Relativistic HFS oscillator strengths for indium and gallium

NUCLEAR

INSTRUMENTS

AND METHODS

IIO ( I 9 7 3 )

227--230;

© NORTH-HOLLAND

PUBLISHING

CO.

R E L A T I V I S T I C H F S O S C I L L A T O R S T R E N G T H S F O R I N D I U M AND G A L L I U M * C.P. BHALLA

Kansas State University, Department of Physics, Manhattan, Kansas, U.S.A., and Argonne National Laboratoryt , Argonne, Illinois, U.S.A. Oscillator strengths for the diffuse series are calculated with the relativistic Hartree-Fock-Slater model. The effects of retardation are included. Satisfactory agreement is obtained between the

experimental data and the present calculations. It is shown that the relativistic effects can be as much as 15% in certain transitions for large atomic numbers.

1. Introduction

the EL and the M L multipoles where the retardation is included. Here L is the order of the multipolarity, while E and M are electric and magnetic, respectively. These formulas are based on a single-particle relativistic model. We reproduce here the relevant formulas for the electric dipole transitions. The mean transition probability between an upper state Ix'p') and a lower state [xp) is

Recently Andersen and Sorensen l) have reported on the systematic studies of atomic lifetimes in Ga, In and T1 by the beam-foil technique. Earlier Penkin and Shabanova 2-3) measured the oscillator strengths by the hook method. It was pointed OUt 1-3) that the calculations of Gruzdev 4) give typically a factor two larger oscillator strengths for the diffuse series as compared to the experimental f-values. We had reported earlier 5) a reasonable agreement between our relativistic theoretical results and experiments of Penkin and Shabanova for diffuse transitions in TI. It should be noted here that for the resonance transitions (n2p~/2,3/2-n2D3/2,5/2) the experimental results obtained by the beam-foil technique and by the hook method are the same within 10%. It is the purpose of this paper to present new calculations based on the relativistic H a r t r e e - F o c k Slater model 6) for the diffuse series, and to investigate the effects which arise solely from the relativistic considerations. Section 2 contains the relevant theoretical formulas and a brief discussion of the theoretical model, followed by numerical results in sec. 3.

w(E1) = 4c~k ( t c ' - x ) Ig- + I o - ~ x 5e(J i, 1, Jf),

(1)

where the relativistic units are used: h = m = c = 1. The fine structure constant is designated b y ~ = 1/137 and the statistical factor is given as follows: 3 2Ji+3 4 Ji+l , ~ ° ( J i , 1, J f )

=

/3

,

1

for

Jr = Ji +1 ,

for

Jf = Ji,

for

Jr = J ~ - 1,

(2)

4 Y~(d~ + 1) 2J i- 1 '

2. Theoretical model

We summarize the relevant formulas of the relativistic radiative transition probabilities in sec. 2.1. Sec. 2.2 contains the details of the numerical calculations.

( x ' - ~) I~ + I 2

,

Ji

with the triangular relation A (Ji 1 Jf) and (/i + l f-t-1) an even integer. The final and the initial total angular m o m e n t u m values, denoted by Jr and Ji, are related to • and x' by eqs. (3)

2.1. RELATIVISTIC TRANSITION RATES

Jf ~--- IKI - ½,

FOR ELECTRIC DIPOLE

We have published 5) the theoretical formulas for * Work supported by the U.S. Army Research Office, Durham, North Carolina, U.S.A. t Work performed under the auspices of the U.S. Atomic Energy Commission and the Argonne Center for Educational Affairs.

J i = IK'l

(3) 2.

We note that

1,~ = IKI

K -

½ + --.

(4)

2Ix[

227 III. T H E O R Y

228

c.P.

T h e relativistic single-particle w a v e f u n c t i o n s 6) for a c e n t r a l field p o t e n t i a l , V(r), are:

•F '?

if_ K /

/

BHALLA 2.2. NUMERICAL CALCULATIONS

T h e results o f t h e c a l c u l a t i o n s p r e s e n t e d in this p a p e r are o n l y for o n e - e l e c t r o n t r a n s i t i o n s o u t s i d e a c l o s e d core. It s h o u l d be n o t e d h e r e t h a t the o s c i l l a t o r s t r e n g t h is g i v e n by eq. (8) w i t h w(tc' ~ K) as defined in eq. (1). O n e needs to c a l c u l a t e the radial integrals, I L defined in eqs. (7), w h i c h d e p e n d u p o n the radial functions, dF

with

Z~ = ~ C(1½J", / t - M , M) X,MY~ .,-~ M

-

dr ,

~:

F-

[W-

1-V(r)]

G,

(9a)

- V ( r ) ] F - -~: G .

