THEORETICAL ATOMIC TRANSITION PROBABILITIES!

By DAVID LAYZER Harvard College Observatory, Cambridge, Massachusetts

AND Roy H. GARSTANG Joint Institute for Laboratory Astrophysics

University of Colorado, Boulder, Colorado INTRODUCTION

by R. H. Garstang

The total world effort on the measurement and calculation of atomic transition probabilities has grown so extensive that it is no longer practicable to list in a review every paper that has been published in the field during a period of years. Accordingly, users of transition probabilities are becoming more dependent on the output of the data center at the National Bureau of Standards, Washington. This data center is engaged in a continuing program of collection and compilation not only of bibliographic information but also of every available individual transition probability. The latter are compiled into "best" values whenever possible. Two publications of the data center are of wide utility (1, 2) :

(a) Bibliography on Atomic Transition Probabilities, by B. M. Glennon & W. L. Wiese (1966, ) . This is a complete revision of an earlier publica­tion under the same title. It is a list, arranged according to elements and stages of ionization , of all the papers that contain transition probability data for each ion. No attempt is made to be selective; on the contrary, every effort is made towards completeness. The list now includes over 1200 papers and probably omits few if any papers containing numerical data of significance in this field. The bibliography is complete through 1967.

(b) Atomic Transition Probabilities, Vot. I, Hydrogen Through Neon, by W. L. Wiese, M. W. Smith & B . M. Glennon (1966, 1968). This contains a complete compilation of all known transition probabilities for both allowed and forbidden lines for all stages of ionization for the first ten elements, H through Ne. Data for some 4000 lines are included, complete to 1963 with some later additions. Although this compilation also aims at completeness, some selectivity has been used in reporting individual numerical values. When several determinations were available for a particular line an attempt was made to provide a "best" value--a weighted average taking into considera­tion whatever is known about the reliability of particular methods, and so on. For some important transitions for which only relative transition probabili-

1 The survey of literature for this review was concluded in March 1968. 449

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450 LAYZER & GARSTANG

ties were available, the Coulomb approximation was used to provide an abso­lute scale, in this way making the published results of much greater value to users. A further compilation containing data for the next ten elements, Na through Ca, is now in preparation. Inquiries as to the availability of data not referenced in the above publications may be addressed to Atomic Physics Data Center, Atomic Physics Division, National Bureau of Standards, Washington, D. C. 20234. Anyone who has unpublished data on transition probabilities (including theses) is urged to send a copy to the data center so that it may take due account of the work in future compilations.

An excellent review article (3) by Foster (1964) covers nearly all aspects of the experimental determination of transition probabilities. It discusses lifetime techniques, emission arcs, absorption and dispersion measurements, and other techniques; it is highly recommended as an introduction to the subject. A recently published book on atomic spectra by Shore & Menzel (4) contains an introduction to the theory of atomic line strengths from a modern viewpoint. Garstang (5) has recently reviewed a number of topics in atomic physics (including transition probabilities) relevant to the inter­pretation of atomic spectra of astrophysical interest.

A. PERMITTED TRANSITIONS

by David Layzer

The present discussion of electric-dipole transitions concentrates on the problem of calculating absolute J values for complex atoms in all stages of ionization. For two-electron atoms, transition probabilities and other physi­cal quantities can be evaluated with great precision by a variety of methods (6)-none of which, however, is practical for atoms with as many as four electrons. For the rare earths and heavier atoms, current mathematical re­sources do not in general permit strictly quantitative predictions of atomic parameters that are sensitive to the wavefunctions of the outer electrons. Be­tween these two extremes lies a large and astrophysically important group of atoms for which it is now possible to make quantitative (though not usu­ally highly precise) estimates of electric-dipole transition probabilities. This review will concentrate on methods that apply to this group.

As has been mentioned, complete and up-to-date lists of calculated J values can be obtained from the Atomic Physics Data Center of the National Bureau of Standards. Equally important to the prospective user of such in­formation are error estimates. It is a truism that predictions not capable of being contradicted by experiment have no scientific value; and a calculated J value unaccompanied by an error estimate is such a prediction. Compari­sons between different calculations of the sameJ value and comparisons be­tween predicted and measuredJ values indicate that calculatedJ values vary widely in accuracy from transition to transition and from method to method. What makes one method of calculation better than another for a particular transition? For which transitions does a particular method give reliable re­sults, and for which not? The scientific worth of calculatedJ values depends

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ATOMIC TRANSITION PROBABILITIES 451

largely on how well questions of this kind can be answered. The problem of estimating the accuracy of predicted I values accordingly forms one of the two principal themes of the present review.

From a formal point of view, transition probabilities can be regarded as byproducts of energy-level calculations. To evaluate a transition matrix ele­ment, one must have approximate solutions of the eigenvalue problem for the initial and final states. But while it is true that the evaluation of transition matrix elements presupposes the possibility of finding appropriate approxi­mate solutions to the eigenvalue problem, it is also true that the appropriate­ness of these solutions depends on the use to which they are put. Wave­functions well suited to energy calculations are not necessarily well suited to I-value calculations. At any given level of approximation, the problem of selecting the best wavefunctions to use in evaluating a transition matrix element cannot be solved simply by applying the variation principle, nor is any other principle of comparable generality known to be applicable. The problem of devising suitably specialized approximation methods for the evaluation of transition matrix elements forms the second main theme of this review.

1. ONE-CONFIGURATION ApPROXIMATIONS

Reduction of the transition matrix element.2-The transition probability A, the j value (or oscillator strength) j, and the strength S, of an electric­dipole transition connecting states k (of energy E) and k' (of energy E') of an N-electron atom are defined by the relations

where N

X = LX;, 1_1

w= IE'-EI

By virtue of the equation of motion

.du 1-=Hu-uH dt

1.2.

1.3.

where u is any dynamical variable and H is the Hamiltonian of the system, the transition matrix element can be written in the equivalent forms

(k'lxl k) = ic,,-l(k'I*1 k) = - ",-'W Ixl k) 1.4.

In a nonrelativistic approximation, the total dipole velocity X and the total dipole acceleration X are given by

• N X = - iL Vi,

,_1

• Atomic units (m=e=1i=l, c=137) are used throughout this review.

1.5.

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452 LAYZER & GARSTANG where Z is the nuclear charge. Thus sums of one-electron operators occur in all three forms of the transition amplitude.

An electron configuration, defined by N pairs of one-electron quantum numbers n/;E(X, will be denoted by the symbol K. The state vector for an atom in a definite configuration can be written in the form

I KI') = (t<I>(r, ... rN» I {"'} r)..··N 1.6.

Where <I> is a function of the radial coordinates only and the ket I {(X} r)l. .. N refers to the spin and angle coordinates of the N electrons. The label r is used here to indicate a coupling scheme in which the Hamiltonian H is diagonal

within the configuration K:

("r'l HI Kr) = Iir'rE(Kr) 1.7.

The antisymmetrizing operator (t is given by

1.8. where C is a normalization factor, Op = ± 1 according as the permutation Pis even or odd, and the summation runs over the N! permutations of the N elec­tron' indices.

With the state vectors for the initial and final states given by Equation 1.6, the transition matrix element vanishes unless precisely one of the N azimuthal quantum numbers, which we may take to be.eN in the representa­tion 1.6, changes by unity between the initial and final states. If this condi­tion is satisfied, the transition matrix element reduces to

(K'r'l xI'I Kr) = (NICC') (<I>'r(L op,P')<I» X (,/r'l C/I)(04)) I "r) 1.9.

We have written r, 0, rP for rN, ON, rPN; C(I ) denotes the unit irreducible tensor of rank 1, and the operators P' permute the indices of electrons with a fixed value of (and different values of n. The matrix elements of the dipole-velocity and the dipole-acceleration factor in a similar way:

and

where

(,,'r'l xI'I "r) = - i(N!CC')(<I>'d,lZ'(L op,P')<I» X (,,'r'l CI'(I) I Kr)

(K'r'l xI'I KI') = Z(NICC') (<I>'r-2(L op,P')<I» X (,,'r'l Cp(l) I "r)

a (,I, - ,1,')(,1, + ,1,' + 1) dTH' -= - + ------"'-ar 2r

1.10.

1.11.

1.12.

I t is customary to approximate <I> by a product of one-electron radial func­tions:

N <I>h . . . rN) = II pend,; r,)

,,-1 1.13.

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ATOMIC TRANSITION PROBABILITIES 453 Then the radial matrix elements in formulae 1.9-1.11 reduce to sums of prod­ucts of N one-electron radial integrals. N-l of the integrals in a given prod­uct are overlap integrals of the type (P'(n' !')P(n!'»; the remaining integral is a transition integral of one of the three following types :

lent, n't') = (P'(n't')rP(nt»

l(n', n't') = (P'Cn't')d,WP(nt»

K(nt, n't') = (P'(n'(')r-2P(nt»

1.14.

1.15.

1.16.

If the passive electrons occupy the same orbitals in the initial and final states of a transition, the radial matrix elements in formulae 1.9-1.11 reduce approximately to the one-electron transition integrals I, J, and K, respec­tively, multiplied by identical products of (N-1) overlap integrals, each of order 1:

N-l (q,'rCL opP)q,)'" lent, n't') II (P'(ndi)PCn.ti»

i_I N-l

WdTll'(L IipP)q,)�J(nt) n't') II (P'Cniti)PCniti» i=l N-l

(<I>'r-'(L opP)<I»�K(nt, n't') II (P'(ni!,i)P(niti»

1.17a.

1.l7b.

1.17c.

If the passive electrons do not occupy the same set of orbitals in the ini­tial and final states, one or more of the overlap integrals is of the type (P'(n'!,)P(nf» and is normally small in magnitude.

Relative transition probabilities in pure coupling schemes.-The condition 1. 7 may be satisfied approximately in a pure coupling scheme (SL, jj, or j!'). The matrix element (K'I" I C,.(l) I KI') in formulae 1.9-1.11 can then be evalu­ated by means of standard techniques involving applications of the Wigner­Eckart theorem and the use of irreducible tensor algebra. An extensive col­lection of formulae, together with references to the original literature, may be found in the recent text by Shore & Menzel (4).

The one-electron radial functions vary only slightly from state to state within a configuration. If this small variation is neglected, all transition ma­trix elements connecting states belonging to a fixed pair of configurations have the same radial factor. Relative transition probabilities (the relative intensities of Zeeman components of a spectral line, of lines in a multiplet, and of m ultiplets in a transition array) are then governed by the spin-angu­lar matrix element.

Relative transition probabilities in intermediate coupling.-Even in a non­relativistic approximation, the matrix II (KI" I H I Kr)II ==H. is usually not diagonal in any pure coupling scheme. If relativistic effects are taken into account, H. is not even diagonal with respect to Sand L but only with re­spect to J. To construct a basis in which condition 1. 7 is satisfied, one must first calculate the energy matrix in some convenient pure-coupling scheme and then diagonalize it. The energy matrix elements are linear combinations of certain radial integrals, which can be evaluated if accurate radial wave-

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454 LAYZER & GARSTANG

functions are available. Experience has shown, however, that this procedure usually leads to rather poor agreement between the predicted and the ob­served energy levels, even when Hartree-Fock wavefunctions are used in the calculation.

An alternative procedure, which involves much less labor and leads to a better representation of the energy spectrum, is to treat the radial integrals as adjustable parameters to be determined by fitting the eigenvalues of the energy matrix to the observed energy levels. The goodness of fit can be sub­stantially improved through the introduction of "effective interactions" which represent certain effects of configuration mixing.3

Having diagonalized the energy matrices for a given pair of configura­tions, one can transform the transition matrix S'/2 == II (K'r'l ell(!) I Kr)11 from a pure coupling scheme to a representation in which the energy is diagonal within both configurations. If V and V' denote the unitary matrices that diagonalize the energy matrices in the configurations K and K', the transition rna trix in the diagonal represen ta tion is (V') t S'/2 V.

Recent intermediate-coupling calculations are reviewed in Part B.

2. MANy-CONFIGURATION ApPROXIMATIONS The meaning of configuration mixinf!.-To illustrate the distinction be­

tween pure and mixed configurations, let us consider the ground state (1S)2 'S of a two-electron atom. In the one-configuration approximation we may write

2.1.

where cp is a symmetric function of its arguments and the two kets that follow it refer to the angle and spin coordinates respectively. The best radial func­tion cp in the sense of the variation principle satisfies the linear partial dif­ferential equation

(��_ � � � � � Z � Z +�) <I> = E<I> 2 ar,2 2 ar22 r, r2 r,. 2.2.

