source, characteristics of, 150- 152 Gallium arsenide source, negative electron affinity and. 151-152 Gaunr factor, 215 “Good” model atoms, 57 Ground state hydrogen, parity and, 32032 1 Ground state potentials. in interatomic potential determination, 84 Ground state problem. in frozen core approximations. 41-43
H
Hamiltonian of composite system, 165 effective, 4 eigenvalues of, 161 IPM. 329 master equation and. 173 model. 60, 63 nonlocal HF, 335 nonrelativistic, SO perturbation, 23 relativistic, SO single-particle. 320-32 I spin-orbit term in, 128 stochastic component of, 169 of unperturbed system, 167 valence, 60-61 Hanbury-Brown and Twiss experiment. 186
Handbook of Spectroscopy, 224-225 HartreeFock approximation applications of, 2 time-dependent, 332-333, 336. 340 HartreeFock energy, derivation of, 6-8
INDEX
Hartree-Fock method, 4- 12 central field spin orbitals in. 4-6 shells of equivalent electrons in. 5 Hartree-Fock potentials. 340 in PNC. 329-331 Hartree- Fock theory atomic, see Atomic Hartree-Fock theory excited state calculations in. 13- 14 extended. 3 orbital energies in. 10 radial equations in. 8- I2 Hartree-Fock wave functions Brillouin's theorem in, 14- 16 ionization potentials of, 12- 13 orthogonality of excited state functions and. 13-14 parity nonconservation and. 332-333 properties of. 12- 16 Heavy particle scattering. model potential and. 78-80. see also Scattering Helium. chemi-ionization of. 107- 112 Helium isoelectric sequence, total energies for, 29 Helium-like ion resonance, intercombination lines and.219-220 Helium-like ion transitions, satellites to, 220-223 HF, see Hartree-Fock approximation; Hartree- Fock theory High-density effects, in ionization, 209212 High energy incident particles, Mfller theory for. 282-286 Holtzmark function, 229-230 Hot electrons absolute energy deposition by, 264-266 preheating by. 249, 268 Hydrogen4 ke ions. 21 8-268 satellites to. 221-222
I
ICF, see Intertially confined fusion Impact parameters in atomic collisions theory, 286-288 formulation of in electron capture. 307309
379
Implosion core spectroscopy, 269-272 Implosions ablatively driven, 251 exploding pusher, 250, 270, 280 in laser-produced plasmas, 250-25 1 Independent particle model potential. 340, see also Model potential Independent particle model wave functions. 328-329 lnertially confined fusion. 202 Inglis-Teller limit. in line broadening. 233-234 Inner-shell ionization theory. Mfller interaction in, 294-304 Intensity fluctuation spectra, in resonance fluorescence. 183- 186 Interaction potentials, line-shape experiments in. 80-83 Interatomic potentials comparison of from different experiments. 87-91 determination of, 55-97 electronic model potentials and. 58-70 experiments used for. 67-70 ground state potentials and, 84 model potentials and, 91-96 phenomenological approach to. 66-69 problem of, 56-57 quantum-mechanical scattering theory and. 85 for sodium-argon system, 87-91 spectral distributions in, 82-83 standard determination of, 56 lntercombination lines, helium-like ion resonance and, 219-220 Interparticle Coulomb interaction energy, 209 Ion dynamic model, in line broadening, 23 1-233 Ionization. 203-213 in atomic collisions theory, 282-306 collisional-radiative model of, 206207 collision process rates and, 21 1-212 coronal model of, 204-206 degeneracy and, 210-21 1 high-density effects in, 209-212 inner-shell, 294-304 ionization potential reduction and, 208210
380
INDEX
Ionization (Cow.): K shell, 291-294 local thermodynamic equilibrium model and. 204 transient, 212-213 Ionization impact parameter formulation in, 307-309 relativistic effects and, 304-306 Ionization potentials in HF wave functions, 12-13 reduction of, 208-209 IMP wave functions, see Independent particle model wave functions Iron (XVI). dipole transition wavelengths for. 22-23
