Effects of nuclear decay on atomic electron rearrangements

Nuclear Instruments and Methods in Physics Research A280 (1989) 151-160 North-Holland, Amsterdam

151

Section L Fundamental processes in radiation physics EFFECTS OF NUCLEAR DECAY ON ATOMIC ELECTRON REARRANGEMENTS Yasuhito I S O Z U M I

Radioisotope Research Center, Kyoto University, Sakyo-ku, Kyoto 606, Japan

Experimental and theoretical research on K-shell internal ionization and excitation (K-IIE) accompanying 13decay, X-ray internal conversion and nuclear electron capture is briefly reviewed. New studies to get a better understanding of these phenomena are required both in experiment and theory.

1. Introduction

Since every nuclear transition takes place in the whole system of nucleus plus atomic electrons, the atomic electron cloud is perturbed in most of the nuclear decays. For example, some vacancies in inner shells are produced through internal conversion (IC) or electron capture (EC). These nuclear decays are regarded as first-order phenomena caused by mixing terms between atomic and nuclear variables in electromagnetic or weak interactions. The perturbation of the electron cloud during nuclear decays generally results in additional ionizations and excitations of inner shells, but with very small probabilities, e.g. typically 5 × 10 -4 per 13-decay for the K-shell ionization of 9Oy ( Z = 39). This process, called internal ionization and excitation (liE), is one of the various second-order phenomena during nuclear decays, such as internal bremsstrahlung, internal Compton effect and internal pair creation, liE is maybe the most interesting subject among those phenomena, because it is much concerned with the interaction of the nucleus and the atomic electrons. Therefore, the study of l i E can provide valuable information about the atomic correlation effect. Decay modes discussed in this review are limited to the K-shell l i E (K-IIE) accompanying 13 decay, K internal conversion (K-IC) and K electron capture decay (K-EC). Previous reviews for K-lIE are listed in ref. [1].

2. Experimental work Some measurements are discussed here to show the difficulties in observing the rare phenomena of K-IIE during nuclear decay.

2.1. f l - and fl + decay Since K-lIE accompanying 13- or 13+ decay is signified by the emission of K X-rays of daughter elements, the phenomenon can be observed by the detec0168-9002/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

tion of these K X-rays. A measurable quantity to be compared with theory is the K-hole creation probability per decay, PK, which is defined by the ratio of the total X-ray emission rate to the total 13 activity of the source. There are not many 13 emitters suitable for the measurement of PK. Most 13-active nuclides decay to excited states of the daughter and K X-rays are emitted after K internal conversion of the excited states. Because of the small probability PK (10- 3-10-4/decay), the K X-rays caused by K-lIE are usually buried in identical and more intense K X-rays caused by K-IC. Boehm and Wu first studied K-lIE during 13- decays of 147pm by observing K X-rays of the daughter element [2]. Since their work, the PK-values for 23 nuclides have been measured with various methods of X-ray detection. As a typical example, the spectrum of X-rays emitted from a 99Tc source [3] is shown in fig. 1. 99Tc decays only to the ground state of 99Ru with a half-life of 2.1 × 105 yr. The Ru K X-rays arise from K-lIE while the Tc K X-rays result from ionizing collisions of 13 particles with other Tc atoms in the source. High-resolution X-ray spectrometry, using e.g. a solid-state detector or a curved-crystal spectrometer, is thus necessary to separate K X-rays of the daughter element from those of the parent. The absolute countings of both K X-rays and 13 particles have often been performed to determine PK experimentally. However, absolute 13 counting is not always necessary for some nuclides. For example, when a 13- emitter has a decay scheme with only one "t transition in the daughter and its transition energy is not enough to convert in the K shell or the transition is very weakly converted, the total 13 activity can be determined by the measurement of the v-ray intensity and decay scheme parameters (13 branching ratio, K and total conversion coefficients). A typical example of this case is 151Sm [4]. Measured PK(13-) values are listed in table 1. The measurement for 13+ emitters is more difficult than that for 13- emitters, because the K capture decay (K-EC) inevitably competes with positron emission and I. FUNDAMENTAL PROCESSES

-I

X IX

152

l&

IJJ Iz 104

99Tc

K~r

_~

0

n, UJ 10 3 Z LU n. LU Q.