Now, among the excited configurations that could be mixed with (1S)2 are all configurations of the type (ES E'S) . The state vectors for the 'S states of these configurations all have the form 2.1; they differ only through the radial fac­tor <P. Hence any linear combination of these state vectors also has the form 2.1. Since cP is the solution of Equation 2.2 that corresponds to the lowest eigenvalue E, there can be no mixing between the ground configuration and excited configurations of the type (ES E'S).

Conversely, if cP is not an eigensolution of Equation 2.2, there must be some mixing between (ls)2 and excited configurations of the type (ES E'S) .

3 The most important of these effective interactions gives rise to a term propor­tional to L(L+l), first introduced by Layzer (7) and Trees (8). The physical sig­nificance of these interactions was explained by Layzer (7) and Racah (9). For a modern discussion, see Judd (10) and references cited therein.

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ATOMIC TRANSITION PROBABILITIES 4 55

1£.p is approximated by a product of identical one-electron wavefunctions,

2.3. then the mixing coefficients (ES E'S/ (1S)2) with E¢1¢E' are all finite. But all mixing coefficients of the type (1SES / (1S)2) can be made to vanish simultane­ously through an appropriate choice of the function P. A necessary and suffi­cient condition for this to happen is that P should be the best one-electron radial function in the sense of the variation principle; that is, P must be a solution of Hartree's integrodifferential equation.

More generally, if a many-electron radial function is approximated by a product of one-electron radial functions, each orbital (n!) being associated with a single radial function, then there will be no mixing between configura­tions that differ only through a single principal quantum number if the radial functions satisfy the variation principle (Brillouin's theorem).

A less restrictive approximation than 2.3 results from representing tP by the symmetrized product of two different one-electron radial functions:

1 <p(r" r2) = y'2 [P(r,)Q(r2) + P(r.)Q(y,) 1 2.4.

This approximation is equivalent to a two-configuration approximation:

<p(r" r2) = a'FhlF(r2) + b'G(r,)G(r,) 2.5.

[Set P= 1/ y'2 (aF+ibG), Q= 1/ y'2 (aF-ibG).] By letting P, Q or F, G be determined by the variation principle, one can allow for most of the effects of mixing with doubly excited configurations of the type (es E's) . The "con­figuration" represented by the radial function G(rl)G(r2) is a composite of the excited configurations (fS f'S) . For convenience it may be designated as (2s)2, but G does not in any sense represent a 2s wavefunction.

The exact state vector / (ls)2 'S) also contains components of the type / (de't) IS) with C=I, 2, . . . . Thus Equation 2.1 should be replaced by

! (ls)2 IS) = t CI<PI(r" Y2) ! (U)OO)12! (!!) 00)12 l-o 2.6.

where the functions .p are symmetric in their arguments. The preceding dis­cussion of approximations to tP ""'tPo appJies also to the functions <PI with t>O.

The mixing coefficients (probability amplitudes) CI have a definite physi­cal meaning: / cll' is the probability that a two-electron atom known to be in its ground state will be found in a configuration of the type (ele'C). On the other hand, the assignment of probability amplitudes to specific values of e and e', for a fixed value of t, is purely a matter of convention. One should bear this fact in mind when assessing calculations that claim to allow for the mix­ing of very large numbers of configurations.

The preceding discussion applies, with obvious modifications, to more complex many-electron systems and states.

The many-configuration Hartree-Fock approximation.-Let I K) denote a

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456 LAYZER & GARSTANG

one-configuration state vector whose radial part, before antisymmetrization, is a product of one-electron radial functions. The many-configuration ap­proximation uses finite linear combinations of such one-configuration state vectors:

I k) = t C. I K), 2.7 . • _1

Suppose that the mixing coefficients C. have all been specified. Then the variation principle yields a system of simultaneous nonlinear integrodifferen­tial equations, one for each orbital that figures in the description.4 Each equation has the usual Hartree-Fock form (11 )

L� �+ t(t+ 1) _ Z-S"_.)Pa= L' XafJPfJ 2.8. I 2 dr' 2r' r � {3 where Sa and Xa{3 are certain functionals of the radial functions Pa. The ef­fects of exchange and of configuration mixing are both represented by the terms on the right side of this equation. The practical difficulties encountered in solving systems of this kind are caused mainly by these terms.

Suppose now that we are given a complete set of radial functions P" for a given mixture of configurations. The best set of mixing coefficients C. for this set is determined by the variation principle, which yields the following system of linear, homogeneous, algebraic equations:

t (K I HI K' )(C.' - a.,.X) = 0, K = 1, . • . ,r 2.9. 1('=1

where X is the Lagrange multiplier associated with the normalization condi­tion 2.7. The appropriate solution of the secular equation associated with this system is evidently the energy of the state under consideration.

In practice it is convenient to use an iteration procedure to calculate the radial functions and mixing coefficients. Starting, for example, with values of the mixing coefficients given by perturbation theory, one solves the cor­responding system of generalized Hartree-Fock equations. With the wave­functions so obtained, one can evaluate the matrix elements in Equations 2.9, whose solution yields an improved set of mixing coefficients, and so on.

The first calculations of this general character were made by Hartree, Hartree & Swirles ( 15) in 1939. Among modern many-configuration Hartree­Fock calculations, those of Charlotte Froese (13, 14, 16-19) are especially relevant to problems of astrophysical interest. See also (20-24).

Perturbation-theoretic calculations with a Hartree-Fock basis.-Given a Hartree-Fock function P and the corresponding energy parameter E, we IIlay define a central field S(r) through the equation

[_�.:!....+(t+l)_Z-S(r)_.Jp=o 2.10. 2 dr2 2r' r

4 For very highly ionized atoms it is essential to add correction terms for rela­tivistic contributions to the Hamiltonian; these are discussed in (29).

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ATOMIC TRANSITION PROBABILITIES 457 To S(r) there corresponds a complete set of radial functions. Using complete sets of this kind one can apply the techniques of perturbation theory to evaluate energies, transition probabilities, and other atomic parameters. Tn such calculations the number of mixed configurations is essentially infinite, but only the state vector for the principal configuration satisfies the variation principle. This method has been employed with conspicuous success by Kelly [(25) and references cited therein}, who used the Brueckner-Goldstone per­turbation formalism to simplify and rationalize the calculations.

Two-electron atoms.-Following the lead of Hylleraas, many writers have

carried out variational calculations for two-electron systems using wave­functions in which the radial and angular coordinates are not separated as in Equation 2.5. Since an S wavefunction is invariant under spatial rotations, it must be a function of the variables rl, r2, and r12, the only invariants that can be constructed from the coordinates of the two electrons. The explicit occurrence of rl2 in wavefunctions of the Hylleraas type prevents separation of the radial and angular variables. Similar symmetry considerations enable one to construct nonseparable two-electron wavefunctions for L> O. For example, Breit showed in 1930 that P wavefunctions must have the form

2.11 .

The use of such nonseparable wavefunctions has not proved practicable for a wide class of atomic states. However, the extremely accurate calcula­tions for two-electron systems that have been carried out by Hylleraas's method and its extensions (6, 26) afford a valuable standard for comparisons with less accurate but more widely applicable methods.

Transition amplitudes in the many-configuration approximation.-The matrix element (k'i Xl k) connecting two mixed configurations of the form 2.7 is a linear combination of matrix elements connecting pure configurations:

(k' Ixl k) = t t C.,C.(K' Ixl K) 2.12. «:'=1 K_l

It often happens that some, or even most, of the matrix elements in this sum vanish by virtue of the selection rules for matrix elements of one-electron vector operators. For example, consider the transition (1S)2 IS-t(1s2p) Ip in the many-configuration approximation indicated schematically by

k = (1S)2 + (ES)' + p' + d', k' = 1s2p + pd + dJ 2.13. Six of the 12 matrix elements in the sum 2.12 vanish and four of the remain­ing six probably do not contribute significantly to the sum. Often one needs to retain only the matrix element connecting the two principal configurations [(1S)2 and ls2p in the above example]. This does not mean, however, that configuration mixing is unimportant in such cases. It still affects the radial functions that figure in the transition matrix element connecting the princi­pal configurations; and as will be discussed in Section 5, radial transition in­tegrals are sometimes extremely sensitive to small changes in the radial functions.

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458 LAYZER & GARSTANG Practical considerations.-The difference between the exact nonrelativis­

tic energy of a many-electron system and the energy calculated in the Har­tree-Fock approximation is called the correlation energy. The effectiveness of a given many-configuration approximation may be measured by the fraction of the correlation energy that it picks up. Generally speaking, any attempt to include some effects of configuration mixing entails a considerable expenditure of time and effort. Experience has shown, however, that the re­sUlting improvement in the predicted energy is not always commensurate with the effort needed to obtain it. The reason is that the part of the state vector that arises from configuration mixing, though it may be small, always has a very complicated structure. To represent it accurately, one needs to use either a moderate number of variational components or a very large number of analytic components of predetermined form. Consequently, ac­curate calculations of the kind we have been discussing are necessarily quite elaborate, and have been carried out for only a small number of compara­tively simple many-electron systems. Fortunately, there exists a compara­tively simple approximation method-actually, a collection of methods em­bodying a common principle-that applies to a wide variety of problems of astrophysical interest and yields results that are often considerably more accurate than those that can be obtained with the one-configuration approxi­mation. This method is described in Section 3.

Thefixed-core approximation.-Both the one-configuration and the many-, configuration Hartree-Fock approximations are designed for the evaluation of energy levels and other atomic parameters represented by diagonal matrix elements. The following approach is tailored to the evaluation of off-diagonal matrix elements cif one-electron operators.

Instead of seeking to construct an accurate N-electron state vector, we begin by constructing state vectors I r 1 )N-l for the (N -1) -electron core (the subsystem of passive electrons) . The exact (N -I)-electron state vectors satisfy the eigenvalue equations

2.14.

Let I 'Yh denote one member of a complete set of one-electron state vectors. An energy eigenvector for the N-electron system can then be written in the form

I r) = a L I r1)N-11 ")I(r!'Y I r) 2.15. r,'Y

where [ N-l ] a = c 1 - t; FiN 2.16.

Here PiN denotes the operator that interchanges the electron indices i and N, and C is a normalization constant.

In practice, of course, one can use only a finite number of terms in the expansion 2 .16, and one must approximate the state vectors Ir1)N-I. The

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ATOMIC TRANSITION PROBABILITIES 459 distinguishing features of the fixed-core approximation are that (a) the state vectors I r1)N-l are approximate eigenvectors of the (N - i) -electron Hamil­toni an; and (b) the same (N - I) -electron state vectors are used to construct the N-electron state vectors for the initial and final states of the transition.

The fixed-core approximation is used routinely in scattering calculations. It has also been used as a computational device to simplify transition-prob­ability calculations. The simplest form of the fixed-core approximation uses a fixed central field to define the initial and final wavefunctions of the active electron. Approximations of this type have been used in transition-probabil­

ity calculation since the twenties. However, the possibility of basing more accurate variational calculations of f values on the fixed-core approximation does not seem to have been explored.

Consider the application of the fixed-core approximation to an alkali-like atom, made up of closed shells plus a valence electron. For such atoms the independent-orbital approximation is known to give accurate results. We first solve the Hartree-Fock problem for the ( N -I)-electron core. Using the core wavefunctions so obtained, we then solve the Hartree-Fock equation for the valence electron. The resulting approximate description is not self-consis� tent in Hartree's sense of the term; the valence electron moves in the field

of the core electrons, but the core electrons are oblivious to the valence elec­tron. Thus the fixed-core approximation must give a poorer value for the total energy of the atom than the ordinary Hartree-Fock approximation. However, the fixed-core approximation has certain compensating features. (a) Once the core wavefunctions have been calculated, one can obtain the wavefunction for the valence electron in any of its possible states by solving a single Hartree-Fock equation instead of a system of N equations. (b) The overlap factors in formulae 1.17 are exactly unity in the fixed-core approxi­mation. (c) In the fixed-core approximation, the energy parameter E of the active electron represents the negative binding energy of that electron. In the conventional Hartree-Fock approximation, E represents the difference between the energy of the N-electron atom and the (N -I)-electron core when all N electrons are present; to calculate the binding energy of an elec­tron in the ordinary Hartree-Fock approximation, one must calculate the energies of the N-electron and (N -I)-electron systems separately and take their difference. Thus in the ordinary Hartree-Fock approximation the bind­

ing energy of a single electron is represented by a small difference between two large quantities.

