Advances in
ATOMIC A N D MOLECULAR PHYSICS
VOLUME 16
CONTRIBUTORS TO T H I S VOLUME R. J. CELOTTA M. COHEN
R. DURE...
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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
New York
London
Toronto
Sydney San Francisco
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.
ACADEMIC PRESS, INC.
111 Fifth Avenue, New York, New York 10003
United Kiirgdom Edition pirblislied b y ACADEMIC PRESS. INC. ( L O N D O N ) LTD. 24/28Oval Road, London’NWI 7 D X
LIBRARY OF
CONGRESS CATALOG CARD
NUMBER:65-18423
ISBN 0- 12-003816-.1 PRINTED IN TH E UNITED STATES OF AMERICA
80818263
987654321
Contents
ix
1.IST OF CON.1RlBUTORS
Atomic Hartree-Fock Theory M . Cohen and R. P. McEachran 1. Introduction 11. The Hartree-Fock Method 111. Properties of Hartree-Fock Wave Functions
IV. V. VI. VII.
Properties of the Frozen Core Approximation The Extended Frozen Core Approximation Improved Frozen Core Approximations Conclusions Appendix: Relativistic Corrections to the Energy Levels References
2 4 12 16
23 34 49 50 52
Experiments and Model Calculations to Determine Interatomic Potentials R . Diiren 1. Introduction 55 11. Electronic Model Potentials and Interatomic Potentials 58 I l l . Experimental Sources 70 IV. Interatomic Potentials Determined with Model Potentials 91 V. Conclusions 96 References 97 Note Added in Proof 100
Sources of Polarized Electrons R. J . Celotta and D . T . Pierce 1. Introduction 11. Source Characteristics 111. Chemi-ionization of Optically Oriented Metastable Helium V
I02 I 04 107
CONTENTS
vi
IV. Photoionization of Polarized Atoms V. The Fano Effect Source VI. Field Emission from Ferromagnetic Europium Sulfide on Tungsten VII. Low-Energy Electron Diffraction VIII. Photoemission from GaAs IX. Summary References
112 1 I6 120 i27 134 152 154
Theory of Atomic Processes in Strong Resonant Electromagnetic Fields S . Swain I . Introduction 11. Master Equations 111. Resonance Fluorescence
IV. The Optical Autler-Townes Effect V. Conclusion References
159 165 171 190
196 196
Spectroscopy of Laser-Produced Plasmas M . H . Key and R , J . Hutcheon
I. Introduction 11. Ionization 111. Population Densities of Bound Levels IV. Intensity of Line Radiation V. Line Broadening VI. Continuum Emission VII. Radiative Transfer VIII. Structure and Spectroscopic Characteristics of Laser-Produced Plasmas IX. Spectroscopic Diagnostics of Laser-Produced Plasmas References Note Added in Proof
202 203 213 217 225 234 238 246 25 1 272 280
Relativistic Effects in Atomic Collisions Theory B . L. Moiseiwitsch I. Introduction 11. Excitation and Ionization 111. Electron Capture
References
28 1 282 307 3 I6
CONTENTS
vii
Parity Nonconservation in Atoms: Status of Theory and Experiment
E . N . Fortson and L . Wilets I. Introduction 11. The Neutral Current Interaction in Atoms 111. Observable Effects IV. Atomic Calculations V. Optical Rotation Experiments: Bismuth VI. Stark Interference Experiments: Cesium and Thallium VII. Atomic Hydrogen Experiments VIII. Conclusions References
INDEX C O N T E N T S O F PREVIOUS V O L U M E S
3 I9 32 1 324 328 338 357 367 3 70 37 1 375 387
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List of Contributors Numbers in parentheses indicate the pages on which authors’ contributions begin.
R. J. CELOTTA, United States Department of Commerce, National Bureau of Standards, Washington, D. C. 20234 (101) M. COHEN. * Institute for Advanced Studies, The Hebrew University, Jerusalem, Israel ( I ) R. DUREN, Max-Planck-Institut fur Stromungsforschung, D3400 Gottingen, West Germany (55) E. N. FORTSON. Department of Physics, University of Washington, Seattle, Washington 98195 (319) R. J. HUTCHEON,+ Physics Department, University of Leicester, Leicester, England (201) M. H. KEY, Science Research Council, Rutherford and Appleton Laboratories, Chilton, Didcot, Oxfordshire 0x1 1 OQX, England (201)
R. P. McEACHRAN,~Institute for Advanced Studies, The Hebrew University, Jerusalem, Israel ( I ) B. L. MOISEIWITSCH. Department of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Belfast BT7 INN, Northern Ireland (281). D. T. PIERCE, United States Department of Commerce, National Bureau of Standards, Washington, D. C. 20234 (101)
S. SWAIN, Department of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Belfast BT7 INN, Northern Ireland (159) L. WILETS. Department of Physics, University of Washington, Seattle, Washington 98195 (319) * Permanen[ address: Department of Physical Chemistry. The Hebrew University, Jerusalem. Israel. .pPresent address: Nuclear Power Company (Whetstone) Ltd.. Cambridge Road, Whetstone, Leicester. England. Permanent address: Department of Physics, York University, Toronto. Canada. IX
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li
.