J Johannes Gutenberg University. 135
K Kolbenstvedt theory. of K shell ionization, 291 - 294 K shell electrons ionization energy of, 291 photoionization of, 223-224 K shell ionization cross sections for, 300-304 Kolbenstvedt theory of, 291-294 Mfller interaction in, 294 K, emission. 267-269
L Laser, neodymium. 207 Laser linewidth effects, master equation and. 169-171 Laser-produced plasma ablation, ionization temperature of, 2 13 Laser-produced plasmas ablation-front plasma and. 207, 247-249
ablatively driven implosions in, 251 bound-level population densities and. 21 5-217 collisional-radiative solutions in, 241245 continuum emission and. 234-238 expansion plume spectroscopy in. 251-258 exploding pusher implosions in, 250-25 1 hot-electron preheating zone in, 249 line broadening in, 225-234 line radiation intensity and. 217-225 local thermodynamic equilibrium model and. 216-217 parameter space for. 210 physics of, 203 plane targets and, 247-249 radiative transfer and, 238-246 as sources for classification of XUV spectra, 203 spectroscopic diagnostics of, 25 1-272 spectroscopy of, 201-272 spherical shell targets in, 249-250 structure and spectroscopic characteristics of, 246-251 Laue condition, scattering potentials and, 130 LEED. see Low-energy electron diffraction Lennard-Jones potentials, 68. 91-92 LILAC code, 240 Line broadening Doppler, 226 impact and quasi-static broadening in. 228- 229 Inglis-Teller limit in. 233-234 interatomic potentials and, 55 ion dynamic model in. 231-233 in laser-produced plasmas, 225- 234 natural. 226 Stark, 227-234 Line radiation intensity characteristic X-ray K lines and. 223224 hydrogen-like ions and. 218-219 intercombination lines and. 219-220 Line-shape experiments, interaction potentials and, 80-83 Line shapes, theory of, 82 Liouvillean operator, defined, 166 Local thermodynamic equilibrium model of ionization. 204
38 I
INDEX
LTE limit and, 214-215 radiative transfer and, 240-241 Low-energy electron diffraction, 127I34 modulation of polarization i n , 132 polarized electron gun and, 143 source in, 131-134 source disadvantages in. 134. 153 Low-ionization impact energies. relativistic effects at. 304-306 LPPs. see Laser-produced plasmas L, shell ionization. by relativistic electrons, 300 LTE model. see Local thermodynamic equilibrium model of ionization Lyman lines. splitting of, 233
M M1-E2 transitions, 324 M I matrix element. 32s-326 Mach number. in alkali-rare gas molecule measurement, 76- 77 Magnetic field geometry. for polarized atom beam. 112 Markoff approximation, 168 Master equation for atom field problem. 168- 171 defined. 166 derivation of, 165- 168 MCFC. set' Multiconfiguration frozen core approximation MCHF calculation. see Multiconfiguration Hartree-Fock calculation Metastable helium atoms, collisional ionization between. 107 Microballoons, laser-imploded, 248-250. 280 Model potentials. electronic. see Electronic model potentials Mgiller interaction, in inner shell ionization. 294-304 Mfller theory. for high-energy incident particles. 282-286 Morse potential, 68 Mott scattering, 104, 129 Multiconfiguration extended frozen cores. 34-36. 41-42
Multiconfiguration frozen core approximation. 3 Multiconfiguration H a r t r e e Fock calculations. 3 multiconfiguration frozen cores and. 3435 Multilevel atoms experiments with, 193-194 resonance fluorescence in. 186- 189 Multiphoton processes, 171
