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Hindrance phenomena in unique first- and third-forbidden β-decay
ANNALS
OF PHYSICS:
66, 674-696 (1971)
Hindrance in Unique
Phenomena
First- and Third-Forbidden
P-Decay
I. S. TOWNER, Nuclear
Physics
E. K.
Laboratory,
Oxford,
England
WARBURTON,*
Nuclear Physics Laboratory, Oxford, England, Brookhaven National Laboratory, Upton, New
and York
AND
G. T. GARVEY+ Nuclear Physics Laboratory, Oxford, England, and Princeton University, Princeton, New Jersey
Received September 9, 1970
The characteristic hindrance of all unique n-forbidden beta decays (with n odd) among light nuclei (A < 50) can be traced to correlations introduced by the repulsive T = 1 particle-hole force. These correlations are examined in perturbation theory. The results explain the experimentally observed hindrance, and in a large proportion of the cases, good numerical agreement is also obtained.
I. INTRODUCTION
The “Israeli School” of nuclear theory in which the late Amos de-Shalit figured so prominently, provides nuclear physicists with a useful formalism for generating the dominant piece of the wavefunction for low-lying nuclear states where the nucleon numbers are too far from the so-called magic numbers. In this model [l] a * National Science Foundation senior post-doctoral fellow, 1968-1969. Permanent address: Brookhaven National Laboratory, Upton, New York. Work partially supported by the U. S. Atomic Energy Commission. + A. P. Sloan Foundation Fellow, 1967-1969. Permanent address: Princeton University, Princeton, New Jersey.
674
HINDRANCE
PHENOMENA
615
system of nucleons is described in terms of appropriate low seniorityj” configurations. These configurations are then diagonalized, using a set of phenomenologically determined two-body matrix elements, to obtain the eigenfunctions and eigenenergies. This prescription does very well in determining binding energies and reproducing the observed spectroscopic factors associated with single-nucleon stripping and pick-up. However, due to the presence of many small but correlated terms in the wavefunctions, the shell model described above has difficulty in reproducing certain observed y- and P-decay rates even in regions where it reproduces the energy levels quite well. A familiar short coming of this model involves the large B(E2) values observed between low-lying levels, for example, the l/2+ first-excited state of 170 has a lifetime the order of lo3 times shorter than would obtain in the simple shell model. This fast decay rate is attributed to a polarization of the I60 core. That is, the initial and final wavefunctions are more complicated than simply having a single neutron beyond the doubly magic core of 160. Some of the electromagnetic consequences of this core polarization can be attended to by assigning an “effective charge” to the neutron and proton [2]. In so far as the observed transition rates of a particular multipolarity are proportional to predicted shell-model rates, the notion of “effective charge” is useful. In this paper we address ourselves to the problem of determining an “effective axial-vector coupling constant” for the unique first- and third-forbidden beta decays involving nucleons in the lf7,2 and Id,,, shells. The “effective coupling constant” that would be extracted from the data in this case is smaller than the isolated nucleon value. We show this effect to be a consequence of small configuration admixtures, induced into thej’” wavefunctions by the repulsive T = 1 part of the particle-hole force, which have a coherent effect on the beta-decay rate. This study was initiated as a result of our work [3] on the unique higher-order forbidden decays in nuclei with A < 50. As opposed to the second-forbidden decay of loBe and 32Na, the third-forbidden decay of 40K to either 40Ar or 4oCa is observed to be considerably retarded relative to that predicted with simple lf,,2-ld3,2 configurations for the 40K and “OAr ground states and a doubly closed 40Ca ground state. Using an extended particle-hole wavefunction for 40K from the work of Gillet and Sanderson [4] or Perez [5] reduces the disagreement in the case of 40K + 40Ca from a factor of 5.1 to 2.6. Our attention was then called to the earlier work of Oquidam and Jancovici [6]. They had shown that the then known (1959) first-forbidden unique decays between the Id,,, and lf7,2 shell nucleons were slower by a factor of - 5 than could be obtained from wave functions diagonalized over this ld3,2-lf,l, space. The constancy of this inhibition factor is convincingly demonstrated for a class of these unique forbidden decays in Section II of the present paper using more recent experimental results. The regularity of this inhibition factor for the first-forbidden decays, along with a similar factor for third-forbidden unique decays, leads one to suspect a common origin for both these effects.
676
TOWNER, WARBURTON,
AND GARVEY
There has been little theoretical work on these beta decays apart from that of Oquidam and Jancovici [6]. Acalculation involving projections fromNilsson orbits was made by Cohen and Lawson [7]. This picture accounted rather nicely for the extremely slow decay rates observed for 39Ar(p-) 39K, 40K(EC) 4oAr, and 41Ca(EC) 41K; however, it will be shown that the very strong inhibition in these three cases follows directly from a shell-model calculation, and, in any case, these decays are atypical. The most detailed treatment of retardation among n = 1 decays to date is that recently reported by Bertsch and Molinari [8] who consider the decay of 17N, 39Ar, and 41Ca from a point of view similar to ours. However, as will be seen later in the text, 39Ar and 41Ca are unfortunate cases to treat as most of the inhibition results simply from the effects of the residual force within the
lf7dd312 space. The general problem of inhibition in first-forbidden nonunique beta decays in heavy nuclei has been discussed in a series of papers by Ejiri, Fujita and Ikeda [9]. They extend the schematic model, as employed by Brown and Bolsterli [lo, 111 in their treatment of the giant-dipole resonance, to show how inhibited decays to lowlying states can be qualitatively understood. In Section IV, we also carry out a schematic model calculation and include correlations in the final state which are shown to be quite important. In fact, quantitative agreement with experiment results only when this latter effect is included. Our work is presented in the following format. In Section II, the relevant experimental data are presented and the appropriate operators for discussing unique n-forbidden decay are defined. In Section III, simple shell-model estimates of the decay rates are made which exhibit the magnitude of the hindrance phenomena to be explained. In Section IV, we present a perturbation calculation and employ some reasonable approximations to obtain a simple formula which gives an understanding of the origin of the observed hindrance. In Section V, the perturbation calculations are carried out without the simplifying assumptions of Section IV and the results obtained are shown to be in excellent accord with the experiment. The conclusions are presented in Section VI.
