High-energy internal conversion coefficients

3.B

1

Nuclear Physics A132 (1969) 374--384; (~) North-Holland Publishiny Co., Amsterdam N o t to be reproduced by photoprint or microfilm without written permission from the publisher

HIGH-ENERGY INTERNAL CONVERSION COEFFICIENTS O. D R A G O U N t

Max-Planck-lnstitut fiir Kernphysik, Heidelberg, Germany Received 7 March 1969 A b s t r a c t : T h e K-shell internal conversion coefficients were calculated for every fifth value o f the

atomic n u m b e r Z from Z = 30 to Z : 100 for lowest four electric and m a g n e t i c transition multipoles. T h e finite nuclear size effect a n d screening by atomic electrons were considered precisely in the energy region f r o m 6 to 9.5 mc 2 a n d approximately for energies from 12 to 20 mc 2. T h e K / ( L + M ) conversion coefficient ratios equal roughly 8.7, 6.6 a n d 4.3 for Z = 30, 60 and 95, respectively, independently o f the transition energy a n d multipolarity. In the considered energy region, the internal conversion in the s-subshells is f o u n d to be d o m i n a n t .

I. Introduction

A comparison of experimental internal conversion coefficients ([CC) with theoretical predictions for different transition multipolarities has proved to be a powerful method for a spin-parity assignment of low-energy excited nuclear states [see e.g. ref. 1)]. The ICC as well as their separation for different multipolarities are known to be a decreasing function of the transition energy. In the high-energy asymptotic limit [ref. 2)], the conversion coefficients lose completely their dependence on the transition multipolarity. It was believed that the ICC can hardly be measured for the transition energies above 2.5 MeV and also present tabulations 3-6) of the theoretical values do not exceed this region. Recent neutron-capture experiments, however, led to the ICC of transition energies up to 9 MeV [see e.g. refs. 7, 8)]. Smither 9) concluded on the basis of his experimental K-shell ICC for gamma rays of 1-9 MeV energy studied in the reaction 113Cd(n ' 7)t 1 4 C d that the separation of the coefficients for different multipolarities at high energies should enable one to assign multipolarity to high-energy transitions. However, the approximate calculations 1o) of high-energy K-ICC for the cadmium (Z = 48) is in disagreement with this conclusion. The experimental reinvestigation 8) led to the K-coefficients in 1a4Cd which agree with the theoretical values 1o). The precise high-energy coefficients for Z = 48 were calculated recently by Carroll and O'Connell ~1) using the relativistic self-consistent-field electron wave functions. The results again disagree with the experimental values of Smither 9). Very recently, Church xz) explored the properties of the high-energy K-ICC using the approximate t On leave from the N u c l e a r Research Institute, I~e~, Czechoslovakia. 374

HIGH-ENERGY ICC

375

corrections for the effects of the finite nuclear size and atomic screening. The results were illustrated by the case of 2°°Hg (Z = 80). It is the aim of this work to perform the calculations of the high-energy K-shell coefficients for the wide Z-region as well as to investigate the internal conversion of high-energy gamma rays in the L- and M-atomic subshells.

2. Description of calculations The ICC calculations of the present work were based on the two physical models. Precise K-shell ICC for transition energies 6.0, 7.5 and 9.0 m c 2 were calculated with the computer program 13) adapted for the CDC 3300 computer. The finite nuclear size was considered in the no-penetration model of Rose 4) assuming the nucleus to be a homogeneously charged sphere of radius R = 1.2A ~ fm. The atomic screening was expressed by the non-relativistic Hartree-Fock-Slater screening functions as tabulated by Herman and Skillman 14). The same computer program and physical assumptions were used recently 15) to calculate the ICC for the N-subshell electrons. The K-shell coefficients for transition energies k < 5 m c 2 were found to be in agreement with the values of Sliv and Band 3). In order to get the K-ICC for the transition energies 12, 15 and 20 m c z, as well as all L and M subshell coefficients considered in this work, the following procedure was applied: The ICC in the point nucleus without screening approximation were calculated with the computer program 16) based on formulae derived by O'Connell and Carroll 17). The approximate correction for the finite nuclear size effect was performed according to ref. 17), i.e. the unphysically large contribution of the Coulomb electron wave functions and transition electromagnetic potentials inside the nucleus to the conversion matrix elements was substituted by zero. The atomic screening was approximately taken into account by renormalization of the bound-electron wave functions near the nucleus. This method was suggested originally by Hinman 28) for the K-ICC where the screening represents only a few percent correction. Investigations i9) verified its applicability also for the ICC of higher atomic shells, where the screening plays an important role. The screening correction factor is defined as the ratio of the two bound-state electron densities calculated at nuclear vicinity from the electron wave functions with and TABLE 1 Correction screening factor for different atomic subshells

