The Z-independence of oscillator strengths

NUCLEAR

INSTRUMENTS

AND METHODS

IIO

(1973)

183--188;

© NORTH-HOLLAND

PUBLISHING

CO.

T H E Z - D E P E N D E N C E OF O S C I L L A T O R S T R E N G T H S * A. D A L G A R N O

Harvard College Observatory and Smithsonian Astrophysical Observatory, Cambridge, Massachusetts, U.S.A. Nuclear charge expansion theories of oscillator strengths are reviewed and a comparison is made with experimental data. Ways in which the expansion theories can be carried to higher order are discussed.

1, Introduction

HylleraasS), expands E and ~v(r) according to

Because beam-foil spectroscopy provides data on highly stripped atomic systems, an analysis of transition probabilities and oscillator strengths as a function of nuclear charge Z along an isoelectronic sequence has a particular value, as indeed has been explicitly demonstrated by Wiese~'2), Wiese and Weiss3), and Smith and Wiese4). Their semi-empirical analyses can be placed on a more secure foundation with the help of nuclear charge expansion theories in which the transition matrix element is developed as an explicit function of Z. Such theories probably cannot hope to compete as a reliable predictive tool with refined calculations [cf. NussbaumerS), Weiss6), Nicolaides and Sinanot~luT)] but the simplicity and generality of Z-dependent theories offer unique advantages in systematizing laboratory data. Z-dependent theories are conveniently developed by introducing Zao as a unit of length and Ze2/2ao as a unit of energy. The Schr6dinger equation for an N-electron system becomes /-/q,(r) =

('

Ho + Z H ~ - E

)

E = ~ Z-mE.,

~u(r) = L Z-"ffm(r)' (4)

m=v

m=0

Then

(s)

( H o - E o ) ~o(r) =- O, ( H o - E o ) ~ a ( r ) 4- ( H 1 - E l )

~o(r) = 0,

(Ho-Eo)

I//l(r) - ~ 2 I~o(r) = 0 .

~2(r) + (Hx-El)

(6)

(7) The matrix element of a single-electron operator N

Q(N) = ~ qi(rl),

(8)

i=1

connecting two eigenfunctions ~P")(r) and ~u(f~(r) with eigenvalues E (i) and E (f) respectively can also be expanded in the form

l ( m ) = (~u(i)lQlT~f)) = ~, Z -m lm(N ).

~U(r) = O,

Then

(1)

Io(N ) = @~)lQl~(of)> , where Ho =

L Ho(i)=

i=1

L [ - ! V 2 - 1 ] '\ 2 i 77/

i=1~

(2)

i< j

1

Iri--rjl

,

(10)

is the hydrogenic limit obtained as Z tends to infinity. The gradient of I(N) for large Z is

I~(N) = (ff(oi)lo[~(~f)>4- <¢~i)]Ol~b(or)>.

and N H1 = ~

(9)

n=0

(3)

(11)

If no zero-order degeneracy occurs, interchange theorems [Cohen and Dalgarno9"l°)] can be used to simplify the evaluation of/,,. Thus introduce functions )~(i) and Z(f) such that

r denoting collectively the position vectors r i of the electrons measured from the nucleus.

(¢,g, iQiCof)) ~/gi),

(12)

(Ho-E(o°) Z(° + Q@~oi) = (~b(of)lQlO(oi)) ~,(or' .

(13)

(Ho_Egi)) z(i)+ Q~jgf) = and

2. The Z-expansion method There are many ways of developing Z-dependent theories. The simplest, originating in the work of * This work supported in part by the United States Air Force under AFOSR Grant 71-2132.

It is convenient to require that (l~gi)l~/]i) > = (~/gf)l~/(lf)> = (0(0i)l)~(i)> ~--- (~kgf)lz(f)> = 0

(14)

183 III. T H E O R Y

184

A. DALGARNO

Then we may write 11 in the alternative form I(N~ 1

--

<¢oi)iHIz¢i)> + .

