ft values of superallowed 0-0 transitions

- - ~

NuclearPhysics AI06 (1968) 433

4~1; ( ~ North-Holland Publishino Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

f t VALUES OF S U P E R A L L O W E D

0-0 TRANSITIONS

H. B E H R E N S

lnstitut fiir Experimentelle Kernphysik der Technischen Hochschule und des Kernforschungszentrums Karlsruhe and W. B O H R I N G

II. Physikalisches lnstitut der Universitiit Heidelberg Received 22 July 1967 Abstract: T h e ft values for s o m e superallowed 0+-0 + transitions are calculated. T h e influence o f second forbidden and finite nuclear size terms is discussed. A slightly modified definition o f the ft value is r e c o m m e n d e d , which is m o r e suitable for c o m p a r i n g different nuclei.

1. Introduction

Pure Fermi beta decays play an important role for determining the properties of the weak interaction. From the measured half-life t and transition energy W o, the reduced half-life or f t value is calculated. The quantity contains the relevant information as its inverse is proportional to the square of the product of the vector coupling constant g and the Fermi matrix element Mr. The present paper is concerned with the calculation of t h e f t values for the carefully measured 0+-0 + transitions of some nuclei in the range from 140 to 54C0. Even though the f t values of these transitions have already been calculated by Freeman et al. 1), by Nair 2), by Matese and Johnson 3), by Bhalla 4) and by Suslov 5), an independent investigation seems interesting for the following reasons: (i) The ft values derived from different tables of electron radial wave functions (ERWFs) or Fermi functions showed small but significant discrepancies. (ii) T h e f t values of different Fermi transitions are compared with each other in order to see whether or not the product g M F of the vector coupling constant and the Fermi matrix element is the same for different nuclei. We wish to point out that for this purpose the usual definition of the f-function is not the most appropriate one, in particular if heavier nuclei are considered. In sect. 2, we adopt the usual definition of the f-function in order to compare our results with those of other authors. In sect. 3, we introduce a slightly different definition of the f-function and treat the higher order terms of the transition probability, which are usually called second-forbidden corrections, in a transparent way. Also, we discuss the dependence on the nuclear radius R. The final results are collected in sect. 4. 433

434

H. BEHRENS AND W. B/.JHRING

2. Usual ft values For pure Fermi transitions, the following relation holds:

ft

=

27z 3

In 2 2 _," g My

(1)

Here, t is the half-life of the transition, g the coupling constant for the beta interaction and the Fermi matrix element MF = .( I. In the case of the superal|owed 0-0 fl-transitions considered in this paper, one has M 2 = 2. The usual definition o f f is

f =

/?

F(Z, We)PcWe(Wo

-

We)2 d ~ ; ,

(2)

where the momentum and energy of the electron, Pe and We, are related to each other by W 2 = p2 + l, if the usual units are used t. The end-point energy of the fl-spectrum is denoted by W o. TABLE I Comparison of]/values (scc) calculated by different authors (usual definition off, including screening without radiative correction)

~'0 ~6mAl

s*CI ~2Sc '6V 5°Mn 64Co

BB

N

MJ

S

Bh

3078 3038 3092 3067 3075 3063 3065

3081 3037 3092 3066 3072 3059 3063

3078 3035 3089

3075 3034 3090 3068 3075 3063 3067

3075.7

3069 3059

T h e n u c l e a r r a d i u s is R = roA~, w h e r e BB, N , S a n d B h h a v e u s e d r o = 1.2 f m a n d M J ro = 1.5 fro. O n l y t h e v a l u e s o f S a r e c o r r e c t e d f o r K - c a p t u r e (see t a b l e 5).

The Fermi function F is 1

F(Z, We) = 2p~ (g2 l(R)+f2 I(R)}.

(3)

Here the ERWFs g _ l ( R ) and f + t ( R ) are evaluated at the nuclear radius R. The E R W F s are solutions of the Dirac radial equations with a potential corresponding to the extended uniform nuclear charge distribution of radius R and the charge distribution of the electron cloud. Our calculations are based on the model of the screened field 6) which has been used for our tables 7) of fl-decay functions. In table 1, we compare our results (BB) with h - m c = C ~ I.