N National Bureau of Standards. 135 spin-polarized electron scattering apparatus used by.143-144. 149 Natural broadening. 226. see also Line broadening NEA. see Negative electron affinity Negative electron affinity. in GaAs photoemission, 135- 143 Neodymium laser. 207 Nickel (XVIII), dipole transition wavelengths for, 22-23 Nonconservation, parity, see Parity nonconservation Nondegenerate two-level systems collisions in, 170- 171 radiative dumping of, 172 resonance fluorescence in. 173- 183 Nonrelativistic Hamiltonian. SO
0
OBK approximation. see OppenheimerBrinkman-Kramers approximation One- and two-electron ions, satellites to resonance in. 220-223 One-electron central field Dirac wave function, 359 Oppenheimer-Brinkman- Kramers approximation. in electron capture. 30731 I Optical Autler-Townes effect. 164. 170. 190- 196
382
INDEX
Optical rotation experiments, 338-356 angle revolution in, 342-343 Faraday rotation and, 343-344 general features of, 342-344 results in, 355-357 Orbital energies. in Hartree-Fock theory, 10 Orbitals, central field. 4-6
P Panty conservation, 319-320 Panty nonconservation, 3 19-37 I atomic calculations in, 328-338 atomic hydrogen experiments and, 36737 1 atomic number Z in, 323 for bismuth, 339-342 bismuth rotation experiments and. 342357 cesium experiments and, 358-363 circular dichroism and optical rotaion in, 326- 327 dipole transition operator in, 335 El-MI interference in. 325-327 h, value in. 330-331 in heavy atoms, 324 independent particle approximations in, 328- 330 many-body perturbation expansion in. 336 nearby configuration mixing and. 331 observable effects in, 324-328 optical rotation experiments in. 338-357 shielding in, 331-334 Stark interference experiments and, 327- 328. 357- 367 thallium experiments in, 358-363 Parity Revolution (1956), 319-320 Pauli approximation. 23 PEA. see Positive electron affinity PEGGY polarized electron source. 1121 I6 beam produced by, 155 Perturbation Hamiltonian. 23 Perturbation theory, for model potentials. 62
PFC approximations, see Polarized frozen core approximations Photoelectric energy distributions, AutlerTownes splitting and, 195 Photoemission, from GaAs, 134- 152 Photoionization. of polarized atoms, 112116, 153 Photon antibunching, 164, 184 Planck function, 240 Plane target LPP, 248-249 Plasmas. laser-produced, see Laser-produced plasmas Plasma satellites, Stark effect and. 233 Plasma spectroscopy, 203, see also Spectroscopy PLEED, see Polarized low-energy electron diffraction PNC. see Parity nonconservation Polarizabilities determination of. 71 model potentials and. 71-72 Polarization energy-selective spin, 123 in low-energy electron diffraction, 127I34 Zeeman degeneracy and. 188 Polarization modulation, in LEED source, 132 Polarization potential. in frozen core approximations. 43-48 Polarization shift, in line broadening, 231 Polarization source. characteristics of, 104- I07 Polarized atoms, photoionization of, 112116. 153 Polarized-electron-polarized proton scattering, 112, see also Scattering Polarized electrons emittance of. 105- 106 Fano effect and. 116- I19 PEGGY source in. 112-1 16 Polarized electron sources, 101- 154 flowing afterglow. 109 Polarized electron technology. future application of. 103 Polarized frozen core approximations, 4348 Polarized low-energy electron diffraction. I28 LEED source and. 131
383
INDEX
Positrons. scattering of by free electrons. 289- 290 Positive electron affinity, in GaAs photoemission. 137- 143 Pseudopotential theory. for model potentials. 62
Q Quantum mechanical scattering theory. 85-86. see also Scattering Quantum regression theorem. 174 Quark-parton models. 102 Quasi-static broadening. 228- 229, see also Line broadening