(15)

Because Q is a single electron operator, the determination of g(i) and X(r) is much simpler than that of ~]~) and qJ]f); indeed eqs. (12) and (13) for X°) and ~((f) can be solved exactly. For a two-electron system described in zero order by the configuration (nol o, n'ol~), say, the interchange function Z can be written as an anti-symmetrized sum of products of orbitals (nolo) (n'fl'O and (nil1) (n'ol6) [Cohen and Dalgarnol°), Laughlin11)]. The perturbed orbitals (n~ll) and (n'fl~) satisfy simple one-electron equations that can be solved exactly [Dalgarno and Parkinsona2)], by expansion in the set of unperturbed hydrogenic orbitals (nl) [Laughlin, Lewis, and HoraklS)] and by numerical integration [Laughlin~)]. Exact values of 11(2) for several states of helium-like systems are given by Cohen and Dalgarnol°), by Ali and Crossley 14'15), by All and Schaad16), by Laughlin and Dalgarnol7), and by CrossleylS). The solution 4J~ (r]N) of eq. (7) for a many-electron system can be expressed in terms of the two-electron helium-like solutions ~kl (r[2) [Sinano~lu19), Chisholm and Dalgarno2°)]. It follows that the first order matrix element I~ (N) can be written as a linear combination of the two-electron first order matrix elements 11 (2) [Dalgarno and Parkinson2a)]. As an example consider the transition ls22s2p ~P-ls22p 2 ~D of the beryllium isoelectronic sequence [Laughlin and Dalgarno22)]. We may write, in the notation of Chisholm and Dalgarn02°), tP(ls22s2p lp) = ~

1

[ - ~ ( l s 2 1S) ' 7J(2s2p lp)] _

1

,/6

E~(ls2p ~P), tP(ls2p ip)]

_

1

- - - [~U(ls2p 3p), ~(ls2p ap)] -4-

,/2

+ ~

1

DU(2p z 'D), ~ ( l s 2 1S)3. (17)

The reduced matrix element of the dipole operator Q is given in zero-order by <¢o(lsZZsZp lp)JIQIJ ¢o(ls22p 2 'D)> = = ,,/15 [ ~ 2

<¢'o(ls2s 1S) IIQII¢'o(ls2p 'P)> +

1

+ ~-~ <¢,o(ls2s as)IIQIIqJo(ls2p + 3~

ap)> +

(¢o(2s2p ~P)IIQIi ~bo(Np2 1D)> •

(18)

Evaluating the hydrogenic matrix elements, we obtain (~o(lsZ2sZp lp)ILQ[] qJo(ls22p 2 aD)) = 3,/10. (19) The first order contribution is <¢o(ls22s2p Xp)IIQil ffl(ls22p 2 1D)> + + (~1(ls22s2p 'P)]lQil @0(Is22p 2 XD)> = 6 x/15 { l - ~ 2 [ + + (~P~Kls2s ls)IIQ[I ~o( ls2s 'P)>] +

_ _ _ Eq,(ls2 s 1S)' tP(ls2p 1p)] _ 243

1

[<¢o(ls2s 3s)ilQII ¢',(ls2s 3p)> +

_ ½171(ls2s 3S) ' t/J(is2 p 3p)] __ + (tPl(lS2s 3S)liQII t/Jo(ls2p Zp)>] + - -[tp(ls2p 'P), 7J(ls2s ~S)] 243 _ ½[tP(ls2p 3p), tP(ls2s 3S)] + + ~1 [~(2s2p ap), ~ ( l s 2 1S)], (16) and ~(ls22p 2 1D) = ~1 [-tP(ls 2 1S), tp(2p2 ~D)]

-

1 [<0o(2s2p Xp)IIQll 6,(2p 2 1D)> + +6--73 + ]~. )

(20)

The two-electron matrix elements in square brackets are respectively 49991 x/3/39, 44823x/3/39 and - 73x/10 / /640 [Laughlin and Dalgarno~7)]. Thus, correct to first order in Z -1,

Z-DEPENDENCE

OF O S C I L L A T O R

STRENGTHS TABLE 1

( ~ ( l s 2 2 s 2 p 1p)IIQli ~(IsE2p 2 1D)) = _ 3~_10 I1

185

Line strength parameters according to first-order Z - e x p a n s i o n theory in the exact a n d H a r t r e e - F o c k description for transitions ( 1s22sa2pb)-(1S22Sa-Z2pb+t). S = S o / ( Z - - r) g.