ADVANCES IN ATOMIC A N D MOLECULAR PHYSICS VOL . 16
ATOMIC HARTREE-FOCK A4. COHEN* and R . P . M c E A C H R A N t Institute for Advanced Studies The Hebrew University Jerusalem, Israel
I . Introduction . . . . . . . . . . . . . . . . . . . . . . 2 I1. The Hartree-Fock Method . . . . . . . . . . . . . . . . . 4 A. Central Field Spin Orbitals . . . . . . . . . . . . . . . . 4 B . Derivation of the Hartree-Fock Energy . . . . . . . . . . . . 6 C . Derivation of the Radial Equations . . . . . . . . . . . . . 8 D . Integrals of the Radial Equations . . . . . . . . . . . . . . 10 E. Solution of the Radial Equations . . . . . . . . . . . . . . 11 I11. Properties of Hartree-Fock Wave Functions . . . . . . . . . . . 12 A . Ionization Potentials . . . . . . . . . . . . . . . . . . 12 B . Orthogonality of Excited State Functions . . . . . . . . . . . 13 C . Brillouin’s Theorem . . . . . . . . . . . . . . . . . . . 14 IV . Properties of the Frozen Core Approximation . . . . . . . . . . . 16 A . Ionization Potentials . . . . . . . . . . . . . . . . . . 16 B . Orthogonality of Frozen Core Functions . . . . . . . . . . . 17 C. Some Results of Frozen Core Calculations . . . . . . . . . . . 18 V . The Extended Frozen Core Approximation . . . . . . . . . . . . 23 A . Derivation of the Energy Expression . . . . . . . . . . . . . 24 B. The EFC Valence Radial Equation . . . . . . . . . . . . . 26 C . Orthogonality of the EFC Wave Functions . . . . . . . . . . . 28 D . Some Results of EFC Calculations . . . . . . . . . . . . . . 29 VI . Improved Frozen Core Approximations . . . . . . . . . . . . . 34 A. Multiconfiguration Frozen Cores . . . . . . . . . . . . . . 34 B . Valence Orbitals for Excited States . . . . . . . . . . . . . 37 C . The Ground State Problem . . . . . . . . . . . . . . . . 41 D . Polarized Frozen Core Approximations . . . . . . . . . . . . 43 E. More Elaborate Procedures . . . . . . . . . . . . . . . . 48 VII . Conclusions . . . . . . . . . . . . . . . . . . . . . . 49 Appendix: Relativistic Corrections to the Energy Levels . . . . . . . 50 References . . . . . . . . . . . . . . . . . . . . . . . 52
Permanent address: Department of Physical Chemistry. The Hebrew University. Jerusalem. Israel . Permnenf address: Department of Physics. York University. Toronto. Canada.
.
