Capture of outer orbital electrons

Nuclear Physics 64 (1965) 197--208; (~) North-Holland Publishiny Co., Amsterdam Ngt to be reproduced by photoprint or microfilm without written permission from the publisher

CAPTURE OF O U T E R ORBITAL E L E C T R O N S BEROL L. ROBINSON

Western Reserve University *, Cleveland, Ohio, USA and

Soreq Research Establishment tt, Israel Atomic Eneryy Commission, Rehovoth Received 10 July 1964 A b s t r a c t : Theoretical electron capture rates are given for all electrons captured in unique second

and third forbidden transitions. The MI/L I capture ratios and M subshell capture ratios are tabulated for 55 ~< Z --< 90, and capture from the N, O, and P shells is estimated. The available experimental data on La 138 and Po ~°~ are re-interpreted in the light of these results, and M/L capture ratios are given as functions of transition energy for the low-energy capturers Pt l°3, Pb ~°~ ~and Pb 2°5. Comparative half-lives of unique second and third forbidden transitions are tabulated and discussed in connection with La x38 decay.

1. Introduction

This work is an extension to more unusual situations of the basic results of Brysk and Rose on the theory of orbital electron capture 1), which contained numerical results on the L and K wave functions as well as analytic expressions for capture rates and a discussion of a number of pertinent points concerning branching ratios and the wave functions of bound electrons. In view of the importance of capture from the L m subshell in low-energy forbidden transitions, notably the unique first-forbidden transitions in K 4° and T12°*; it seemed desirable to calculate the capture rates for electrons of larger orbital angular momentum. Since these electrons occur in shells of higher principal quantum number, it was necessary to obtain capture rates for all the M subshells at least. The MI, MII and M m capture rates can be found by using suitable wave functions in the expressions given in ref. 1) for allowed, first-forbidden and second-forbidden transitions; but an extension of that work was needed to account for the capture of electrons of higher angular momentum. Capture rates have been calculated for all electrons captured in unique second and third forbidden transitions; and using relativistic Thomas-Fermi-Dirac screened wave functions for atoms with finite nuclei, the capture ratios were evaluated for some examples. The present and potential availability of more detailed experimental data on some less common nuclear species has made it seem worthwhile to present here results and some related calculations at this time. t Permanent address. The work at Western Reserve University was supported in part by the National Science Foundation. ** On leave of absence from Western Reserve University, 1961-1962. 197

198

B. L. ROBINSON

2. Theoretical Electron Capture Rates for Unique Forbidden Transitions

The method outlined by Brysk and Rose 1) for obtaining electron capture probabilities from the published beta-decay shape factors has been used to extend their results to all electrons which may be captured in unique second and third forbidden transitions. These rates are given in table 1, and for the sake of comparison, the TABLE 1 Theoretical electron capture rates for allowed and unique forbidden transitions ')

2

=

a)(n_,g_lq_,+n,f,ql)

S(2) =

~2 4 x q _ 9 R - 2 / n , - 2 2x.n 3(3!!)2 aAfl2(1, a)[(n-~gE~q4-~ + n 1Jlql) ( - 2 g - 22 q - 22- t - " n 2J2 q2)l

4

S~3) _

a a A I 3 ( 2 , a)[(n-lgZ-lq6~+nlfP q l6) + 3 0 R - -4 3(5li) 2 2 4+n2f~q2)+225R

St34) _

-2

2 4 2 (n_2g_2q_

--4.