{32410

(21)

N=a+b+2

a

b

Transition

S0

T

/"

(Hartree- (Exactp Fock) a

and S = 90/(Z-

1.6086) 2 .

(22)

The corresponding formulas for transitions (ls22s a

3 4

1 1

0 1

5

1

2

7

2

3

8

2

4

9

2

5

2 p b ) - ( 1 S2 2 S a - 1 2 p b + ~) h a v e b e e n w o r k e d o u t b y L a u g h l i n

and Dalgarno w) for all cases where zero-order degeneracy does not occur. Some gain in accuracy appears to be achieved by replacing the first order expression ( I o + I , / Z ) 2 / Z 2 by l o / ( Z - I , / I o ) 2 to which it is equivalent to first order in Z - ' [Dalgarno and Stewart23)]. The values of the parameters that determine the line strengths in formulas like eq. (22) are listed in table 1. The values that are obtained in the Hartree-Fock approximation [Cohen and Dalgarno24)] are also tabulated. In fig. 1, the Z-expansion oscillator strengths for the (ls22s2p)'P-(ls22p 2) ' D transitions are compared with experimental data and with the results of more elaborate calculations. A similar comparison for the I

2S _ 2p up _ 3p t p _ 1D 4p-4s 2D-2D 2p-2D

4s-4p

54 162 90 108 135 135

1.694 2.066 2.165 2.437 2.511 2.563

108

3.251

1.610 1.837 1.609 2.027 1.913 1.684

1.912

2D - 2D 2D-'2P

135 135

3.260 3.270

2.026 2.254

3p 3p 1D-1P 2p_ 2S

162 45 54

3.663 3.636 4.065

2.102 2.331 2.292

a C o h e n a n d DalgarnoZa). b Laughlin a n d Dalgarnoa7). ( l s 2 2 s 2 p 2) 4 p - ( l s 2 2 p 3 ) 4S transitions is presented in fig. 2. The simple formulas are remarkably successful although since they are correct only to first order in

I

I

I

0,24

x

0.20

Jx

0.16

x

x \\\ \\

f 032

x

\

\\ \\\

0 08

0.04

0 001

I 0.05

I O. I0

I O, 15

I 0.20

L25

I/Z

Fig. 1. Oscillator strengths for the (lsZ2s2p) 1p-(ls22p2) 1D transition in the beryllium isoelectronic sequence. T h e crosses are experimental data f r o m A n d e r s e n et al.a°), Poulizac et al.~t), Pegg et al.a2), Ceyzeriat et al.3a), B e a u c h e m i n et al.34), Bickel et al.aS), and DruettaZ6). T h e circled value for Be I is a theoretical value o f Nicolaides et al.37). T h e solid curve is the Z - e x p a n s i o n curve; the dashed curve is a suggested extrapolation to the theoretical value for the neutral system. III. T H E O R Y

186

A. DALGARNO

Z , meaningful results are n o t to be expected, except b y chance, for neutral a n d singly-ionized systems. The f o r m u l a s u n i f o r m l y overestimate the oscillator strengths. Similar f o r m u l a s have been derived for o t h e r transitions in the lithium [ D a l g a r n o and P a r k i n s o n 2 ' ) , Crossley'8)] a n d s o d i u m sequences [Laughlin et al.~3)]. I

The calculations for the transitions in the s o d i u m isoelectronic sequence p r o v i d e a g o o d e x a m p l e o f the v a l u e o f a k n o w l e d g e o f I, in the e x t r a p o l a t i o n o f d a t a f r o m values o f Z. Fig. 3 is a c o m p a r i s o n o f an e x t r a p o l a t i o n to the h y d r o g e n i c limit by Smith and Wiese 4) with one t h a t e m p l o y s b o t h lo and I , . I

I

I

0.24

0.20

×~

0.16

x

f 0.12

0.08

0.0'4

0

I 0.05

0

I 0.10

I O.t5

I 0.20

I/Z

Fig. 2. Oscillator strengths for the (ls22s2p 2) 4p-(ls22p3) 4S transitions in the boron isoelectronic sequence. The crosses are experimental data from Martinson and BickelSS), Martinson et al.39), and HerouxaO). The solid curve is the Z-expansion curve. The values are close to the theoretical results of Weiss41). I

I

I

I

0.25

0.2(

fo.I

0.1(

005

,.----

(a) I

0.02

I

004

I

I

0.06

0.08

0,10

I/Z

Fig. 3. The oscillator strengths of the 3d 2D~p zp transitions of the sodium isoelectronic sequence. Curve (a) is reproduced from Smith and Wiese4); curve (b) is an extrapolation with the correct gradient at large Z (ref. 13).