04) TRANSITIONS

435

the results of Nair 2) (N), Matese and Johnson 3) (M J), Bhalla 4) (Bh) and Suslov 5) (S). The experimental values for the end-point fl-energy and the half-life are from the work of Freeman et al. l). The agreement is satisfactory, the very slight differences might be due to the use of different screening models and, in the case of M J, of a different nuclear radius and the use of the ROSE WKB screening formula. We conclude from this section that there are essentially no discrepancies between f-values obtained by different authors. It should be noted, however, that t h e f t values derived earlier 1) from the various available tables * of Fermi and related functions show some discrepancies compared with the results of table I. 3. Modified f i values

The usual definition of the f-value is based on the approximation that the ERWFs, which enter the Fermi function F(Z, W e), are evaluated at the nuclear radius R. But actually the ERWFs still depend on the radial coordinate r and should contribute to the radial integration over the nucleon coordinates. The ERWFs inside the nucleus can exactly be represented by power series in the variable (r/R) 2. Therefore additional nuclear matrix elements occur, J" (r/R) 2, S (r/R) 4, etc. The usual treatment is equivalent to the assumption that these additional nuclear matrix elements are all equal to the Fermi matrix element Mr = S 1. We recall that the ERWFs inside the nucleus t t, 12) are

g-t(r) = ot-i (l +a12 (R)2+ ...), f+l(r)

= ~+1

+ ....

(4)

--~[(W,R+.}~Z) 2 - R~],

(5)

where

a,2

=

((J) l+atz

if a uniform nuclear charge distribution of radius R is used**. Here, Z is the nuclear charge of the daughter nucleus and ~ the fine structure constant. The usuaift value therefore determines the quantity

Ill+all2-f(R)2+...12

~ ( f l ) 2 II+2a~12f__(RS, f l

(6)

where the bar denotes the appropriate average over the beta spectrum. For the decay of laO, the quantity 2a-~2 is only 8 • 10 -4, but it increases for heavier nuclei and is equal to 1 • 10 -2 for 54Co. Without detailed calculations using nuclear models, • It might be useful to mention that the tables of electron radial wave functions by Bhalla and Rose s) are incorrect as far as positons arc concerned, see ref. 9.10). ,t The modification of eq. f5) due to screening by the atomic electrons is unimportant, see ref. s).

436

H. BEHRENS AND W. B(JHRING

nothing can be said about the magnitude of the ratio x = S (r/R)2/S 1 except that 0 __
1

±

/~(Z, We) = 2P 2 {0 2 l ( 0 ) + f 21(0)} = 2P 2 {Ct2l +ct 2 l}.

(7)

Then the term a12 ~ (r/R)2/$ 1 has to be put into the spectrum shape factor and considered together with other small correction terms which are usually called secondforbidden terms but include, among others, also contributions with the same nuclear matrix element. TABLE 2 Relation between form-factor coefficients and reduced nuclear matrix elements xs,ss)

v~(o) * 0 0 0 = Cv f l

l" ,~',( n )

l ' / r \ 2~

v,~,O,_o,, = Cv f i ~ R"" "to'?, = c,, f i ~'r/r~ ~- ~1

2

Indeed this new definition o f f follows naturally from the formulation of the betadecay theory given by Stech and Schuelke 13, 14) and has been adopted in chapter 10 of ref. 15). We are now going to investigate the contributions of the higher-order correction terms contained in the spectrum shape factor C(W). For this purpose we introduce the following definitions: 2re3 In 2 (1 +6)C( w ) f t - gZ(VFtoO)o)2 ,

(8)

0-0 TRANSITIONS

f =

437

f;o~(Z, W,)p, W,(Wof/o

(9)

W,)2dW,,

)~" C(W

p, We(Wo - We)2 d H,~

c(w) =

(1o)

fW°Fp, W¢(Wo-W,)2d W, Here 6 is the radiative correction. We have replaced the Fermi matrix element M F = S 1 by the corresponding form factor coefficient VFto°o) o and we shall also do so for the other nuclear matrix elements [refs. 13, 15)]. For convenience, the correspondence between the form factor coefficients and the nuclear matrix elements is given in table 2. The spectrum shape factor C ( W ) can be expressed in terms of ratios of form factor coefficients and well-known, slightly energy-dependent functions. (See appendix A.) By taking the average of these functions over the beta spectrum according to eq. (10), one obtains

C(W) =

I+A 1 /~(0)! - -ooo-

+A2

+'" \1/F~o°~o !