2 2 (n-ag-aq-3+nafEq2)]

4 ctaaI4(3, a)[(n -1 g2-1 qS-1 + nl H q l s) 3(7!!)2 2 6 2 6 + 6 3 R --2 (n_2g_2q_2+n2fiq2)+1575R

-4

2 4 (n-3g-aq-a

..~.-n ~,2

4x aJaq3)

+ (105)2R- 6(n_,, g 2 ,*q 2 ,, + n4 f2q~)]

capture rates previously obtained for allowed (Gamow-Teller) and for unique firstforbidden transitions are included. The total electron capture probability for the Ath atomic shell is aSr~l, but the shell index A = K, L, M, etc., has been suppressed throughout; the righthand subscript refers to the degree of forbiddenness and the superscript to the change of nuclear angular momentum. The quantities n, q, f a n d g bear the suppressed shell index A and the subshell index ~c, the Dirac quantum number; the correspondence between ~c and the Bohr-Coster notation for the subshells is t¢ ~---

- 1 for the K shell and the LI, MI . . . . subshells;

1 for the Lii , M n , . . . subshells; - 2 for the Lui , Mll I . . . . 2 for the Miv, N w . . . .

subshells; subshells;

- 3 for the My, Nv, • • • subshells; etc. The functionsf~ and g~ are the two relativistic radial wave functions for an electron in the state x, evaluated at some position in the nucleus, traditionally at the surface

ELECTRON CAPTURE

199

of the nucleus. For positive ~, f~ >> g~ ; and for negative x, f~ << g~; and we speak of the large and small components. Only the large components enter into the transition rates given in table 1, but (non-unique) forbidden transitions depend upon both small and large components. The quantity n~ is the fraction of the maximum number of places in the subshell which are filled; that is, n~ = 1 if the subshell is filled and n~ = 0 if it is empty or non-existent: for example, for the K shell all n~ = 0, except n_ 1 = 1. The nuclear radius R = 1.2.4 ~ fm is expressed in units of the electron Compton wavelength as R = 0.426 ~ A ~, where ~ denotes the fine structure constant. Although 50 < 1/R < 100, the small density of high angular momentum electrons in the vicinity of the nucleus serves to keep down the capture of such electrons. Finally, q~ is the momentum (in units of mc) of the neutrino emitted when a A~ electron is captured. The neutrino momentum is practically a constant for a given shell, since the subshell energy differences are negligible except for very small neutrino energies, that is, except when the transition energy is only slightly greater than the X-ray edge of the daughter. The terms corresponding to the capture of higher angular momentum electrons have successively lower powers of energy dependence; thus in very low energy transitions higher angular momentum electrons are much more likely to be captured. Table 1 presents only the leading contributions to electron capture. Odiot and Daudel 2) and Bahcall in a more explicit fashion 4) have shown that there are additional contributions to electron capture attributable to the indistinguishability of the atomic electrons (exchange) and to the fact that the final atomic states are states of different atoms (overlap). The exchange contributions interfere with the contributions given here, and change the capture rates from different shells by as much as twenty percent for low Z.

3. Theoretical Capture Ratios 3.1. THE MI/LI CAPTURE RATIOS Odiot and Daudel 2) included exchange and overlap effects in a calculation of electron capture rates in A 37 and found that these effects would make the L / K capture ratio about 20 ~ greater than the ratio of the squares of the wave functions evaluated at the nuclear radius (taking into account, of course, the different neutrino energies for L and K capture). This was in accord with earlier experimental results on A 37) and is supported by later precision experiments a). Bahcall4, S) independently studied these questions, introduced certain simplifying assumptions and justified their use. Performing detailed numerical calculations, Bahcall found that the overlap effect tends to reduce all capture rates slightly, and that the exchange effect reduces K capture rates and enhances L~ and M~ capture rates, with the M~ enhancement being greater than the I a enhancement; all the corrections are decreasing functions of atomic number 6). Bahcall also obtained numerical values for the MI/LI capture ratios for 14 __< Z < 37.

200

B. L. R O B I N S O N

Using the relativistic Thomas-Fermi-Dirac-screened wave functions for finite nuclei given by Brysk and Rose 1) and by Brewer, Harmer and Hay 7), we have computed M~/L~ wave function ratios for 55 < Z < 90. The squares of these wave function ratios and Bahcall's results are plotted on fig. 1. An extrapolation of Bahcall's corrections 6) suggests that the MI/LI capture ratios may be about 9 % larger than the square of the wave function ratio for Z = 55 and about 6 % for Z = 90. The application of these corrections places the capture ratios upon the dashed curve of fig. 1, which connects smoothly to the more detailed results o f Bahcall for lower atomic number.