Next let us drop the assumption that the core is made up of closed shells, but retain for the moment the one-configuration approximation. In general two or more core states r1 belonging to the same core configuration will occur in the expansion 2 . 15 . Moreover, the active electron may belong to the same subshell as one of the core electrons; but if that is the case we shall not as­sume that the active electron has the same radial wavefunction as its fellows. Thus in describing the ground state of the helium atom in the fixed-core ap­proximation, we would use a hydrogenic wavefunction to describe the core

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460 LAYZER & GARSTANG

is electron and a variational wavefunction characterized by a considerably smaller effective charge to describe the active 1s electron. This additional degree of flexibility in the fixed-core approximation may well compensate, or more than compensate, for the approximation's neglect of screening by the active electron.

The energy of the N-electron atom is given by

E = L 1 (r11 r> IZE(r1) + • 2.17. r,

The quantity I <rr/ r>lz is the probability of finding the core in the eigenstate rl when the whole atom is known to be in the state r. The energy parameter € therefore admits the following interpretation . When the active electron is removed from an atom in the state r, the resulting ion may be in any of the states rl. The ionization energy associated with the state r1isE(rl) -E. The energy parameter € is the expectation value of the negative binding energy of the active electron ; that is, it is an average of the possible negative binding energies weighted according to the probabilities of occurrence of the various ionic states ri.

Finally, in the most general case one obtains the following formula for the total energy:

E = L 1 (ril r> 12E(rI) + L 1 (1' 1 r) 12,(1') 2.18. r, �

Here the ionic states r1 need not belong to the same configuration. The quan­tity 1< 'Y 1 r) 12 represents the probability of finding the active electron in the one-electron state 'Y. If, as is often the case, the second sum on the right side of Equation 2.18 can be approximated by the value of e corresponding to the nominal one-electron state of the active electron, it remains true that -IE is approximately equal to the expectation value of the energy required to re­move the active electron. The probabilities I (rIlr>12 can usually be calcu­lated. Hence if the appropriate ionization energies are known from experi­ment, an approximate value for € can be found. An important application of this result will be discussed in Section 5, where it will be shown that for cer­tain types of transitions the radial transition integral I depends largely o n the values of E associated with the initial and final states of the active electron.

3. Z-EXPANSION ApPROXIMATIONS The physical idea underlying the one-configuration approximation is that,

in a first approximation , each electron in a many-electron atom moves in a central field. This field is the same for all electrons in the same subshell but is different for different subshells. The physical idea underlying Z-expansion approximations is that, in a first approximation , each electron in a many­electron atom moves in a screened Coulomb field. This implies that the energy is given, in a first approximation, by the formula

E = _ t. (Z - S;)2 ;-1 2n,2

3.1.

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ATOMIC TRANSITION PROBABILITIES 461

where the ni are integers and the Si are either constants or, more generally, functions of Z that asymptotically approach constants in the limit Z---7 00. The most important aspect of this formula is that the coefficient of Z· de­pends only on the set of principal quantum numbers n. It follows that if one wishes to calculate energy levels in an approximation that is accurate up to terms of order Z, one must allow for the effects of mixing between con­figurations of like parity belonging to the same set of principal quantum numbers. Thus to calculate the energy levels of first-row atoms correctly to first order in Z-l, one must allow for the mixing between configurations of the form (1s)2(2s)2(2p)m and (1S)2(2p)mH_as was first done by Hartree, Hartree & Swirles in their classic paper (15) on configuration mixing. The aggregate of states characterized by a given set of principal quantum numbers and a given parity is sometimes called a complex.

I t is an empirical fact, exploited by experimental spectroscopists since 1924 (27), that energy levels in isoelectronic sequences are usually well rep­resented by a formula of the type 3.1, which can also be written as

N 1 Eo=-L: -i�l 2ni' 3.2.

where (T is a slowly varying function of Z. This representation is especially successful for moderately and highly ionized atoms and has been used with conspicuous success by Edlen (28) in his studies of such atoms.

The empirical success of Equation 3.2 in representing energy levels in isoelectronic sequences indicates that the first three or four terms of the per­turbation expansion of E in powers of Z-l afford an accurate approximation to the nonrelativistic energy levels of moderately and highly ionized atoms­and in some instances to the energy levels of neutral atoms as wei\.5 This fact can be exploited in various ways. Conventional calculations often neglect interactions that contribute to the leading terms in the Z expansion of an energy level or other physical quantity while retaining contributions of higher order in Z-I. Such calculations can often be simplified without appreciable loss of accuracy or made more accurate without appreciable increase of complexity.

Atomic parameters other than the energy have useful Z expansions. How­ever, as we shall see presently, the application of this approach to the evalua­tion of electric-dipole transition probabilities is not quite straightforward.

First-order calculations.-In the Z-expansion scheme, a first-order calcu­lation is one that correctly predicts the two leading terms in the expansion 3.2 of E and the leading term in the Z expansion of any other quantity. A first-order calculation must allow for the mixing of configurations belonging to the same complex, and the radial functions that figure in a first-order cal­culation must differ from hydrogenic functions by quantities of relative order Z-I. These conditions are both necessary and sufficient (32).

6 Formula 3.2 has even been used with success to represent energy levels (elec­tron affinities) of �egative ions (30, 31).

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462 LAYZER & GARSTANG

The simplest first-order calculations are based on pure hydrogenic func­tions. These suffice to give the coefficient El in Equation 3.2 and the constant (Z-independent) part of the mixing coefficients connecting configurations in the same complex. Values of E1 and of the mixing coefficients for atoms in the first row of the periodic table were given in (32); Godfredsen (33) has ex­tended these calculations to atoms with up to six electrons or six holes in the M shell.

The simple hydrogenic scheme with constant mixing coefficients affords a convenient starting point for first-order many-configuration Hartree-Fock calculations. Charlotte Froese (13, 16, 17, 19) has carried out a number of such calculations and has studied the effects on electric-dipolc transition probabilities of mixing between configurations in the same complex. Table I , taken from her work (16) illustrates these effects.

There is an important practical and theoretical distinction between Har­tree-Fock calculations that allow only for configuration mixing within a given complex and those that allow for other kinds of configuration mixing. In the former, the radial functions associated with the subordinate configura­tions approach hydrogenic functions as Z--+ 00 ; this fact greatly simplifies the calculations. If the subordinate configurations do not belong to the same complex as the principal configuration, the radial functions associated with them do not have any prescribed asymptotic form. For the sake of brevity we shaH refer to the first-order, many-configuration Hartree-Fock approxi­mation that allows for configuration mixing within a given complex as the extended Hartree-Fock approximation.

The extended Hartree-Fock approximation to the energy, E&3CfJ, has a Z expansion of the form 3.2, in which the two leading coefficients are exact. Now, the coefficient E2 is given exactly by second-order perturbation theory in a representation based on hydrogenic radial functions:

3.3.

where the state vector 10) is the zero-order state vector (in general a linear combination of one-configuration state vectors) and the virtual intermediate states k may conveniently be taken to be pure-configuration eigenstates of Ho. It can be shown (34) that in the extended Hartree-Fock approximation, E2 is given by the same formula with the summation restricted to configura­tions that differ from those labeling the state. 0 through a single principal quantum number. For example, let 10) = a I (ls)2(2s)2 IS)+b I (ls) 2(2P)2 IS). Then the virtual intermediate configurations that contribute to E2&3CfJ are of the type (1m) (2S)2, (ls)2(2sEs), (ls)2(2pep), and (lses) (2p) 2 . In the ordinary Hartree-Fock approximation (in which, incidentally, E1 is not in general given correctly) the permitted intermediate configurations are of the form (lsEs) (2S)2 and (ls)2(2SES).

To calculate E2 in the Hartree-Fock .or extended Hartree-Fock approxi­mation, one need only solve the appropriate variational equations to first

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ATOMIC TRANSITION PROBABILITIES 463 TABLE I: Absolute multiplet strength S(a->a') for several

transitions in Fe xv and Fe XVI (16)

TRANSITIONS IN FE XVI

a'

Without configuration With configuration a interaction interaction

3d 2D 3s 2S 3d 2D 3s 2S

3p 2PO ................... 1.4628 0.9039 1.4628 0.9039

TRANSITIONS IN FE xv

a'

Without configuration With configuration a interaction interaction

3p3d .po 3s3p 'po 3p3d 'po 3s3p 'po

3s3d aD .................. 1.3988 2.3017 1.6326 2 .0678

, a

Without configuration With configuration a interaction interaction

3p3d lpO 3s3p lpO 3p3d lpO 3s3p lpO

3d2 'D .................. 0.5374 0.0 0.6590 0.0006 3s3d 'D .................. .4702 0.7677 .6889 1.4384

3p2I D .................. .0256 1.5636 .0028 0.5748 3d2 IS .................. .3148 0.0 .3752 0.0007 3p2 IS .................. 0.5134 0.3137 .6050 0.3409 3s2 IS ........ . . ... . .... 0 0 .9372 0.0002 0.7573

order in Z-l. Thus if H denotes the radial Hamilton and P one of its eigen­functions, we may write

P = Po + Z-IPI + ... 3.4.

where Po is an eigenfunction of the hydrogenic Hamiltonian Ho. The func­tion Pl satisfies the first-order perturbation equation

3.5.

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464

where

LAYZER & GARSTANG

EO = - - , 2n2

f' = (PoH,po) 3.6.

Equation 3.5 is an ordinary inhomogeneous differential equation. Charlotte Froese (35,13) has programed its numerical solution (for the Hartree-Fock case) and carried out extensive calculations with the resulting wavefunctions. Cohen (36) and Constantinides (37) have independently derived analytic solutions for 1s, 2s, and 2p orbitals of first-row atoms.

Having obtained P" either numerically or analytically, one can calculate both E2 and E3 from the well-known formulae

f2 = (PoH,P,), fa = (P,(H, - E,)P,) 3.7.

The coefficient of Z- k+2 in the expansion of the total energy E is given by N L fik = (k + l)Ek 3.8.

where Eik is the coefficient of z- kH in the Z expansion of the radial energy parameter Ei for the ith electron. Equation 3.8 follows at once from the well­known formula

N L fi = (if'Hoif') + 2Z-'(if'H,if')

= (if'Hif') + Z-l(if'H,if')

and an application of the Hellman-Feynman theorem :

aE aH - = (if'-if') = (if'H,if') aZ-1 aZ-1

From Equations 3.9 and 3.10 we obtain N aE Lfi = E+Z-'­i�' aZ-1

which is equivalent to formula 3.8.

3.9.

3.10.

3.11.

One can also calculate transition matrix elements to the same approxima­tion:

3.12.

I t is clear, however, that the first-order wavefunctions contain more infor­mation than is actually needed to evaluate the first-order corrections to the transition matrix element. Thus in the perturbation expansion

if', = L (0 I HII k)if'k

k Eo - Ek 3.13.

the only terms that contribute to the transition element ('Jt'ol XjLl'Jt,) are those for which the state k satisfies the selection rules for electric-dipole tran­sitions. These, of course, are much more selective than those that define the

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ATOMIC TRANSITION PROBABILITIES 465 states k that occur in Equation 3 . 13. The same thing is true of the matrix elements of any one-electron operator: the evaluation of one-electron matrix elements to first order requires only partial information about the first-order wavefunctions.

This fact is made explicit by the following result, sometimes called the in­terchange theorem;6

Let U be any real-symmetric N-electron operator. Then the first-order contribution to the matrix element ('¥' U'¥) connecting two eigenstates of the Hamiltonian H satisfies the identity

where

(Ho - Eo)if>'1 + U>Jt'o - >Jto(>JtoU>Jt'o) = 0t (Ho - Eo)if>1 + U>Jto - >Jto'(>Jt'oU>Jto) = 05

3.14.

3.15.