99Tc

9/2*

(/) IZ

2.1x105y\

8

\

0_ ' 292KEY

99Ru s/2 o 12

14

16

ENERGY

t 20

18

,

,-

22

24

(keV)

Fig. 1. X-ray spectrum from a 99Tc source obtained by Watson, Chulick and Howard [3]. The spectrum was measured using a 30 m m 2 × 30 m m Si(Li) detector with a resolution ( F W H M ) of 250 eV for 26.4 keV 241Am y rays. A 0.25 m m thick Be absorber was used to eliminate background created by the interaction of 13 particles with the X-ray detector. The intensity of Tc K X-rays is about 6 times greater than the intensity of Ru K X-rays. Table 1 Measured P K ( ~ - ) and available theoretical predictions Nuclide

Parameters a)

Experiments b)

Z

BK [keVl

Eo [keV]

36C1 45Ca 6°Co 63Ni 64Cu

18 21 28 29 30

3.2 4.5 7.7 9.0 9.7

714 261 320 67 571

89Sr 9°Sr

39 39

17.0 17.0

1463 546

9Oy

40

18.0

2273

95Nb 99Tc

42 44

20.0 22.1

160 292

n4In 141Ce 143pr

50 59 60

29.2 42.0 43.6

1989 444 933

ta7pm

62

46.8

224

151Sm

63

48.5

76

169Er ]85W

69 75

59.3 71.7

340 430

2°3Hg

81

85.5

214

2°4T1

82

88.0

766

2mBi

84

93.1

1160

Calculations c) (PK X 10 4)

PK × 104

Ref.

CNTM

LC/ISM

LS

22.1 + 3.8 24.3 -t-3.9 14.2 + 1.4 4.6 +0.4 11.9 +0.8 11 _+2 8.6 +0.7 5.4 _+1.0 6.5 +0.8 7.0 + 1.0 7.4 +1.5 3.4 -+0.4 3.89 -+0.16 3.9 +0.3 5.4 _+0.4 1.79 +0.11 2.88 -+0.20 2.92 +0.16 0.81 -+0.09 0.98 + 0.08 0.036 + 0.005 0.023 -+ 0.003 1.0 + 0.2 1.0 +0.3

[9] [10] [11] [12] [13] I14] [10] [15] [10] [15] ll0] [16] [3] [10] [17] [18] [19] [12] [10] [20] [20] [21] [4] [20]

30.0 21.5 12.3 11.5 10.8

17.6 12.9 * 2.60 *

46.09 28.57 14.7 5.54 14.28

6.54 6.54

4.32 3.43

8.97 7.30

6.24

4.38

8.89

5.66 5.18

1.41 1.57

2.88 3.88

4.09 3.11 3.03

2.71 * 1.46

5.42 1.81"* 2.90

2.88

0.39

0.78

2.82

0.012

0.020

2.47 2.25

0.418 0.412

0.81 0.78

2.10

0.073

0.13

2.08

0.57

1.05

2.04

0.74

1.38

1.0

_+0.2

[101

0.11 0.15 1.0 1.12 1.23

'+0.035 -+0.045 +0.1 _+0.11 5:0.10

[22] [231 [24] [12] [4]

a) Z : atomic n u m b e r of daughter element; BK: K-shell binding energy of daughter element; E0: m a x i m u m 13-ray energy.

b) Selected data by high-resolution X-ray spectrometry, such as by Ge or Si detectors or curved crystal spectrometers. c) C N T M : SCF overlap theory by Carlson et al. [5]; L C / I S M : relativistic hydrogenic one-step theory by Law and Campbell [6] and Isozumi, Shimizu and M u k o y a m a [7]; LS: O D F S approximation in SCF one-step theory by Law and Suzuki ( * *: L D A approximation) [8].

Y. Isozumi / Effects of nuclear decay on atomic electron rearrangements

153

Table 2 Measured PK(13+ ) and available theoretical predictions Nuclide

Z

Calculations b) (PK X 10 4)

Experiments

Parameters a)

PK

BK

E0

[keV]

[keY]

5sCo

26

8.3

474

64Cu

28

9.7

657

6sZn 6SGa

29 30

10.4 11.1

325 1880

x 10 4

Ref.

CNTM

LS

I

13.8 4- 2.4 16.6+2.1 13.24-0.8 13.3 4-1.1 16.1 + 3.0 10.3 + 1.0

[26] [271 [13] [28] [29] [30]