"'

and the numerical values of the constants AI, A2 . . . . are given in table 3. With the help of the CVC theory we have a relation between 1/r~")-oooand [see, for example, refs. 13,15)] 2n v~,~,-1) -oll

=

--

{(14,o~2.5) R +-e~lZl} 6 vFooo ~.)

VF~lqt) (12)

(upper and lower sign for fl- and fl+ decay, respectively). Eq. (12) yields * 1/17'(0) -oll<

1.1 • 10 -2 vr-tl) .t 0 0 0 ,

VF(II) 1 <

5 " 10-3

VL-,(2) JO00

for all the transitions considered in this paper. Then it can be seen from table 3 that we might neglect all the correction terms contained in C ( W ) except the contribution from A i. In order to show the dependence on the nuclear radius, we give in table 4 the ft values for an extended nucleus with uniform charge distribution, without screening as a function of the nuclear radius. It should be noted that actually the radius dependence is considerably smaller than is suggested by an investigation of the point charge Fermi function. Comparison of the first column of table 4 and the third column of table 5 allows us to extract the magnitude of the screening effect. * It should be noted that eq. (12) is not exact as emphasized by Damgaard and Winter 16).The error might be rather large in our case where the terms on the right-hand side ofeq. (12) interfere destructively. It is not serious for our conclusion, however,since we only need an orderof magnitude estimate.

-7.31

42Sc

• 10 ..3

--I.18-

64Co

10 .2

10 ~

7 . 2 4 " 10 -2

7.01 • I 0 -'~

6 . 6 7 " IO -2

6 . 4 0 " 10 2

5 . 3 0 " I 0 -~

4 . 4 7 • 10 - z

3 . 0 6 " 10 -~

A2 10 2

10 -2

I 0 -~

lO -2

I0 2

I0 2

0.4258/137.0388

- 3.82'

--3.53-

--3.23'

--2.93"

-2.34.

1.76 • 10 e

--1.02'

As

values of the constants

T h e n u c l e a r r a d i u s is R ~ r o A ~ , w h e r e ro -

--1.02'

8 . 6 7 " lO -3

S°Mn

46V

-4.70"

34C1

10 - a

--2.75 • 10 a

s6mAI

10 4

--9.78"

140

A~

Numerical

TABLE 3

1 . 3 2 " 10 -a

1 . 2 4 " 10 -3

1 . 1 3 " 10 -3

1 . 0 4 " 10 s

7 . 1 4 " 10 -4

5 . 0 9 " 10 4

fm has been used.

10 -s

10 a

10- ~

10 s

10 - a

10 ' ~

2 . 3 9 " 10 -4

As

3.65

3.11

2.61

2.15

1.37

7.70

2.61

A6

10 4

10 -4

10 .-4

10 -4

10 4

10 - s

10 s

A s in e q . (I 1) f o r d i f f e r e n t s u p e r a l l o w c d

10 7

(3:1.2

3.89

2.88

2.07

1.45

5.96

2.00

2.46'

A4

At...

- 4.27"

--3.58'

--2.90"

--2.34"

--1.25"

--6.17'

--1.50"

10 -5

10 .4

10 -4

10 -4

10 -4

10 4

10 - s

A7

0-0 transitions

10 - s

10 -4

2.25"

10 -4

1 . 8 0 " 10 ..4

1 . 4 0 " 10 4

1.07'

5 . 5 0 " 10 - s

2.42"

5 . 0 0 " 10 -e

A8

10 4

--1.38'

-1.24'

--1.08"

"-9.38"

10 3

10 3

10 -3

10 -4

- - 6 . 2 1 • 10 4

- 3 . 9 3 " 10 4

--1.56"

As

7'

> Z

-r

oo

439

0=0 TRANSITIONS

TABLE 4 Dependence o f the f t values (without screening) on the nuclear radius (R = r o A ~ ) used in the calculation r0(fm)