(9b)

r

(6a)

dG__ = [ W + I

and

dr

W e use the n o t a t i o n o f R o s e 7) for the C l e b s c h - G o r d o n coefficient in eq. (6a). T h e m a t r i x e l e m e n t s in eq. (1) are d e f i n e d in t e r m s o f t h e l a r g e c o m p o n e n t , G a n d t h e small c o m p o n e n t , F o f t h e u p p e r a n d the l o w e r states.

I[ = ~gL(kr) (GF' + FG') dr,

(7a)

I~ =- ~gL(kr) (GF'-FG') dr.

(7b)

T h e s p h e r i c a l Bessel f u n c t i o n o f o r d e r L is d e n o t e d by T h e w a v e v e c t o r k in t h e relativistic units is e q u a l to the t r a n s i t i o n e n e r g y in mc 2 units. T h e o s c i l l a t o r s t r e n g t h , f , is g i v e n by"

g~.(kr).

f_

2x' w(x'--* h') 22 , 2K

(8)

87r2e 2

where the mean transition probability and the wavel e n g t h o f t h e r a d i a t i o n , 2 are in units o f mc2/h a n d h/mc a n d ~ = 1/137.

r

T h e eqs. (9) w e r e s o l v e d f o r a c e n t r a l p o t e n t i a l in a self-consistent m e t h o d , so t h a t the final V(r) leads to the v a l u e s o f F a n d G f o r e a c h o r b i t a l w h i c h give the s a m e p o t e n t i a l V(r) w i t h i n a c e r t a i n t o l e r a n c e . W e u s e d the S l a t e r a p p r o x i m a t i o n for the e x c h a n g e c o n t r i b u t i o n in o b t a i n i n g the s e l f - c o n s i s t e n t s o l u t i o n f o r the n e u t r a l a t o m s , Z = 49 a n d Z = 31. T h e excited states w e r e t h e n c a l c u l a t e d u s i n g the s e l f - c o n s i s t e n t p o t e n t i a l . T h i s p r o c e d u r e , t h o u g h it i n v o l v e s a p p r o x i m a t i o n , has the m e r i t t h a t all t h e excited o r b i t a l s are o r t h o g o n a l to e a c h o t h e r as well as to the g r o u n d state. T h e r a d i a l integrals, IL, w e r e t h e n c o m p u t e d n u m e r ically. T h e f - v a l u e s w e r e o b t a i n e d by u s i n g eq. (8) w i t h the e x p e r i m e n t a l l y m e a s u r e d v a l u e s o f 2 a n d the t r a n s i t i o n energies. T a b l e 1 c o n t a i n s the relativistic m a t r i x e l e m e n t s for i n d i u m in t h e case o f several transitions. T h e v a l u e s o f I ~ a n d 12 are t y p i c a l l y 10 - 6 t i m e s I ~ o r I o for the r e s o n a n c e transitions. T h e r e f o r e , we c o u l d h a v e i g n o r e d the radial m a t r i x e l e m e n t s f o r L = 2.

TABLE 1 Relativistic matrix elements for indium a.

Upper state

Lower state

I+

1o

I+

I~-

52D3/2 62Da/2

52P 1/2 -

-- 0.1007E-- 2 - 0.6620E - 3

- 0.1301E - 2 - 0.7341E - 3

-- 0.4917E - 9 - 0.3568E - 9

- 0.2104E - 8 - 0.8948E - 9

52Da/2 52D 5/2 62D~/2 62D5/2

52p3/2 -

0.1622E- 7 0.1752E -- 2 0.1432E- 8 0.1131E-2

- 0.2359E- 0.6005E - 0.1383E-0.2493E-

2 3 2 3

0.9045E0.3159E 0.6499E0.2271E-

a With the inclusion of retardation and the assumption of the frozen core for the excited states.