13.2

6.7

13.2

11.5

5.8

10.6

10.8 9.7

4.8 5.54

12.4 *

a) Z: atomic number of daughter dement; BK: K-shell binding energy of daugther element; E0: maximum 13-ray energy. b) CNTM: SCF overlap theory by Carlson et al. [5]; LS: ODFS approximation in SCF one-step theory by Law and Suzuki [8]; I: hydrogenic one-step theory including direct collision by Intemann [25]. p r o d u c e s K X-rays identical to those caused b y K - l I E . Since the ratio of p o s i t r o n emission to K - E C increases with increasing decay energy a n d decreasing atomic n u m b e r , 13+ emitters practically suitable for the PK m e a s u r e m e n t are limited to those with high decay energy in light elements. T h e m e a s u r e m e n t s have so far b e e n p e r f o r m e d o n 58Co, 64Cu, 65Zn a n d 6SGa, as listed in table 2. T h e m o m e n t u m spectra of electrons emitted in K - I I E d u r i n g 13- decays of 89Sr, 143pr a n d 147pm were measured using the A r g o n n e double-lens magnetic spect r o m e t e r [31,32]. In fig. 2 the s p e c t r u m of 143pr is given. T h e s p e c t r u m consists of atomic a n d nuclear electrons simultaneously ejected in K-lIE. Detailed c o m p a r i s o n s b e t w e e n those spectra a n d the recent one-step theory with S C F a n d D i r a c wave functions have been perf o r m e d by Law a n d Suzuki [8].

2.2. K internal conversion (K-IC) and K electron capture (K-EC) A d o u b l e K-hole ( K - 2 ) state is created in K - I I E d u r i n g IC transitions or E C decays. T h e d e t e r m i n a t i o n

of the double K-hole creation p r o b a b i l i t y per decay, PKK, relies usually o n m e a s u r e m e n t s of the K X-rays associated with the filling of the e m p t y K shell. T h e first emitted K X-rays are called hypersatellite lines, d e n o t e d b y ctrt a n d 13H, c o r r e s p o n d i n g to the transitions ( K -2 --, K - 1 L - 1 ) a n d ( K -2---, K - I M -1 or K - 1 N - 1 ) , respectively. T h e next emitted K X-rays are satellite lines, e.g. ~xS(K-1L -1 --, L - 2 ) , 13S(K-1L-~ ~ M - 1 L -1 or N - 1 M - 1 ) , etS'(K-1M -1 ~ L - 1 M - 1 ) a n d 13S'(K-1M-1 -o M - 2 a n d K - 1N - 1 __. M - 1N - 1). T h e hypersatellite a n d satellite lines are shifted to higher energies with respect to d i a g r a m lines; the shift of the hypersatellite is a b o u t ten times larger t h a n t h a t of the satellite. B r i a n d et al. first observed the hypersatellite line in the E C decay of 71Ge by a K X - r a y - K X-ray coincidence experiment [33]. M e a s u r e d PKK(IC) a n d PKK(EC) are listed in tables 3 a n d 4, respectively. PKK is usually m e a s u r e d b y c o u n t i n g coincidences between satellite a n d hypersatellite lines. It is essential to avoid false events in such X - X coincidence measurements, e.g. coincidences between two K X-rays emitted by cascade K-IC transitions or coincidences between a K X-ray b y K - E C a n d

Table 3 Measured PKK(IC) and available theoretical predictions Nuclide

Experiments

Parameters a) Z

BK [keV]

Ev [keV]

PK × 105

Ref.

l°9mAg

47

25.5

88

114mIn 137mBa

49 56

27.9 37.4

192 662

139La 141pr

57 59

38.9 42.0

166 145

203Ti

81

85.5

279

15.3 -/-2.4 13.0+ 1.1 10.2 4- 0.6 7.1 + 3.5 10.04-0.9 6.0 4-1.4 6.3 4- 4.1 3.4 + 0.3 4.0 + 1.5

[35] [36] [37] [38] [39] [40] [41] [421 [43]

Calculations MS

b)

(PK)< 105)

0.916 2.06 3.76 * 4.32 3.63

atomic number; BK: K-shell binding energy; Ev: v-transition energy. b) MS: Relativistic one-step theory by Mukoyama and Shimizu [34]. a) Z :

I. FUNDAMENTAL PROCESSES

154

Y. Isozumi / Effects of nuclear decay on atomic electron rearrangements

143p

20

.

°

~-, 9 3 3

143pr

ke~

~-.

15

143Nd

N(P)

1o

.L NUCLEAR B K ORBITAL

][

o

o

1.o 2.0 I I I I [ I ~(mc t~nits) i 3.516 38 75 125 200 300 450 600 keV

3.0 I 900

Fig. 2. Momentum plot of the composite electron spectrum for 142pr measured by Fischbeck, Wagner, Porter and Freedman [31]. The spectrum was obtained in coincidence with K X-rays of the daughter element Nd. The steep rise in the low-momentum part is caused by the atomic electrons, while most of the nuclear electrons form the high-momentum part.