Z

Ll

Lz,3

M1

Mz,3

M4,5

30 60 95

0.743 0.846 0.894

0.71 0.77 0.85

0.371 0.594 0.705

0.26 0.51 0.64

0.06 0.29 0.46

without screening, respectively. We obtained the screening factors by comparison of the non-relativistic screened wave functions 14) with those for the pure Coulomb field

376

o. DRAGOUN

at the smallest value o f the electron c o o r d i n a t e given in ref. 14). F o r the K-shell electrons, the screening factors varied f r o m 0.973 to 0.990 for Z = 30 a n d Z = 100, respectively. T h e screening correction factors for the L- a n d M-subshell electrons are given in table 1. The K - I C C calculated for the four lowest electric a n d m a g n e t i c nuclear t r a n s i t i o n multipoles for every fifth value o f the a t o m i c n u m b e r Z fror~ Z = 30 to Z -- 100 can be f o u n d in the appendix. In tables 2-4, we present the L2+a/L~, K / ( L + M ) a n d K/L~ conversion coefficient ratios for the transition energies 6 a n d 20 m c 2 a n d a t o m i c n u m b e r s Z = 30, 60 a n d 95.

3. Discussion The K - c o n v e r s i o n coefficients calculated in this w o r k for transition energies from 6 to 20 m e z are shown in figs. 1 a n d 2 for Z = 40 a n d 95, respectively. The K - I C C o f

IJ i

i

i

10-2

M4~L M3 ~ E4 ~ ', M2 \x i E 3 ~a~\',

(~ 10-3

M,.\,~

I t K-shell '!, Z : 4 0

I

M4 Q 10_1 M3 ~ M 2 X'\ ~ E4 \\\~,'

t_ sliv

!

" ' " ~ !

10-4 f

2 5 10 20 transition energy (me 2)

Fig. 1. The K-shell internal conversion coefficients for Z = 40 versus transition energy. The results of this work are given together with the values of Sliv and Band a) for the transition energy k -<_-5 m c 2. Four lowest electric and magnetic multipoles are considered. The letter A denotes the high-energy asymptotic limit 2). Points on the two curves indicate the energies for which the coefficients were calculated.

I

K-shell Z=95

\\\R,',,

\\%

Shv i and Band iPre;ent w°rk

I andj Band!presentll I wOrkl 1

I

E 3 ~ , kx \ \ \\,~1 ,

'°"F 10-5 L

I

1

2

5 10 20 transition energy (mc 2)

Fig. 2. The K-shell internal conversion coefficients for Z = 95 versus transition energy. See caption to fig. 1.

Sliv a n d B a n d 3) for t r a n s i t i o n energies k < 5 m c 2 are also given together with the high-energy a s y m p t o t i c limit 2) for p o i n t nuclei w i t h o u t screening. It can be seen t h a t o u r coefficients f o r m a s m o o t h c o n t i n u a t i o n o f the Sliv-Band values. As the t r a n s i t i o n energy increases, the m o s t o f the I C C a p p r o a c h slowly the a s y m p t o t i c limit. F o r

HIGH-ENERGY

377

ICC

magnetic dipole transitions in heavy nuclei, however, the approximate ICC of this work are obviously too small. This is apparent also from fig. 3, where the K-ICC for the transition energy 20 m c 2 are plotted versus atomic number. The effect is not observed for the transition energy 6 m c 2 for which the precise coeffÉcients were calculated (see fig. 4). The failure of the high-energy M1 coefficients for heavy nuclei is very probably caused by the correction for finite nuclear size ~7). In fact, this is a rather crude apI

I

I

I

I

I

I

I

I

I

o~

I

I

I

i

i

I

I

// / /

M4 .,~ M 3 p M2

OL

10-3

K - shell

10 -2

/M4 / . M3

k=20 mc 2

/ /. E4

,;y.

K-shell k mc 2

/ / /

/////./ /..

M2

,//// ///-<

10-4

E4

/

/"

I

/ E3 M1 E2

E2 10-3

///

~

/// / / // ~ / ///,*" ~,7 / /

M1

-~--

E1

/////

10-5

10-6

10 - 4

I

I

40

I

I

60

I

I

I

I

I

80 100 atomic number Z

Fig. 3. The K-shell internal conversion coefficients for the transition energy 20 mc 2 versus atomic n u m b e r Z. Nuclear transition multipoles El . .., E4 M1 . . . M4 were considered. The finite nuclear size effect and atomic screening were taken into account approximately.