R Rabi oscillations. 171 Radial equations derivation of. 8- 10 integrals of. 10- 1 I solution of, 11- 12 Radiative capture. photoionization cross section and. 315-316 Radiative damping finite bandwidth laser and, 179- 180 in resonance fluorescence. 176- 181 Radiative decay rate. vs. collisional decay. 215 Radiative recombination. 237-238 Radiative transfer, 238-246 collisional-radiative solutions in. 241-245 escape-factor approximation in. 242-243 LTE solutions in. 240-241 Radiative transfer equation. 239- 240 self-consistent solution of. 243-245 Radiative transport, with flow Doppler shifts, 245-246 Raman scattering. 160. see also Scattering Random phase approximation. PNC and, 332. 336 Rate coefficient data, bound level popula. . tion densities and. 216-217
Recombination, dielectronic, 208 Recombination continuum, 237-238 Relative multiplet strength, defined. 33 Relativistic effects in atomic collision theory. 281-316 in electron capture, 307-316 in excitation and ionization, 282-306 at low impact energies, 304-306 Relativistic Hamiltonian, SO Relativistic wave functions, in ionization calculations, 291-299 Resonance fluorescence, 163- 164. 171- 190 coherent and incoherent spectra in, 178 collision broadening and, 178 intensity fluctuation spectra in. 183- 186 in multilevel atoms with monochromatic fields. i86- 189 in nondegenerate two-level systems, 173- 183 radiative damping in. 176- 181 strong collision model in. 181- 182 time-dependent spectra and. 182 Resonance transitions, satellites to, 220223 Restricted Hartree-Fock approximation, 3 Richtstrahlwert. defined, 106
S
Saha-Boltzmann equation. 214 Saha-Boltzmann population. 215 Satellites. to helium-like ion transitions. 220- 223 Scattering cross sections for, 73 direct, 73 double, 312-315 electron-deuteron, 321 heavy particle. 78-80 high-energy electron-proton. 32 I integral cross sections for, 74 polarized-electron-polarized proton. 1 12 resonance. 73-74 Scattering theory, general quantum-mechanical. 8.5 Schrodinger - equation . exact solutions for. 4
384
INDEX
SchrMinger equation (Conr.): and Hamiltonian for fixed internuclear distance. 58 interatomic potentials and. 59 radial wave function as solution for. 74 zeroth-order solution for, 63 Shielding, in PNC. 331-332. 340 SLAC. see Stanford Linear Accelerator Center Slater determinants, 6 Slater integrals two-electron. 7 variation of. 8 SOC calculations. see Superposition of configurations calculations Sodium-argon interaction. ground state potential and, 91. 94 Sodium-argon system calculated vs. experimental values of interatomic potentials for. 87-93 phenomenological basis in. 94 spectroscopic data for, 100 Sodium isoelectronic sequence electric dipole oscillator strengths for. 21 ionization energies of, 20 Sodium-mercury interaction, interaction potentials for. 94 Sodium-neon interaction. interaction potentials for, 95-96 Sodium-neon system. interatomic potentials for. 77 Space-resolved X-ray spectroscopy, 2532.58 Space-time-averaged X-ray continuum emission, 263-265 Spectral distributions, interatomic potentials and. 83 Spectroscopy of ablation-front plasma. 258-262 continuum and K, emission in. 263-269 implosion-core. 269- 272 model potentials and. 76-78 normal incidence VUV. 2.52-253 space-resolved grazing-incidence, 253255 space-resolved X-ray. 255-258 stigmatic visible/UV. 252-553 Spin angular momentum. conservation of. 108 Spin-dependent effects, modulation in, IOS