Copyright 0 1980 hy Academic Press Inc. All rights of reproduction in any form reserved. ISBN 0 -12 -003816-1
2
M . Cohen and R. P. McEachran
I. Introduction The Hartree-Fock (HF) approximation, in its many variations, forms the basis of the overwhelming majority of calculations of atomic, molecular, and crystal structures. The underlying physical assumptions and the main mathematical procedures were worked out in detail some fifty years ago, but it is only during the past twenty years that extensive variational calculations on excited as well as ground states of many-electron atoms have provided adequate data for comparison with experiment and to assess the strengths and weaknesses of the H F procedures. The rapid advances in digital computer technology during this latter period have resulted in the development of the numerical techniques and calculation procedures that have made tractable the solution of sets of equations much more complicated than the original H F integro-differential equations. At the same time, a number of fully automatic H F computer programs have become generally available and have been used to calculate numerous properties of atoms and ions. The recent rapid advances on the computational front have not been matched on the side of the theory, probably due to a widespread belief that explicit treatment of correlation effects will normally yield more accurate wave functions and calculated properties. Furthermore, there has been a natural tendency to prefer general expansion methods which may be applied equally to atoms, molecules, and crystals. This article deals exclusively with applications of atomic H F theory to calculations of atomic properties and in particular to oscillator strengths for electric dipole transitions. Much of the basic theory of the H F method has been described in detail in the monographs of Hartree (1957), Slater (1960), and more recently, Froese Fischer (1977), and we shall make extensive reference to these sources. Consequently, we have made no attempt to give a comprehensive description of the H F method. Rather, we take the opportunity to describe the development and application of a number of particularly simple variants of the H F approximation, namely the frozen core (FC) procedures. These F C approximations share with other H F approximations the ability to predict one-electron properties of atoms (including the rather sensitive oscillator strengths) with remarkable accuracy. The development of the various FC approximations parallels the earlier development of the H F approximations, and we will attempt to underline the similarities as well as the differences at each stage of the development. Conceptually, F C approximations emphasize the physical role of the valence electrons in determining many of the physical and chemical
ATOMIC HARTREE-FOCK THEORY
3
properties of atoms and molecules. For some purposes, the properties of a single valence electron are of crucial interest. For example, the ionization energy required to remove a valence electron from a neutral atom, leaving behind an ionized core, is calculated directly in FC procedures; the famous result of Koopmans (1933) emerges naturally from the FC theory. The formal mathematical properties of the elementary F C approximation, which applies strictly to the alkali atoms (McEachran et al., 1968), and some others with genuinely one-electron spectra (McEachran and Cohen, 1971) have been derived simply from those of the restricted Hartree-Fock (RHF) approximation. [We adopt the nomenclature of Froese Fischer (1977) in describing various H F approximations; we return to these names in the following sections.] For systems with complex spectra, and in particular for the effective description of valence shells containing two or more equivalent electrons, an extension of the theory is required, which closely parallels the extended Hartree-Fock (EHF) theory. The main novel feature of the extended frozen core (EFC) procedure lies in the fact that its wave functions contain nonorthogonal orbitals which therefore introduce overlap integrals as well as new one- and two-electron integrals into the energy expressions, leading ultimately to slightly more complicated equations for the radial functions. Even the derivation of the energy expression is no longer straightforward in the presence of nonorthogonal orbitals, but a general procedure has been developed (Jucys, 1967), and its application within the FC framework yields a description that retains much of the simplicity of the elementary FC model. In the last few years, there has been an increasing number of multiconfiguration Hartree-Fock (MCHF) calculations, whose main aim is to improve the H F treatment of electron correlation. Although the resulting wave functions often yield more accurate calculated properties than the corresponding H F wave functions, the simple orbital description is lost. By contrast, a relatively simple model in which the valence electron is attached to a multiconfiguration core is conceptually more attractive. We have felt it worthwhile to describe one such multiconfiguration frozen core (MCFC) approximation, even though its application presents serious difficulties in some ground state calculations. No such difficulties arise for excited states, however, and we present the results of some MCFC calculations for comparison with the corresponding FC and MCHF results. Efforts to improve the accuracy of H F calculations, either by including explicit correlation terms in the trial wave functions or through the use of several configurations, inevitably lead to a physically complicated model. A simpler procedure is to introduce a semiempirical polarization potential into the equation which describes the active valence orbital. Although such
4
M . Cohen and R. P. McEachran
a polarized frozen core (PFC) procedure is not easy to justify a priori, its results for some simple systems are considerably more reliable than those of the elementary FC procedure. In the following sections, we describe the various FC procedures in some detail, illustrating their results by comparison both with the results of the corresponding H F procedures and with more accurate theoretical or experimental values whenever these are available.