II. THE EXPERIMENTAL
DATA
The experimental data on the unique first-forbidden beta decays we wish to consider are given in Table I. For completeness, this table includes all 22 of the unique first-forbidden decays with A < 50 which are known to us. The experimental data for the two cases known of third-forbidden unique decays are given in [3]. Our method of extraction of the p-moment, or the nuclear matrix element entering into unique forbidden p-decay, from experimental data such as that of Table I has
Transition
312
512 3
S/2+;
l/2-; o+; 2-;2 3/2+; l/2-; 2-; 2-; 2-; 4-; 7/2- ; l/2-; 0+;3 2-;2 2-;2 3/2+; 3/2+; 7/2-; l/2-;
512 l/2 512 l/2
512 312 3 3 3 1 512 l/2
1 1 312
2-; 2-; l/2-;
Ji”; Tai
Summary
3/2+; 2-;2 o+; l/2-; 3/2+; 0+;2 o+; 4+;2 2+;2 3/2+; 3121; 2-;2 o+; o+; 7/2-; 7/2-; 3/2+; 3/2+;
l/2-;
5/2+;
l/2
1 1 312 512 312 512
3/2 312
2
1 312 112
312
l/2
Of;0 o+;o
(3.036 (1.018 (2.228 (3.33 (8.49 (8.28 (8.28 (8.28 (4.027 (6.576 (2.43 (1.04 (4.48 (4.48 (7.83 (1.20 (1.406 (3.917
(2.71
of Experimental
i& * i f + + & * i + f f & f f + f
f
(7.14 (7.14 (4.16
0.006) 0.004) 0.002) 0.01) 0.09) 0.12) 0.12) 0.12) 0.025) 0.024) 0.35) 0.03) 0.01) 0.01) 0.18) 0.06) 0.003) 0.0035)
0.01)
for
x lo2 x 104 x 103 x 103 x lo9 x 10’ x 10’ x 10’ x lOI6 x lo3 x lOI2 x 10s x lo* x 104 x IO* x lOa x 10’ x 106
x 10’
+ 0.02) & 0.02) i 0.01)
(se4
Half-Life
Information
I
5.5 14.2 57.6 7.0
f
:I;)
x 10-Z
26.0 f 2.0 i 0.4) x IO-4 1.6 & 0.5
Ratio
Unique
rir 0.6 + 2.0 i 1.3 f 2.0 100 13.0 * 4.0 0.6 f 0.07 0.8 3 0.12 10.6 zt 1.3 0.83 + .081 100 100 81.5 f 0.6 0.18 i 0.031 1.4 & 0.141 ~8.6 zt 2.0’ (1.7 &- 0.6) x 1O-3 0.09 f 0.03
(4.8
(1.2
Branching 03
First-Forbidden
TABLE
(5.52 (7.15 (3.87 (4.76 (8.49 (6.4 (1.38 (1.04 (3.80 (7.92 (2.43 (1.04 (5.50 (2.49 (5.59 ~(1.40 (8.27 (4.35
I-t 0.58) * 1.0) & 0.88) + 1.4) * 0.09) i 2.0) * 0.16) + 0.16) f 0.47) i 0.77) f 0.35) & 0.03) f 0.41) * 0.42) zk 0.56) + 0.32) i 2.92) f 1.54)
x lOa x 105 x lo3 x lo4 x 109 x lo2 x lo* x lo4 x 10” x lo5 x 1Ol2 x 109 x 104 x 10’ x lo6 x lo4 x IO” x 108
(28 3~ 2) 5 2.0) x 10’ A 0.80) x lo2
A Q 50
57+ 12.3 (. - 1.21 x lo4
(6.0 (2.68
with
Half-Life (se4
Decay
Partial
Beta
f
3 4860 i 40’ 2947 + 20’ 4913 i- 5h 3438 31 18 565 i 5 7410 + 150 5289 i 150 4518 + 150 44.1 f 1.0” 2491.9 31 1.1 412 f 8 598 i 41 3519 f 4h 1691 & 5 1817 + 10 4195 f 13 239.6 i 2.0 1213.8 f 2.7
4709
10422 f 3.5 4369 & 4 8679 + 15
tkW
E, (max)
Refs.
678
TOWNER,
WARBURTON,
AND
GARVEY
been fully described previously [3, 121. We shall only give a brief review here. The first step is the evaluation of the Fermi function fn = ,:” 4-T
W> W,
W P W+‘o -
W)’ d W,
(1)
where Wis the /? energy and W, the disintegration energy both in units of the electron rest mass (and including the rest mass), F(Z, W) is the ratio of the electron density at the nucleus to the density at infinity, and u,(Z, W) is the shape factor. The latter can be written in the form a,(Z, W) = (2n + l)! i: C~,“~2~w~“E;:,n’“~l
l;jz(n-“)
3
(2)
V=O
where, in our normalization, G,o(Z, w> = G*,(O, w> = Cn,,(Z a> = 1, and C,,,(Z, W) was evaluated from results summarized by Konopinski accuracy of the calculation has been discussed [12]. Ref.
to Table
[13]. The
I.
o From the atomic mass table of J. H. E. MATTAUCH, W. THIELE, AND A. H. WAPSTRA, Nucf. Phys. 67 (1965), 1, unless otherwise specified. b E. K. WARBURTON, W. R. HARRIS, AND D. E. ALBURGER, Phys. Rev. 175 (1968), 1275; J. K. BIENLEIN AND E. KALSCH, Nucl. Phys. 50 (1964), 202. c F. AJZENBERG~ELOVE AND T. LAURITSEN, Nucl. Phys. 11 (1959), 1. d M. G. SILBERT AND J. C. HOPKINS, Phys. Rev. B 134 (1964), 16. e P. M. ENDT AND C. VAN DER LEUN, Nucl. Phys. A 105 (1967), 1. IS. WIRJOAMIDJOJO AND B. D. KERN, Phys. Rev. 163 (1967), 1094; B. D. KERN, Phys. Rev. C 3 (1971), 413. g G. A. P. ENGELBER~NK AND J. W. OLNESS, Bull. Amer. Phys. Sot. 15 (1970), 566; and private communication. n D. S. KAPPE, Ph.D. Thesis, Pennsylvania State University, 1965, unpublished. *J. VAN KLINKEN, F. PLEITER, AND H. T. DIJKSTRA, Nucl. Phys. A 112 (1968), 372. j B. D. KERN, R. W. WINTERS, AND M. E. JERRELL, to be published; E. L. ROBINSON, M. S. Thesis, Purdue University, 1958, unpublished. b J. B. MARION, Nucl. Data Sect. A 4 (1968), 301. z No uncertainty quoted in literature; the value given here is assumed. m H. W. TAYLOR, J. D. KING, H. ING AND R. J. Cox, Can. J. Phys. 47 (1969), 1539. n H. MORINAGA AND G. WOLZAK, Phys. Lett. 11 (1964), 148. 0 L. G. MULTHAUF AND W. W. PRAY, Nucl. Phys. A 114 (1968), 476. f H. T. FISCHBECK, Bull. Amer. Phys. Sot. 13 (1968), 697. g S. T. HXJE, M. U. KIM, L. M. LANGER, E. H. SPEJLWSKI AND J. B. WILLETT, Nucl. Phys. A 101 (1967), 688. r J. C. COOPER AND B. CRASEMANN, Phys. Rev. C 2 (1970), 451. 8 K. I&WADE, H. Y~OTO, K. YOSHIKAWA, K. IIZAWA, I. KITAMLJRA, S. AMEMIYA, T. KATOH, AND Y. YOSHIZAWA, J. Phys. Sot. Japan 29 (1970), 43, give results from which (0.36 j, 0.03) % is extracted for this branch.