Spin-polarized electrons. as labeled particles. 107 Spin-polarized electron scattering approximations. 143- 144. 149 Spin-polarized low energy electron diffraction. 102 Stanford Linear Accelerator Center. 112. 135. 143 GaAs polarized electron source of. 144I45 Stark amplitude, 363 Stark broadening, 227-234. see also Stark effect Coulomb effects in. 229-230 elementary theory in. 227-228 impact and quasi-static, 228-229 Inglis-Teller limit in, 233-234 ion dynamic model in. 231-233 plasma satellites and. 233 "standard" method in. 230 Stark effect. 162. 227 in multiphoton ionization, 194 plasma satellites and. 233 Stark interference experiment at Berkeley. 363-367 in Paris. 363-364 thallium and. 358-362 Stark-PNC interference. 327- 328 Stern-Gerlach magnets. 102 Strong collision model. in resonance fluorescence. 181- 182 Supercooled LPP expansion plumes. 21s Super operator. master equation and. 166 Superposition of configurations calculations. FC approximation of. 48-49 Swiss Federal Institute of Technology. 135. 143
T TDHF. see Time-dependent Hartree-Fock Thallium Berkeley experiment and. 365 in Stark interference experiments. 358362 Time-dependent Hartree-Fock. 332-333, 336. 340
385
INDEX
Transient ionization collisional-radiative model of. 213 coronal model of, 212-213 Transition probability, defined, 82 Tungsten, ferromagnetic europium sulfide on. 120-127. 153
X X-ray bremsstrahlung absolute intensity of, 265-266 polarization and isotropy of, 266 X-ray line emission. K lines in, 223-224 X U V transitions, see Extreme ultraviolet transitions
W Washington. University of, atomic hydrogen experiments at. 369-370 Wave formulation. in electron capture, 309- 3 I2 Weinberg-Salam theory. 320, 322-323. 356, 368 Wien filter. 104. I12
2
Zeeman degeneracy Autler-Townes splitting and, 194 polarization effects for. 188 Zeeman splitting, 343
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Contents of Previous Volumes
Volume 1
Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, F. J. de Heer Mass Spectrometry of Free Radicals, S. N. Foner
Molecular Orbital Theory Of the Spin Properties of Conjugated Molecules, G. G. and A. T. A mos Electron Affinities Molecules, B. L. Moiseiwitsch Atomic Rearrangement Collisions, Volume 3 B. H. Bransden The production of ~ ~ and The ~ Quanta1~ CalCUhtion ~ Of Photoi Vibrational ~ ~ in En- ~ ionization ~ Cross Sections, ~ A. iL. counters between Molecules, K. Takayanagi Radiofrequency Spectroscopy of Stored Ions. I. Storage, H. G. The Study of Intermolecular Potentials with Molecular Beams at Dehmelt Thermal Energies, H. Pauh and Optical Pumping Methods in Atomic Spectroscopy, B. Budick J . P. Toennies High Intensity and High Energy Energy Transfer in Organic MolecuMolecular Beams, J. B. Anderson, lar Crystals: A Survey of ExperiR. P. Andres, and J. B. Fenn ments, H. C. Worf AUTHORINDEX-SUBJECTINDEX Atomic and Molecular Scattering from Solid Surfaces, Robert E. Stickney Volume 2 Quantum Mechanics in Gas Crystal-Surface van der Waals ScatThe Calculation of van der Waals Interactions, A . Dalgarno and W. tering, F. Chanoch Beder D. Davison Reactive Collisions between Gas Thermal Diffusion in Gases, E. A. and Surface Atoms, Wise and Bernard J . Wood Mason, R. J. Mum, and Francis J . Smith AUTHORINDEX-SUBJECTINDEX Spectroscopy in the Vacuum Ultraviolet, W. R. S. Garton Volume 4 The Measurement of the Photoionization Cross Sections of the H. s. W. Massey-A Sixtieth BirthAtomic Gases, James A. R. day Tribute, E. H. s. Burhop Samson Electronic Eigenenergies of the HyThe Theory of Electron-Atom Collidrogen Molecular Ion, D. R. sions, R. Peterkop and V. Veldre Bates and R. H. G. Reid 387
~
~
388
CONTENTS OF PREVIOUS VOLUMES