a n o t h e r K X-ray by K-IC. The m e a s u r e m e n t of PKK (IC) is practically limited to the IC transition between m e t a stable a n d g r o u n d states (l°9mAg, lX4mIn a n d 137mBa) or the IC transition between the first excited a n d g r o u n d states in the d a u g h t e r of a 13--active nuclide (139La, 141pr a n d 2°3T1). A special case is 139La, the d a u g h t e r of the E C - d e c a y i n g n u c l i d e 139Ce, a t r i p l e - c o i n c i d e n c e technique was applied to remove the c o n t r i b u t i o n from K - I I E d u r i n g K-EC. PKK (EC) is a b o u t one order smaller t h a n PK(13-) a n d PKK(IC) for the same element, as seen in tables 3 a n d 4. Therefore, the m e a s u r e m e n t of PKK(EC) is generally more difficult; m e a s u r e m e n t s are p e r f o r m e d w h e n E C leads to the g r o u n d state of the d a u g h t e r nucleus (5~Fe a n d 131Cs) or to a m e t a s t a b l e state with a n excitation energy less t h a n the K - b i n d i n g energy of the d a u g h t e r a t o m (165Er a n d 181W). Some groups successfully m e a s u r e d PKK for nuclides with m o r e complicated decay schemes (54Mn, 65Zn, 85Sr, l°3pd, l°9Cd a n d 2°7Bi). As a typical example, in fig. 3 we show m e a s u r e d spectra of K X-rays emitted in the E C decay of ~31Cs [52]. These spectra were o b t a i n e d b y K X - r a y - K X-ray coincidence experiments. Two b u m p s a p p e a r o n the s p e c t r u m RT: a K a hypersatellite (et~ + et2H) o n the high-energy side of the K a d i a g r a m line a n d a KI32 hypersatellite o n the high-energy side of the K I32 dia g r a m line. Thus, hypersatellite lines of h i g h - Z elements can barely b e resolved by solid-state detectors. A s p e c t r u m of electrons ejected in K - l I E d u r i n g IC has n o t b e e n observed yet because of the great difficulty of such measurements. There is only one truly valid

Table 4 Measured PKK (EC) and available theoretical predictions Nucfide

Parameters a)

Experiments b)

Z

PK x 105

Ref.

SL

I

ILS

36 _+3 12 _+4 22 _+2 6.0 +0.5 3.13 + 0.31 1.02 + 0.36 1.33 + 0.33 1.4 _+0.1 0.67 + 0.39 0.82_+0.28 0.24 _+0.06 0.6 _+0.25

[47] [38] [48] [49] [50] [36] [51] [521 [51] [531 [53] [54]

24.3** 18.80 15.3 9.38 5.59 0.83 2.99

11.25 9.42 * 3.38 1.74 0.34 0.75

11.42 9.04 6.82 3.14 1.32 0.19 *

1.58

0.26

*

0.14 * * 1.81

0.022 0.11 d)

* .

BK

Calculations c) (PK × 105)

[keV] S4Mn 5sFe 65Zn SSSr 103Pd l°9Cd 131Cs

24 25 29 37 45 47 54

" 6.0 6.5 9.0 15.2 23.2 25.5 34.6

165Er

67

55.6

aSlW 2°7Bi

73 82

67.4 88.0

a) Z: atomic number of daughter element; BK: K-shell binding energy of daughter element. b) Selected data by high-resolution X-ray "spectrometry, such as by Ge or Si detectors or curved crystal spectrometers. c) SL: ODFS approximation in SCF theory in variational approach by Law and Suzuki (**: LDA approximation) [44]; I: semi-relativistic hydrogenic theory in Coulomb-propagator approach by Intemann [45]; ILS: hybrid theory of variational and propagator approaches by Intemann, Law and Suzuki [46]. d) Contribution from 10% 2 + branch not included.

155

Y. lsozumi / Effects of nuclear decay on atomic electron rearrangements

5./'2" 131Cs

104

~

~

~Z103 -

l

z

¢n

350ECkeV~

Kdr

.J

131Xe

R

RT

~K~2 -" ! ~

HYPE~3;~E

~/

2°°r

~T(=RT--R)

1 ~°°1- ~tt I

P/3//2 .

~---K~

10]00

260

9.7d

i HYPERSATELLI-I-E ~

10

131/'~s

I 280

%/ I

I I 300 CHANNEL

,

N

~

__1

I I I 320 340 NUMBER

I

experiment may be helpful to understand the difficulty in the determination of PK and PKK" (1) X-ray detection efficiency: Commercially available photon sources are usually used to calibrate the efficiency of y-ray detectors. However, such sources are not available for X-ray detectors because of the uncertainty in the X-ray emission rate. In ref. [52] one absolute X-ray counting measurement is necessary for the determination of PK and PKK" In fact, the overall efficiencies of Ge detectors for Xe K X-rays were determined with a ]3]Cs source, carefully calibrated in the Laboratoire de Mrtrologie des Rayonnements Ionisants ( C E N Saclay) with an uncertainty of 2%. We found more than 15% systematic deviation between the efficiency determined by the calibrated source and that by commercially available photon sources. (2) Coincidence efficiency: The coincidence efficiency is often determined by a different source which emits two coincident X-rays having similar energies. However, this is very dangerous because the coincidence curve strongly depends on the energies of the detected photons. To estimate the efficiency in the experiment with 13]Cs, we used directly the time spectrum obtained