9 8 9 8

-- 0.3212E- 0.9562E - 0.1648E-0.3118E-

8 9 8 10

RELATIVISTIC

HFS

OSCILLATOR

3. Numerical results

The calculated f-values for the diffuse series of indium are compared with the experimental data in table 2. The quoted f-value 1) for t h e (5p21/2-52D3/2) transition is 0.44. The contributions of other transitions from the upper s t a t e 52D3/2 were not included in ref. 1 in converting the mean lifetime of this state to the f-values. The theoretical f-value for t h e (52p3/2 52D5/2) transition is in excellent agreement with the beam-foil measurement1). The experimental f-value for the (52pl/2-52D3/2) transition is ~ 15% larger than our theoretical results. We do not show in table 2 the theoretical f-values of Gruzdev4), which differ from the experimental values typically by a factor of 2. Table 3 contains a comparison of the present TABLE 2 Comparison off-values for the diffuse series in In 1. Transition

Beam foil a

52p1/2 - 52D3/2 62D3/2 52P8/2 - 52D5/2 - 62D5/2 -- 52D3/2

0.37°4-0.03 0.08 0.31 4-0.02 0.07

-

--

H o o k method b

Theory (RHFS)

0.364-0.02

0.30 0.09 0.30 0.085 0.033

0.374-0.02 0.044-4-0.004

a T. Andersen and G. Sorensen, Phys. Rev. A5 (1972) 2447. b N. P. Penkin and L. N. Shahanova, Optika i Spektrosk. 14 (1963) 167 [Opt. Spectry. 14 (1963) 87]. e The f-value quoted by Andersen and Sorensen is 0.44. We have corrected this f-value for all the other transitions from the upper state 5eDa/2.

229

STRENGTHS

theory with the experimental f-values for gallium. The beam-foil results were again corrected by us for the branching ratios of the 42D3/2 state. Fig. 1 shows the various branching ratios for the de-excitation of t h e 52D3/2 state in indium and t h e 62D3/2 state in thallium. The results for thallium are in agreement with the experimental data of Gallagher and Lurio 8) who used the level-crossing technique. 4. Discussion and conclusions

We have presented the relativistic H a r t r e e - F o c k Slater calculations off-values for the diffuse series in indium and gallium. These calculations were performed without ignoring the retardation effects (long wavelength approximation was not used) and with the frozen core potential. In contrast to the previous calculations, we obtain a reasonable agreement with the experimental f-values for the resonance transitions. It was found that the (nZP3/2-nZD3/2) transition contributes significantly to the mean lifetime of the upper state n2D3/2 (n = 4 for Z = 31 and n = 5 for Z = 49). We discuss now the ratios of reduced transition probabilities to ascertain the relativistic effects. Table 4 contains a summary of the results. The radial matrixBRANCHING

RATIOS OF VARIOUS TRANSITIONS (n 2D ) 3/2

FROM UPPER STATE

5 203/2 6 PI/2

5 195

TABLE 3

IO00

5 2P3/2

C o m p a r i s o n off-values for diffuse series in Ga 1.

5 2pI/2 Transition

42p1/2 42p3/2 - -

42D8/2 52D3/2 42Da/2 4ZD5/2 52D5/2

- 52D3/2

Beam foil a

H o o k method b

0.25 + 0.02 c 0.32 4-0.02 0.544-0.0050 0.0384-0.004 0.344-0.03 0.29 4-0.02 0.08 -0.057 -

Theory (RHFS)

0.27 0.09 0.03 0.25 0.078

In I

6 2D 2 3/2 7 PI/2 183

I000

0.009

a T. Andersen and G. Sorensen, Phys. Rev. A5 (1972) 2447. b N. P. Penkin and L. N. Shabanova, Optika i Spektrosk. 14 (1963) 167 [Opt. Spectry. 14 (1963) 87; 18 (1965) 504]. e The f-value quoted by Andersen and Sorensen is 0.30. We have corrected this f-value for all the other transitions from the upper state 42D3/2.

6 2P3/2 T,E I

6 2Pf/2

Fig. 1. Calculated branching ratios for the de-excitation o f the upper state n2D3/2(n = 6 for Z = 81, n = 5 for Z = 49). Ili. THEORY

230

C.P. TABLE 4

initial a n d t h e final v a l u e o f the t o t a l a n g u l a r m o m e n t u m . T h i s is c a u s e d b y the s p i n - o r b i t c o u p l i n g , w h i c h is i n c l u d e d in t h e relativistic c a l c u l a t i o n s .