Applications of Quantum Theory to the Viscosity of Dilute Gases, R. A. Buckingham and E. Gal Positrons and Positronium in Gases, P. A . Fraser Classical Theory of Atomic Scattering, A. Burgess and I. C. Percival Born Expansions, A. R. Holt and B. L. Moiseiwitsch Resonances in Electron Scattering by Atoms and Molecules, P. G. Burke Relativistic Inner Shell Ionization, C. B. 0. Mohr Recent Measurements on Charge Transfer, J. B. Hasted Measurements of Electron Excitation Functions, D. W. 0. Heddle and R. G. W. Keesing Some New Experimental Methods in Collision Physics, R. F. Stebbings Atomic Collision Processes in Gaseous Nebulae, M. J . Seaton Collisions in the Ionosphere, A . Dalgarno The Direct Study of Ionization in Space, R. L. F. Boyd AUTHORINDEX-SUBJECTINDEX
The Meaning of Collision Broadening of Spectral Lines: T h e Classical-Oscillator Analog, A . Ben-Reuven The Calculation of Atomic Transition Probabilities, R . J . S. Crossley Tables of One- and Two-Particle Coefficients of Fractional Parentage for Configurations s‘s’”p4, C. D. H. Chisholm, A. Dalgarno, and F. R. lnnes Relativistic Z-Dependent Corrections to Aomic Energy Levels, Holb Thomis Doyle AUTHORINDEX-SUBJECTINDEX Volume 6
Dissociative Recombination, J. N. Bardsley and M. A. Biondi Analysis of the Velocity Field in Plasma from the Doppler Broadening of Spectral Emission Lines, A . S. Kaufman The Rotational Excitation of Molecules by Slow Electrons, Kazuo Takayanagi and Y u k i k a z u Itikawa The Diffusion of Atoms and Molecules, E. A. Mason and R. T. Marrero T h e o r y a n d A p p l i c a t i o n of Sturmain Functions, Manuel RoVolume 5 tenberg Flowing Afterglow Measurements Use of Classical Mechanics in the of Ion-Neutral Reactions, E. E. Treatment of Collisions between Fer uson, F. C. Fehsenfeld, and A . Massive Systems, D. R. Bates and L. 8chmeltekopf A . E. Kingston Experiments with Merging Beams, AUTHORINDEX-SUBJECTINDEX Roy H. Neynaber Radiofrequency Spectroscopy of Stored Ions 11: Spectroscopy, H. Volume 7 G. Dehmelt Physics of the Hydrogen Master, C. Audoin, J . P. Schermann, and P. The Spectra of Molecular Solids, 0. Grivet Schnepp
CONTENTS OF PREVIOUS VOLUMES
Molecular Wave Functions: Calculation and Use in Atomic and Molecular Processes, J . C . Browne Localized Molecular Orbitals, Harel Weinstein, Ruben Paunez, and Maurice Cohen General Theory of Spin-Coupled Wave Functions for Atoms and Molecules, J. Gerratt Diabatic States of MoleculesQuasi -St a t i o n a ry Electronic States, Thomas F. O’Malley Selection Rules within Atomic Shells, B. R. Judd Green’s Function Technique in Atomic and Molecular Physics, Gy. Csanak, H. S. Taylor, and Robert Yaris A Review of Pseudo-Potentials with Emphasis on Their Application to Liquid Metals, Nathan Wiser and A. J. Greenfield AUTHORINDEX-SUBJECTINDEX
Volume 8 Interstellar Molecules: Their Formation a n d Destruction, D . McNally Monte Carlo Trajectory Calculations of Atomic and Molecular Excitation in Thermal Systems, James C. Keck Nonrelativistic Off-Shell Two-Body Coulomb Amplitudes, Joseph C. Y. Chen and Augustine C. Chen Photoionization with Molecular Beams, R. B. Cairns, Halstead Harrison, and R. I. Schoen The Auger Effect, E. H. S. Burhop and W. N. Asaad AUTHORINDEX-SUBJECTINDEX
389
Volume 9 Correlation in Excited States of Atoms, A. W. Weiss The Calculation of Electron-Atom Excitation Cross Sections, M. R. H, Rudge Collision-Induced Transitions Between Rotational Levels, Takeshi Oka The Differential Cross Section of Low Energy Electron-Atom Collisions, D. Andrick Molecular Beam Electronic Resonance Spectroscopy, Jens C. Zorn and Thomas C. English Atomic and Molecular Processes in the Martian Atmosphere, Michael B. McElroy AUTHORINDEX-SUBJECTINDEX Volume 10 Relativistic Effects in the ManyElectron Atom, Lloyd Armstrong, Jr. and Serge Feneuille The First Born Approximation, K. L. Bell and A. E. Kingston Photoelectron Spectroscopy, W. C. Price Dye Lasers in Atomic Spectroscopy, W. Lange, J. Luther, and A . Steudel Recent Progress in the Classification of the Spectra of Highly Ionized Atoms, B. C. Fawcett A Review of Jovian Ionospheric Chemistry, Wesley T. Huntress, Jr. SUBJECT INDEX Volume 11 The Theory of Collisions Between Charged Particles and Highly Ex-