HINDRANCE
679
PHENOMENA
The p-moment, (G,Jz, is then given in terms offat, where t is the partial half-life for the transition in question, (G,)‘J
= (‘n 2) (--$$)
[(:n++1)l;!12&,)2n(fnt>-‘,
(3)
where A,, is the Compton wavelength of the electron. Giving fnt and (In 2)(2rr3/g2CA2) the units of time, (G,)2 has the units of A:: . We shall use (In 2)(2.rr3/g2CA2)= (4.04 & 0.07) x lo3 set [14, 151 and X,, = 386.M with the result that (G1)2 = 1.81 x log (fit)-‘F2, (G2)2 = 4.04 x 1015Cfit)-’ F4,
(4)
(G3)2 = 2.11 x 1O22(f3t)-l F6. TABLE Extraction
of the fit
Transition
and
from
11 the Experimental
Data
h&2)
fit
(1.26 + 0.10) x lo9 (9.2 j, 3.1) x lo8 (3.6 & 1.1) x log (1.5
“‘s(p-)“‘Cl ““S(j?)““Cl 38C1(/+)38Ar
;I;)
x 10’0
9.10 + 0.04 9.96 & 0.15 9.56 5 0.13
(2.13 (1.18 (1.64
+ 0.26) * 0.17) & 0.04)
x lo9 x 109 x lOa
9.33 9.07 9.22
+ & ?E &
o’51 0.10 0.05 0.06 0.01
(2.11
& 0.61)
x lo8
9.32
f
0.12
* f + + & * i k * + * i
0.02 0.15 0.10 0.11 0.07 0.04 0.07 0.16 0.005 0.07 0.05 0.10
(1.25 (3.90 (9.42 (2.57 (3.46 (4.97 (2.28 (2.05 (2.98 (1.60 (5.46 >(2.29
:
10.16
10.10 9.59 9.97 9.41 11.54 9.70 10.36 9.31 9.474 10.20 9.74 29.36
I
(se4 16N(p-)160 ‘“N(p-)‘“O* “N(/F)“O
of Table
I-t & & f & i f i * + + *
0.06) x 10”’ 1.3) x 108 2.1) x lo9 0.7) x lo9 0.55) x 10” 0.48) x 10s 0.36) x 10”’ 0.80) x lOa 0.03) x lo9 0.27) x 1O’O 0.60) x lOa 0.55) x LO9
(1.90
*
0.67)
x 10”’
10.28
* 0.15
(4.29
+
1.43)
x 1O’O
10.63
& 0.14
1.43 0.20 0.50 0 13 ’ 0.85 1.53 1.10 0 84 ’ 0.145 0.46 0.19 0.70 0.0052 0.37 0.079 0.88 0.607 0.11 0.33 GO.79 o 1o . o o4 .
* 5 i + rt i + + f * * + & & If * + 5 * & + + -
0.11 0.07 0.15 0.03 0.09 0.10 0.22 0.03 0.34 0.20 0.007 0.15 0.04 0.20 0.0008 0.03 0.013 0.34 0.014 0.02 0.035 0.19 0.05 0.02
0.02 0.01
680
TOWNER, WARBURTON,
AND GARVEY
Table II summarizes our evaluations of fit, log(f, t), and (G1)2 for the data shown in Table I. The evaluation for the two cases of electron capture and the two cases of IZ = 3 decay were discussed previously [3]. For our shell-model calculations, we shall use an isospin formalism so that the p-moment has the form [3] (G,) = (- l)Tt-=3t ( -‘--ff
;
;,,
(J,T{;~~;;!,~T~)
,
(5)
where the definition of the Wigner-Eckhart theorem follows de-Shalit and Talmi [l] and the matrix element is reduced with respect to both J and T. GF+l’ is assumed to be a sum over all A nucleons of a one-body operator. For a single-particle ;;i;3jTansition, the isospin dependence gives unity and evaluation of Eq. (5)
=
(jf 11P[P) X c~](~+l)11ji) (2ji + l)lj2
= 61j2(2n + 3)li2 (2jf + 1)1/2 (n,l, 1rn 1nJi)(l~ 1)Ccn) 11li)
2
(6)
where a 9j-coefficient appears on the far right; (lf II Cfn) 11Ii) = (-)“’
[(2lf + 1)(2/l + l)]“”
(,”
z
k);
(7)
our convention being to couple I + s = j.
III. SIMPLE SHELL MODEL
ESTIMATES
The beta decays in question are of the following type lb, 4-M (p,fY;
T) + lb, W-)
(~,f>~-‘;
T - I>,
09
that is, a state of isospin T and a configuration of N particles-M holes beta decays to a state of isospin T - 1 and a configuration of (N - 1) particles444 - 1) holes. The magnitude of the isospin T is &(M + N). The situation for positron emission or electron capture is given by reversing the direction of the arrow in Eq. (8). The spins of the initial and final nuclei for the unique first-forbidden decays which we shall consider are given in Table III. An exception to this scheme is the electron
681
HINDRANCE PHENOMENA
capture of the 40K 4-, T = 1 ground state to the 40Ar 2+, T = 2 ground state.l Furthermore, 40K has two branches for unique third-forbidden decay which we also consider below. TABLE III The Spins of the N Particle-A4 Hole Initial State and the (N - 1) Particle-@4 - 1) Hole Final State in Unique First-Forbidden p- Decay M
N
Initial Spin
Final Spin
even odd even odd
even odd odd even
o+ 27/23/2+
20+ 3/2+ 7/2-
We define the extreme single-particle estimate to be the transition rate obtained in the following calculation. Consider all M holes of the initial state to be in the l&,-orbit in a seniority zero (one) configuration for M even (odd). Further, consider all N particles to be in the If,,,-orbit again, in the lowest seniority configuration. With similar assumptions for the final nuclei, algebraic expressions for the p-moment are easily derivable for the transition l&/P
Tl , ‘3112)~ T, ; T) + I@s/P’-~)
Tl - 4; (h/F’
T, - ;; T -
I>,
(9)
where T = Tl + T, , M = 2T, and N = 2T, . They are obtained by standard techniques of fractional parentage expansions. Using algebraic formulae for the cfps as given in de-Shalit and Talmi [ 1] one finds there are four cases: (i)
M = even, N = odd, i.e., l/2- -+ 3/2f (G1)& = $9
(ii)
M = odd, N = even, i.e., 3/2+ + 7/2(G,):,
(iii)
- 2TJ T,(G, ; (7/2- + 3/2+))2;
= Q(5 - 2T,) T,(G, ; (7/2- -+ 3/2+))2;
M = even, N = even, i.e., O++ 2(G)&
= T,T,“;
1 We do not consider the decays of ““Cl and 4aK to excited states of 4oAr and %a. The experimental evidence for the former was reported after this work was completed while the latter, like the mass 4.5 and 47 decays of Tables I and II (also not considered further), is forbidden in first order and thus is not of direct interest to the present work.
682
TOWNER,
(iv)
WARBURTON,
AND
GARVEY
M = odd, N = odd, i.e., 2- + O+
&(5 -
2T,)(9 - 2T,)(G,
; (7/2- + 3/2+))2;
where (G, ; (7/2- -+ 3/2+))2 is the single-particle (sp) p-moment which from Eq. (6) is (G; (7/2- + 3/2+))” = j+ (r)2 = 8F2. Similar expressions for the two cases of unique third-forbidden (i)
M = even, N = even, i.e., Of + 4
(ii)
decay are as follows:
; (7/2- -
3/2+)>“;
M = odd, N = odd, i.e., 4- + 0+ (G3)&, = &5
- 2T,)(9 - 2T,)(G,
; (7/2- + 3/2+))“;
where (G3 ; (7/2- --f 3/2+))” = &- (r3)” w 300F6. All the above formulae hold for electron emission. Expressions for positron emission are obtained by multiplying the appropriate formula by [(2JY + 1)/(2J, + l)], where Ji and Jf are the initial and final nuclear spins in the positron emission process. In Table IV we list these single-particle estimates together with a hindrance factor, h, defined as
It is immediately obvious from Table IV that the experimental p-moments,
HlNDRANCE
683
PHENOMENA
TABLE
IV
Single Particle Estimates of the Beta Moment Initial P;
Decay
T
Final Jm; T
h”
Unique First Forbidden “‘s(p-)“‘Cl 3qp--)w1 38Cl(,P)38Ar
7/2-; 512 Of;3 2-;2
3]2+ ; 312 2-; 2 Of; 1
0.85 & 0.10 1.53 f 0.22 1.10 f 0.03
39Cl(,+)39Ar
3/2+;
512
7/2-;
312
0.84
7/2-; 2-; 4-; 7/2-; 7/2-; o+; 22; 312’;
312 3 1 512 l/2 3 2 5/2
3/2+; 0+;2 2+;2 3/2+; 3/2+; 2-;2 Of; 7/2-;
l/2
0.145 0.46 0.0052 0.37 0.079 0.88 0.607 0.33
3/2 3/2 1 3/2
+ * i i &
o’34 0.20 0.007 0.15 0.0008 0.03
8.0 16.0 6.4
9.4 i 1.1 10.5 * 1.7 5.8 i 0.2
2.0
2.4 A 0.7
4.0 4.8
28 & 1 10.4 & 3.4
3.0 2.0 16.0 9.6 8.0
l 0.013 * 0.34 + 0.014 * 0.035
8.1 25 18 15.8 24
& + * 3~ i
0.6 4 6 0.4 2
Unique Third
Forbidden
40K(p +)aoAr “OK(fl-)‘OCa ah =
4-; 4-;
~:G,)&/
Of;2 o+;o
1 1
10.6 & 1.3 51.6 k 1.3
33.0 266.0
3.1 5 0.4 5.1 * 0.1
.