Z-DEPENDENCE

OF O S C I L L A T O R

The Z-dependent formulas are asymptotically correct only when relativistic effects are ignored. It appears that in the dipole length formulation relativistic effects modify more strongly the transition frequency than the matrix element and measurements of the probabilities of transitions in which there is no change in the principal quantum number of the active electron would be instructive for Z not greatly larger than ten. For such transitions the non-relativistic transition energy increases only linearly with Z. The Z-dependent theory is less accurate for transitions in which there is a change in principal quantum number and high values of Z must be reached for it to be quantitatively reliable. The value of the Z-independent theory would be much enhanced if it could be carried to higher order without loss of computational simplicity and accuracy. Laughlin u) has discussed some semi-empirical methods for improving the estimate of the higher order contributions but it is also possible to proceed with a purely theoretical analysis. For two-electron systems, the sequence of perturbation equations can be solved by variational methods and excellent approximations to the entire Z -1 series expansion can be readily obtained [Dalgarno and Drake25), Drake and Dalgarn026), Drake27)]. Table 2 reproduces the results of some unpublished calculations by Drake and Dalgarno of oscillator strengths f expressed as series expansions f = ~ f,, Z-re.

(23)

m=O

The variationally determined values of I 1 for the ls 2 lS_ls2p 1p, ls 2 1S_ls3p 1p, and ls 2 as-ls4p IP transitions are respectively - 0.16329, 0.15360 and 0.12524 TABLE 2 co

Z fm Z -m, of the oscil-

Coefficients a in the Z-expansion, f =

m--0

lator strengths of the ls z l S - l s 2 p 1p, ls3p 1p and ls4p 1p transitions of the helium isoelectronic sequence b. n

ls z aS_ls2p 1p

ls 21S_ls3p 1p

ls z 1S_ls4p 1p

0 1 2 3 4 5 6 7 8 9

0.83239 - 1.06852 -0.33560 0.40059 0.30452 -0.15157 -0.27475 0.06248 0.31577 0.16247

0.15820 -0.06693 -0.23624 -0.06664 0.24897 0.18606 -0.22712 -0.28787 -0.00266 2.90501

0.05798 - 0.01107 -0.09228 -0.05598 -0.02362 0.21007 0.16419 4.64943 0.22611 11.37725

187

STRENGTHS TABLE 3

Oscillator strengths of the l s 2 1 S - l s 2 p , ls3p and ls4p 1p transitions of the helium isoelectronic sequence by high order Z-expansion theory a and by variational calculations b. ls 2 1S_ls2p 1p Z-exp. Var.

Z 2 3 4 5 6 7 8 9 10 11 12

0.27614 0.45661 0.55153 0.60889 0.64705 0.67418 0.69444 0.71012 0.72262 0.73280 0.74127

0.2762 0.4566 0.55155 0.60891 0.64707 0.67420 0.69445 0.71013 0.72263 ---

ls 2 lS_ls3p 1p Z-exp. Var.

is 2 1S_ls4p 1p Z-exp. Var.

0.0733 0.1106 0.1267 0.1353 0.1404 0.1437 0.1461 0.1478 0.1491 0.1501 0.1510

0.0291 0.0427 0.0487 0.0517 0.0533 0.0544 0.0550 0.0555 0.0559 0.0562 0.0564

0.073 0.1106 0.1269 0.1354 0.1405 0.1438 0.1461 0.1479 0.1492 ---

0.030 0.0437 0.0492 0.0520 0.0535 0.0545 0.0552 0.0556 0.0560 ---

a Drake and Dalgarno (unpublished). u Schiff et al.29).