390
CONTENTS OF PREVIOUS VOLUMES
cited Atoms, I. C. Percival and D. Study of Collisions by Laser SpecRichards troscopy, Paul R. Berman Electron Impact Excitation of Posi- Collision Experiments with Laser tive Ions, M. J. Seaton Excited Atoms in Crossed Beams, I. V. Hertel and W. Stoll The R-Matrix Theory of Atomic Process, P. G. Burke and W. D. Scattering Studies of Rotational and Robb Vibrational Excitation of Molecules, Manfred Faubel and J . Role of Energy in Reactive MolecuPeter Toennies lar Scattering: An InformationTheoretic Approach, R. B. Low-Energy Electron Scattering by Bernstein and R. D. Levine Complex Atoms: Theory and Calculations, R. K. Nesbet Inner Shell Ionization by Incident Nuclei, Johannes M. Hansteen Microwave Transitions of Interstellar Atoms and Molecules, W. B. Stark Broadening, Hans R. Griem Somerville Chemiluminescence in Gases, M. F. AUTHORINDEX-SUBJECTINDEX Golde and B. A. Thrush AUTHORINDEX-SUBJECTINDEX Volume 14 Volume 12 Nonadiabatic Transitions between Ionic and Covalent States, R. K. Janev Recent Progress in the Theory of Atomic Isotope Shift, J. Bauche and R.-J. Champeau Topics on Multiphoton Processes in Atoms, P. Lumbropoulos Optical Pumping of Molecules, M. Broyer, G. Gouedard, J. C. Lehmam, and J. Vigue Highly Ionized Ions, Ivan A. Sellin Time-of-Flight Scattering Spectroscopy, Wilhelm Raith Ion Chemistry in the D Region, George C, Reid AUTHORINDEX-SUBJECTINDEX Volume 13 Atomic and Molecular Polarizabilities-A Review of Recent Advances, Thomas M. Miller and Benjamin Bederson
Resonances in Electron Atom and Molecule Scattering, D. E. Golden T h e Accurate Calculation of Atomic Properties by Numerical Methods, Brian C. Webster, Michael J. Jamieson, and Ronald F. Stewart (e, 2e) Collisions, Erich Weigold and Ian E. McCarthy Forbidden Transitions in One- and Two-Electron Atoms, Richard Marrus and Peter J. Mohr Semiclassical Effects in HeavyParticle Collisions, M. S. Child Atomic Physics Tests of the Basic Concepts in Quantum Mechanics, Francis M. Pipkin Quasi-Molecular Interference Effects in Ion-Atom Collisions, S. V. Bobashev Rydberg Atoms, S. A. Edelstein and T. F. Gallagher UV and X-Ray Spectroscopy in Astrophysics, A. K. Dupree AUTHORINDEX-SUBJECTINDEX
CONTENTS OF PREVIOUS VOLUMES
Volume 15
39 1
Aspects of Recombination, D. R. Bates The Theory of Fast Heavy Particle Collisions, B. H. Bransden Atomic Collision Processes in Controlled Thermonuclear Fusion Research, H. B. Gilbody Inner-Shell Ionization, E. H . S. Burhop Excitation of Atoms by Electron Impact, D. W. 0. Heddle
Negative Ions, H. S. W. Massey Atomic Physics from Atmospheric and Astrophysical Studies, A . Dalgarno Collisions of Highly Excited Atoms, R. F. Stehbings Theoretical Aspects of Positron Collisions in Gases, J. W. Humberston Experimental Aspects of Positron Collisions in Gases, T. C. GriSfith Reactive Scattering: Recent Ad- Coherence and Correlation in Atomic Collisions, H. Kleinpoppen vances in Theory and Experiment, Richard B. Bernstein Theory of Low Energy ElectronIon-Atom Charge Transfer ColliMolecule Collisions, P. G. Burke sions at Low Energies, J . B. Hasted AUTHORINDEX-SUBJECTINDEX
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