I 360

Fig. 3. K X-ray-K X-ray coincidence spectra for 131Cs obtained by Isozumi, Brianc;on and Walen [52]. RT: photon spectrum observed by the pure Ge detector in coincidence with Xe K X-rays detected by the second Ge detector; R: corresponding random coincidence spectrum; T ( - - - R T - R ) : true coincidence spectrum obtained by subtracting the spectrum R from the spectrum RT; F: unwanted-coincidence spectrum.

I

2. 3. On the possible sources o f systematic errors

In order to obtain PK and PKK of high reliabilities, we have to be very careful for non-negligible systematic errors often remaining in experimental data. The author went through tedious efforts on removing such errors from the determination of PKK for the EC decay of 13]Cs [52]; the preliminary result, (2.3 -t- 0.3) × 10 -5 for PKK, was corrected to (1.4 -t- 0.1) x 10 -5 after thorough re-examinations of the data analysis and the whole instrumentation. Discussion of error sources in this

l

|

I

t

I

i

f

i

f

I

|

lo-2

I

1

I

5~Fo

,

10-3 ~.

55 F e

/

EC 232

/

keY

°tMn /5/2 -

~-. lo.4

o ,~

measurement of electrons emitted in K - I I E during KEC. Fig. 4 shows the measured spectral distribution of the ejected K electrons during the EC decay of 55Fe [55]. The comparison of the spectrum with theory in ref. [55] is rather out of date in the present situation. A refined analysis of this spectrum with recent theoretical approaches [44-46] is clearly required.

I

\ x-x~x~,_

lO-~

\

10

I 1-,

~ ~%~~

-6

10 "e 0

~ ~ ~ ~ I , 50

t

~ , I , 100

,

,

T

,

I ,k ,I 155

EK(keV)

Fig. 4. Spectral distribution of the ejected K electrons during EC of 55Fe measured by Kitahara and Shimizu [55]. The spectrum was obtained by the triple-coincidence (K X-rayejected electron-K X-ray) experiment using a gas proportional counter for ejected electrons and two gas proportional counters for Mn K X-rays. For the higher energy region of ejected electrons the confinement of the electrons within the sensitive volume of the electron counter was achieved by applying a 4 kG magnetic field. I. FUNDAMENTAL PROCESSES

156

Y. lsozumi / Effects of nuclear decay on atomic electron rearrangements

by classifying the original data, which had been recorded event by event on magnetic tapes through the list-mode operation of data processors. The discrepancy between efficiencies differently determined was surprisingly large; the efficiency determined by a ]33Ba source is 65% of that estimated from the original time spectrum. (3) Unwanted coincidence counts: Counts caused by unwanted events are sometimes included in the true coincidence spectrum. The following coincidence process mainly contributes to the Ket peak of the true coincidence spectrum obtained by the experiment with ~31Cs (spectrum T in fig. 3): (a) Xe K X-ray ® Xe K X-ray during the neutralization of the double K-hole initial state; (b) Xe K X-ray ® IB (internal bremsstrahlung) during EC; and (c) Xe K X-ray ® Ta K X-ray due to the interaction of high-energy IB photons with Ta diaphragms in the source holder. Coincidences due to (b) and (c) are the unwanted counts in this experiment. It is possible to reproduce a spectrum caused by (b) and (c) from the original data on magnetic tape, as given by spectrum F in fig. 3. It is clear that peak counts of spectrum F may cause systematic errors in the X - X coincidence counts.

3. Theoretical work The perturbation of the atomic electron cloud in 13 decay, internal conversion and electron capture appears instantaneously, since the time scale of those decay modes is usually much shorter than the period of motion of orbital electrons. Therefore, the ordinary theory for sudden perturbation can be employed to treat the K-shell internal ionization and excitation during those decays, while the full time-dependent perturbation theory (e.g. adiabatic approximation) is applied to K-lIE during cx decay because of the small velocity of et particles. In the simplest approach for K-lIE during 13, 1C and EC decays, the perturbation is approximately expressed by a change in nuclear charge or in effective charge including the shielding effect by inner electrons. According to the sudden approximation [56], the matrix element for the monopole transition from an initial state to a possible final state with a different nuclear charge is given by the overlap integral between the initial and final states. The monopole transition of a K electron to a continuum state is called K-electron shakeoff (K-SO) and the transition to bound states is called K-electron shakeup (K-SU). The simple picture based on the overlap integral is not enough to express the K-IIE phenomena quantitatively. Recent theoretical works have revealed that many-body effects, e.g. electron-electron correlation and screening by other electrons, should explicitly be taken into account for a quantitative comparison of

measured PK and PKK data of high precision. Actually, the sensitivity to correlations or configurations makes K-lIE a valuable probe of the atomic inner-shell structure. Mechanisms of K-lIE, so far established or proposed for each nuclear decay, are summarized in this chapter. Historical sketches of previous investigations are also given. 3.1. f l - decay