1.2

1.3

1.4

1.5

140 ~mA1 34C1 42Sc 46V 6°Mn 54Co

3075 3038 3096 3077 3087 3077 3082

3074 3037 3094 3074 3083 3073 3078

3074 3036 3092 3072 3081 3070 3074

3074 3035 3091 3069 3078 3067 3071

TABLE 5

ft values (sec) E0(keV) '40 ~6mA1

s4CI 42Sc 46V 6°Mn 5~Co

1812.6-t-1.4 3207.8=1.9 4459.7,4.0 5409.0--2.3 6032.1-t:2.2 6609.0-!:2.6 7227.7-3.8

t(msec) 71360 + 9 0 6376 :- 6 1565 ± 7 683.0:- 1.5 425.91- 0.8 285.7_k 0.6 1 9 3 . 7 , 1.0

ft

f t ( l +t))

3080--11 3044~: 9 3103,19 3084, 9 3095, 8 3086-: 9 3091 --18

3142.-11 3099-: 9 3156~19 3133, 9 3145, 8 3132, 9 3137,18

Since the partial half-life due to fl" decay is needed rather than the total half-life as given in the second column, including K-capture, the last three columns should be multiplied by a further correction factor (1 -i-e) where e, -- 0.086 ~o, 0.079 ~o, 0.071 ~o, 0.089 ~ , 0.091 ~ , 0.095 ~ and 0.098 ~ for ~40, zSraAl, 34CI, 4~Sc. 4sV, 5°Mn and s~Co, respectively.

4. Conclusions

In table 5 we give the quantities ft and ft(l +~), where the radiative correction 6 is taken from the recent work by K/illen 17)t Eqs. (8) and (11) together with tables (3) and (5) show the dependence of the desired quantity g2(F°oo)2 on the unknown ratios of nuclear matrix elements. We see iv(I) iv iv(o) that the unknown ratio v ~oooJ -ooo = J" (r/R)2/J" 1 induces an uncertainty in the determination of #2 M 2 increasing from 0.0978 ~o in the case of 140 to 1.18 ~ in the case of 5aCo. Consequently, for an accurate determination of the beta-decay coupling constant g, the decay of ~40 is more suitable than the decays of heavier nuclei. Applying a small correction of 0.086 ~o for competition from electron capture and assuming M 2 = 2 for 140, we obtain g = (1.3986+0.0024)

•

10 - a 9

erg. c m 3.

It should be noted, however, that the values o f the radiative correction given in a very recent paper by Chern et al. is) differ from those given by Kallen.

440

H. BEHRENS AND W. BUHR1NG

Finally we would like to draw attention to the fact, that the modified f t values for the different nuclei (table 5) are in better agreement with each other than is the case with the usual f t values * The authors are indebted to Professor H. Schopper for his interest in this work.

Appendix A According to refs. ]*' t s), the shape factor C ( W ) for 0-0 transitions has the form

(VF(o°o~o)2C(W)

Mo2(1, l)+m~(1, 1 ) - 2 / q 7 Mo(1 ' l)mo(1 ' 1),

=

(A.1)

with Mo(l, 1) = vr(o) -ooo-

vt~(o) - o , 1 , , -tm -,

+

N,)+VF
N2)

-VFtoli'i(Oa + N i H2 - N2DI - N3),

(A.2)

too(l, 1) = - VF~o°',~a, -- v~(i),ooo"~ai d x - V F t o l i ) l ( d 3 - N 2 d l ) . The energy-dependent quantities in formula (A.I) and (A.2) are H2 = -~(P
D, = ~W,R,

N, = ~p,R,

D3 = - ~ o ( W , R ( p < R ) 2 ) - ~ o

N 2 = ~ ( p v R ) 2,

d, = ~R,

Ns = ~6o(p~R)3,

d3 = -~o(R(p,R)2),

W, = We + ~aZ/R, ~

Z,

(A.3)

p, = B"o - We,

(A.4)

= W~--I, 2

~,

-

We

2

~<-'-~<+' 2 2 ~ - 1 +0~+1

~ =

(A.5)

,/1-(~
For positon decays, the sign of Z is negative. V~,(O)

V L-( 1)

C( W ) :

l+

Vl:qt)