10"5

I

I

4O

I

I

6O

I

I

I

I

I

8O 100 atomic number Z

Fig. 4. The K-shell internal conversion coefficients for the transition energy 6 m c 2 versus atomic n u m b e r Z. Nuclear transition multipoles El . . . E 4 , MI . . . M4 were considered. The finite nuclear size effect and atomic screening were taken into account precisely.

proximation giving an upper limit of the correction, as the contribution of the internal conversion inside the nucleus is completely neglected. In many cases, however, the predictions of the simple approximation 17) are comparable with the results of the more sophisticated approach. The separation of the K-shell ICC for different multipolarities seems to be too small to allow unique spin-parity determinations of high-energy gamma rays with present experimental accuracies. This is demonstrated in fig. 5, where the results of our calculations for Z = 48 are compared with the recent experimental values 20). Most of the experimental coefficients group around the curves for the M1, E1 and E2 multipolarity, but usually one cannot determine the transition multipolarity uniquely. This is particularly apparent for the four transitions above 7 MeV (shown by open circles in fig. 5) which are known to be of magnetic dipole character. The situation should improve for heavier elements as emphasized also by the results of Church 12) for 2°°Hg

( Z = 80).

378

o. DRAGOUN

It is known that the conversion coefficient ratios can be measured with a substantially better accuracy than the absolute ICC. Unfortunately, the subshell intensity ratios which are most sensitive to the transition multipolarity cannot be measured in the high-energy region, and the use of the K/L intensity ratios only would often lead to ambiguities even in the case of the low-transition energies. Especially, it is difficult to distinguish between the El and M1 multipolarities [see e.g. ref. 21)]. Let us consider the [CC subshell ratios in the energy and Z-region of table 2. The contribution of the L 2 + 3 [CC to the L-total coefficient is very small. The same holds for the M 2 + 3/M1 ICC ratios which never deviated from the corresponding L-subshell ratios more than by 20 ~ . The M4+ 5/M~ ICC ratios are in the mentioned region smali

i

I

10-3 _ M2 E3 \ ~

I

I

I

I

I

I

I

I

I

I

I

K - shell Z = 48 "

~'x

10 .4

I

10-5 1.5

I 2

I 3

I

I

4 6 8 10 transition energy [-Me G

Fig. 5. The comparison of the K-shell internal conversion coefficients calculated in this work for Z = 48 with the experimental values 20). The group o f known MI transitions above 7 MeV is s h o w n by open circles.

TABLE 2 The Lz+3/Lt conversion coefficient ratios (in percent) for different values of the atomic n u m b e r Z and transition energy k Z

k (me 2)

Multipolarity E1

E2

E3

E4

M1

M2

M3

M4

30

6 20

1.3 1.0

1.3 0.9

1.4 0.9

1.7 0.9

1.1 0.9

1.1 0.8

1.3 0.8

1.6 0.8

60

6 20

4.0 3.0

5.2 2.8

5.8 2.8

7.6 2.9

3.6 3.1

4.1 3.4

5.0 2.8

6.2 3.0

95

6 20

10 7.1

16 8.0

23 9.6

30 11

13 11

14 10

17 11

20 12

HIGH-ENERGY ICC

379

let than 6 × l0 - 4 . We thus conclude that the internal c o n v e r s i o n o f the high-energy g a m m a rays proceeds mainly t h r o u g h the s-subshells. Th e small c o n t r i b u t i o n o f the p-electrons can be observed for the transitions o f high m u l t i p o l a r i t y in heavy elements. T h e K / ( L + M ) ratio for m e d i u m - w e i g h t elements is a l m o s t i n d e p e n d e n t o f the TABLE 3 The K/(L+M) conversion coefficient ratios for different values of the atomic number Z and transition energy k z

k (mc 2)

Multipolarity E1

E2

E3

E4

MI

M2

M3

M4

30

6 20

8.7 8.9

8.7 8.9

8.6 8.8

8.6 8.8

8.7 8.9

8.7 8.8

8.6 8.8

8.5 8.8

60

6 20

6.7 6.9

6.5 6.8

6.4 6.7

6.1 6.7

6.6 6.8

6.4 6.7

6.3 6.6

6.1 6.5

95

6 20

4.9 5.0

4.2 4.7

3.8 4.5

3.4 4.4

4.4 4.4

4.2 4.3

3.9 4.2

3.6 4.2

transition energy and m u l t i p o l a r i t y (see table 3). This fact could help to establish the pairs o f the K an d ( L + M ) c o n v e r s i o n lines b e l o n g i n g to the same transition in the extremely c o m p l i c a t e d internal c o n v e r s i o n spectra m e a s u r e d in the n e u t r o n capture experiments. TAaLE 4 The K/L1 internal conversion coefficient ratios for different values of the atomic number Z and transition energy k Z

k

Multipolarity

(me 2)