In 1941 Feinberg [57] and, independently, Migdal [58] for the first time studied theoretically K-SO and K-SU during 13 decay. The approach based on the overlap integral was followed by Schwartz [59], Levinger [60], Winther [61], and Green [62]. Following the scheme of Levinger, Carlson et al. [5] computed the monopole transition probability including the contributions from K-SO and K-SU: PK = 2(1 - PR

-

-

PF),

(1)

where PR is the probability for a K electron to remain in the K shell, PF is the probability for excitations to occupied shells which are forbidden by the Pauli principle, and the factor of 2 accounts for two electrons in the K shell. They used self-consistent-field (SCF) wave functions for bound electrons, i.e. nonrelativistic Hartree-Fock ( Z < 30) and relativistic Hartree-FockSlater ( Z _> 30) wave functions. Two important factors are ignored in the above overlap approaches, i.e. the phase-space sharing between three leptons (13, atomic electron and neutrino) and the quantum exchange between two continuum electrons in the final state. A one-step approach including the phase-space sharing was first developed by Stephas and Crasemann [63] and followed by several other workers [64,65]. Law and Campbell elaborated the one-step approach taking account of the quantum exchange [66,20,6]. A counting error in the approach (an overestimate by a factor of two) was corrected by Isozumi, Shimizu and Mukoyama [7] and soon confirmed by Cooper and .Aberg [67], after a considerable controversy [68]. The one-step theory based on the relativistic hydrogenic wave functions was established in this way. As long as the same wave functions are employed, PK deduced from the one-step theory is less than that based on the overlap approach. As shown in fig. 1 of ref. [7], the decrease can be approximately expressed by a function of B K / B o, where B K is the binding energy of the daughter element and B0 is the maximum energy of the 13 particle. In table 1 are listed the overlap calculations by Carlson et al. (CNTM) and the calculations based on the one-step theory (LC/ISM), respectively. As seen in table 1, there is an evident discrepancy between experiment and the one-step theory; PK values estimated from the theory (LC/ISM)

Y. lsozumi / Effects of nuclear decay on atomic electron rearrangements

~ro roue,ldy ono-half the experimentally determined values. In addition to K-SO and K-SU, there is another possibility of K-lIE by means of a direct collision (DC) between the emerging 13 particle and an atomic electron, i.e. 13-e correlation in the final state. Feinberg [57] first discussed this process, suggesting that the DC contribution to the probability PK is negligibly small; the relative importance of SO and DC was estimated as PK(DC)/PK(SO)-BK/Eo. Stimulated by the 13-e coincidence experiments [69], Feinberg [70] re-examined his DC theory and pointed out that the ratio P K ( D C ) / PK(SO) may not be less than unity for high Z and low decay energy. A similar theoretical result was obtained also by Weiner [71]. Thereafter, in order to search for the DC contribution, several groups [72] measured the K-shell ionization probability as a function of electron energy. Although most of the measurements did not result in a reliable conclusion on the DC contribution, the precise electron spectra measured with the Argonne double-lens magnetic spectrometer indicates that the lowest limit on the contribution of the DC process is of that predicted by the Feinberg theory [31,32]. In order to explain the discrepancy between measured PK values and calculations from the one-step theory (corrected for the SCF values by Carlson et al. [5]), Isozumi, Shimizu and Mukoyama [7] again suggested that the DC contribution to P t may be much larger than that assumed from Feinberg's estimate so that a consistent theory of K-IIE must take both SO and DC mechanisms fully into account. A satisfactory theory including the DC process has been developed by Intemann [73-75]; through numerical calculations with nonrelativistic hydrogenic wave functions it has been concluded that the DC contribution is not large even under the circumstances which ought to be favorable for the DC mechanism. A similar theoretical approach was performed by Batkin et al. [76]. Many-body effects in K-IIE during 13 decay were discussed by Cooper and .~berg [67] and by Mukoyama and Shimizu [77]. Using SCF functions, Law and Suzuki [8] performed systematic calculations according to the one-step approach for K-SO and K-SU. The DiracFock-Slater model with optimized potentials for the exchange interaction (ODFS) [78] was employed for the ground state of the parent atom. The configuration in the final state was carefully reconsidered in their SCF calculations. It is implicitly assumed in the SCF calculation by Carlson et al. [5] that the final continuum electron moves in the SCF potential defined from the solution of the final ground-state ion. The K-shell occupation in this case is 2. However, the actual final state compels the K-shell occupation to be 1, and one should find the SCF functions for the configuration with a hole in the K-shell. They used SCF wave functions for the final state with such a configuration, which were