M.~ Lz 0.25 f / / ~ /

0.20

•

•

•

•

•

•

1

f o•@@Qtoo@O@Q 0@@

0.15 0.10 005

,b

2b

3'o 4'o

~'o

6'o

zb

8'o

90

z Fig. 1. MI/LI capture and wave function ratios, omitting energy dependence. The points for 14 _~Z _~37 are~Bahcall'sresults and include exchange and overlap effects. The points for 55 ~ Z ~ 90 are ratios of screened relativistic wave functions for finite nuclear size: if Bahcall's exchange and overlap corrections are extrapolated to this region, the capture ratios lie on the dashed curve.

The above considerations apply to allowed transitions, but it is plain that they are purely atomic effects having to do only with the nature of the electronic configuration. In unique forbidden MI and LI (and K) captures, the greater change of nuclear angular momentum appears only in the increased power of the dependence of the capture rate upon the neutrino energy q. Since the two effects are clearly separated, it is permissible to take over into the theory of unique forbidden transitions the exchange and overlap corrections for the allowed transitions, which have the form of multiplicative factors accompanying the wave functions. 3.2. THE M SUBSHELL CAPTURE RATIOS Relativistic Thomas-Fermi-Dirac screened wave functions for atoms with finite nuclei 1, 7) have been used to obtain wave function ratios for the M subshells in the region 55 < Z __< 90, as well as MI/LI and M1/K wave function ratios. The results are shown in table 2 and applied below to the decay of La 1as and Po 209. To obtain capture ratios, the wave function ratios must be corrected to account

201

ELECTRON CAPTURE

TABLE 2a Wave function ratios for the M subshells Z

R

fI~

g~ .~I2

g2 55 60 65 70 75 80 85 90

0.01606 0.01644 0.01691 0.01732 0.01778 0.01818 0.01862 0.01899

0.0325 0.0410 0.0500 0.0605 0.0719 0.0833 0.1008 0.1180

0.0380 0.0462 0.0558 0.0672 0.0795 0.0920 0.1104 0.1201

30 if21

fl 2

g?V

225 g~

R2 gl2

gI~!

.Ill2

R4 g?

0.00318 0.00409 0.00510 0.00619 0.00748 0.00880 0.01040 0.01200

0.00364 0.00464 0.00573 0.00694 0.00835 0.00980 0.01145 0.01326

0.841 × 10-a 1.152 1.485 1.860 2.24 2.62 3.04 3.30

0.268 0.298 0.324 0.346 0.364 0.384 0.401 0.396

The MIII/M1 and Mv/M I ratios are multiplied by factors appropriate for unique second forbidden capture ratios. TABLE 2b Wave function ratios and capture ratios including extrapolated exchange correction (energy dependence omitted) Z

55 60 65 70 75 80 85 90 2 2 ,.,~fzx/f'" • t

eMi _-T-

eM,

eL I

PL t

g~

PK

0.207 0.213 0.219 0.225 0.231 0.237 0.243 0.249

0.226 0.231 0.237 0.242 0.247 0.253 0.258 0.264

0.0263 0.0284 0.0303 0.0324 0.0350 0.0375 0.0405 0.0432

0.0302 0.0323 0.0342 0.0364 0.0390 0.0415 0.0445 0.0471

2 2 is about 10 ~o less t h a n .qMx/9'Li..