between the orbits fT12and ds12. This is estimated from the binding energies (BE) of the calcium isotopes, viz., AE = 2BE(40Ca) - BE(3gCa) - BE(41Ca) = 7.2 MeV. The diagonalizations were performed using the Oxford shell-model program [17], and the resulting p-moments are given in Table V. In several cases a very small /?-moment results from the calculation. Consider, for example, the 40K(EC) 40Ar decay. The 40K wavefunction must be a pure If,,,-Id,,, particle-hole state in this model, whereas the 40Ar wavefunction may have as many as four terms: I 40Ar; 2+T
=
2)
=
~o(d3di=fo,~=l +
44/&,~
(f7~2)~=2,~=1 (f&1
+
+
4(d3&
4(4&2,
.
(11)
The p-moment amplitude for this decay is (G,) = 3 dz b[A, - 7 d$A1
- 6 d$4,
+ &
A3],
(12)
Forbidden
4oK@+)40Ar 4oKQ3-)40Ca
Third
4-i 4-;
l/2- ; 2-;3 4-; l/2-; l/2-; o+; 2-;2 3/2+;
39Ar(j-)38K 4~C1@-)4~Ar 4oK(EC)40Ar* 41Ar@-)41K *Q(EC)“K 4aAr(p-)42K 42K(/?)42Ca 4sKt’J-)4SCa
1 1
512
1 512 l/2 3
312
512
3/2+;
Wl@-)=Ar
Forbidden
l/2-; 512 Of; 3 2-;2
First
0+;2 o+;o
1 312
312 312
l/2
; 312
3/2+; 0+;2 2+;2 3/2+ ; 3/2+; 2-;2 o+; l/2-;
l/2-
3/2+; 312 2-;2 Of; 1
Final
J”; T
Jn; T
Moment
Initial
of the Beta
Yqp-y’cl ~~s(p-)=Cl 38C1(/-)48Ar
a Ref. [16]. *Ref. [5]. ’ htheo =
Unique
Unique
Decay
Calculations
using
TABLE
i f f + + 6 i * f f * &
Interaction
V
0.10 0.22 0.03 0.34 0.20 0.007 0.15 0.0008 0.03 0.013 0.34 0.014 0.035
10.6 & 1.3 51.6 * 1.3
0.85 1.53 1.10 o 84 * 0.145 0.46 0.0052 0.37 0.079 0.88 0.607 0.33
the Residual
61. 266.
0.2 3.4 0.01 2.8 0.04 6.9 4.9 1.4
8.0
8.1 16.3 6.5
of Kuo
1.2 1.7 0.2
3.5 0.2 0.4
0.6
2.4
5.8 & 0.8 5.1 & 0.1
7.9 f 8.1 f 4.3 *
7.4 f 7.6 f -
-
9.5 + 3.0
9.5 f 10.6 i 5.9 *
and Browna
19. 266.
4 x IO-5 2.9 0.015 2.4 0.004 6.4 4.3 1.0
8.2
8.1 16.3 6.5
and of Perez”
7.4 f 5.1 *
0.8 0.1
7.4 + 3.5 7.1 + 0.2 3.3 ,. 0.4
6.5 * 0.6
6.3 h 2.1 -
-
9.8 i= 3.0
9.5 * 1.2 10.6 & 1.7 5.9 f 0.2
HINDRANCE
685
PHENOMENA
where the radial integral (Y) has been evaluated using oscillator wavefunctions of length parameter b. The computed 40Ar wavefunction, using the residual interaction of Perez [5], gives A, = 0.748, A, = 0.512, A, = -0.300 and A, = 0.299. It is immediately obvious that there is a large degree of cancellation in Eq. (12), which is responsible for the small p-moment. Similar arguments explain all the very small ((G$ < 0.15F2) /3-moments and thus we see that the three most inhibited transitions observed yield very small p-moments with wavefunctions diagonalized within the l~71p-ld3,2 space. A small variation of the residual interaction could probably give complete agreement with experiment for these three cases. The remaining cases, however, still show significant hindrances which are not yet explained by the theory. For these the hindrance factor, hKuo or h,,,,, , is much more uniform than the corresponding values obtained from the extreme singleparticle estimates given in Table IV. For the unique first-forbidden decays the hindrance factor appears to be 7 f 1, and for the two cases of unique thirdforbidden decays it appears to be a little smaller at 6 + 1. The near constancy of the hindrance factor for first-forbidden decay in this region of the periodic table has been noticed before in the work of Oquidam and Jancovici [6]. It gives weight to the procedure of obtaining p-moments from simple shell-model calculations, but employing renormalized coupling constants. In this case the axial vector coupling constant C, has to be divided by approximately 2.5 f 0.3 (i.e., something in the range 45-d@. This is exactly analogous to the common practice of introducing an effective charge into shell-model calculations of EL y-ray transitions. The remainder of this paper will be concerned in investigating the origin of these hindrance factors.
IV. FIRST ORDER PERTURBATIONTHEORY The calculations of Section III are extended in the following way. To the initial nuclear wavefunction terms are added which connect via a one-body operator with the dominant terms in the final state wavefunction; and similarly to the final state wavefunction terms are added which connect with the dominant terms in the initial wavefunction. Figure 1 shows a picture of typical configurations involved. Schematically the calculation can be viewed in the following way: l(h.)-M
(f7dN;
T) + c c&ae+ah l(d~J(~-~)
(f,,JNel;
T -
1)
Ph
- WW~)-(~-~) (.f,~~)~-~; T- 1)+ cDh db,+ahl(4d-“(f7dN;T).
(13)
686
TOWNER,
WARBURTON,
INITIAL CONFIGURATIONS
AND
GARVEY
FINAL CONFIGURATIONS
leading term
leading term
final state
FIG. 1. Diagram of some of the relevant configurations entering for first forbidden tions. Arrows indicate the configurations linked by the beta decay.