157

evaluated from three different approximations for the exchange interaction part in the SCF potentials, i.e. the unmodified potential (ODFS), the modified SCF potential using the local density approximation (LDA) and the modified SCF potential by the average Fermi momentum approximation (AKF). The agreement between their results (ODFS) and experimental values are excellent, as seen in table 1. The discrepancy between the measured PK and the relativistic hydrogenic onestep calculation (LS/ISM) was thus resolved. 3.2. fl + decay

Except for the SO and DC processes, there is another lowest-order mechanism for K-IIE during 13+ decay: the virtual annihilation and creation of electronpositron pairs. However, the amplitude of this process is expected to be negligibly small, because the average 13+ particle energy for almost all 13+-decaying nuclides is much less than 2 m c 2. There appears a certain puzzle when the SCF calculations from Law and Suzuki [8] are compared with PKI,: measured for 13+ emitters; the measured values are larger than the predictions (LS) by factors of 2-3, as seen in table 1. To solve this striking discrepancy, Law discussed the possibility that, while the DC contribution will be small for 13- decay, it can be quite large for 13+ decay [79]. Stimulated by this, Intemann studied the relative importance of the DC contribution to PK(I3+) using a theory similar to that developed for the study of K-lIE during EC decay [25]; the theory is based on nonrelativistic hydrogenic wave functions and a Coulomb Green function originally developed by Glauber and Martin [80], and it yields results whose relative accuracy is of order Zcc As seen in table 1, the predictions by Intemann satisfactorily agree with measured PK(13+) values for 58Co, 64Cu and 65Zn. Thus, contrary to the case of 13- decay, the DC process plays an essential role in K-lIE during 13+ decay. 3.3. K-shell internal conversion (K-IC)

The principal mechanisms for K-lIE during IC transitions are K-SO and K-SU, which is similar to that observed in the 13 decay. There are other possible mechanisms, e.g. direct collision (DC) between the converted and unconverted electrons, double internal conversion [81] and internal conversion of the internal Compton effect [82], which probably does not make a large contribution to the double K-hole creation in IC transitions. K-SO and K-SU result from the sudden change in the effective nuclear charge rather than from the charge itself. PKK(IC) was calculated by Mukoyama and Shimizu [83,34], using screened relativistic hydrogenic wave functions; the screening constants were determined from the relativistic SCF calculations and the I. FUNDAMENTAL PROCESSES

158

Y. lsozumi / Effects of nuclear decay on atomic electron rearrangements

presence of a vacancy resulting from IC was taken into account. In table 2 their results are listed; the agreement between theory and experiment is not good except for the case of 2°4T1.

3.4. K-electron-capture decay (K-EC)

The theoretical treatment for K-lIE during EC decay is more difficult than that for 13 decay, because the correlation between two K electrons in the initial state, in addition to the change of nuclear charge, should be taken into account. The theory for this phenomenon has been developed in two ways, i.e. the variational method and the Coulomb propagator method, which are distinguished by how to treat the initial K - K correlation. In the variational method the electron-electron interaction is included in the unperturbed Hamiltonian. The calculation by this method becomes too sophisticated if eigenfunctions of the Hamiltonian are deduced by introducing adjustable parameters, e.g. a screening constant and a spatial correlation constant. Then its accuracy may not be so reliable that it can be compared with recently measured data of PKK" This method was first employed in the pioneering work by Primakoff and Proter [84] and was developed by several groups [85-87]. In the Coulomb propagator method, the electron-electron interaction is regarded as a perturbation on the nuclear Coulomb interaction. Since the initial K - K correlation can be included in two-electron wave functions derived from the conventional perturbation theory, this approach is free from adjustable parameters. This approach was first developed by Intemann and Pollock [88], subsequently by Intemann [89,90,45] and independently by Law and Campbell [91]. Intemann expressed the two-electron wave function using Coulomb Green functions similar to those of the theory of internal bremsstrahlung accompanying EC decay [80]. In his semi-relativistic calculation, he employed the solution of the symmetric Hamiltonian of Biedenharn and Swamy [92] for the Coulomb Green functions, which differs from the exact Dirac-Coulomb eigenfunctions by terms of order ( a Z ) 2. As proved by Intemann [90], his approach is identical with that of Law and Campbell [91]. SCF calculations in the variational approach have been performed by Suzuki and Law [44], applying the same procedure as used in their study for K-lIE during 13 decay [8]. Under the SCF scheme, the initial state was given by the bound state produced by the SCF one-body potential which emerges from the Dirac-Fock-Slater theory. The continuum electron state was given as the scattering state produced in a similar SCF potential of the daughter atom, which has an atomic configuration with a double hole in the K shell. They found that screening of the nuclear charge by remaining bound electrons results in an increase of PKK"