for the interference between direct capture a n d other modes o f electron capture which differ f r o m the direct m o d e by the exchange of a pair of electrons. Bahcall has p o i n t e d o u t that LII a n d MII capture will be affected in a b o u t the same way as LI a n d MI capture 5), b u t exchange corrections have n o t been m a d e for higher a n g u l a r m o m e n t u m electrons. I n view o f their i m p o r t a n c e in at least one case, one should look f o r w a r d to o b t a i n i n g a n estimate o f the exchange corrections for L n b MIH a n d p r o b a b l y M v capture. 3.3. CAPTURE OF ELECTRONS FROM OUTER SHELLS E l e c t r o n capture f r o m shells o f principal q u a n t u m n u m b e r greater t h a n three is n o t negligible, a n d it can be estimated f r o m the self-consistent field wave functions

202

B.L. ROBINSON

of Hartree and others or by the use of Slater screening constants 8). Table 3 presents estimates obtained from self-consistent field wave functions. TABLE 3

Capture of electrons from outer shells Z 40 42 55 59 69 80

4s+5sq3s 0.16 0.17 0.25 0.27 0.27 0.30

4p-l- 5pq3p

4dq-5d 3d

0.14 0.15 0.23 0.24 0.25 0.29

0.19 0.21 0.23 0.28

The entries are the ratios of the squares of wave functions. Mo ÷ E. C. Ridley, Proc. Cambr. Phil. Soc. 51 (1955) 702 Cs+ D. R. Hartree, Proc. Roy. Soc. A143 (1934) 506 Pr+a, Tm ÷a E. C. Ridley, Proc. Cambr. Phil. Soc. 56 (1960) 41 Hg÷ (relativistic), D. F. Mayers, Proc. Roy. Soc. A241 (1957) 93 Au +, T1+ Douglas, Hartree and Runciman, Proc. Cambr. Phil. Soc. 51 (1955) 486; in agreement with Mayers' relativistic treatment of Hg +. Zr +4 L. S. Altmann, Proc. Phys. Soc. A68 (1955) 987

4. Some Applications C a p t u r e ratios were c o m p u t e d for the two cases for which there are some exp e r i m e n t a l data. The theoretical results o f table 1 were used a n d the wave f u n c t i o n ratios o f table 2; the c o n t r i b u t i o n s of higher shells were estimated a n d exchange corrections were m a d e in the MI/L~ ratio, b u t wave f u n c t i o n ratios were used for the higher shells. 4.1. THE NUCLEUS LaT M

La 138 decays 9) by both fl- emission (30 ~ ) and electron capture (70 ~). The spin of La 1as is five lo). The final states of the beta transitions are the first excited states of the even-even daughters, which are almost surely 2 + states. The spin changes in these transitions are therefore 3h. The parity o f La 13s is uncertain, a n d hence the degree of forbiddenness of the beta-decay transitions; if the parity is odd, the transitions are third forbidden, b u t if the parity is even, the transitions are u n i q u e second forbidden. Some arguments pertaining to the parity assignment are set forth here. (i) The comparative half-lives do n o t discriminate between the two possible values of the parity of La 13s, as shown in sect. 5 below. (ii) I f the parity o f La 13s were even, the f l - spectrum w o u l d display the second f o r b i d d e n u n i q u e shape. The energy d i s t r i b u t i o n for this shape is relatively flat f r o m