The coefficients aD,hare found from first-order perturbation
cd;;= <(4/P’
(fv,dN; T I ~G,+G WNP-~)
transi-
theory
(f,,zF;
T - 1) ,
(14)
AE;;
with a similar expression for cy. ‘,’ . We shall take the energy denominator, AE,, , to be approximately given by the single-particle energies, E, of the orbits involved, viz., AE(~) = Ef - Ed + Eh - ED, Db (15)
A&
= Ed -
Ef +
ch
-
Ep
*
The contribution to the p-moment from these admixtures depends on %h where , N, - n,
(16)
where N, represents the number of oscillator quanta in the orbital j, and similarly for Nh . Thus, for unique first-forbidden beta decay, j, and jh may differ by only one oscillator quantum, hence in Eq. (13) the sum over p and h will only involve particles in the (If, 2p)-shell and holes in the (2s, Id)-shell. For unique third-
HINDRANCE
687
PHENOMENA
forbidden decays we shall also consider particles in the (3s, 2d, Ig)-shell and holes in the (lp)-shell. The admixtures in the initial wavefunction o$; are induced through the particlehole force. That this force will give rise to a hindrance in the beta decay can be seen most simply from schematic model calculations [3] and from the calculations of Bertsch and Molinari [8]. The admixtures in the final-state wavefunction, ala;, that are connected to the dominant term in the initial state via a one-body operator have N + 1 particles and M + 1 holes. They are often termed ground-state correlations. Such correlations are known to be important when treating electromagnetic transitions in doubly magic nuclei in so far as they increase the transition probability to “collective” partile-hole states by a factor of two or so. At the outset of this study it was not evident to us that these ground-state correlations would have an important effect on retarded transitions. As will be shown in what follows their effect is appreciable, in some cases as large as the effects arising from the extended diagonalization of the initial state. This large effect, due to the final-state correlation, comes about even though the energy denominator, d Eih , is larger than the corresponding quantity in the initial state, because there are more configurations possible which taken in total give rise to an appreciable correlation. Numerical examples are presented in Section V. It is instructive to evaluate the retardation of the decay due to configuration mixing under certain simplifying assumptions. We calculate the ratio of the contribution to the transition matrix element from a particular particle-hole admixture in the initial wavefunction, viz., A:; = a$
G, ; I aa+ah / (d3,2)-(M-1) (f,,#‘-l; ( + I(&&(-) (f7/dN-? JtW)>
relative to the contribution
J,T,); JiTi) (17)
from the dominant terms, viz.,
D = ( G, ; l(4,P
(f,dN; JiTi) - IQ&,& (“-lYf,,zF1; J,W).
(18)
Here Ji , Ti are the spin and isospin of the initial state and Jf, Tf that for the final state. Summing (A$ + A$/0 over all particle and hole orbits, p and h, and squaring will give a measure of the hindrance. After some Racah algebra we find that A(i) -
uh
D
zzz
1
2.L
Jq-
x
+
2/g
1
(G
; (j,
+
jh)>
; (n + 1) 1 I VI d;if7,2 ; (n + 1) 1).
(19)
688
TOWNER,
WARBURTON,
AND
GARVEY
Note that the only particle-hole matrix elements involved are those with T = 1 and spin equal to the multipolarity of the transition. In the derivation we have assumed j(d3,2)-(M-1) (f,,2)N-1; JfTf) that j, # hi2 and jh f G2 , i.e., the configuration can be taken as an inert core of spin Jf and isospin Tf, and the creation (a,+) and annihilation (ah) operators have their usual commutation relations. For the case j, = fCj2 or jh = daj2 the above expression is obtained on neglecting antisymmetrization. In the numerical results given in the next section, antisymmetrization is correctly taken into account using the methods of fractional parentage coefficients. Notice that the hindrance in the p-moment is independent of Ji , Ti , Jf , Tt , N and A4 and depends only on the orbits involved and the multipolarity of the transition. The hindrance can further be shown to depend only on orbital angular momenta. This is achieved by assuming that the energy denominators AEfi and all radial integrals depend only on orbital angular momenta and summing over particle and hole orbits j, and j, , assuming that both members of a spin-orbit pair j = 1 f l/2 are present in the summation over p and h. First the particle-hole matrix element is expressed in L-S coupling:
x ((l,-l1,); LST = 1 1 I’ j (d-y);
LST = 1).
(20)
Then substituting this expression into Eq. (19), evaluating the single-particle /?-moments from Eq. (6), and summing over the two possible j values, 1 =t l/2, it is seen that the orthonormality conditions on the 9j-symbols selects L = n and S = 1, in which case the sum over all particle-hole admixtures becomes
x ((&‘Z,); L = n, S = 1, T = 1 1 V 1(d-lf);
L = n, S = 1, T = 1). (21)
This expression can be further simplified on evaluating these particle-hole matrix elements using a zero-range residual interaction, and assuming that the radial integrals can be replaced by a constant average value, 1, independent of the orbits.
HINDRANCE
PHENOMENA
689
In which case
(22) Here I’,, is the strength of the residual particle-particle interaction (negative for attractive particle-particle forces) and (B + H) is the proportion of spin and isospin exchange forces in the interaction. Repeating the entire procedure for admixtures in the final nucleus wavefunction, we find, for an analogously defined quantity Afph, the identical expression:
For attractive particle-particle forces the energy denominators dEPh defined in Eq. (15) are negative as well as V,, . Furthermore, the value of B + H is certainly between 0 and 1 from what we understand about the nuclear force. The assumption that the radial integrals, Z, are approximately constant and are all positive implies a choice of sign convention for the radial wavefunctions, namely that the functions are positive asymptotically. This convention also makes the integrals (P) all positive, thus we can assert that for attractive particle-particle forces Eqs. (22) and (23) are negative definite and hence the admixtures shown schematically in Eq. (13) will produce a hindrance in the beta-decay rate. Although a number of drastic approximations have been made in deriving Eqs. (22) and (23) they enable a number of conclusions to be drawn which should hold for more realistic calculations. We summarize these results as follows: (i) The hindrance residual particle-particle
in the beta-decay rate increases as the strength of the interaction, V, , increases for attractive forces.
(ii) The hindrance decreases as the proportion forces (B + H) increases in the residual interaction.
of spin and isospin exchange
(iii) For a transition of multipolarity, IZ + 1, predominantly described by the matrix element (G, ; (ji -+ j,)), the hindrance introduced by particle-hole admixtures is approximately independent of the particular nuclei involved (i.e., independent of Ji , Ti , Jf , T, , Nand 44). This justifies the procedure of renormalizing the one-body operator, G(“+l), the renormalization depending only on the multipolarity and the orbital angular momenta Ii and If . 595/w2-19
690
TOWNER, WARBURTON,
AND GARVEY
(iv) The hindrance introduced by admixtures in the initial wavefunction are greater than those introduced by admixtures in the final wavefunction (correlations of the core) since the energy denominator AEiA is less than AE$ . However the core correlations are not negligible as will be seen by the computations in the next section.