El 10 10

E2

E3

E4

MI

M2

M3

M4

10 10

I0 10

10 10

10 10

10 10

10 10

9.9 10

30

6 20

60

6 20

8.5 8.6

8.3 8.4

8.1 8.3

8.0 8.3

8.3 8.4

8.1 8.3

8.0 8.2

7.9 8.1

95

6 20

6.6 6.7

6.0 6.2

5.7 6.1

5.4 6.0

6.1 6.1

5.9 5.8

5.6 5.8

5.3 5.8

T h e K / L I I C C ratios given in table 4 are very similar to the ratios o f the K- and L1bou n d - s t at e electron densities at the nuclear surface as calculated by Sliv and Band a nd used for the d e r i v a t i o n o f the L I - I C C for transition energies 3, 4 an d 5 m e 2 f r o m the c o r r e s p o n d i n g K-shell coefficients 22).

380

0. DRAGOUN

4. Conclusion The precise K-shell internal conversion coefficients, presented in the appendix for the transition energies 6, 7.5 and 9 m c 2 extend the energy region of the existing tabulations 3- 6). Figs. 1-4 indicate that an interpolation in both energy and Z-scales should introduce no difficulties. The analytical estimates for the K - I C C between 12 and 20 m c 2 can be too small especially for the M1 transitions in heavy elements where large finite nuclear size corrections are to be expected. The K / ( L + M) coefficient ratios for medium-weight elements are almost independent of the transition energy and multipolarity. This could be useful to identify the pairs of the K and ( L + M ) conversion lines of the same nuclear transition. In order to apply the process of internal conversion to the spin-parity assignments of the high-energy transitions, extremely precise experiments as well as calculations are necessary. The region of the heavier elements seems to be more promising. Thanks are due to Professor W. Gentner for his interest and to Dr. H. C. Pauli for helpful discussions. The author is indebted to the Max-Planck-Gesellschaft for a postdoctoral fellowship.

Appendix TABLE OF THE K-SHELL INTERNAL CONVERSION COEFFICIENTS

The ICC are presented for every fifth Z-value from Z = 30 to Z --- 100 for six transition energies k given in m c 2 units. All coefficients in the table are written in the following convention: 1 . 9 3 E - 5 = 1.93x 10 -5 . Z= k 6.0 7.5 9.0 12.0 15.0 20.0