ELECTRON CAPTURE

203

zero to 100 keV, and falls off more or less linearly from 100 keV to the end point at 200 keV; this is a distortion of the allowed spectrum toward higher energies. If on the other hand the parity were odd, t h e n the fl- spectrum might be quite different; the prototype third forbidden transition is Rb aT, the energy spectrum of which is distorted toward low energies 11). The fl- spectrum of La la8 has been observed only with a relatively thick source 12), and it is not clear whether the excess low energy electrons are due only to source thickness and back-scattering, or whether they are evidence of a spectrum similar to that of Rb aT. It is clear however that the fl- spectrum of La 13s is by no means as strongly distorted as that of Rb s7. (iii) The shell model favours even parity for La 13s with the 57th proton in a g~ or d~ state and 81st neutron in a d÷ or s÷ state: however an odd parity state, h ~ , is also available for the neutron although it lies about 700 keV above the d~ state in neighbouring odd-mass nuclei. (a) The s~ neutron state cannot couple with either proton state to form the observed spin of 5. The combination o f the d~ neutron state and the d~ proton state is likewise rejected. (b) The participation of the h~ neutron state would be expected to give rise 13) to spins 8 or 9 (in +Jp) or 2 or 3 (Jn-Jp). The observation o f an intermediate value would thus tend to speak against the h~ state for the neutron. (c) The magnetic moment of La ~as is + 3.7073 n.m. I o). This is in fair agreement with the magnetic moment of +2.863 n.m. calculated from the Schmidt moments for g~ proton and d~ neutron states, and in better agreement with a value o f + 3.373 n.m. which is obtained using empirical g factors given by the magnetic moments of nearby odd-mass nuclei 14). If the neutron state were h ~ , the magnetic moment would be expected to be about - 1 n.m. in the Schmidt approximation (and there are no data on the magnetic moments of h~ nuclei). Therefore the observed magnetic moment strongly contradicts the assumption of odd parity for La 1as. I f the parity of La ~aa is even, then it decays by unique second forbidden betadecay transitions. One can find an electron capture transition energy, Qec = 190__ 15 keV, for which the theoretical K capture fraction agrees with the observation that only 4 2 ~ of the electron captures occur via K capture 9). For this energy, the capture from the various atomic shells and subshells is given in table 4. Note that there is considerable LIH and MHI capture, but that Mv capture is still quite small. Although the evidence given above strongly favours even parity for La 13s, it may be worthwhile to explore the consequences of the assignment of odd parity. Then the fl- and electron capture transitions are third forbidden. The transition rates involve mixtures of the large and small components of the wave functions, terms containing higher powers of the lepton energies, and a forbidding combination of matrix elements. In the electron capture transition rates, the coefficients of the higher powers of the neutrino energy are very small, and the principal effect of the assumption of a parity change is that the main contributions to the transition probabilities now arise not

204

B. L. ROBINSON

from terms containing the large components of the Dirac wave function, but rather from terms containing the quotient of the small component and the nuclear radius. I f the transition energy is large, most of the captured electrons are from the K and LI shells, and for a given transition energy the L / K capture ratio is about 20 ~o larger for the third forbidden transition: (fLJfK) 2 = 0.102, (gLr/ffK) 2 = 0.084, for lanthanum. However there is relatively less L m capture in the third forbidden transition: (fLixi/fLi) 2 = 0.64X 10 -6, a n d (ffLiii/gLi)2 = 1.7x 10-6; and for a given small transition energy, the L / K capture ratio is smaller for the third forbidden transition. This assumes that the exchange effects are similar for both components of the Dirac wave functions. No estimate of the beta-decay matrix elements is available. I f however one makes the rather unrealistic assumption that the matrix elements 15) for Rb 87, the only certified third forbidden beta-decay, apply to the supposed third forbidden transitions in Lala8; then one can compute capture rates and ratios as functions of transition energy. Under this assumption, the observed K capture fraction corresponds to a transition energy is about 150 keV. U p o n examining the way in which the ratios o f the matrix elements enter into the capture ratios, one finds that major changes in the ratios of the matrix elements would be needed in order to make significant changes in the estimated transition energy. TABLE 4

Electron capture fractions in unique second forbidden transitions Laxaa

D a t a : PK =

Totals K 424-4

Po ~°9 Totals

K 72~

024-4)%

L 42q:3 LI 15.6% Ln 0.5 % LIII 25.7 ~.~,

Data:

Qee =

1.0 MeV

L 21 ~ LI 18 ~ Ln 1.6~ LIII 1.7 ~

Conclusion: M

13q:l

Mi+n MIII Mxv Mv

4.2 ~o 8.5 0.03 ~ 0.21 ~

Conclusion: M MI Mn Mni Miv My

Oee =

6% 5~ 0.5 0.5 0.005 0.001

PK

(1904-15)keV

N,O 3.3q:0.3 N, OI+n 1.1 NIII+IV 2.2 Nv 0.04% should be 72 N,O

1

4.2. THE NUCLEUS PO209 In Po 2°9 decay, the transition energy is obtained from a closed cycle involving known alpha- and beta-decay energies. The spins and parities of the initial and final states can be assigned unequivocally 16) on the basis o f the measured spins of parent and daughter, the shell model, and the internal conversion of the ground-state gamma-