/Vl:ql) \ 2

VUoOo~oft(W) + V- o~ , , 321, ¢ t W '~"t, VF~o)o - o 1 1 ¢33~.i W ~' + ,,(V~-)--OOO" -ooo f 4 ( W ) -1-00-0

IVK,(O) \ 2 "t Oil

/vlT(t) \ 2 --Oil

\

'* 0 0 0 "

\

'~ 0 0 0 "

VK,(0) Vl.(t ) v~.(l) VK,(I) _ a O I l * O0 t. [ W ~ + aOll -00o¢ \

(IM'I

"t 0001

Villi) • 011 - - 0 1 1 ¢ [ W ~

V 1~(0)

+

v 7o~(Fooo)

Jgt

s,

• We o b t a i n f t ( l + 6 ) = 3140, 3093, 3145, 3116, 3124, 3109 a n d 3111 for l*O: ~eAI, aiCI, a~Sc, 4*V,

6°Mn and 5~Co, respectively.

0-0 TRANSITIONS

441

with f t ( W ) = 2 ( H 2 - N I D I - N 2 ) + 2 I~L~ N t d 1, w,

A ( w ) = -2(D1 + N1)+2 _~2~a,, wo f3(W) = -2(D3 + Nt H 2 - N E D I - N3)+2/~1 "2( d 3 - N2 dl), f.(w)

= (H~ - N 1 0 x - N~) ~ + ( N , d i Y + 2 . i ~, ( H . - N , O, - N~)~', d , ,

f s ( W ) = (D, + N i ) 2 + d ~ - 2 PtT (DI + Nl)dt

f6(W)

=

(ma + N t H2

-- N2

U~ - N3) 2 + (d 3 - N2 d l)2

- 2 tIlT(D3+ N1H2- N 2 D I - N3)(d3-N2d,), w; f~(w) = - 2 ( o l + N I ) ( n ~ - Iv, oi - N~)+ 2NI d~ --2/21 "y[(D 1+ Ni)NI d , - ( H a - N, O , - N2)dl"], we fs(W) = --2(//2 -N1D~ - N2)(D s + N I t t 2 - N2D I - Ns)+ 2Ntdl(d s - N2dl) - 2 #'7 [(Ds+NI H 2 - N E D I - N s ) N i d I - ( H 2 - N 1 D 1 - N 2 ) ( d s - N 2 d l ) ] , we f9(W)

=

2(DI + Nt)(D3 + N, H 2 - N2D1- N s ) + 2 d t ( d 3 - N2d,) --2 p'7 [(D, + Nl)(d s - N 2 d I ) + ( D 3+ N I H 2 - N 2 D I - N 3 ) d l ] . References

1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18)

J. M. Freeman, J. G. Jenkin, G. Murray and W. E. Burcham, Phys. Rev. Lett. 16 (1966) 959 S. C. K. N a i l quoted by R. J. Blin-Stoyle and S. C. K. Nair, Advan. Phys. 15 (1966) 517 J. J. Matese and W. R. Johnson, Phys. Rev. 150 (1966) 846 C. P. Bhalla, Phys. Lett. 19 (1966) 691 Yu. P. Suslov, lad. Fiz. 4 (1966) 1187; Soy. J. Nucl. Phys.4 (1967) 854 W. Btihring, Nuclear Physics 61 (1965) 110 H. Behrens and J. Jiinecke, Landolt B6rnstein, to be published C. P. Bhalla and M. E. Rose, ORNL-2954 (1960), ORNL-3207 (1961) W. Bfihring, Report Kernforschungszentrum Karlsruhe Nr. 559 (1967) J. N. Huffaker and C. E. Laird, Nuclear Physics A92 (1967) 584 W. Bfihring, Nuclear Physics 40 (1963) 472 M. E. Rose, Relativistic electron theory (John Wiley, New York, 1961) B. Stech and L. Schiilke, Z. Phys. 179 (1964) 314 L. Schiilke, Z. Phys. 179 (1964) 331 H. Schopper, Weak interactions and nuclear beta decay (North-Holland, Amsterdam, 1966) J. Damgaard and A. Winther, Phys. Lett. 23 (1966) 345 G. K~illen, Nuclear Physics B! (1967) 225 B. Chern, T. A. Halpern and L. Logue, Phys. Rev. 161 (1967) I 116