V. NUMERICAL
RESULTS
We now compute A!;, A$ and D as defined in Eqs. (17) and (18) and hence obtain a theoretical estimate for the @moment (24) where Ni and Nf are normalization
constants,
(25) Nf2 =
1 + c / 01 nh
and the o19%are the admixture coefficients defined in Eq. (14). The particle-hole matrix elements were calculated using a simple phenomenological finite-range residual interaction Jqr) = ~Op+ro2 [w + BP, + MP, + HP,],
(26)
where W, B, M and H are the strengths of the Wigner, Bartlett, Majorama, and Heisenberg exchange forces, V,, is the overall strength of the interaction, PH = -P, and Y,,is the range. Their values were taken from the calculations of Perez [5] in which the parameters had been adjusted until a good fit to the experimental particle-hole energy levels in 40Ca was obtained. This procedure gave V, = -40.84 MeV, W = 0.13, B = 0.26, M = 0.57, and H = 0.04. Perez, however, used a Yukawa shape cur/(w) for the residual interaction with 01 = 0.65F-l. For simplicity we have used a Gaussian shape, the range r. was chosen by requiring that the Gaussian and Yukawa shapes should have the same volume integral, in which case r. = (+‘+P)W
a-1
HINDRANCE
691
PHENOMENA
giving rD = 2.018F. All radial integrals were evaluated using harmonic oscillator wavefunctions, the length parameter was chosen such that b = 1.006A1J6F, which for mass A = 40 gives b = 1.86F, this being the value used in our previous calculations of unique third-forbidden beta decay [3]. First-order perturbation theory was used to calculate A$ and AFL as discussed in the last section. The cases for which j, = fT12or j, = dsj2 are correctly evaluated using the methods of fractional parentage expansions [l]. The energy denominators AE,, were approximated by Eq. (15). Note, however, that the energy denominator AE$ can be quite small; in fact on neglecting spin-orbit splitting and using oscillator energies AE;; = 0. Thus the use of first-order perturbation theory for the initial wavefunction must be viewed with some suspicion. The procedure adopted was to make the full matrix diagonalization calculation for the initial wavefunction, and then to repeat the calculation in perturbation theory using the single-particle energies given in Table VI. A constant was then added to the energy denominator AE$ which would bring the perturbation theory calculation into agreement with the full diagonalization for the important admixtures. The required constant varied from case to case but ranged from zero to 3 MeV in the worst cases (these typically being Of states). We therefore conclude that the use of perturbation theory was not so inaccurate as might be suspected. There is no (f) this is typically of the order problem with the final state energy denominator A Eyh, of 2 hw. TABLE
VI
Single Particle Energies, E, used in the Evaluation the Energy Denominators AE,,
ORBIT
C&V)
IP,h IPI/? Ids/z
-31.0 -28.0 -20.74 -18.11 -15.64 -8.36 -6.30 -2.70 -4.22 -3.38 0.60 2.60 6.00 7.00
2% M/s lh13 2P3/2
If512 2P1/9 kd2 2&z 3.%/z 2&z I&/z
of
692
TOWNER,
WARBURTON,
AND
GARVEY
The results of the computations are given in Table VII. For the transitions with very small experimental /I-moments, (G1)2 < 0.15Fs, the calculations also predict very small values due primarily to cancellations among various dominant terms in D. This was discussed in Section III. In these cases the results are very sensitive to the parameters of the calculation, and with the simple phenomenological residual interaction used here, we do not expect to obtain a good numerical agreement. We are content to note that while the hindrance phenomena under consideration are present in these cases, the small &moment can be well explained by a shell-model calculation in the highly truncated space involving only ds12and f,,2 orbits. Two of the decays considered by Bertsch and Molinari [8] are of this category, and their failure to obtain good quantitative agreement with experiment is the same as ours, namely that the computations predict a near perfect cancellation in the transition matrix element. TABLE Theoretical
Beta Moments Calculated A!; and A$ Which Interfere
VII
using Perturbation Theory for the Admixtures with the Dominant Terms, D
.Z,,,,A”’ (F$?
Decay
Unique First Forbidden 3’S@-)3’Cl ““s(p-)““Cl 38Cl(/+)38Ar
0.92 0.81 0.95
0.96 0.98 0.91
2.84 4.04 2.56
-0.99 -1.50 -0.84
-0.75 -1.06 -0.67
0.96 1.56 0.83
38Cl(g-)38Ar
0.94
0.98
2.86
-1.21
-0.75
0.69
3eAr(fi-)38K %1(/3-)4OAr 40K(EC)40Ar* 41Ar(,!-)41K “Ca(EC)“‘K 4zAr(p-)4aK 42K(fl-)42Ca 43K(,3-)43Ca
0.96 0.96 0.99 0.95 0.97 0.93 0.95 0.96
0.97 0.95 0.95 0.98 0.98 0.98 0.93 0.98
0.00 1.72 0.12 1.57 0.05 2.54 2.07 1.01
-0.08 -0.38 -0.16 -0.72 -0.12 -0.99 -0.72 -0.49
0.00 -0.45 -0.03 -0.42 -0.01 -0.66 -0.55 -0.27
0.0065 0.66 0.003 3 0.16 0.0054 0.62 0.51 0.056
Unique Thud Forbidden 4oK@f)40Ar 40K(j-)40Ca
0.96 0.99
0.99 0.89
8.90 16.32
-3.14 -4.55
-0.83 -1.54
21.7 81.0
<‘=&a (F2’9
0.85 1.53 1.10 o 84 . 0.145 0.46 0.0052 0.37 0.079 0.88 0.607 0.33
f * + + f f +
0.10 0.22 0.03 0.34 0.20 0.007 0.15 0.0008 l 0.03 i- 0.013 * 0.34 & 0.014 f 0.035
10.6 51.6
f f
1.3 1.3
Of more interest are the decays in which simple shell-model calculations fail. In these cases the hindrance introduced by perturbing the initial and final wavefunctions is sufficient to bring theory and experiment into almost complete agreement.
HINDRANCE
693
PHENOMENA
Note in particular that the contribution from &ADh (f) , this being due to correlations in the core, is sizable despite the fact that the energy denominator AELf,’ used in this case is larger than that used when perturbing the initial wavefunction. It is quite clear that these core correlations cannot be neglected. It is of interest to see which particle and hole orbits are responsible for the hindrance. In Table VIII, a breakdown of the contributions are given for the decay 38C1(p-) 3*Ar. For admixtures in the 38C1wavefunction the largest values of Ati occur for j,, = dsj2 , j, = fT12 and j, = 2~~,~, j, = 2p,,, and for both these casesj, = Zh + l/2 and j, = Z, + l/2 with the consequence that in Eq. (6) for the single-particle p-moment, (G, ; (j, --f j,)), the 9j symbol is stretched and correspondingly has a large value. For the 4 hole-2 particle admixtures in 38Ar almost the entire contribution to the hindrance comes from the configuration (d3,J4 (f,,2)2. Again this is to be expected since this configuration has a large seniority zero component which is energetically favored, whereas all other configurations considered break the pairing energy by having seniority four or six.
Values of the Individual
TABLE VIII A$! and A$ for the Decay 38C1(~-)38Ar 5
Hole Orbit
l&z
2w2 l&z
Particle Orbit
Db
-0.353 -0.387 -0.131
2P3/2
lf5la
2PliZ
$0.033 -0.004 -0.324 -0.094 -0.029 -0.013
-0.008 -0.023 -0.087 -0.041
-0.001 -0.0001 -0.033 -0.015
a The upper entry gives A:,! and the lower entry ALfl in F. b D is the dominant term, in this case D = 2.555 F.