Ratios of the reduced transition probabilities a. Ratios

Statistical factor b

BHALLA

Z = 31 Z = 49 Z = 81 n = 4 n = 5 n = 6

(n2P3/2 - n2D 5/2) (n~P1/2 - n2D3/~)

1.20

1.22

1.25

1.38

(n2P3/2 - n2D5/2) (n2p3/2 - ngD312)

6.00

5.99

5.97

5.90

(n2P3/2 - n2D3/2) (n2P1/2 - n2D3/2)

0.200

It is a p l e a s u r e to t h a n k D r Y o n g - K i K i m a n d Dr Mitio Inokuti of the Argonne National Laboratory f o r several s t i m u l a t i n g discussions. I a m g r a t e f u l f o r t h e f i n a n c i a l s u p p o r t u n d e r the S u m m e r F a c u l t y R e s e a r c h P a r t i c i p a t i o n P r o g r a m o f the A r g o n n e C e n ter f o r E d u c a t i o n a l Affairs.

References 0.204

0.209

0.234

a Transition probability divided by the transition energy factor. (Present relativistic calculations.) b This corresponds to the non-relativistic approximation, wherein the dipole matrix element is the same for all these transitions. e l e m e n t d o e s n o t d e p e n d o n t h e J - v a l u e in t h e n o n relativistic a p p r o x i m a t i o n i.e. the r a d i a l m a t r i x - e l e m e n t is t h e s a m e f o r t h e t r a n s i t i o n s : n2pa/2-n2D3/2, n2p1/2n2D3/2 a n d n2pa/2-n2Ds/2 . W e define the r e d u c e d t r a n s i t i o n p r o b a b i l i t y as t h e c a l c u l a t e d t r a n s i t i o n p r o b ability d i v i d e d by all f a c t o r s w h i c h d e p e n d u p o n the t r a n s i t i o n energy. C o l u m n 1 o f t a b l e 4 gives t h e ratios in t h e n o n r e l a t i v i s t i c a p p r o x i m a t i o n a n d these are the same for Z-31, 49 a n d 81. T h e last t h r e e c o l u m n s give o u r t h e o r e t i c a l results. It s h o u l d be n o t e d t h a t t h e d e v i a t i o n s b e t w e e n the n o n r e l a t i v i s t i c a n d t h e relativistic c a l c u l a t i o n s i n c r e a s e as a f u n c t i o n o f i n c r e a s i n g a t o m i c n u m b e r . T y p i c a l l y d e v i a t i o n s are ~ 16% f o r Z--- 81. N o t e t h a t t h e r e a s o n f o r the d e v i a t i o n is t h a t t h e relativistic r a d i a l m a t r i x e l e m e n t s d e p e n d o n the

1) 2) 3) 4) 5) 6) 7)

T. Andersen and G. Sorensen, Phys. Rev. A5 (1972) 2447. N. P. Penkin and L. N. Shabanova, Opt. Spectry. 14 (1963) 87. N. P. Penkin and L. N. Shabanova, Opt. Spectry. 14 (1963) 5. p. F. Gruzdev, Opt. Spectry. 20 (1966) 209. C. P. Bhalla, Nucl. Instr. and Meth. 90 (1970) 149. C. P. Bhalla, Phys. Rev. 157 (1967) 1136. M. E. Rose, Relativistic electron theory (J. Wiley, New York, 1961). 8) M. E. Rose, Elementary theory of angular momentum (J. Wiley New York, 1957). 9) A. Gallagher and A. Lurio, Phys. Rev. 136 (1964) A87.

Discussion CROSSLEY: YOU refer in your abstract to theoretical values of Gruzdev. Were these by the Bates-Damgaard method, or by the Burgess-Seaton method? BHALLA: I think they were with the Burgess-Seaton method and I tried to figure out what they had done, but with very little success. Those theoretical values do differ by a factor of 2 from the experimental f-values for the diffuse series. CROSSLEY: Yes. Well, I set one of my students to work on the Burgess-Seaton paper to find the formula that Gruzdev had used. She did this in the end and recalculated some of his numbers for magnesium and beryllium and was unable to reproduce his results. So it does seem possible that perhaps his calculations are in error. ANDERSEN: I think it should be pointed out that Gruzdev's calculations for the sharp series, the s-p transitions, seem to be in order, so some of his numbers may be correct.