When U is a one-electron operator, the inhomogeneous differential Equa­tions 3.15 may have simple analytic solutions, from which, with the help of formula 3.14, the first-order contributions to the matrix elements of U can be calculated, either analytically or numerically. Cohen & Dalgarno (39, 40) used this method to make first-order Hartree-Fock and extended Hartree­Fock calculations of the probability amplitudes of 2s-2p transitions in atoms belonging to the first row of the periodic table.

First-order correlation effects.-In the perturbation scheme that proceeds from the independent-orbital approximation, it is useful to distinguish among effects that depend on the virtual excitation of one, two, etc., electrons. The Hartree-Fock approximation allows for all effects associated with single virtual excitations. Correlation effects are associated with double and higher­order virtual excitations. In the Z-expansion scheme a somewhat different classification is more natural. Because the electrons in open shells do not in general occupy definite orbitals in the zeroth approximation, we cannot in general speak of singly excited virtual states. We can, however, distinguish between states that differ from the unperturbed state in one, two, etc., principal quantum numbers, since the unperturbed state does belong to a definite set of principal quantum numbers. On this basis, we write the first­order state vector \[I 1) i n the form \[1'1) +\[1" 1), where the states k that occur i n the expansion 3.13 differ from the unperturbed state through a single princi­pal quantum number, while those that occur in the expansion of '¥" 1) differ from the unperturbed state through two principal quantum numbers. Similarly we may write E2 as a sum E'2+E"2' Clearly \[1'1 includes '¥loJClf

6 This theorem seems to have been first stated by Dalgarno & Stewart (38) for the special case of diagonal matrix elements. Particular cases of the nondiagonal formula for specific Hartree-Fock operators H were subsequently given by Cohen & Dalgarno (39). The theorem was stated and proved in the above form in (34, 40-42). Dalgarno and his colleagues have made the most extensive use of the interchan'ge theorem.

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466 LAYZER & GARSTANG

TABLE I I : EiJCfJ, E28Xif, E2', and E2 for some two-, three-, and four-electron states (34, 34a).

State - EiJCfJ -E28Xif -E2' -E, Refs.

ls' lS 0 . 1 1 10 0 . 1577 1 1 1 1s2s IS 0 . 1036 0 . 1072 (34a) 0 . 1 145 1 1 1 1s2s 3S 0 . 0453 0 . 0469 (34a) 0 . 0474 1 1 1 1s2p Ip 0 . 1466 0 . 1503 0 . 1570 1 12 1s2p 3p 0 . 0679 0 . 0698 0 . 0730 1 1 2 1a22s 2S 0 . 3545 0 . 3595 0 . 4081 34a 1s'2s' 1S 0 . 8055 0 . 825 1

and E'2 includes ElJCfJ. In general, however, there exist states k' that occur in the expansions of '1" 1 and E', but not in those of 'I'18XfJ and E,8Xif-for example, the states (2ptp) IS when the unperturbed state is (1s2s) IS. . Table I I , taken from (34) , compares the values of E 2, E 2XfJ, E 2SXif.

and E' 2 for a few simple states. I t is noteworthy that in some instances E' 2 contains a large part of the correlation energy. I nspection of the table sug­gests that this is true of two-electron atoms when the electrons belong to different shells but not when they belong to the same shell. It can be shown (34) that E", for an N-electron atom can be expressed as a linear combina­tion of the values of E" 2 for its two-electron subsystems. The results shown in Table II suggest that the contributions from electron pairs with different principal quantum numbers may in general prove to be comparatively insignifican t.

It is easy to write down the form of a variational state vector that coin­cides with \fI'1 up to terms of order Z-l and belongs to a variational energy whose Z expansion contains the term E' 2. For the state (ls2s) IS, for example, we can write (omitting the spin kets)

'¥(1s2s 'S» = 2-lI'[Ul(r,)u,(r2) + u2(rl)u,(r,)] , . (00)00)"

+ 2-;-I12Z-I[v,Crt)w(r,) + w(rl)v,(r,)] I (1 1)00)12 3.16.

where Ul and u, go over asymptotically into Is and 2s hydrogenic functions and v , goes over into a hydrogenic 2p function ; the function w has no pre­scribed asymptotic form. Application of the variation principle leads in the usual way to a system of Hartree-Fock equations for the radial functions. To carry through a first-order calculation with a state vector of the form 3. 16, one expands Ul and u, to first order in Z-t, drops the terms of order Z-' in the products UIU" and replaces u, by its asymptotic form (since the second term is already of order Z-;-l) . The first-order perturbation equations then lead, in the usual way, to a system of coupled, inhomogeneous, differ­ential equations for the function w and the first-order parts of the functions Ui and u,. If one wishes to calculate matrix elements of operators other than the energy, one can use the interchange theorem exactly as before.

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ATOMIC TRANSITION PROBABILITIES 467 Second-order calculations.-As we have mentioned, the evaluation of

E" 2 for an arbitrary N-electron state can be reduced to calculations of E"2 for two-electron states. Such calculations have now been carried out, by variational methods, for a fair number of two-electron states ; see (6, 43).

We have seen that it is not necessary to have full information about the first-order parts of \[I and \[I' in order to calculate the first-order contributions to the matrix element (\[I' U\[I) of a one-electron operator U. Only states k that differ from the state \[I' through the quantum numbers of a single orbital need be kept in the perturbation expansion 3.13 of \[I I. Consider, for example, the two-electron transition element « ls)2 ls i XII I ( ls2p)lp). To first order in Z-l, we need allow only for mixing between (ls)2 and configurations of the type (1ses) and (2pep) and between ( ls2p) and configurations of the type (1sep) and (2sep) . This can be done by one of the first-order methods described above. Calculations of this kind have been carried out by Dalgarno & Parkinson (44), who calculated first-order transition probabilities for the sequence ( ls)2-+(1snp) .

Although they present no obvious difficulties, full Hartree-Fock calcu­lations of transition probabilities in the approximation under discussion have not yet been reported. Comparisons of such calculations with ordinary or extended Hartree-Fock calculations (as well as with more accurate varia­tional calculations, where available) would give useful information about the effects of correlation on electric-dipole transition amplitudes.

Z dependence of transition matrix elements and related quantities.-From the definition of the radial integral I it follows that

I ", Z-l 3.17 .

That is , the Z expansion of I begins with a term of order Z-l. Recalling that

w = I E' _ E I '" Z2 if (n) ,c. (n')

3. 18. Z if (n) = (n')

We see from the definition 1 . 1 that

constant j ""'" Z-I

if (n) ,c. (n')

if (n) = (n') 3.19.

This equation shows that the relative importance of transitions in which the principal quantum number of the active electron does not change becomes smaller and smaller with increasing Z along an isoelectronic sequence.

From Equations 1 .4, 1 .9-1 . 1 1 , and 1 . 1 7 (or from exact relations given in the next section) we obtain

if (n) � (n') z J ", constant if (n) = (n')

za if (n) ,c. (n') K ,....., Z if (n) = (n')

3.20.

3.21.

Now, the leading term in the Z expansion of I is simply the hydrogenic value of the transition integral ; thus for. sufficiently large values of Z, I is

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468 LAYZER & GARSTANG

accurately known. The same is true of the integrals .T and K for transitions in which the principal quantum number of the active electron changes. For transitions in which the principal quantum number of the active electron does not change, however, Equations 3.20 and 3.21 show that the hydro­genic values of J and K vanish ; moreover, the next term in the Z expansion of K also vanishes identically. (These results can, of course, be verified directly.) In other words, for transitions in which the principal quantum number of the active electron does not change, the leading term in the Z expansion of J depends on the first-order parts of the initial and final wave­functions, while the leading term of K depends on both the first- and second­order parts of these wavefunctions. Whether or not explicitly Z-dependent wavefunctions are used, it is clear that the velocity and acceleration forms of the radial transition integral can be expected to give grossly inaccurate results for transitions in which the principal quantum number does not change.

If R denotes an integral that depends on one or more radial wavefunc­tions, we may define a screening parameter u associated with this integral and with the wavefunctions used to evaluate it by the equation

R = RH(Z - 0') 3.22.

The right side of this equation is the integral that results when all the radial functions are replaced by screened-hydrogenic functions. For example, Hartree (12) associated a screening parameter u with the mean radius (r ) by the equation

(r) == (PrP) = (r)H(Z - 0')-1 3.23.

where (r)H denotes the hydrogenic value of (r). Similarly, we can write

l(nt, nil') = JH(Z - 0')-1 3.24.

The transition screening parameter u(nt, n't') has a formal Z expansion of the type

0' = 0'0 + 0'1Z-1 + . . . 3.25.

A knowledge of the leading term in this expansion is evidently equivalent to a knowledge of the first-order term II in the Z expansion of J. Dalgarno & Stewart (45) have suggested, however, that formula 3.24 with u approxi­mated by Uo is more accurate than the first-order formula I ""'Io+Z-II1•

There is reason to believe that for certain classes of transitions the formal Z expansions of u and I are asymptotic rather than convergent, and give useful results only for very large values of Z. This is true, for example, of transitions for which I changes sign with increasing Z along an isoe1ectronic sequence. The integral itself is then well behaved for all values of Z, but u has a singular part of the form (Z - ZO)-1 with Zo > N.

To test the convergence of the expansion 3.25, Charlotte Froese has calculated the two leading coefficients in the Z expansion of I in the Hartree­Fock approximation, using the numerical method described earlier in this

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ATOMIC TRANSITION PROBABILITIES 469 TABLE I I I : Convergence of the Z expansion of uXfJ(nt, n't') : uoXfJ• U1XfJ, uoXfJ +Z-IU1XfJ, and uXfJ for three transitions in the Na isoelectronic sequence ; see also

Figure 5 .

... 6, Na I Mg JI Al III Fe XVI --- --- ---

3s-3p 6 . 9289 2 . 495 60+Z-10'1 9 . 20 9 . 0 1 8 . 85 7 . 03

0' 8. 18 7 . 78 7 . 5 8 7 . 09 ------ --- --- --- ---

3p-4d 2 . 2600 -24. 2942 O'O+Z-1ql 0 . 05 0 . 24 0 . 39 1 . 33

0' 1 5 . 53 24.98 70.25 0 . 09 --- --- --- ----

3d-4p 24. 2876 -198. 940 O"o+Z-lO] 6 . 20 7 . 7 1 8 . 99 1 6 . 64

0' 1 0 . 86 1 1 . 68 1 2 . 43 1 7 . 65

Section. The results are shown in Table III. For transitions in which thc principal quantum number does not change, the expansions converge quite well. In all the other cases tested, the expansions appear to diverge for small values of Z.

We conclude that the Z-expansion method often cannot be applied directly to the evaluation of radial transition integrals. In Sections 5 and 6 it will be shown, however, that these integrals are sometimes determined by one-electron parameters that do have useful Z expansions.

4. RADIAL TRANSITION INTEGRALS : GENERAL PROPERTIES The preceding sections dealt with the problem of constructing an appro­

priate representation in which to calculate transition matrix elements. Having chosen a representation, one can express the transition matrix element as a linear combination of radial transition integrals. In this and the two following sections we discuss the evaluation of these integrals.

Length, velocity, and acceleration forms of the transition integral.-Of the three equivalent forms of the radial transition integral (see Section 1) , the acceleration form is clearly the least prepossessing. Because of the factor r-2, it is sensitive to the forms of the initial and final radial wavefunctions in a range of r where they are comparatively poorly determined. Numerical calculations by many authors have amply confirmed this judgment.

I t is more difficult to choose between the length and velocity forms. At first sight it might appear that the velocity integral ] ought to be the more accurate because it gives most weight to roughly the same range of r as the Coulomb integrals, whereas the length integral I gives most weight to a range that is comparatively unimportant in the evaluation of energy inte­grals. This argument is valid if, for example, one uses variational radial functions of prescribed analytic form. It does not apply, however, to numeri­cal Hartree-Fock wavefunctions, or, more generally, to radial functions whose asymptotic form is properly determined. The shape of a radial func­tion at large values of r depends only on the value of the energy parameter in the radial wave equation ; in a variational calculation the energy parameter,

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4 7 0 LAYZER & GARSTANG

and hence the asymptotic shape of the radial wavefunction, is determined to second order. The amplitude of the wavefunction is determined by the normalization condition and is accordingly known to first order. Thus the dipole-length form is not inherently inferior to the dipole-velocity form if variational wavefunctions with the correct asymptotic form are used in the evaluation of the radial transition integral.