VI. CONCLUSIONS The fact that all unique n-forbidden beta decays, with n odd, among light nuclei (A < 50) show a characteristic hindrance relative to truncated shell-model calculations has been traced to the repulsive nature of the T = 1 particle-hole interaction. This repulsive particle-hole interaction (attractive particle-particle) causes thef,,, particles and d3,2 holes to move further apart Cf1,2 particles and d3,2 particles to move closer together) than can be obtained with simplef& dGy wavefunctions and hence considerable configuration mixing is required. The fact that
694
TOWNER,
WARBURTON,
AND
GARVEY
the particle and hole move further apart leads to a retardation of the decays in which the particle decays into the hole. In this paper we have examined these correlations using a microscopic shell-model description. The degree of success we have had can be gauged from Table VII where the computations are compared with the experimental p-moments. The results are exceptionally good considering that there has been no adjustment of parameters to fit the data, and that the hindrance of - 7 has been explained with terms making up about 10% in intensity of the wavefunctions. The calculations can however be criticized on a number of points. First, the spurious center-of-mass motion has not been removed. Second, we have used simple oscillator wavefunctions and phenomenological residual interactions rather than more realistic forces. Finally, our choice of energy denominator, AE,, , which has a crucial bearing on the perturbation theory calculations, was rather arbitrarily chosen. Related to this is the choice of single-particle energies given in Table VI. We have used values taken from the positions of the single-particle and single-hole states in 41Ca and 3gCa, in particular the gap between the d3,2 and fT12 orbits was taken to be 7.2 MeV. Gerace and Green [19] reason that the shell-model f,,2-d3,2 separation energy should be 5.4 MeV rather than 7.2 MeV. This is because the contributions to the binding energy due to nuclear deformations are different for 40Ca, 3gCa, and 41Ca. One alternative procedure which avoids some of these problems is to first diagonalize the l-particle-l-hole states in 40Ca. Then these correlated particle-hole vibrations are coupled to the states l(d3d-cM-1) (hdN-l
; T - 1) and l(d3d-hf (hdN;
0
to give the admixtures shown schematically in Eq. (13). The particle-hole energies required in the energy denominator for perturbation theory are now given in terms of the eigenvalues of the particle-hole diagonalization. This technique has been used by Shukla and Brown [20] in discussing transitions in the 2-hole-l-particle system at mass A = 15, and by Mavromatis and Singh [21] for the 2-particlel-hole system at mass A = 17. To improve our calculations in the light of some of these comments would increase the complexity of the computations by an order of magnitude. It is doubtful whether these improvements will cast any further light on the nature of the correlations involved in the hindrance phenomenon, but they may well improve on the numerical agreement with experiment. Finally, we wish to repeat some of the conclusions reached in Section IV from some simple arguments, namely that the unique first-forbidden beta-decay hindrance (i) increases as the strength of the particle-hole force is increased; (ii) decreases as the proportion of spin and isospin exchange forces are increased;
HINDRANCE
(iii)
is approximately
PHENOMENA
695
independent of the particular nuclei involved;
(iv) depends essentially on orbital angular momenta and the multipolarity of the transition only.
ACKNOWLEDGMENTS We are indebted to Dr. H. G. Benson for many fruitful discussions. This work was initiated while two of us (G.T.G. and E.K.W.) were guests at the Weizmann Institute in the winter of 1969. While there, stimulating discussions with Professors Igal Talmi and Amos de-Shalit were very important in focusing our attention on the long-standing problem of these first-forbidden unique decays. The hospitality extended to us at the institute during our short visit allowed us to become better acquainted with Prof. de-Shalit and thereby to feel most keenly the loss of this able physicist and generous person.
REFERENCES 1. A. DE-SHALIT AND I. TALMI, “Nuclear Shell Theory,” Academic Press, Inc., New York, 1963. 2. S. SIEGEL AND L. ZAMICK, Nrrcf. Phys. A 145 (1970), 89. 3. E. K. WARBURTON, G. T. GARVEY, AND I. S. TOWNER, Ann. Phys. New York 57 (1970), 174; I. S. TOWNER, Nucl. Phys. A 151 (1970), 97. 4. V. GILLET AND E. A. SANDERSON, Nucl. Phys. A 91 (1967), 292. 5. S. M. PEREZ, Nucl. Phys. A 133 (1969), 599. 6. B. OQUIDAM AND B. JANCOVICI, Nuouo Cimenfo 11 (1959), 578. 7. S. COHEN AND R. D. LAWSON, Phys. Lett. 17 (1965), 299. 8. G. BERTSCH AND A. MOLINARI, Nucl. Phys. A 148 (1970), 87. 9. H. EJIRI, K. IKEDA, AND J. FUJITA, Phys. Rev. 176 (1968), 1277; J. FUJITA AND K. IKEDA, Nucl. Phys. 67 (1965), 145; J. FUJITA, S. Funr, AND K. IKEDA, Phys. Rev. B 133 (1964), 549. 10. G. E. BROWN AND M. BOLSTERLI, Phys. Rev. Left. 3 (1959), 472. 11. G. E. BROWN, “Unified Theory of Nuclear Models and Forces,” North-Holland Publ. Co., Amsterdam, 1967. 12. E. K. WARB~TON, W. R. HARRIS, AND D. E. ALBURGER, Phys. Rev. 175 (1969), 1275. 13. E. J. KONOPINSKI, “The Theory of Beta Radioactivity,” Oxford University Press, Oxford, England, 1966. 14. C. J. CHRISTENSEN, A. NIELSEN, A. BAHNSEN, W. K. BROWN, AND B. M. RUSTAD, Phys. Lett. B 26 (1967), 11; C. J. CHRISTENSEN, V. E. KROHN, AND G. R. RINGO, Phys. Left. B 28 (1969), 411. 15. J. M. FREEMAN, J. G. JENKINS, G. MURRAY, AND W. E. BURCHAM, Phys. Rev. Left. 16 (1966), 959; J. M. FREEMAN, J. G. JENKINS, D. C. ROBINSON, G. MURRAY, AND W. R. BURCHAM, Phys. Left. B 27 (1968), 156. 16. T. T. S. Kuo AND G. E. BROWN, Nucl. Phys. A 114 (1968), 241; and private communication from T. T. S. Kuo. 17. I. S. TOWNER AND W. G. DAVIES, Oxford Nuclear Physics Theoretical Group Report No. 43, unpublished.
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AND
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18. S. G. NILSSON, Kgl. Danske Videnskab. Selskab, Mat. Fys. Medd. 29, No. 16 (1955). 19. W. J. GERACE AND A. M. GREEN, Nucl. Phys. A 113 (1958), 641. 20. A. P. SHUKL.A AND G. E. BROWN, Nucl. Phys. A 112 (1968), 296; A. P. SHUKLA, Thesis, Princeton University, 1967. 21. H. A. MAVROMATIS AND B. SINGH, Nucl. Phys. A 139 (1969), 451.
OF PHYSICS:
66, 674-696 (1971)
Hindrance in Unique
Phenomena
First- and Third-Forbidden
P-Decay
I. S. TOWNER, Nuclear
Physics
E. K.
Laboratory,
Oxford,
England
WARBURTON,*
Nuclear Physics Laboratory, Oxford, England, Brookhaven National Laboratory, Upton, New
and York
AND
G. T. GARVEY+ Nuclear Physics Laboratory, Oxford, England, and Princeton University, Princeton, New Jersey
Received September 9, 1970
The characteristic hindrance of all unique n-forbidden beta decays (with n odd) among light nuclei (A < 50) can be traced to correlations introduced by the repulsive T = 1 particle-hole force. These correlations are examined in perturbation theory. The results explain the experimentally observed hindrance, and in a large proportion of the cases, good numerical agreement is also obtained.
I. INTRODUCTION
The “Israeli School” of nuclear theory in which the late Amos de-Shalit figured so prominently, provides nuclear physicists with a useful formalism for generating the dominant piece of the wavefunction for low-lying nuclear states where the nucleon numbers are too far from the so-called magic numbers. In this model [l] a * National Science Foundation senior post-doctoral fellow, 1968-1969. Permanent address: Brookhaven National Laboratory, Upton, New York. Work partially supported by the U. S. Atomic Energy Commission. + A. P. Sloan Foundation Fellow, 1967-1969. Permanent address: Princeton University, Princeton, New Jersey.