ELECTRON

CAPTURE

205

ray which follows electron capture. In this case, we expect 72 % K capture and capture f r o m the outer shells as shown in table 4. The contributions of the outer atomic shells is smaller than in the case of La 1as chiefly because the transition energy is much larger.

I0

i

i

i

\

P,

i

i

i

ALLOWEDm;TANSITIONS

I

0.11

2'0

'

4'0

'

6'0

'

Qee (keV)

80

Fig. 2. M/L capture ratios as a function of transition energy for allowed transitions in platinum.

pMI° FIRSTFORBIDDENUNIQUE

~

3 I

0.30

i

....~ TRANSITIONIN LEAD

i

/

i

i

I

i

20 40 60 80 I

I

I

I

I

I

I

I

l

l

I

i

i

i

I00 120 14.0 I

I

I

(b)

II

logftt I0 9 8 7

I

l

l

20

l

l

l

l

l

l

40 60

I

I

80 I00 !20 140

Qec (keY) Fig. 3. (a) M/L capture ratio as a function of transition energy for unique first-forbidden transitions in lead. (b) Comparative half-life (log fit) for the unique first-forbidden electron capture transitions in lead-202 and -205: most unique first-forbidden transitions have log fit between 7.5 and 9.5.

206

B.L.

ROBINSON

4.3. THE NUCLEI Pt l°3g, Pb ~°2and Pb~°5 There are several cases of electron capture in which no K X-rays have been observed. In these activities the transition energy must be presumed to be less than the K binding energy of the daughter atom, and one m a y make use of the fact that the M / L capture ratio is energy dependent through the neutrino momenta to obtain an estimate of the transition energy. For example, fig. 2 shows the energy dependence of the M / L capture ratio for allowed transitions in platinum. Pt 193g has been observed to emit no radiation more energetic than L X-rays, and the shell model predicts that the transition is first forbidden (½- to {+). The MI/LI capture ratio depends mainly upon the ratio of the small components of the Dirac wave functions rather than the large components. However, the M / L wave function ratios are even less affected by this interchange than are the L I / K wave function ratios discussed above in connection with La 13s, and if exchange effects are not large one m a y expect that fig. 2 will be useful as a good first approximation. Unfortunately there are no data on the M / L capture ratio in Pt 193*. Fig. 3 (upper) shows the predicted M / L capture ratios for the unique first-forbidden transitions in the long-lived lead radioisotopes. This curve was calculated using wave function ratios without exchange corrections. The lower part o f the figure shows the comparative transition probabilities (log f l t) for these activities. The curves have different shapes because the reported half-life for Pb 2°2 is the total half-life, whilst that for Pb 2°5 is the partial half-life for L capture. The curves end at about 98 and 108 keV; these are the upper limits for the transition energies as set by the reported lower limits on the intensities of the K X-rays. Systematically, unique first-forbidden beta-decays have 17) log f i t between 7.5 and 9.5, and one m a y use this rule t to draw the dashed line in the figure and thereby to estimate upper limits for the transition energies of about 75 keV for Pb 2°2 and about 22 keV for Pb 2°s. An analysis of mass spectroscopic data and neutron separation energies in this region 18) shows that the electron capture energy of Pb 2°s m a y be expected to be 33 _ 12 keV, which is not in disagreement with the conclusion drawn from the systematics off1 t values.