Dalgarno & Lewis (46) discussed the relative merits of the length and velocity forms of the transition matrix element from another point of view. From Equations 1 .4 and 2.7 we obtain

(k' i xI'I k) = L: CwCk« K' 1 xI' I K) = (iVk'k)-1 L: Ck,.,CkK(K' 1 xI' I K) � ��

= (illk'k)-2 L: Ck,.,Ck.(K' I xI' I K) 4. 1. 1(.,,'

where the summations extend over all the one-configuration states that occur in the expansions of the exact state vectors I k) and I k' ) . Each of the sums in Equations 4.1 consists of a main term, in which the coefficient of the matrix element is close to unity, and a remainder term. To compare the remainder terms, Dalgarno & Lewis used the approximate relations

4.2.

where II.'. is the energy difference between the states K' and K. These relations are approximate because the one-configuration state vectors I K) and I K' ) are not eigenvectors of the same Hamiltonian. From Equations 4. 1 and 4.2 it is clear that the remainder terms i n the formulae for the velocity and acceleration matrix elements are likely to be small when the transition energy 1Ik'k is much greater than II.'. for all pairs of states K', K that make substantial contributions to the length form of the matrix element.

The conclusions that can be drawn from this argument clearly depend on the nature of the radial functions associated with the two principal configura­tions as well as on the energy differences between the various states. For example, if Hartree-Fock wavefunctions are employed, singly excited con­figurations K and K' are automatically excluded from the sums in Equation 4.1 . The argument does suggest, however, that it is unwise to use either the velocity or the acceleration form of the transition matrix element when the transition energy is especially small. This conclusion is consistent with that reached in Section 3 through considerations of the Z dependence of the transition matrix elements and energy differences. We saw there that for transitions with An = 0 the velocity form of the transition matrix element is likely to be less accurate than the length form, and the acceleration form still less accurate ; and we noted that the transition energy increases linearly with Z along an isoelectronic sequence if An �O, but remains bounded if Lln = O.

It is possible to derive exact relations connecting the transition integrals I, J, and K, provided that the energy parameters I: and 1:' associated with the initial and final wavefunctions of P and P' are known. Given the radial wavefunction P and the associated energy 1:, we may define a screening func�

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ATOMIC TRANSITION PROBABILITIES 4 7 1 tion S(r) through the radial wave equation

[_ 2. � + e(e + 1) _ Z - S(r) _ eJ Per) = 0 2 dr' 2r' r 4.3.

If P is a Hartree-Fock function, the function S(r) may have singularities, but the function S(r)P(r)/r is regular everywhere. The function P' and the associated energy parameter E' define a function S'(r) , which in general differs from S(r). Denoting the Hamiltonian operator that appears in Equation 4.3 by H and that which appears in the analogous equation for P' by H', one can derive mixed commutation relations

and

rH - H'r == d/I' + S - S' 4.4.

4.5.

From Equations 4.4 and 4 . 5 we obtain

where

J = (e - E')1 + (P'(S' - S)P) K = (. - .')J + (P'(S' - S)d,Il'P)

K = _ (P, � [Z - S'(r) J p) dt" r

4.6.

4.7.

4.8.

Note the occurrence in this definition of the screened nuclear charge Z - S'(r) , and the difference between Equation 4.8 and the definition 1 . 16 of K.

If the independent-orbital approximation were exact, it would follow from Equations 1 .4 and 1 .5 that

J = (E - E')1 4.9.

and

K = (E - E')J 4.10. where E and E' denote the total energies of the initial and final states. In fact, Equations 4.9 and 4. 10 are only approximate. The exact Equation 4.6 is not inconsistent with the approximate Equation 4.9, however. Subtracting the former from the latter, we obtain

[(E - .) - (E' - .')]1 = (P' (S' - S)P) 4.11.

The difference (E-E) represents the energy of the passive electrons in the initial state; thus the quantity in square brackets on the left side of Equa­tion 4.11 represents the change in the energy of the passive electrons between the initial and final states; This change is finite because the screening of the passive electrons by the active electron is not the same in the two states. On the right side of Equation 4. 1 1 , (S' - S) represents the change in the screen­ing of the active electron by the passive electrons between the initial and

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4 7 2 LAYZER & GARSTANG

final states. These physical interpretations indicate that the two sides of Equation 4.11 at least have the same sign. Moreover, it is clear on physical grounds that as S-4S', E-4E' and �-4l.

By contrast, Equation 4.7 fails to reduce to Equation 4.10 when S-4S', since K does not then reduce to K. This indicates that the approximate relation 4. 10 may be quite inaccurate. Now, there is no reason why the independent-orbital approximation should not be reasonably accurate for evaluating the matrix element of X and X, as both these quantities are sums of one-electron operators. On the other hand, the operator X is a sum of both one- and two-electron operators in which the two-electron contributions cancel. The independent-orbital approximation is obviously inconsistent with this kind of cancellation ; that is, cancellation does not occur when the forces Fi are approximated by central forces : hence the screening term in the definition of K. We conclude that the approximate relation 4.10 breaks down because the acceleration form of the transition matrix element is basically

inconsistent with independent-orbital approximation.

S. RADIAL TRANSITION INTEGRALS : STABILITY

A symptoticform of P.-It has been recognized since the twenties that the radial transition integral I is sensitive to the asymptotic forms of the radial wavefunctions P, P', which in turn depend largely on the energy parameters �, �'. The radial wave equation has the asymptotic form

The solution that is regular at infinity is

P = A q1l2W • .l+l/2(2qr/v)

where q and II are defined by

q = Z - N + l

The Whittaker function W has the asymptotic expansion

5.1.

5.2.

5.3.

5.4.

This series does not terminate unless v is an integer, which could happen only by accident. To obtain a function that is regular at the origin, we terminate the asymptotic expansion S.4 at the lowest positive power (v- k > O) of x. The resulting radial function, suitably normalized, will be denoted by pc.

Normalization.-The normalization of pc presents a problem. Hartree

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ATOMIC TRANSITION PROBABILITIES 4 7 3 suggested using the hydrogenic normalization factor AH with n replaced b y v :

A H = v-' [ (v - t - l) !/(v + t) !j1l2 5.5.

This prescription, which was used by Bates & Damgaard (47) to construct their tables of Coulomb transition integrals, is accurate for small values of the quantum defect A = n - v, but not for large values (corresponding to highly penetrating orbits) .

Seaton (48) used the so-called quantum defect method, developed by Ham (49) , to derive a formula relating the normalization factor to the spec­trum I Enl } associated with a fixed value of t and with the effective potential S(r) defined by E and P through Equation 2. 10. Although Seaton's method is very elegant, its practical value is limited because information about the spectrum { Enl l is not always available. In some cases an approximate value for E itself can be derived from measured ionization energies (see Section 6) , but the same procedure must give less accurate results for the other eigen­values enl, because while they are defined for a fixed central field, the effective central fields associated with the various orbitals in a real atom vary with n for a fixed value of t. This theoretical difficulty and the paucity of appropriate experimental information greatly limits the scope and reliability of the quantum defect method.

An alternative normalization procedure (50) defines the constant A in Equation 5.2 through the condition

(PC,PC) = (PrP) 5.6. This definition emphasizes the range of r that is most important for the transition integral I-except when the numerical value of I is abnormally small because of cancellation ; then, however, the Coulomb approximation evidently fails anyway. In the following discussion the normalization rule 5.6 will be assumed to be part of the definition of the Coulomb approxima­tion.

Range of validity of the Coulomb approximation (51) .-Except in cases of strong cancellation, the value of the transition integral I(nt, n' t') is deter­mined mainly by the outermost loops of the radial functions Pent) and P(n't') . These will lie well outside the core if n and n' exceed the principal quantum numbers of the core electrons. For such transitions the Coulomb approximation should be at its best, because the outermost loops of P and P' are well represented by the asymptotic formula. Transitions between excited states are normally of this type ; so are transitions to and from the ground state of alkali-like atoms. The Coulomb approximation should also improve, for any kind of transition, with increasing Z along an isoelectronic sequence, because screening by the outer electrons becomes less and less important. These expectations are confirmed by the numerical results shown in Figure

1 and Table IV. For all transitions but one in atoms of the Na sequence, the Hartree-Fock and Coulomb approximations agree to within a few per cent. The exceptional transition, 3s-4p, is one for which considerable cancellation

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4 7 4

4.0

3.0 2.5

2.0

1 .0 .9

LA YZER & GARSTANG

x/

J:+

/�.

/. . 2� -L�' f) B. '" B � c • N ? 0

3. �3. X No + Mg o Al

1.5 2.0 25 3.0 40 5.0

I HF

FIG. 1 . Coulomb versus Hartree-Fock values of the transition integral I for transi­tions with An = 0 (for transitions with An �O, see Table IV). The Coulomb wavefunc­tions belong to the same values of • and (r) as the corresponding Hartree-Fock func­tions. All the transitions labeled 2s-2p (except for the transition ls'2s-1s'2p in the Li sequence) are of the type ls2Zs2zpm_ls2Zszpm+1. The 3s-3p transitions in the Mg and Al sequences are of the types 3s2-3s3p and 3s23p-3s3p' respectively. The value of I for a given transition decreases along the isoelectronic sequence. Note that the largest discrepancies between Ie and JXff occur for Be and Mg, where the active elec­tron is equivalent to one of the passive electrons in the initial state but not in the final state, and that the smallest discrepancies occur for Li and Na, where the active electron is not in the same shell as the passive electrons in either the initial or final state.

occurs. For the transition (1s)2(2s)2_ ( 1S)2(2s2p) in the Be sequence, (ls)2(2s)2(2p)m_(ls)2(2s)2(2p)m-l(3s) in the C, N, 0, and F sequences, and (ls)2(2s)2(2p)6(3s)2_ (is)2(2s)2(2p)6(3s3p) in the Mg sequence, the Coulomb approxi�ation is significantly less accurate, though still quite good, and improves rapidly along the isoelectronic sequence.

I n the Ar transition (3p)6_(3p)54s, the active 3p electron belongs to the same shell as seven other electrons and the 4s orbital is highly penetrating ; these circumstances militate against the Coulomb approximation-which indeed breaks down rather badly, as can be seen from Figure 2. Note that, in marked contrast to previous cases, the approximation improves very slowly with increasing Z along the isoelectronic sequence.

Two questions now arise. How sensitive is Ie to small errors in E and (r ) ? What methods are available for estimating these parameters, and what are their relative merits? The remainder of this section deals with the first ques­tion. The second question will be discussed in the next section, which also discusses approximate methods that go beyond the Coulomb approxima­tion.

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ATOM IC TRANSITION PROBABILITIES 4 7 5

Dependence of I o n (r) and (r').-It follows from Schwarz's inequality that

I" == 1 (PrP') I " .::; (PrP)(P'rP') == (r)(r') 5.7. Let

Q(nt, n't') = lent, n't') 5.8. rnZrn,l'

where rnl = (r) , rn'l' = (r' ). Then I QI � 1. The parameter Q (52) is closely related to Bates's cancellation measure

1 (P'rP) 1 R = ( I P'rP I )

but is somewhat easier to obtain and interpret .

5.9.

. When �1, the transition integral I is largely determined by the param­eters (r), (r' ) . For such transitions any approximation that correctly pre­dicts (r) and (r' ) ought to predict I accurately. The simplest approximation of this kind employs screened-hydrogenic wavefunctions PH(Z - S) , the screening parameter s being determined by the condition

rH (r) = -- 5.10. Z - s

where rH is given by the hydrogenic formula

rH = nO [� _ t(t + 1)J 2 2n2 5.11.

The screened-hydrogenic and Hartree-Fock approximations are compared in Table IV for representative transitions with a wide range of Q values (53) . The screened-hydrogenic approximation appears to be quite accurate for Q�0.7 but not for small values of Q. When �1, the screened-hydrogenic approximation to Q itself is also accurate. The applicability of the screened­hydrogenic approximation can therefore be tested without going outside the framework of the approximation.

Table IV shows that the largest values of Q characterize transitions in which the principal quantum number of the active electron does not change; for all such transitions in the material under discussion �1. We saw in Section 4 that when t.n = 0 the transition integral and the transition screen­ing parameter have useful Z expansions, but not in general when t.n = O. Thus the direct Z-expansion method and the screened-hydrogenic approximation have roughly the same range of validity.