E1 1.93E--5 1.39E--5 1.01E--5 8.34E--6 6.45E--6 4.68E--6

E2 2.96E--5 2.01E--5 1.40E--5 1.08E--5 7.97E--6 5.51E--6

E3 4.33E--5 2.79E--5 1.87E--5 1.36E--5 9.75E--6 6.47E--6

30

E4

M1

6.14E--5 3.78E--5 2.45E--5 1.70E--5 1.18E--5 7.54E--6

2.89E--5 1.96E--5 1.36E--5 1.04E--5 7.70E--6 5.33E--6

M2 4.50E-- 5 2.88E--5 1.91E--5 1.38E--5 9.75E--6 6.43E--6

M3 6.56E--5 4.00E--5 2.57E--5 1.75E--5 1.20E--5 7.60E--6

M4 9.28E-- 5 5.40E-- 5 3.35E--5 2.19E--5 1.45E--5 8.87E--6

Z=35 k 6.0 7.5 9.0 12.0 15.0 20.0

E1 2.93E--5 2.12E--5 1.56E--5 1.25E--5 9.63E--6 6.96E--6

E2 4.62E--5 3.14E--5 2.21E--5 1.64E--5 1.21E--5 8.31E--6

E3

E4

6.86E--5 4.44E-- 5 3.01E--5 2.12E--5 1.51E--5 9.90E--6

9.82E--5 6.08E--5 3.99E--5 2.68E--5 1.84E--5 1.17E--5

MI 4.56E--5 3.07E--5 2.14E--5 1.58E 5 1.16E--5 7.93E--6

M2 7.34E--5 4.67E--5 3.12E--5 2.15E--5 1.51E--5 9.83E--6

M3 1.09E--4 6.60E--5 4.26E--5 2.80E--5 1.89E--5 1.18E--5

M4 1.55E--4 9.00E-- 5 5.64E-- 5 3.54E--5 2.32E--5 1.40E-- 5

381

H I G H - E N E R G Y ICC

Z=40 k 6.0 7.5 9.0 12.0 15.0 20.0

E1 4.21E--5 3.04E--5 2.26E--5 1.76E--5 1.36E--5 9.76E--6

E2 6.80E--5 4.62E--5 3.27E--5 2.37E--5 1.74E--5 1.18E--5

E3

E4

M1

M2

M3

M4

1.03E--4 6.65E--5 4.54E-- 5 3.12E--5 2.20E--5 1.43E--5

1.49E--4 9.23E--5 6.10E--5 4.01E--5 2.73E--5 1.71E--5

6.79E--5 4.53E--5 3.17E--5 2.25E--5 1.64E--5 1.11E--5

1.14E--4 7.17E--5 4.80E--5 3.20E--5 2.22E--5 1.42E--5

1.T1E--4 1.03E--4 6.67E--5 4.24E--5 2.84E--5 1.74E--5

M1

M2

M3

M4

9.74E--5 6.41E--5 4.45E--5 3.08E--5 2.21E--5 1.48E--5

1.69E--4 1.06E--4 7.05E--5 4.58E--5 3.13E--5 1.97E--5

2.58E--4 1.55E--4 1.00E--4 6.20E--5 4.09E--5 2.48E--5

3.73E--4 2.15E--4 1.35E--4 8.04E-- 5 5.17E--5 3.02E-- 5

M2

M3

M4

2.47E--4 1.53E--4 1.01E--4 6.36E--5 4.28E--5 2.65E--5

3.81E--4 2.27E--4 1.46E--4 8.81E--5 5.75E--5 3.41E--5

5.53E--4 3.17E--4 1.99E--4 1.16E---4 7.36E--5 4.23E--5

M2

M3

M4

3.52E--4 2.16E--4 1.42E--4 8.68E--5 5.76E--5 3.49E--5

5.50E--4 3.26E--4 2.09E--4 1.23E--4 7.92E--5 4.62E--5

8,01E--4 4.59E--4 2.87E--4 1.64E--4 1.03E--4 5.83E--5

2.45E--4 1.42E--4 8.92E--5 5.43E--5 3.53E--5 2.09E--5

Z = 45 k 6.0 7.5 9.0 12.0 15.0 20.0

E1 5.79E--5 4.18E--5 3.10E--5 2.39E--5 1.83E--5 1.31E--5

E2 9.61E--5 6.51E--5 4.60E--5 3.29E--5 2.39E--5 1.62E--5

E3 1.48E--4 9.56E--5 "6.53E--5 4.42E--5 3.09E--5 1.99E--5

E4 2.16E--4 1.34E--4 8.90E--5 5.76E--5 3.90E--5 2.42E--5

Z= k 6.0 7.5 9.0 12.0 15.0 20.0

El

E2

E3

E4

7.72E--5 5.55E--5 4.13E--5 3.13E--5 2.38E--5 1,70E--5

1.32E--4 8.90E--5 6.30E--5 4.42E--5 3.19E--5 2.13E--5

2.06E--4 1.33E--4 9.12E--5 6.07E--5 4.21E--5 2.68E--5