5. Comparative Haft-lives of Unique Second- and Third-Forbidden Transitions Davidson showed that fnt, the comparative half-life including the shape factor, was a rather better criterion for the classification of unique first-forbidden transitions than t h e f t value without the shape factor 19). At that time, the data for more highly forbidden transitions were very meager, although Davidson tabulated Be 10 and the ground-state transition in Na zz as unique second-forbidden transitions and the betadecay of K 4° as a unique third-forbidden transition. t The exceptions to this rule are mainly cases of incomplete experimental data: very weak fl-branches or electron capture transitions for which the transition energy is not well known. It should be remarked that Aul~s (log fxt = 11.2) is the only firmly established exception to this rule.

207

ELECTRON CAPTURE

The a c c u m u l a t i o n o f m o r e data confirms the i m p o r t a n c e o f D a v i d s o n ' s criterion, i n some cases in a spectacular fashion. The c u r r e n t l y available data for u n i q u e second a n d third f o r b i d d e n t r a n s i t i o n s are s u m m a r i z e d in table 5. The f o t values were o b t a i n e d f r o m M o s z k o w s k i ' s n o m o g r a m , a n d the f , t were calculated f r o m the a p p r o x i m a t i o n s given by D a v i d s o n ; except f2 t for t h e / ~ - decay o f La 13s which was calculated f r o m the exact C2A spectrum. The electron capture entries were o b t a i n e d b y the m e t h o d of M a j o r a n d B i e d e n h a r n 20). TABLE 5 Comparative half-lives of unique second and third forbidden transitions Nucleus and mode

Energy (MeV)

(Partial) half-life (y)

K 4° /~8+

Unique third forbidden (AI = 4, yes) 1.321 1.4x 109 18.46 0.49 1.2 × 1014 20.75

15.60 16.00

Be1° Na s2 A1~6 fl+ ec Co 6° Tess Lalss flec Po 2°9

Unique second forbidden (AI = 3, no) 0.556 2.7 × 10e 14.50 e) 1.830 4.3 × 104 12.95 1.16 7.4× 105 14.1 1.04 1.8 x l0 T 12.8 1.48 5.3 × 104 14.1 0.30 1.5 X 10° 13.1 0.205 2.8 × 1011 18.2 a) 0.190 1.8 × 1011 18.0 a) 1.0 2× 104 13.9 e)

12.08 e) 12.2 12.6 11.0 12.7 10.9 14.44-0.1 13.04-0.2 11.9 c)

logf0t n)

logf.t b)

4) A. Moszkowski, Phys. Rev. 82 (1951) 35; reprinted in an enlarged format in the Table of isotopes, Strominger, Hollander and Seaborg, Revs. Mod. Phys. 30 (1958) 585 b) Jack P. Davidson, Jr., Phys. Rev. 82 (1951) 48 e) Contains the factor (2I~4-1)/(2It+1); see, e.g. Eugene Feenberg, Shell theory of the nucleus (Princeton University Press, Princeton, New Jersey, 1955) p. 81 a) Compare with Rb s~, the typical third forbidden decay, logft ~ 17.9. The branches o f K 4° decay have been discussed 21) b y Tilley a n d M a d a n s k y , a n d b y Engelkemeir, F l y n n , a n d G l e n d e n i n , i n terms of the ratio o f the squares o f the m a t r i x elements c o n n e c t i n g K 4° with the g r o u n d states of Ca 4° a n d A 4°. This ratio is o f course j u s t the reciprocal o f the ratio o f t h e f a t values, a n d the result given i n table 5 differs slightly f r o m that o f Engelkemeir et al. o n l y because of the approxim a t i o n used here i n getting the f3 t. A similar analysis can be m a d e for the two b r a n c h e s in La 13s decay if the transitions are u n i q u e , that is, if the p a r i t y of La 13s is even. I n this case we are dealing with the transitions to the first 2 + states i n Ba 13s a n d Ce 138. M,~ _ (f2 t)p- _ 2.60 x 1014 sec ( + 25 ~o) = 25 + 10. M~-

(f2 t)~¢

1.05 x 1013 sec (___20 ~ )