Variation of I with (z/ z') in the screened-hydro genic approximation.­Although the screened-hydro genic approximation cannot provide quantita­tive estimates of the integral I except when Q�l, it gives useful qualitative information about the variation of transition integrals along isoelectronic sequences. In the Coulomb approximation, I is a function of four variables. By considering Q instead of I, we can reduce the number of variables to

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476 LAYZER & GARSTANG

TABLE IV: Transition integrals and reduced transition integrals in the Hartree-Fock, Coulomb, and screened-hydrogenic approximations for atoms in the Na

and Ar isoelectronic seq uences.

ARGON SEQUENCE (3p6) lS_(3p5, 4s) Ipo

IXff' Ie ISH Q1Cil' QC QSH

3p 4s Ar I . 731 . 252 - . 396 . 24 . 08 - . 13 3p 4s K II . 634 . 256 - . 324 . 27 . 1 1 - . 14 3p 4s Ca III . 546 . 262 - . 2 19 . 27 . 13 - . 1 1 3p 4s Sc IV . 474 . 252 - . 142 . 27 . 14 - . 08 3p 4s Ti V . 416 . 237 - .089 . 26 . 15 - . 06 3p 4s V VI . 369 . 224 - . 053 . 26 . 16 - . 04 3p 4s Cr VII . 33 1 . 209 - . 027 . 25 . 16 - . 02 3p 4s Mn VIII . 299 . 196 - .008 . 25 . 16 - .01 3p 4s Fe IX . 273 . 184 .006 . 24 . 16 . 01 3p 4s Co X . 250 . 1 72 . 016 . 24 . 16 . 02 3p 4s Ni XI . 231 . 162 . 024 . 23 . 16 . 02 3p 4s Ga XIV . 187 . 1 39 . 037 . 22 . 17 . 04 3p 4s Kr XIX . 140 . 1 10 .044 . 2 1 . 16 . 07

SODIUM SEQUENCE

[Xff' IC ISH greff' QC QSH

3d 4fNa I 1 0 . 180 10 . 138 10 . 160 . 74 . 74 . 74 3d 4f Mg II 4 . 971 4 . 970 4 . 929 . 73 . 73 . 73 3d 4f Al III 3 . 2 18 3 . 2 15 3 . 164 . 72 . 72 . 7 1

3d 4f Fe XVI . 581 . 578 . 569 . 70 . 70 .69

3p 4d Na I - 1 . 67 1 - 1 .671 - 2 . 123 - . 15 - . 15 - . 19

3p 4d Mg II - . 584 - . 584 - . 822 - . 10 - . 10 - . 14 3p 4d Al I I I - . 131 - . 130 - . 27 1 - . 03 - . 03 - . 06

3p 4d Si IV . 103 . 104 .0 12 . 03 . 03 . 00

3p 4d P V . 227 . 228 . 163 . 09 . 09 . 06

3p 4d S VI . 293 . 293 . 244 . 1 3 . 13 . 1 1

3p 4d Fe XVI . 292 . 292 . 28 1 . 3 2 . 32 . 3 1

3 s 4p Na I - . 376 - . 370 - 1 . 3 15 - . 05 - . 05 - . 17

3s 4p Mg I I . 037 . 026 - . 771 .01 . 01 - . 16

3s 4p Al III . 166 . 155 - .424 . 05 . 04 - . 12

3s 4p Si IV . 2 19 . 216 - . 2 19 . 08 . 08 - . 08

3s 4p P V . 242 . 238 - . 093 . 10 . 10 - . 04

3s 4p S VI . 250 . 246 - . 0 13 . 12 . 12 - .01

3s 4p Fe XVI . 196 . 194 . 140 . 2 1 . 2 1 . 15

4d 4f Na I - 15 . 899 - 15 . 958 - 15 . 914 . 82 . 82 . 82

4d 4f Mg II - 8 . 004 - 8 . 004 - 7 .998 . 83 . 83 . 83

4d 4f Al III - 5 . 375 - 5 . 374 - 5 . 362 . 84 . 84 . 84

4d 4f Fe XVI - 1 .014 - 1 . 0 1 1 - 1 .003 . 85 . 85 . 85

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ATOM IC TRANSITION PROBABILITIES 4 7 7

TABLE I V-(Continued)

IX,'"f 10 ISH rjiCff QO QBH

SODIUM SEQUENCE

4p 4d Na I - 10 . 703 - 10 .642 - 10 . 975 .63 .62 .64 4p 4d Mg I I - 7 . 032 - 7 . 032 - 7 .219 . 79 . 79 . 81 4p 4d Al I I I - 5 . 388 - 5 . 385 - 5 . 480 . 88 .88 .90 4p 4d Si IV - 4.358 - 4 . 354 - 4 . 395 .93 . 93 .94 4p 4d P V - 3 .643 - 3 . 638 - 3 . 650 .96 . 96 .96 4p 4d S VI - 3 . 120 - 3 . 112 - 3 . 112 .98 . 97 .97 4p 4d Fe XVI - 1 . 256 - 1 . 252 - 1 . 243 . 99 . 98 .98

4s 4p Na I - 10.513 - 10 . 553 -9 .678 . 86 . 86 . 79 4s 4p Mg I I - 6 . 602 -6.606 - 6 . 437 .93 .93 .90 4s 4p Al I I I - 4 . 960 - 4 .954 - 4 . 909 .96 . 96 . 95 4s 4p Fe XVI - 1 . 283 - 1 . 283 - 1 . 282 1 . 00 1 . 00 1 . 00

3p 3d Na I - 6 . 052 - 6 . 051 - 5 . 543 . 76 . 76 . 70 3p 3d Mg I I - 3 .698 - 3 .691 -3 .539 . 87 . 87 . 83 3p 3d Al I I I - 2 . 737 - 2 . 726 - 2 . 654 .93 .93 .90 3p 3d Fe XVI - . 60S - .594 - .588 . 94 .93 .92

3s 3p Na I - 4 . 519 - 4 . 526 - 3 . 930 . 90 .90 . 78 3s 3p Mg I I - 3 . 015 - 3 . 019 - 2 . 887 .96 .95 . 9 1 3 s 3 p Al I I I - 2 .346 - 2 . 350 - 2 . 301 .97 .98 . 96 3s 3p Si IV - 1 . 943 - 1 . 937 - 1 .922 .98 .98 .97 3s 3p P V - 1 . 666 - 1 . 664 - 1 . 655 . 99 . 99 .98 3s 3p 5 VI - 1 .463 - 1 . 461 - 1 . 456 . 99 . 99 . 99 3s 3p Fe XVI - .672 - . 671 - .670 1 . 00 . 99 . 99

3d 4p Na I 9 . 418 9 .450 7 . 822 . 78 . 78 . 65 3d 4p Mg I I 4 . 040 4 . 039 3 . 224 .64 . 64 . 5 1 3d 4p Al I I I 2 . 27 1 2 . 268 1 . 735 .53 . 53 . 40 3d 4p Fe XVI . 156 . 156 . 108 . 18 . 18 . 12

3p 4s Na I 4 . 408 4 . 400 1 . 868 . 55 . 55 .23 3p 4s Mg 1 1 2 .087 2 . 074 . 890 .44 .43 . 19 3p 4s Al I I I 1 . 339 1 . 326 . 608 . 38 . 3 7 . 17 3p 4s Fe XVI . 197 . 195 . 136 . 2 1 . 21 . 15

two, namely, E and E' . I n the screened-hydrogenic approximation, however, Q depends on only one parameter, the ratio zlz' = (Z - s)/(Z - s') :

QBH EO (ZZ')112QH(Z - 0')-1 = !(z!z') 5.12.

where (j is the transition screening parameter defined by Equation 3.24 and QII is the hydrogenic value of Q. In Figure 3 QSH is plotted against (z'lz)

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4 7 8 LAYZER & GARSTANG

.4

.3 (3p·) IS - (3p' 45) ' p

.Z HF __________

Q . 1 C � 0 -

- . 1 SH� -.2

3 5 7 9 I I 13 15 17 19

q FIG. 2. The reduced transition integral Q in the Hartree-Fock, Coulomb, and

screened-hydrogenic approximations, for the transition (3p6) IS_(3p54s) lp. in the Ar sequence. The Coulomb wavefunctions are characterized by the same values of E and (r) as the Hartree-Fock wavefunctions, and the screened-hydrogenic functions by the same values of (r). Note the slow convergence of QC and QSH to rjlCIT'.

1 .0

0.8

0.6

SH . Q 0.4

O.Z

O r-�--------��r-��----------------------�

-0.2

.4 .5 .6 .7 .8 .9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7

l 'f l FIG. 3 . The reduced transition integral Q in the screened-hydrogenic approxima­

tion as a function of the ratio (z'/z) between the effective charges characterizing the final and initial wavefunctions of the active electron. For transitions with An = O, QSH is positive for all values of (z'/z) ; for other transitions Q811 may change sign. Note the wide range of Q values and of slopes.

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1.6

1.4

. 6

ATOMIC TRANSITION PROBABILITIES

b o 3d - 4p

� • 3p-4. " 31-4p + 3p-4d

------.-. 0-.-

1I.._G� ............ +

III .....

..---: . .-----A"--f� ' . .

/1lJ� /.,

I � 3 3.-4p 3p-4d 4 5 6 7 8 9

4 7 9

r-3d-4p -3s-4

3p-4

FlG. 4. Variation of the effective-charge ratio (s'lz) along the Na isoelectronic sequence. The effective charges are chosen so that (r)flH = (r)3C5'. Values of (z'lz) for which QBH = O are indicated by horizontal lines. The values of q( = Z - N+ll at which QSH(3s_4p) and QBH(3p_4d) vanish are indicated by downward-pointing arrow­heads ; for comparison, the values of q at which QC(3s-4p) and QC(3p-4d) vanish are indicated by upward-pointing arrowheads.

for some representative transitions. The range of (z'lz) includes all realistic values. The transitions fall into two main groups : those for which Q is posi­tive over the whole range of allowed values of (z'lz) , and those for which it

changes sign somewhere in this range. For a given transition in an isoelectronic sequence, one can relate the

ratio (z'lz) to the value of Z or q( = Z - N+ l) . Examples of such relations are shown in Figure 4, where the zeros of ISH are indicated by horizontal lines. The values of q at which I changes sign can be read off (in this approxi­mation) from these figures.

Variation of Q and I along an isoelectronic sequence in the Coulomb appro xi­mation.-For transitions in the Na sequence, the variation of Ie with q is similar to that of ISH. In particular, the distinction between transitions for which ISH changes sign along an isoelectronic sequence and those for which it does not persists in the Coulomb approximation. This is most easily verified by calculating 1° for the neutral atom in an isoelectronic sequence and noting whether its sign is the same as that of the hydrogenic value to which I tends as q-+ oo .

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480 LAYZER & GARSTANG

- .I

-.2 L.......L...J2'--

3�...l4--'5L,-..,.6�7,-,8!-..,.9-JIO�l.:-1I-l:12,...-:1'="3�14,........1 5::--116·

<t FIG. 5. QC versus q for transitions in the Na isoelectronic sequence.

Of course, the actual value of q at which I changes sign will not in general be given accurately by the screened-hydrogenic approximation. When Q is numerically small, its value depends mainly on E and f/ , not on (r) and (r'). The zeros of 10(3s, 4p) and lC(3p, 4d) for the Na sequence are marked in Figure 4; they differ substantially from those of ISH(3s, 4p) and ISH(3p, 4d) . Figure 5 shows QO versus q for some transitions in the Na sequence. For comparison, plots of QSH versus q for these and some other transitions are shown in Figure 6. Note the qualitative similarities and quantitative differ­ences-the latter are especially marked for small values of Q-between the two figures.

Sensitivity of QC to small changes in f and f' .-To learn more about the sensitivity of calculated transition integrals to small errors in E and fl, we consider the quantities

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ATOMIC TRANSITION PROBABILITIES

.9 3p-3d 4d-4f

.8

.7 3d -4t

.6

.5

.3

.2

. 1

O �------��--��-------------------1

-. I

-.Z L-.���-':--':-"--�-:-8 ---:-9 ---':IO---L.I I---L.12-:'::13,-:L14:-:'::15�16 ct.

FIG. 6. QBH versus q for transitions in the Na isoelectronic sequence. Compare with Figure S.

d in Q 7] = -- ' d Jn . d ln Q' 7]' = d Jn .f

48 1

5.13.