3.05E--4 1.90E--4 1.26E--4 8.03E--5 5.40E--5 3,31E--5

50 MI 1.36E--4 8.81E--5 6.07E--5 4.07E--5 2.88E--5 1.91E--5

Z=55 k 6.0 7.5 9.0 12.0 15.0 20.0

E1 1,00E--4 7.19E--5 5.35E--5 4.00E--5 3.03E--5 2.15E--5

E2 1,77E--4 1.19E--4 8,41E--5 5.81E--5 4.16E--5 2.75E--5

E3 2.82E--4 1.82E--4 1.25E--4 8.17E--5 5.62E--5 3.54E--5

E4 4.22E--4 2,62E--4 1,75E--4 1,10E--4 7.33E--5 4.44E--5

M1 1.86E--4 1.19E--4 8.08E--5 5.23E--5 3.65E--5 2.38E--5

Z-----60 k 6.0 7.5 9.0 12.0 15.0 20.0

E1 1.28E--4 9.14E--5 6.81E--5 5.00E--5 3.77E--5 2.66E--5

E2 2.33E--4 1.56E--4 1.10E--4 7.49E--5 5.31E--5 3.47E--5

E3

E4

3.79E--4 2.45E--4 1.68E--4 1.08E--4 7.37E--5 4.58E--5

5.73E--4 3.57E--4 2.39E--4 1.48E--4 9.79E--5 5.87E--5

M1 2.51E--4 1.57E--4 1.06E--4 6.58E--5 4.51E--5 2.88E--5

M2 4.98E--4 3.02E--4 1.98E--4 1.17E--4 7.62E--5 4.52E--5

M3 7.85E--4 4.62E--4 2.96E--4 1.70E--4 1,08E--4 6.17E--5

M4 1.15E--3 6.54E--4 4.09E--4 2.29E--4 1.42E--4 7.92E--5

382

0 . DRAGOUN

Z=65 k 6.0 7.5 9.0 12.0 15.0 20.0

El 1.61E--4 1.14E--4 8.53E--5 6.16E--5 4.62E--5 3.23E--5

E2 3.02E--4 2.03E--4 1.43E--4 9.54E--5 6.70E-- 5 4.33E--5

E3

E4

MI

M2

M3

M4

5.04E--4 3.25E--4 2.23E--4 1.42E--4 9.57E--5 5.87E--5

7.70E--4 4.81E--4 3.23E--4 1.97E--4 1.29E--4 7.67E--5

3.36E--4 2.07E--4 1.37E--4 8.15E--5 5.46E--5 3.41E--5

6.99E--4 4.20E--4 2.73E--4 1.56E--4 1.00E--4 5.79E--5

1.11E--3 6.50E--4 4.15E--4 2.33E--4 1.46E--4 8.17E--5

1.62E--3 9.23E--4 5.78E--4 3.17E--4 1.95E--4 1.07E--4

MI

M2

M3

M4

Z = 70 k 6.0 7.5 9.0 12.0 15.0 20.0

E1 2.00E--4 1.42E--4 1.06E--4 7.50E--5 5.59E--5 3.88E--5

E2

E3

E4

3.90E--4 2.61E--4 1.84E--4 1.20E--4 8.37E--5 5.33E--5

6.65E--4 4.29E--4 2.95E--4 1.84E--4 1.23E--4 7.46E--5

1.03E--3 6.43E--4 4.33E--4 2.60E--4 1.70E--4 9.97E--5

Z= k 6.0 7.5 9.0 12.0 15.0 20.0

E1

E2

E3

E4

2.46E--4 1.74E--4 1.29E--4 9.05E--5 6.69E--5 4.60E--5

5.00E--4 3.33E--4 2.34E--4 1.51E--4 1.04E--4 6.51E--5

8.73E--4 5.63E--4 3.87E--4 2.39E--4 1.59E--4 9.45E-- 5

1.36E--3 8.55E--4 5.75E--4 3.44E--4 2.23E--4 1.29E--4

Z=

k 6.0 7.5 9.0 12.0 15.0 20.0

E1

E2

3.02E--4 2.12E--4 1.57E--4 1.08E--4 7.95E--5 5.40E--5

6.40E--4 4.24E--4 2.98E--4 1.90E--4 1.29E--4 7.92E--5

E3 1.15E--3 7.39E--4 5.07E--4 3.10E--4 2.04E--4 1.20E--4

E4 1.81E--3 1.13E--3 7.65E--4 4.53E--4 2.93E--4 1.67E--4

4.50E--4 2.71E--4 1.77E--4 9.96E--5 6.50E--5 3.95E--5

9.79E--4 5.84E--4 3.77E--4 2.09E--4 1.31E--4 7.37E--5

1.56E--3 9.10E--4 5.80E--4 3.18E--4 1.96E--4 1.08E--4

2.27E--3 1.29E--3 8.11E--4 4.36E--4 2.66E--4 1.43E--4

M3

M4

75 MI

M2

6.03E--4 3.54E--4 2.27E--4 1.21E--4 7.63E--5 4.49E--5

1.37E--3 8.10E--4 5.19E--4 2.79E--4 1.72E--4 9.38E--5