compared to the exact values [Sando and Epstein28), Dalgarno and Parkinson I 2)] of respectively - 0.163270, 0.153998, and 0.125851. Table 3 gives a comparison of the Z-expansion oscillator strengths with those calculated by Schiff et al. 29) using highly sophisticated variational eigenfunctions determined separately for each nuclear charge. For three-electron systems, similar calculations are possible that should not involve any serious loss of accuracy nor require extensive computer time. Calculations for many-electron systems can then be carried to order Z -2 by expressing the many-electron eigenfunctions in terms of the three-electron eigenfunctions. A similar procedure can be found for further application of an interchange theorem. The secondorder matrix element/2 (N) is defined as Iz(N) = (0~oi)[Q[0C2f)) + (0]i)[QlO~tf)) + (¢~)lQlO(or)) • (24) Now using eqs. (7) and (12), we may obtain the identity

a F o r high m, the coefficients become unreliable, but the summations are still valid. b Drake and Dalgarno (unpublished).

(~,gi>iQl~,g) > = ( z ( i ) l g i - E I I ¢ i *)) - lo(N ) <¢~ol)]~9(zi'>. (25) Orthogonality requires that (,/,(i)h/,(i)\ "F0 IW2 /

=

--l(l~(i)]l//(i))'

(26)

so that IE(N ) _- (z(i)[H1 - - ~l~,(f)ld,(i)\ 1 IW1 / - { - ( Z ( f ) I H 1 - E 1 + <¢~)lQlO~f)> --½Io(N) [<¢(~i)l¢~i)> 4-

(i) 1¢1(f)) 4 -

(~([)lff/(f)>].

(27) III. THEORY

188

A. D A L G A R N O

an expression that involves only first-order functions that have already been determined. However, no c a l c u l a t i o n s o f e q . (27) h a v e y e t b e e n c a r r i e d t h r o u g h , although the simplicity of the structure suggests that Iz (N) is a c o m b i n a t i o n o f t h e s e c o n d - o r d e r t w o - e l e c t r o n m a t r i x e l e m e n t s 12 (2).

References 1) W. L. Wiese, Appl. Opt. 7 (1968) 2361. e) W. L. Wiese, in Beam foil spectroscopy (ed. S. Bashkin; G o r d o n and Breach, New York, 1968). 3) W. L. Wiese and A. W. Weiss, Phys. Rev. 175 (1968) 50. 4) M. W. Smith and W. L. Wiese, Astrophys, J. Suppl. 23 (1971) 103. 5) H. Nussbaumer, Monthly Notices Roy. Astron. Soc. 145 (1969) 141 ; Astron. Astrophys. 16 (1972) 77. 6) A. W. Weiss, Nucl. Instr. and Meth. 90 (1970) 121. 7) C. A. Nicolaides and O. Sinano~lu, Nucl. Instr. and Meth. 90 (1970) 133. 8) E. Hylleraas, Z. Physik 65 (1930) 209. 9) M. Cohen and A. Dalgarno, Proc. Roy. Soc. A275 (.1963) 492. 10) M. Cohen and A. Dalgarno, Proc. Roy. Soc. A293 (1966) 359. 11) C. A. Laughlin, in press. 12) A. Dalgarno and E. M. Parkinson, Proc. Roy. Soc. A301 (1967) 253. 12) C. A. Laughlin, M. N. Lewis and Z. Horak, J. Opt. Soc. Am. (1973) in press. 14) M. A. Ali and R. J. S. Crossley, Intern. J. Quant. Chem. 3 (1969) 17. 15) M. A. Ali and R. J. S. Crossley, Intern. J. Quant. Chem. 4 (1970) 541. 16) M. A. Ali and L. J. Schaad, Molec. Phys. 17 (1969) 441. 17) C. m. Laughlin and A. Dalgarno, Phys. Rev., in press. 18) R. J. S. Crossley, in press. 19) O. Sinano~,lu, Phys. Rev. 122 (1961) 493. 20) C. D. H. Chisholm and A. Dalgarno, Proc. Roy. Soc. A290 (1966) 264. 21) A. Dalgarno and E. M. Parkinson, Phys. Rev. 176 (1968) 73. 22) C . A . Laughlin and A. Dalgarno, Phys. Letters 35A (1971) 61. 23) A. Dalgarno and A. L. Stewart, Proc. Roy. Soc. A257 (1960) 534. 2~.) A. Cohen and A. Dalgarno, Proc. Roy. Soc. A280 (1964) 258. 25) A. Dalgarno and G. W. F. Drake, Chem. Phys. Letters 3 (1969) 349. 26) G. W. F. Drake and A. Dalgarno, Phys. Rev. A1 (1970) 1325. 27) G. W. F. Drake, Phys. Rev. A5 (1972) 614. '28) K. M. Sando and S. T. Epstein, J. Chem. Phys. 43 (1965) 1620. 29) B. Schiff, C. L. Pekeris and Y. Accad, Phys. Rev. A4 (1971) 885. 30) T. Andersen, K. A. Jessen and G. Sorensen, Phys. Letters 28A (1968) 459; Phys. Rev. 188 (1969) 76. 31) M. C. Poulizac, M. Druetta and P. Ceyzeriat, Phys. Letters 30A (1969) 87. 22) D. J. Pegg, E. L. Chupp and L. W. Dotchin, Nucl. Instr. and Meth. 90 (1970) 71.