The fractional error in Q (or I) resulting from errors in E and E' is given by

al aQ OE aE' - = - = n - + 7]' ­I Q f E'

5.14.

In Figure 7 the coefficients 7}. 7}' are plotted against q for some transitions in the Na sequence. Especially noteworthy is the behavior of 'fJ and 'fJ' in the neighborhood of the singularities.

The integral I is comparatively insensitive to errors in (r) and (r' ) ; for such errors the coefficients analogous to '11 and '11' are each equal to - to

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482

40

30

20

1 0

- 1 0

- 20

-30

- 40

LAYZER & GARSTANG

3p - 4, A, 3p - 40 No 3p - 35 No 3p - 4d No

I 2 3 4 !! 6 7 8 9 10 " 12 13 14 15 16

q

FIG. 7. The stability indices 71 = a In QC / a In E and 71' = a In QC / a In .' for repre­sentative transitions in the Na and Ar isoelectronic sequences. The 3s-3p integral is highly stable. The 3p-4s integral is moderately stable in Na but much less so in Ar. The 3p-4d integral is highly unstable in a fairly well-defined range of q (2"Sq"S 7) . Note that the scale of J(3p-4d) is one-fifth that of the other integrals.

6. RADIAL WAVEFUNCTION S : EVALUATION OF KEY PARAMETERS

The preceding discussion suggests that for a well-defined class of transi­tions the integral I is largely determined by the values of E and (r) for the initial and final states. For such transitions, we may regard the calculation of Hartree-Fock wavefunctions (in either the ordinary sense or one of the extended senses discussed in Section 2) as a means of estimating these param­eters. In some circumstances more economical means, some of which are reviewed in this section, may yield acceptable results. Included under this heading are approximate methods that were not expressly devised for the estimation of l! and (r) but which yield useful results because they do in fact afford accurate estimates of these parameters.

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ATOMIC TRANSITION PROBABILITIES 483 The energy parameter f.-

(a) The semiempirical method : We saw in Section 2 that - E is approximately equal to the expectation value of the energy needed to remove the active electron. If either the (N - i)-electron core or the N-electron

atom itself is made up entirely of closed shells, -E simply represents the ionization energy. Otherwise, the N-electron state may have more than one parent state in the core, even in the one-configuration approximation, and the experimental ionization energies must be combined to form a weighted average, the weights being given by a theoretical calculation.

The semiempirical method has been applied mainly to transitions for which the core is in a definite state. As Bates & Damgaard showed in their classic paper (47), the Coulomb approximation used in conjunction with experimental ionization energies yields excellent results for such transitions­provided that the normalization factor can be satisfactorily estimated. The discussion of the fixed-core approximation in Section 2 suggests, however, that the semiempirical method may actually be applicable to a much wider class of transitions.

(b) Helliwell's method (54) : This method was devised for transitions in which the active electron is one of a pair of valence electrons outside a core made up of filled subshells. It rests on the observation that the nodes of the radial wavefunction of a valence electron in a bivalent atom occur at nearly the same values of r as in the monovalent ion. For example, the nodes of the 4s and 4p wavefunctions for Ca nearly coincide with those of the corresponding wavefunctions for the 4s and 4p states of Ca+. In other words, the nodes of the valence electrons are determined primarily by the charge distribution in the core. For a monovalent atom, however, the position of the outermost node can be accurately estimated by means of the Coulomb approximation, provided that the energy parameter E is known ; and E is

given to a good approximation by the ionization energy of the monovalent atom. Thus the position of the outermost node can be estimated for each of the valence electrons in a bivalent atom.

Helliwell uses this "principle of nodal stability" in the following way. He formulates the Hartree-Fock equations for the two valence electrons, replac­ing the core potential by its asymptotic Coulomb form. The Hartree-Fock equations for the valence electrons are now coupled only to one another. The estimated positions of the outer nodes of the two wavefunctions provide

appropriate inner boundary conditions. The resulting eigenvalue problem can be solved by standard methods; the solution gives the energy parameters for the two valence electrons.

Helliwell has applied this method to atoms in which the subconfiguration of the valence electrons is S2, ss', and sp.

Helliwell's method requires far less computation than the ordinary Hartree-Fock approximation, though in this respect it enjoys a smaller advantage over the fixed-core Hartree-Fock approximation. On the other hand, it is much more time-consuming than the semiempirical method j ust

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484

w u z w � W lL.. lL.. Q � Z W U � w a..

6

2

0

- 2

- 4

- 6

- 8

- /0

LAYZER & GARSTANG

2p

Is -

2$ 2 3 4 . 5 6 7 8 9

q FIG. 8. Accuracy of calculated values of the energy parameter . (solid lines) and

of the mean radius (r) (dashed lines) for the term (1s22sZ2p3) 4S0 in the N isoelectronic sequence. The calculated values are given by first-order perturbation calculations employing screened-hydrogenic wavefunctions; the comparison values of E and (r) are Hartree-Fock values.

described. It would be interesting to compare the results of calculations made with the fixed-core Hartree-Fock approximation, Helliwell's method, and

the semiempirical method. (c) The variational screening approximation : Among the simplest

variational wavefunctions are screened-hydrogenic functions PH(Z - s) , the screening parameter s being determined by the variation principle (32). The screening parameters can be evaluated eithel' in a one-configuration or in a many-configuration approximation.

In a given many-configuration approximation, the energy parameter E can be expressed as a linear combination of radial i ntegrals, each of which

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ATOM IC TRANSITION PROBABI L ITIES 485

can be evaluated with screened-hydrogenic wavefunctions. I n Figure 8 we compare the screening and Hartree-Fock approximations to E for some representative states (43, 54). In general, the screening approximation is not v ery accurate for neutral atoms, but it improves rapidly with increasing Z along an isoelectronic sequence-as, of course, was to be expected, since the two leading terms in the Z expansion of E are given correctly by the screening approximation. Because experimental information about energy levels is most complete for neutral and singly ionized atoms, but is usually much less complete for more highly ionized atoms, the screening approximation may afford a useful supplement to the semiempirical method.

(d) Simplified Hartree-Fock calculations: Until fairly recently, the numerical solution of Hartree-Fock equations for moderately complex atoms presented serious technical difficulties. Attempts were therefore made to simplify the Hartree-Fock equations by replacing the most troublesome terms in them-those resulting from exchange effects and configuration mixing-by "equivalent potentials" of various kinds. None of these at­tempts proved very successful, and all of them are now of historical interest only, because developments in computer technology and, to a lesser extent, in numerical analysis techniques have made such simplifications unnecessary.

Normalization of PC._ (a) Evaluation of (r) : The interchange theorem (Section 3) can be

conveniently used to estimate (r) (50) . The calculations (55) are similar to calculations by Dalgarno and his colleagues of polarizabilities and other atomic parameters represented by diagonal matrix elements [see e.g. (40) and references cited therein] . Figure 8 shows that the perturbation calcula­tion yields satisfactory estimates of (r) even for neutral atoms. As usual, the accuracy of the estimate increases rapidly with increasing Z along an iso­electronic sequence.

(b) Other methods : We have already referred to Hartree's method (valid when the quantum defect is small) and the quantum defect method. These methods and the one just discussed seek to obtain the normalization factor with a minimum of calculation. If none of these methods works, but one wishes to avoid a full Hartree-Fock calculation, an intermediate method can be used. This consists in replacing the Hartree-Fock equivalent potential by a simpler potential. If the energy parameter is known to some approxima­tion, one can then integrate the radial wave equation inward, from r = 00 to r = 0, using an appropriate cutoff procedure in the immediate neighborhood of r = O. The resulting wavefunction may not be v ery accurate at small and intermediate v alues of r, but all that is really necessary is that it should pre­dict with tolerable accuracy the value of the ratio

Jorcp2(r)dr / Jo�P2(r)dr 6.1.

where rc represents the radial distance at which the Coulomb approximation to P becomes valid. On physical grounds it seems likely that this ratio,

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486 LAYZER & GARSTANG

which represents the probability of finding the electron inside a sphere of radius re, is insensitive to the detailed character of the charge distribution.

One convenient simplified potential is the Thomas-Fermi potential, which has been used by Stewart & Rotenberg (56) for essentially this pur­pose. Another procedure, which may be more accurate and is nearly as simple, is to replace the radial functions that occur in the Hartree-Fock potential by variational screened-hydrogenic wavefunctions. This procedure can also be regarded as the first step in an iterative solution of the Hartree­Fock equations.

ACKNOWLEDGMENTS

Sections 5 and 6 draw heavily on unpublished work by Charlotte Froese,

Margaret Lewis, and Carlos Varsavsky, whom I thank for permission to cite individual results. I am greatly i ndebted to Dr. M argaret Lewis for her invaluable assistance in the preparation of Sections 5 and 6.

R. RELATIVE TRANSITION PROBABILITIES I N INTERM EDIATE COUPLING

by R. H. Garstang

Neutral argon has been the subject of several studies. Garstang & Van Blerkom (5 7) studied the 3p54s-3p54p and 3p54s-3pG5p arrays, fitting the theory to the experimental energy levels to determine the electrostatic and spin-orbit parameters. The Coulomb approximation was used to put the final transition probabilities on an absolute scale. The inadequacy of any limiting coupling scheme was clearly shown by the values of the parameters obtained. For the 3p54p configuration, for example, LS coupling would require the three electrostatic parameters (Slater integrals) F2, Go, and G2 all to be much greater than the spin-orbit parameters tap and t4p. The jj coupling would require tap and t4p to be much greater than F2, Go, and G2• The jt coupling requires that tap»F2»t4p, F2»GO, and F2»G2• The actual values (in cm-1) obtained were F2 = 1 77, Go = 69 1 , G2 = 40, fap = 93 1 , and t�p = 33. None of the conditions for the various coupling schemes is even approximately satisfied. I t is not surprising that reliable results can only be obtained if the proper intermediate-coupling scheme is calculated. This is confirmed by calculations of the Lande g factors i n LS, jj, andjf coupling, the latter agreeing much more closely with the observed values than either of the limiting coupling schemes. Johnston (58) extended the Ar I calculations to arrays involving the 6p, 6s, 7s, 4d, Sd, and 6d configurations, and he listed line strengths for thejj-coupling scheme. P. W. M urphy of the University of M aryland has made some similar (as yet unpublished) calculations on various arrays in Ne I, Ar I , and Kr I .

The importance of argon for experimental work has led to a number of attempts to measure its transition probabilities. Those made prior to 1965 are listed by Garstang & Van Blerkom (57) in comparison with the calcu­lated values. High-current arc plasmas, shock tubes, anomalous dispersion,

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ATOMIC TRANSITION PROBABILITIES 4 8 7

and absorption and emission techniques have been used. The experimental values agree more closely with the intermediate-coupling values than with LS, jj, or jf, values. There are, however, some appreciable differences be­tween the results of various experiments, so that further experimental work is needed. This has been supplied in part by Popenoe & Shumaker (59). They used a wall-stabilized argon arc containing a trace of hydrogen. The H{3 lineshape measurement and the assumption of local thermodynamic equilibrium provided the determination of the argon level populations. Other recent experimental work on Ar I includes plasma arc measurements by Bott (60) and by M alone & Corcoran (61), shock tube measurements by Coates & Gaydon (62) , wall-stabilized arc measurements by Shumaker &

Popenoe (63) , and lifetime determinations by Klose (64) . Intermediate coupling can be important for highly ionized atoms, be­

cause spin-orbit interaction increases more rapidly along an isoelectronic sequence than electrostatic interactions. This applies in particular for ions that occur in the solar corona. Cases are known where intercombination lines (as described in LS coupling) have higher transition probabilities than allowed lines. Garstang (65) calculated all the transitions in the arrays 2p6_2p63s, 2p6_2p63d, 2p63s-2p53p, and 2p53p-2p53d of Fe XVI I , a member of the Ne isoelectronic sequence. In Ar I, discussed above, the coupling for the 2p53p configuration does not satisfy any of the limiting coupling schemes, and full intermediate-coupling calculations are essential. The 2p63s and 2p53d configurations are closer to it coupling, but intermediate-coupling calculations are also necessary here. Kastner, Omidvar & Underwood (66) performed calculations for the 2p6_2p53s and 2p6_2p53d arrays in the Ne I sequence up to Fe XVI I .