2.18E--3 1.27E--3 8.07E--4 4.34E--4 2.65E--4 1.42E--4

3.16E--3 1.80E--3 1.13E--3 6.00E--4 3.63E--4 1.93E--4

M1

M2

M3

M4

8.15E--4 4.67E--4 2.92E--4 1.45E--4 8.83E--5 4.98E--5

1.93E--3 1.13E--3 7.19E--4 3.74E--4 2.26E--4 1.20E--4

3.06E--3 1.77E--3 1.13E--3 5.93E--4 3.58E--4 1.89E--4

4.40E--3 2.51E--3 1.57E--3 8.25E--4 4.95E--4 2.60E--4

M2

M3

M4

2.72E--3 1.58E--3 9.99E--4 5.08E--4 3.01E--4 1.54E--4

4.28E--3 2.48E--3 1.57E--3 8.17E--4 4.88E--4 2.53E--4

6.11E--3 3.48E--3 2.19E--3 1.14E--3 6.81E--4 3.53E--4

80

Z=85 k 6.0 7.5 9.0 12.0 15.0 20.0

E1 3.68E--4 2.58E--4 1.90E--4 1.30E--4 9.41E--5 6.32E--5

E2 8.18E--4 5.40E--4 3.77E--4 2.38E--4 1.60E--4 9.64E--5

E3 1.50E--3 9.67E--4 6.63E--4 4.03E--4 2.63E--4 1.52E--4

E4 2.39E--3 1.50E--3 1.01E--3 6.01E--4 3.86E--4 2.18E--4

M1 1.11E--3 6.21E--4 3.79E--4 1.75E--4 1.02E--4 5.44E-- 5

HIGH-ENERGY

383

ICC

Z = 90 g

El

6.0 4.50E--4 7.5 3.13E--4 9.0 2.30E--4 12.0 1.55E--4 15.0 1.11E--4 20.0 7.37E 5

E2

E3

E4

Ml

M2

M3

M4

1.05E--3 6.91E--4 4.81E--4 3.00E--4 1.99E--4 1.17E--4

1.98E--3 1.27E--3 8.73E 4 5.28E--4 3.42E--4 1.95E--4

3.17E--3 2.00E--3 1.35E--3 7.99E--4 5.11E--4 2.86E-4

1.54E--3 8.37E--4 4.99E--4 2.11E--4 1.15E-4 5.81E--5

3.87E--3 2.24E--3 1.41E--3 6.96E--4 4,04E--4 2.01E--4

6.02E--3 3.49E--3 2.21E--3 1.13E--3 6.71E--4 3.42E--4

8.50E--3 4.86E--3 3.06E--3 1.58E--3 9.40E 4 4.83E 4

E4

MI

M2

M3

M4

4.21E--3 2.67E--3 1.81E--3 1.07E--3 6.84E--4 3.80E--4

2.16E--3 1.15E--3 6.70E--4 2.56E--4 1.31E--4 6.09E--5

5.56E--3 3.20E--3 2.01E--3 9.69E--4 5.53E--4 2.66E--4

8.51E- 3 4.93E--3 3.12E--3 1.59E--3 9.35E--4 4.69E--4

1.18E--2 6.79E--3 4.29E--3 2.21E--3 1.31E--3 6.69E--4

M1

M2

M3

M4

1.21E--2 7.03E--3 4.46E--3 2.26E--3 1.32E--3 6.56E--4

1.65E--2 9.53E--3 6.04E--3 3.13E--3 1.85E--3 9.41E--4

Z = 95 k

El

6.0 5.49E--4 7.5 3.81E--4 9.0 2.80E 4 12.0 1.85E--4 15.0 1.32E--4 20.0 8.59E--5

E2 1.36E--3 8.89E--4 6.17E--4 3.83E--4 2.50E--4 1.44E--4

E3 2.62E--3 1.69E--3 1.16E--3 6.99E--4 4.49E--4 2.52E--4

Z= k

El

6.0 7.5 9.0 12.0 15.0 20.0

6.73E--4 4.67E--4 3.41E--4 2.22E--4 1.56E--4 1.00E--4

E2 1.77E--3 1.16E--3 7.99E--4 4.94E--4 3.19E--4 1.80E--4

E3

100

E4

3.50E--3 2.26E--3 1.55E--3 9.39E--4 5.99E--4 3.32E--4

5.64E--3 3.59E--3 2.44E--3 1.46E--3 9.27E--4 5.13E--4

3.11E--3 1.61E--3 9.21E--4 3.15E 4 1.49E--4 6.26E--5

8.09E--3 4.64E 3 2.90E--3 1.38E--3 7.74E--4 3.62E--4

Note added in proof: T h e K - I C C w e r e c a l c u l a t e d r e c e n t l y z3) f o r ~14Cd a n d 1 5 ° S m , m u l t i p o l a r i t i e s E l . . . E4, M 1 . . . relativistic Hartree-Fock-Slater approximate