3a) p. Ceyzeriat, A. Denis, J. Desesquelles, M. Druetta and M. C. Poulizac, Nucl. Instr. and Meth. 90 (1970) 103. 34) G. Beauchemin, J. A. Kernahan, E. Knystautas, D. J. G. Irwin and R. Drouin, Phys. Letters 40A (1972) 194. 35) W. S. Bickel, R. Girardeau and S. Bashkin, Phys. Letters 28A (1968) 154. 36) M. Druetta, Compt. Rend. Acad. Sci. Paris 269B (1969)

1154. 37) C. A. Nicolaides, D. R. Beck and O. Sinano~lu, J. Phys. B 6 (1973) 62. 38) i. Martinson, W. S. Bickel and A. Olme, J. Opt. Soc. Am. 60 (1970) 1213. 39) I. Martinson, H. G. Berry, W. S. Bickel and H. Oona, J. Opt. Soc. Am. 61 (1971) 519. 40) L. Heroux, Phys. Rev. 180 (1969) 2. 41) A. W. Weiss, Phys. Rev. 188 (1969) 119.

Discussion SXNANOOLU: I know you have been, for instance, interested in doing some work on relativistic aspects too. Are you doing more on the high-Z transitions and seeing the relativistic effects and so on a little more? OALGARNO: In some ways the Z theories are a natural way of trying to include relativistic effects and we are trying to do so. One thing that seems clear is that the principal effect is actually on the frequency o f the transition rather than on the matrix element itself. SINANOOLU: That is encouraging. However, we have recently found the relativistic effects to be far more drastic, in the high-Z regions o f t h e f v s 1/Z curves in current use, than had been hitherto believed. MARTINSON: YOU have derived a formula which gives the line strengths for the 2s2p 1P-2p2 aD transition in the Be I isoelectronic sequence. For Be I the f-values has been calculated by Weiss (unpublished work, 1970) who obtained f = 0 . 0 0 1 , and Nicolaides ct al. (J. Phys. B, in press) who found f = 0.020. There is also an experimental upper limit o f 0.035, found by Andersen et al. [Phys. Scripta 4 (1971) 52]. 1 wonder what your theory would give in this particular case. In your paper with Dr Laughlin [Phys. Letters 35A (1971) 61], you did not include the Be I f-value. DALGARNO: Our Z-dependent theory has no significance for the neutral systems. If it gives an answer that agrees with somebody else that is just sheer chance. So to put a number down and call it theory is just misleading. The Z-theory probably gives an upper limit. The values go up too fast. At least, all cases we looked at go up too fast. I do not know what the answer is for Be 1. Weiss or Sinano~lu would know better than I the origin o f their factor o f 20. George Victor will be coming down soon and I am sure he is got yet another value and it will be interesting to see what it is. NICOLAIDES: I just want to say that this is a very sensitive transition probability to calculate and I would not be surprised if our value is not as accurate as the others we have been reporting. Observe, however, that we are still close to experiment. Also, Linderberg's Z-dependent theory calculation is off by orders o f magnitude [J. Linderberg, Phys. Letters 29A (1969) 467] which shows that, as Prof. Dalgarno said, for the neutrals, Z-dependent perturbation theory results must be much less accurate.