Even in some light elements intercombination lines can often be observed, and in low-density conditions (such as the solar corona or the gaseous regions in quasars) intercombination lines from low-lying metastable energy levels may be greatly enhanced relative to allowed lines from levels of higher energy in the same ion. This makes i ntercombination lines worth calculating even in many cases where the departures from LS coupling are small by ordinary standards. The intercombination lines will be relatively weak in the first few members of an isoelectronic sequence and will increase in importance up the sequence. Elton (67) calculated the PSo-23P} oscillator strengths in the He I sequence, and Garstang & Shamey (68) calculated additional members of this sequence and the 21S0-23P1 lines in the Be I sequence. Several ions of these sequences are known in quasars or in the solar corona. Another case where intercombination lines are observable, though of very low probability, occurs in the arrays 2s22p2_2s2p3 and 3s23p2-3s3p3. The intercombination multiplets sp_oS are known in the solar spectrum for C I (69), N I I and 0 I I I (70), and Si I ( 7 1 ) . The transition probabilities for the C I and Si I multiplets were given in these references. These multiplets might be detectable in higher ions, and House (72) has calculated their probabilities (and many other lines) in Ca XV and Fe XIII . Here the metastability of the

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488 LAYZER & GARSTANG 5S2 level should lead to an enhanced population in the level and a stronger emission line than would be expected on the basis of the transition probability itself.

A few other calculations have been made, and we have collected in Table V most of the calculations involving intermediate coupling that have been done in recent years.

I t is worth noting that intermediate coupling causes mixing of collision strengths in a way similar to the mixing of line strengths. I t is therefore possible for the cross section to be much greater than if LS coupling were valid. This is true in particular for intercombination lines, where in LS coupling the cross section derives from exchange processes. An excellent example of this occurs in Fe XIII, where Bely, Bely & Lan (80) showed that 3p2 aP2-3p2 lD2 has a large cross section produced by intermediate-coupling effects.

C. FORBI D DEN TRANSITIONS

by R. H. Garstang

The state of our knowledge of forbidden transitions in both atoms and molecules was very thoroughly reviewed by Garstang (1 10) . We shall mention the more important papers published since that time.

Garstang (86) made an extensive study of [Fe I I ] , in which he calculated all the forbidden lines of this ion which arise from the low even configura­tions. His results were compared by Thackeray (87) with observations of the spectra of the nucleus of 'f/ Carinae, [Ni I I ] lines also being compared. The agreement between the theoretical and observed intensities is very good, and on the basis of the theoretically predicted i ntensities many new identifica­tions could be made. The occurrence, or otherwise, of [Fe I I ] in the solar spectrum and in Arcturus has been a controversial problem, on which the final word has not yet been said. Papers by Swings (88) and by Gasson & Pagel (89) contain other references.

Krueger & Czyzak (90) calculated transition probabilities for forbidden lines of [Fe X] to [Fe XVI ] inclusive ; Krueger & Czyzak (91) studies lines of many other highly ionized ions of S, Ar, K, Ca, V, Cr, M n, Fe, Co, and Ni of coronal interest ; Malville & Berger (92) extended the calculations for the 2p2, 2P4, 3p2, and 3p4 sequences to higher sequence members ; Czyzak & Krueger (93) studied ions of P, S, Cl, and Ar, chiefly of interest in planetary nebulae ; Czyzak & Krueger (94) studied [Fe VIII ] ; Garstang (95) studied [Ti I I I ] , [Cr I I ] , [Cr IV] , [Mn V] , [M n vI ] , [Fe VI ] , [Fe VII] , [Ni I ] , [eu I I ] , and a large number o f heavier elements from G a to Po, in various stages of ionization, of primary interest for laboratory experimental work, and tabulated some electric quadrupole multiplet strengths for arrays in­volving equivalent d electrons ; Garstang & Hill (96) gave results for two lines of [Ba I I ] that could be observed in the solar spectrum; Nikitin (97) and Nikitin & Yakubovskii (98) studied coronal lines from Sp2 and Sp3 con­figurations; Goldwire & Goss (99) obtained the transition probability of the

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ATOM IC TRANSITION PROBABILITIES 489 TABLE V: Intermediate-coupling calculations.

Ions Arrays Refs.

Ar I See text See text Ar I I 3p<4s-3p44p 73

3p6_3p44s Au I I I 5d86s-5d86p 74

Be I sequence 2s2-2s2p sP! 68 (to Ca XVII)

C I 2s22p2_2s2ps 69 Ca XV 2s22p2_2s2p3 72 CI I, Br I, I I p6_p'S 75 Co I I 3d74s-3d74p 76 Eu I I 4f'6s-4f'6P 77 Fe IX 3p6_3p64s 78

3p6_3p53d Fe X I I I 3s23p2_3s3p3 72 Fe XIV 3s23p-3s3p2, 3s23d 79 Fe XVII ZpQp53s 65

Zp6_Zp53d 2p63s-Zp53p 2p53p-2p53d

Fe XVII Zp6-2p53s 78 2p6_Zp53d

He I seq uence ls'-ls2p sP, 67 (to Ne IX)

He I sequence ls'-1sZp sp! 68 (Na X to Si XI I I )

N e I sequence Zp6-2p63s 66 (to Fe XVI I )

2p6_2p63d Ni ll 3d84s-3d84p 81 Ni H 3d84s-3d84p 82 P I , As I p3_p'S 7S S I , Se I p'_p3S 7S Si I, Gel, Sn I, Pb I P'-ps 75 Si I 3s23pz-3s3p3 7 1 Si VI, S VI I I ZS'Zp6_ZSZpG 83 Si VII , S IX ZS'Zp4_ZSZp& 83 Si V I I I , S X Zs22p3_ Zs2p' 83 Si IX, S XI 2s'2p'-2s2p3 83 Si X, S XII 2s'2p - 2s2p' 83 Si X 2s22p-2s2p' 79 Sn I Spz-5p6s 84 Sn I SP'-5p6s 5 V II I 3d3-3d24p 85 Yb I I I 4f'36s-4f136p 77

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490 LAYZER & GARSTANG

magnetic dipole 3.5-cm wavelength line between the hyperfine levels of the ground state of 3He II . All these papers were concerned with theoretical calculations.

When both magnetic dipole and electric quadrupole radiation are pres­ent in a particular transition, interference effects are present in the Zeeman components, and these effects vary with the direction of observation. Hults (100) made a careful study of the 7330 A (lDz_3Pl) line of [Pb I] and the 7099 A (2P! 1/2-2Pl/2) line of [Pb I I ] . The intensities of the Zeeman com­ponents with !:J.M = ± 1 were studied in longitudinal and transverse obser­vations. From these observations it was possible to determine the percent­age contribution to each of the whole lines made by electric quadrupole radiation. Hults obtained 4 ± 1 per cent for 7330 A and 3 ± 1 per cent for 7099 A. These values may be compared with 3.2 and 5.0 respectively, cal­culated by Garstang (95) . This agreement must be considered satisfactory. It shows that there is no major disagreement between the theoretical and experimental ratios of magnetic dipole and electric quadrupole radiation, and is therefore an important general confirmation of the theory.

A second major series of experimental investigations has been concerned with the halogens. In particular, Husain & Wiesenfeld (101) studied the flash photolysis of trifluoroiodomethane and the subsequent time-resolved emission I (52Pl/Z)_I(52Pl/Z) +hll in the ground doublet of iodine. The decay rate was studied as a function of pressure (to eliminate collisional deactiva­tion effects) , and there was a measurable decay rate in the limit extrapolated to zero pressure. This was attributed to the spontaneous radiative decay rate of the 52Pl/2 level. The result was 22 ± 6 sec-1 for the spontaneous transi­tion probability, which may be compared with the value 8 seC! computed by Garstang (95). There is clearly room for more work on this lifetime by ex­perimental techniques. We regard it as a remarkable achievement to measure such a long lifetime experimentally, and we think that further experimental refinements may well reduce the discrepancy between theory and experiment. At this stage, we do not think this discrepancy gives grounds for concern.

Several investigations have studied magnetic quadrupole radiation. Mizushima (102) drew attention to the possible occurrence (for any multi­polarity higher than dipole) of magnetic multipole transitions when !:J.S = ± 1 . He pointed out that an allowed magnetic quadrupole transition may have an order of magnitude the same as transitions produced by spin­orbit interactions. Garstang (103) had studied some transitions that take place by nuclear-spin perturbations. It became necessary to consider the relative importance of the various mechanisms. For individual lines it may be possible to eliminate one or more of the competing mechanisms by means of selection rules. If the selection rules are satisfied for more than one transition mechanism, it is not usually possible to say without detailed cal­culations just which mechanism will predominate. This point was emphasized by Dugan (104) , who showed that spin-orbit interaction usually predomi-

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ATOMIC TRANSITION PROBABILITIES 4 9 1

nates in the optical region o f the spectrum. Mizushima ( 102) and Garstang ( 10S) carried out detailed calculations on magnetic quadrupole radiation. They restricted their work to transitions with LlS = ± 1. Then for transitions with multipolarity 2k(k = 2 for quadrupole) the selection rules are

.

.:lS = ± 1

t,L = O, ± 1, · · · , ± (k - l)

L + L' 2 k - l

iV = O, ± l, · · · , ± k

J + J' 2 k

For magnetic quadrupole radiation, with LlS = ± 1 , the rules become

t,L = 0, ± 1(0 �\->0)

t,J = 0, ± 1, ± 2(0 <--\--> 0, 0 <--\-->1, 1/2 <--\--> 1/2)

I n addition there must be a change of parity. Mizushima (102) investigated a number of transitions, such as 2s22p2 3P2-

2s2p3 .S2 in C I, 2p4 3P2-2p33s ·S2 in 0 I, and for these transitions ordinary electric-dipole radiation made possible by spin-orbit perturbations is far larger than magnetic quadrupole radiation. For transitions with LlJ = ± 2,

spin-orbit-induced transitions are impossible. A search for magnetic quadru­pole transitions is therefore more likely to lead to positive results. Of the many possible transitions s· 'So-sp 3P2 is the most interesting. Here there is a competition between magnetic quadrupole radiation and electric-dipole radiation induced by nuclear-spin interactions. The latter occurs only for isotopes with a nuclear spin, that is, for those with odd masses. The conclu­sion reached by Garstang (105) is illustrated by an excerpt from his results in Table VI. The astronomically interesting case is Mg I. The line 3s2 'So-3s3p 3 P 2 at 4562.48 A was observed by Bowen (106) , and others, in the plan­etary nebula NGC 7027. The Cd I line has been observed in the laboratory, and the laboratory intensity of the line as a function of the odd isotope abundance in the cadmium samples was interpreted by Garstang ( l OS) as confirming the presence of magnetic quadrupole radiation. The Zn I line of the same type has also been observed in the laboratory as long ago as 1925.

Atom

He I Mg I Cd I Hg I

TABLE VI ; Transition probabilities for s' 'So-sp ·P. lines.

Transition

1s' 'So-1s2p 'P2 3s' 'So-3s3p 'P2 5s' 'So-5s5p 'P2 6s' 'So-6s6p • P 2

Transition probability

Magnetic quadrupole

(sec-I)

0 . 22 1 . 8 X lO-4 9 . 6 X I0-4 3 . 6 X 10-'

N uc1ear-spin­induced elec­

tric dipole (sec-')

1 . 6 X 10-6 6 . 9 X I0-' 0 . 15

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492 LAYZER & GARSTANG The Hg I line clearly is largely due to nuclear-spin- induced electric-dipole radiation.

N uclear-spin-induced radiation was considered theoretically by Garstang (103) for the IS0_3PO and ISo_3P2 transitions in the s�-sp arrays of Mg I , Zn I , Cd I , and Hg I , the 0-2 transitions being the ones we have just dis­cussed. The correctness of these calculations has been verified by Bigeon ( 107) , who made absorption measurements o n the IS0_3PO line in 199Hg I and 201Hg I, with results that agreed very closely with the theoretical values.

The two-quantum transition 2s 2S1/2-1s 2S1/2 in H e I I was detected ex­perimentally in a coincidence-counting experiment by Lipeles, Novick & Tolk (108). Dalgarno (109) calculated the transition probability for the two­quantum transition 21S-11S in He 1 .

This work was supported in part by the National Science Foundation.

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