11. The Hartree-Fock Method A. CENTRAL FIELDSPIN ORBITALS The spin-free nonrelativistic Hamiltonian operator that describes the motions of an N-electron atom or ion of nuclear charge Z may be written, in atomic units, as a sum of one- and two-electron operators:
The ith electron coordinate ri is measured from the nuclear position (which we assume fixed) and the interelectronic distances are rq = Iri - I.‘, The presence of the two-electron operators l / r v makes it impossible to obtain exact solutions to Schrodinger’s equation for the N-electron system. Essentially, all the H F methods replace these two-electron terms by approximate one-electron potentials giving an effective Hamiltonian which contains only one-electron operators: N
q2
H eff = I C Hieff= - i = = I
I
I
Z - Vleff vf + (rJ] I
‘i
(2)
The detailed form of V:fr(ri) differs for each variant of the H F method, and it will not be considered further here, except to note that, in general, I/ieff need not be a central field potential (depending only on the scalar distance ri).The exact eigenfunctions of Herrcan now be constructed from products of the single-electron eigenfunctions, ui(ri) of the H:“; they are expected to be approximate eigenfunctions of the physical Hamiltonian H , and should therefore display as many of the properties of the exact eigenfunctions as possible. Since H commutes with the spatial angular momentum operators L2 and L, and also (since it is spin-free) with the spin angular momentum operators S 2 and S,, the eigenfunctions of Herrare required to be eigen-
5
ATOMIC HARTREE-FOCK THEORY
functions of L', L,, s2, a nd S, as well. If H:" is spin-free, it is possible to attach a normalized spin function (conventionally written la) or I p ) ) to each eigenfunction uI(rI),so as to obtain spin orbitals Iu,a,) and IuIpl). Each spin orbital thus requires four coordinates in order to specify it completely (three space coordinates and one spin coordinate), .and those products of N such spin orbitals which are eigenfunctions of the angular momentum operators and are also antisymmetric under interchange of any pair of electrons (Pauli exclusion principle) are the required H F approximation eigenfunctions. The antisymmetry requirement is conveniently met by constructing Slater determinants of the form
I Ul(1)
UI(2)
... udN)
I
and it is understood that each spin orbital contains a specified ( a or p ) spin function. In general, linear combinations of such DN will be required to construct eigenfunctions of L2, L,, S 2 , and S,. For a highly ionized atom ( Z >> N ) , it is clear that the interelectronic terms in H will form only a relatively weak perturbation to the much stronger central field nuclear potential, suggesting that it should be a good approximation to choose yeffas a central field potential. If this can be achieved generally for neutral atoms as well as ions, then the spin orbitals may be written explicitly as products of spherical harmonics Ylm(O,+), spin functions %(a), and radial functions (whose forms are to be determined); thus in the traditional H F method, we find (cf. Hartree, 1957) un/m(r,',+;
0) =
('/r)~n/(r)'/~('~~)%(a)
(4)
We have used m rather than m, for the orbital angular momentum projection quantum number, and s instead of ms for the spin projection quantum number. The traditional form of the radial function, ( l/r)Pn/(r), leads later to some simplification of the radial equations. The dependence on quantum numbers of the radial functions may be more extensive (cf. Froese Fischer, 1977); we return to this point later. The advantage of having spin orbitals of central field type cannot be overestimated. The effect is to reduce the H F problem from N coupled integro-partial-differential equations (one for each orbital) in four variables to N coupled integro-differential equations in a single radial variable. A further substantial reduction results from treating shells of equivalent electrons (having the same values of n and I , but different values of m and s) by means of a single radial function (l/r)Pn/(r). This has the effect of
M. Cohen and R. P. McEachran
6
reducing the number of coupled radial H F equations to M , one for each shell of electrons usually designated by s, p, d, etc. for I = 0, 1,2, etc. The conventional description of the periodic table of the elements is a reflection of this shell structure approximation. As was pointed out by Delbriick (1930), choosing spin orbitals of central field type leads self-consistently to a central potential yeff(r,)whenever an atom consists entirely of closed shells, so that it may be described by a single HF determinant. In other cases, v.ef‘(r,) will not generally be of central field type, but the advantages of a central potential are so considerable that it has become standard procedure to perform a spherical averaging whenever the effective potential is not automatically spherically symmetric, as for open-shell systems, or whenever the MCHF procedure is employed. This averaging is implicit in the H F procedure, which constructs an energy expression out of central field type orbitals, expressing it entirely in terms of radial integrals over the unknown radial functions which are subsequently determined variationally. B. DERIVATION OF
THE
HARTREE-FOCK ENERGY
The orthogonality properties of the spherical harmonics Y/,,,(O, +) and of the spin functions &(a) ensure that two spin orbitals unrm and u,,.~.,,,.~, will be automatically orthonormal, provided that
It is a matter of great convenience to have mutually orthogonal radial functions, for it greatly simplifies the calculation of matrix elements of all types. Since the radial functions are to be determined variationally, the orthonormality conditions such as (5) must be included in the variation by means of Lagrange multipliers. In order to obtain the H F energy expression, it is necessary to calculate a stationary value of EN
where
QN
= 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 njol 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)
(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
+ (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
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
~ , + 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”.
I-
u
W
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.
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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.
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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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. .I73 . . 183 . . 186 . .I90 . . 190 . . 190 . . 193 . . 196 . . 196
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.
ISBN 0-12-003816-1
160
S. Swain
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-
ATOMIC PROCESSES IN STRONG RESONANT E. M. FIELDS
161
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
162
S. Swain
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) = 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/)> 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> 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 >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 (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> 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;,, > 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
194
S. Swain
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.