208

B. L, ROBINSON

Table 5 shows that f2 t for electron capture in La 13s is similar to that for other m e m bers o f this class, while f2 t for the f l - b r a n c h is exceptionally large. P r e l i m i n a r y theoretical estimates suggest that the f l - t r a n s i t i o n is expected to be retarded 22), in agreement with this observation; m o r e detailed calculations o n this p o i n t are in progress. I f the electron capture energy is n o t 190 keV as deduced above, b u t a b o u t 350 keV as suggested by a n interpretation 23) of the data o n radiative n e u t r o n capture in b a r i u m , then the ratio of the squares o f the m a t r i x elements is only 0.3, a n d b o t h branches of La 13a decay have very large comparative half-lives. I a m indebted to the Israel A t o m i c Energy C o m m i s s i o n for the grant o f a research fellowship. J. M. P e a r s o n helped with the calculations s u m m a r i z e d in table 1, a n d provided valuable criticism of the manuscript.

References

1) H. Brysk and M. E. Rose, Revs. Mod. Phys. 30 (1958) 1169; Oak Ridge Nat. Lab. Rep. ORNL1830 (1955) unpublished 2) S. Odiot and R. Daudel, J. Phys. Radium 17 (1956) 60 3) R. B. Moler and R. W. Fink, Phys. Rev. 131 (1963) 821 4) J. N. Bahcall, Phys. Rev. Lett. 9 (1962) 500; Phys. Rev. 129 (1963) 2683 5) J. N. Bahcall, Phys. Rev. 131 (1963) 1756 6) J. N. Bahcall, Phys. Rev. 132 (1963) 362 7) H. R. Brewer, D. S. Harmer, and D. H. Hay, Phys. Rev. Lett. 7 (1961) 319 8) A. H. Wapstra, G. J. Nijgh, and R. van Lieshout, Nuclear spectroscopy tables (Interscience Publishers, Inc. New York, 1959), pp. 60, 88 9) W. Turchinetz and W. R. Pringle, Phys. Rev. 103 (1956) 1000 10) P. B. Sogo and C. D. Jeffries, Phys. Rev. 99 (1955) 613A; Nuclear Data Sheets, compiled by K. Way, et al., (National Academy of Sciences-National Research Council, Washington, D.C.) appendix 1, table E, NRC 5-2-125 11) K. Egelkraut and H. Leutz, Z. Phys. 161 (1961) 13 12) R. N. Glover and D. E. Watt, Phil. Mag. 2 (1957) 49 13) M. H. Brennan and A. M. Bernstein, Phys. Rev. 120 (1960) 927 14) H. M. Schwartz, Phys. Rev. 89 (1963) 1293 15) M. A. Preston, G. H. Keech, and J. M. Pearson, Phys. Rev. 119 (1960) 305 16) Nuclear Data Sheets, ref. 10, NRC-5-3-89,-90 17) C. E. Gleit, C.-W. Tang, and Charles D. Coryell, Beta-decay transition probabilities in Nuclear Data Sheets, ref. 10, NRC 5-5-109 18) W. McLatchie, R. C. Barber, R. L. Bishop, H. E. Duckworth, B. G. Hogg, J. D. Macdougall and P. van Rookhuyzen; Can. J. Phys. 42 (1964) 926 19) Jack P. Davidson, Jr., Phys. Rev. 82 (1951) 45 20) J. K. Major and L. C. Biedenharn, Revs. Mod. Phys. 26 (1954) 321; B. L. Robinson and R. W. Fink, Revs. Mod. Phys. 32 (1960) 117 21) D. R. Tilley and Leon Madansky, Phys. Rev. 116 (1959) 413; D. W. Engelkemeir, K. F. Flynn, and L. E. Glendenin, Phys. Rev. 126 (1962) 1818 22) L. S. Kisslinger and C.-S. Wu, private communication, January 1964 23) Nuclear Data Sheets, ref. 10, Compiler's analysis, A = 138 and A = 139, NRC 61-3-78,-87