The SCF calculation (SL) does not adequately take into account the initial K - K correlation, because it is still based on an independent particle model. In contrast, the calculation by Intemann (SRP) [45] treats well the K - K correlation, but it is not fully relativistic. Recently a hybrid approach has been followed [46], which combines the advantages of SL and SRP calculations; a relativistic correlation-split variational wave function is employed for the initial two-electron state and the final atomic state is given by the same SCF wave functions as used in the SL theory. PKK values predicted by various approaches are listed in table 4. The overall agreement between the measured PKK and each theoretical approach is not so good as in the case of 13- decay. This means that the cancellation of the change of nuclear charge by the initial K - K correlation is very sensitive to atomic wave functions. Further theoretical investigations on K-lIE during EC are required.

4. Concluding remarks As seen in tables 1-4, a lot of experimental PK and PKK values available for comparison with theory have been accumulated with the aid of modern nuclear instrumentation. However, in order to obtain fruitful conclusions from quantitative comparisons with recent theoretical approaches, the precision of the experimental data has to be at least < 10%. Such precise data of PKK(IC) and PKK(EC) for high-Z elements are especially desired, though experiments become extremely difficult. In recent theoretical approaches for K-lIE during 13-, 13÷ and EC decays, many-body effects are reasonably included by using appropriate SCF wave functions for the initial and the final atomic configurations [8,44], and electron-electron correlations in the initial and the final states are treated explicitly [74,25,45]. It is now possible to investigate the atomic structure of inner shells in detail by comparing experimental PK or PKK values with predictions from those recent approaches. Actually, the SCF one-step theory for K-lIE during 13decay (LS in table 1) succeeds in explaining the group of experimental PK(13-) values quantitatively, indicating that the atomic configuration in the final state should be correctly taken into account [8]. It has also been proved that, in addition to the final atomic configuration, the 13-e correlation in the final state (DC mechanism) plays an important role in K-IIE during 13+ decay [25]. There seems to remain a problem to be solved in the existing theories for K-lIE during EC, i.e. some systematic deviations between experimental PKK(EC) and each theoretical approach [44-46], as seen in table 4.

Y. Isozumi / Effects of nuclear decay on atomic electron rearrangements

The theory for K-IIE during IC is not quite satisfactory; the a p p r o x i m a t i o n b a s e d on the s u d d e n c h a n g e in effective nuclear charge is too r o u g h [83,34]. In a more reasonable picture, the initial K - K correlation p r o m o tes the ejection of a n o t h e r K electron in a d d i t i o n to the c o n v e r t e d K electron. The K - l I E d u r i n g IC may b e a n ideal p r o b e to investigate the correlation in the K shell. Clearly, a new a p p r o a c h for the K - K correlation like t h a t b y I n t e m a n n [74,25,45] should b e developed. T h e a p p r o x i m a t e matrix element of K - l I E for each nuclear decay is symbolically expressed as M ( 13- ) = [overlap integral], M ( 1 3 + ) = [overlap integral + final 13-e correlation], M ( I C ) = [initial K - K correlation], M ( E C ) = [overlap integral - initial K - K correlation]. It is desired to p e r f o r m theoretical calculations according to the above formulations, using the same wave functions for all decay processes. Such calculations will b e available to establish a more general aspect o n the m e c h a n i s m of K - l I E in nuclear decay. W i t h the X - r a y - X - r a y coincidence m e t h o d it m a y be possible to observe the double K-hole creation in [3decay. By studying this p h e n o m e n o n , the interference between the term of the overlap integral a n d that of the K - K correlation c a n b e m a d e m o r e clear t h a n the case of K - l I E d u r i n g EC decay. Possible candidates for such a m e a s u r e m e n t are 63Ni a n d 147pm.

Acknowledgements T h e a u t h o r acknowledges the c o n t i n u i n g encouragem e n t b y S. Shimizu. He expresses his t h a n k s to R.J. W a l e n a n d Ch. Brian~.on for the stimulating experiment with 131Cs at C e n t r e de Spectrom&rie N u c l t a i r e et de S p e c t r o m & r i e de Masse, Orsay, France. T h e invaluable help of R.L. I n t e m a n n , J. Law, G. Schupp, B. Crasem a n n , M.S. F r e e d m a n , T..Aberg, W. Bambynek, H.H. H a n s e n a n d C.W.E. v a n Eijk is also acknowledged.

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