M 4 a n d f o r t r a n s i t i o n e n e r g i e s u p t o 17 M e V . T h e e l e c t r o n w a v e f u n c t i o n s w e r e used. W e c a l c u l a t e d

a n a l y t i c a l I C C f o r t h e t w o i s o t o p e s a n d t r a n s i t i o n e n e r g i e s l l , 14 a n d

17 M e V a n d f o u n d t h e m t o b e i n a v e r a g e b y 4 % l a r g e r t h a n t h e p r e c i s e v a l u e s 23). T h e g o o d a g r e e m e n t will p r o b a b l y b e c o m e w o r s e f o r h e a v y e l e m e n t s . T h e c o n t r i b u t i o n o f t h e h i g h e r o r d e r t e r m s i n p R (p - c o n v e r s i o n e l e c t r o n m o m e n t u m , R - n u c l e a r r a d i u s ) t o finite n u c l e a r size c o r r e c t i o n is n o w i n v e s t i g a t e d 24). R e f s . 1 o - l a, 23) i n c l u d e s u c h effects t o first o r d e r in pR, w h i l e a s i m p l e c o r r e c t i o n 27) o n l y removes Coulomb singularities for point nucleus. The author appreciates gratefully communications R. F. O ' C o n n e l l .

f r o m Drs. E. L. C h u r c h a n d

References 1) J. H. Hamilton, in Spin-parity assignments, ed. by N. B. Gove (Academic Press,New York, 1966) p. 31 2) E. Church, A. Schwarzschild and J. Weneser, Phys. Rev. 133 (1964) B35; C. Carroll and R. O'Connell, Nucl. Phys. 80 (1966) 500

384

O. DRAGOUN

3) L.A. Sliv and I. M. Band, in Alpha-, beta- and gamma-ray spectroscopy, ed. by K. Siegbahn, (North-Holland Publ. Co., Amsterdam, 1965) p. 1639 4) M. E. Rose, Internal conversion coefficients (North-Holland Publ. Co., Amsterdam, 1958) 5) H. C. Pauli, COO-1420-137, Department of Physics, Purdue University, Lafayette, Indiana (1967) 6) R. S. Hager and E. C. Seltzer, Nucl. Data A4 (1968) 1 7) Th. W. Elze, T. v. Egidy and E. Bieber, Z. Phys. 184 (1965) 229; Th. W. Elze, Z. Phys. 194 (1966) 280; T. v. Egidy, E. Bieber and Th. W. Elze, Z. Phys. 195 (1966) 489 8) J. A. Moragues, W. Gelletly and M. A. J. Mariscotti, Phys. Lett. 27B (1968) 44l 9) R. K. Smither, Phys. Lett. 25B (1967) 128 10) E. L. Church and J. Weneser, Bull. Am. Phys. Soc. 13 (1968) 893 11) C. O. Carroll and R. F. O'Connell, Phys. Lett. 28A (1968) 105 12) E. L. Church, Phys. Lett. 28B (1968) 168 13) H. C. Pauli, The Niels Bohr Institute Report, Copenhagen (1968) 14) F. Herman and S. Skillman, Atomic structure calculations (Prentice-Hall, Englewood Cliffs, N.J., 1963) 15) O. Dragoun, H. C. Pauli and F. Schmutzler, Max-Planck-Institut ffir Kernphysik Report MPIH 1968-V14, Heidelberg; Nucl. Data (to be published) 16) N. Dragounovfi, O. Dragoun and G. Heuser, to be published 17) R. F. O'Connell and C. O. Carroll, Phys. Rev. 138 (1965) B1042 18) G. W. Hinman, Phys. Rev. 104 (1956) 1332 19) E. L. Church, Bull. Am. Phys. Soc. 12 (1967) 904; H. C. Pauli, Nucl. Phys. A109 (1968) 94; O. Dragoun, P. Jahn, H. Allers, M. Vobeck3~ and H. Daniel, Max-Planck-Institut fi.ir Kernphysik Report MPIH-1956-Vll (Heidelberg); Phys. Lett. 28B (1968) 251; O. Dragoun, CI. Ribordy and O. Huber, Nucl. Phys. A124 (1969) 337 20) J. A. Moragues, W. Gelletly and M. A. J. Mariscotti, Brookhaven National Laboratory Internal Report 12654 (1968) 21) R. L. Graham, in ref. ~), p. 53 22) L.A. Sliv and I. M. Band, in G a m m a luchi (Gamma rays), ed. by L. A. Sliv (AN SSSR, Moscow, 1961) p. 318 23) C. O. Carroll and R. F. O'Connell, Nucl. Phys., to be published and private communication, May 1969 24) E. L. Church, private communication, March 1969