196
S. Swain
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
Abate, J. A. (1974). Opt. Commun. 10, 269. Ackerhalt, J. (1978). Phys. Rev. A 17,47 1. Agarwal, G. S. (1973). Prog. Opf. 11, 1. Agarwal, G. S. (1974). Springer Tracts Mod. Phys. 70, 1. Agarwal, G. S. (1976). Phys. Rev. Leu. 37, 1383. Agarwal, G . S. (1978). Phys. Rev. A 18, 1490. Agarwal, G. S. (1979). Z. Physik B33, 1 11. Agarwal, G. S., and Navrayana, P. A. (1979). Opt. Commun. 30,364. Agarwal, G. S., and Saxena, R. (1978). Opt. Commun. 26, 202. Allen, L., and Eberly, J. H. (1974). “Optical Resonance and Two Level Atoms.” Wiley, New York. Apanasevich, P. A. (1964). Opt. Spectrosc. (USSR) 16, 387. Apanasevich, P. A., and Kilin, S.-Ja. (1979), J. Phys. B 12, L83. Armstrong, L., and Feneuille, S. (1975). J . Phys. B 8, 546. Armstrong, L., Jr., and ONeill, S. V. (1980). J. Phys. 813, 1125. Autler, S. H., and Townes, C. H. (1955). Phys. Rev. 100, 703. Avan, P., and Cohen-Tannoudji, C. (1977). J. Phys. B 10, 155 and 171. Averbukh, B. B., and Sokolovskii, R. I. (1977). Opt. Spectrosc. 42, 587. Baklanov, E. V. (1974), Sov. Phys.-JETP (Engl. Transl.) 38, 1100. Ballagh, R. J., and Cooper, J. (1977). Astrophys. J. 213, 479. Bayfield, J. E. (1977). Phys. Rep. 51, 317. Bialynicka-Birula, Z., and Bialynicka-Birula, I. (1977), Phys. Rev. A 16, 1318. Bjorkholm, J. E., and Liao, P. F. (1977). Opt. Commun. 21, 132. Bonch-Bruevich, A. M., and Khodovoi, V. 1. (1968). Sov. Phys.-Usp. (Engl. Traml.) 10, 637. Bonch-Bruevich, A. M., Kostin, N. N., Khodovoi, V. A., and Khromov, V. V. (1969). Sov. Phys.-JETP (Engl. Transl.) 29, 82.
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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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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
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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
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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 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.
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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
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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.
’
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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M. H. Key and R. J. Hutcheon
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.)
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
249
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,
M. H. Key and R. J . Hutcheon
250
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.
-
-
-
’
-
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
25 1
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.
-
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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M. H . Key and R. J. Hutcheon
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
'
A
SPECTROSCOPY OF LASER-PRODUCED PLASMAS
253
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
10’O
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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M . H . Key and R . J . Hutcheon
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.
M . H . Key and R. J. Hutcheon
260
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,
'
I
ABLATION
NoX l s 2 - 1 s 2 p
PLASMA'
I
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)
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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.
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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
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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
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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
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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
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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.
272
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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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 . . . . . . . . . . . . . . . . .
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.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
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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
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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 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.
RELATIVISTIC EFFECTS IN ATOMIC COLLISIONS THEORY
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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.
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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 +$) (15) where q = ? 1 gives the two states of circular polarization, the positive sign denoting right-hand circularly polarized light by the standard optical convention, which corresponds to negative helicity. The transition amplitude is proportional to
A,(fi)
= (fPNC
I(
*
E, +
A
’ Bq)
I iPNC)
The degree of circular polarization in emission is (again suppressing the indices f,i )
PARITY NONCONSERVATION IN ATOMS
327
where we have used Eqs. (13), (14), and (15), and in the last expression we have assumed GPNC Ei - En
1
(20)
where the Stark perturbed states are given a subscript S. For analyzing the geometry of experiments, it is helpful to use the vector Ci defined after Eq. (13):
E. N . Fortson and L. Wilets
328
The Stark-EpNc form of interference is utilized in experiments with heavy atoms at Berkeley and Paris, and in many experiments with hydrogen. We defer to a later section the discussion of hydrogen. Here we present a qualitative sketch of the scheme used at Paris and Berkeley. The basic idea, originally pointed out by Bouchiat and Bouchiat (1979, is that an electronic polarization (i.e., a nonzero expectation value of the electronic angular momentum (J)) in the excited state of the atom is induced by absorption of a circularly polarized photon directed perpendicular to an applied static electric field. For definiteness, let the static field be in the x direction, Es = Es$ 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 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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