674
HINDRANCE
PHENOMENA
615
system of nucleons is described in terms of appropriate low seniorityj” configurations. These configurations are then diagonalized, using a set of phenomenologically determined two-body matrix elements, to obtain the eigenfunctions and eigenenergies. This prescription does very well in determining binding energies and reproducing the observed spectroscopic factors associated with single-nucleon stripping and pick-up. However, due to the presence of many small but correlated terms in the wavefunctions, the shell model described above has difficulty in reproducing certain observed y- and P-decay rates even in regions where it reproduces the energy levels quite well. A familiar short coming of this model involves the large B(E2) values observed between low-lying levels, for example, the l/2+ first-excited state of 170 has a lifetime the order of lo3 times shorter than would obtain in the simple shell model. This fast decay rate is attributed to a polarization of the I60 core. That is, the initial and final wavefunctions are more complicated than simply having a single neutron beyond the doubly magic core of 160. Some of the electromagnetic consequences of this core polarization can be attended to by assigning an “effective charge” to the neutron and proton [2]. In so far as the observed transition rates of a particular multipolarity are proportional to predicted shell-model rates, the notion of “effective charge” is useful. In this paper we address ourselves to the problem of determining an “effective axial-vector coupling constant” for the unique first- and third-forbidden beta decays involving nucleons in the lf7,2 and Id,,, shells. The “effective coupling constant” that would be extracted from the data in this case is smaller than the isolated nucleon value. We show this effect to be a consequence of small configuration admixtures, induced into thej’” wavefunctions by the repulsive T = 1 part of the particle-hole force, which have a coherent effect on the beta-decay rate. This study was initiated as a result of our work [3] on the unique higher-order forbidden decays in nuclei with A < 50. As opposed to the second-forbidden decay of loBe and 32Na, the third-forbidden decay of 40K to either 40Ar or 4oCa is observed to be considerably retarded relative to that predicted with simple lf,,2-ld3,2 configurations for the 40K and “OAr ground states and a doubly closed 40Ca ground state. Using an extended particle-hole wavefunction for 40K from the work of Gillet and Sanderson [4] or Perez [5] reduces the disagreement in the case of 40K + 40Ca from a factor of 5.1 to 2.6. Our attention was then called to the earlier work of Oquidam and Jancovici [6]. They had shown that the then known (1959) first-forbidden unique decays between the Id,,, and lf7,2 shell nucleons were slower by a factor of - 5 than could be obtained from wave functions diagonalized over this ld3,2-lf,l, space. The constancy of this inhibition factor is convincingly demonstrated for a class of these unique forbidden decays in Section II of the present paper using more recent experimental results. The regularity of this inhibition factor for the first-forbidden decays, along with a similar factor for third-forbidden unique decays, leads one to suspect a common origin for both these effects.
676
TOWNER, WARBURTON,
AND GARVEY
There has been little theoretical work on these beta decays apart from that of Oquidam and Jancovici [6]. Acalculation involving projections fromNilsson orbits was made by Cohen and Lawson [7]. This picture accounted rather nicely for the extremely slow decay rates observed for 39Ar(p-) 39K, 40K(EC) 4oAr, and 41Ca(EC) 41K; however, it will be shown that the very strong inhibition in these three cases follows directly from a shell-model calculation, and, in any case, these decays are atypical. The most detailed treatment of retardation among n = 1 decays to date is that recently reported by Bertsch and Molinari [8] who consider the decay of 17N, 39Ar, and 41Ca from a point of view similar to ours. However, as will be seen later in the text, 39Ar and 41Ca are unfortunate cases to treat as most of the inhibition results simply from the effects of the residual force within the
lf7dd312 space. The general problem of inhibition in first-forbidden nonunique beta decays in heavy nuclei has been discussed in a series of papers by Ejiri, Fujita and Ikeda [9]. They extend the schematic model, as employed by Brown and Bolsterli [lo, 111 in their treatment of the giant-dipole resonance, to show how inhibited decays to lowlying states can be qualitatively understood. In Section IV, we also carry out a schematic model calculation and include correlations in the final state which are shown to be quite important. In fact, quantitative agreement with experiment results only when this latter effect is included. Our work is presented in the following format. In Section II, the relevant experimental data are presented and the appropriate operators for discussing unique n-forbidden decay are defined. In Section III, simple shell-model estimates of the decay rates are made which exhibit the magnitude of the hindrance phenomena to be explained. In Section IV, we present a perturbation calculation and employ some reasonable approximations to obtain a simple formula which gives an understanding of the origin of the observed hindrance. In Section V, the perturbation calculations are carried out without the simplifying assumptions of Section IV and the results obtained are shown to be in excellent accord with the experiment. The conclusions are presented in Section VI.
II. THE EXPERIMENTAL
DATA
The experimental data on the unique first-forbidden beta decays we wish to consider are given in Table I. For completeness, this table includes all 22 of the unique first-forbidden decays with A < 50 which are known to us. The experimental data for the two cases known of third-forbidden unique decays are given in [3]. Our method of extraction of the p-moment, or the nuclear matrix element entering into unique forbidden p-decay, from experimental data such as that of Table I has
Transition
312
512 3
S/2+;
l/2-; o+; 2-;2 3/2+; l/2-; 2-; 2-; 2-; 4-; 7/2- ; l/2-; 0+;3 2-;2 2-;2 3/2+; 3/2+; 7/2-; l/2-;
512 l/2 512 l/2
512 312 3 3 3 1 512 l/2
1 1 312
2-; 2-; l/2-;
Ji”; Tai
Summary
3/2+; 2-;2 o+; l/2-; 3/2+; 0+;2 o+; 4+;2 2+;2 3/2+; 3121; 2-;2 o+; o+; 7/2-; 7/2-; 3/2+; 3/2+;
l/2-;
5/2+;
l/2
1 1 312 512 312 512
3/2 312
2
1 312 112
312
l/2
Of;0 o+;o
(3.036 (1.018 (2.228 (3.33 (8.49 (8.28 (8.28 (8.28 (4.027 (6.576 (2.43 (1.04 (4.48 (4.48 (7.83 (1.20 (1.406 (3.917
(2.71
of Experimental
i& * i f + + & * i + f f & f f + f
f
(7.14 (7.14 (4.16
0.006) 0.004) 0.002) 0.01) 0.09) 0.12) 0.12) 0.12) 0.025) 0.024) 0.35) 0.03) 0.01) 0.01) 0.18) 0.06) 0.003) 0.0035)
0.01)
for
x lo2 x 104 x 103 x 103 x lo9 x 10’ x 10’ x 10’ x lOI6 x lo3 x lOI2 x 10s x lo* x 104 x IO* x lOa x 10’ x 106
x 10’
+ 0.02) & 0.02) i 0.01)
(se4
Half-Life
Information
I
5.5 14.2 57.6 7.0
f
:I;)
x 10-Z
26.0 f 2.0 i 0.4) x IO-4 1.6 & 0.5
Ratio
Unique
rir 0.6 + 2.0 i 1.3 f 2.0 100 13.0 * 4.0 0.6 f 0.07 0.8 3 0.12 10.6 zt 1.3 0.83 + .081 100 100 81.5 f 0.6 0.18 i 0.031 1.4 & 0.141 ~8.6 zt 2.0’ (1.7 &- 0.6) x 1O-3 0.09 f 0.03
(4.8
(1.2
Branching 03
First-Forbidden
TABLE
(5.52 (7.15 (3.87 (4.76 (8.49 (6.4 (1.38 (1.04 (3.80 (7.92 (2.43 (1.04 (5.50 (2.49 (5.59 ~(1.40 (8.27 (4.35
I-t 0.58) * 1.0) & 0.88) + 1.4) * 0.09) i 2.0) * 0.16) + 0.16) f 0.47) i 0.77) f 0.35) & 0.03) f 0.41) * 0.42) zk 0.56) + 0.32) i 2.92) f 1.54)
x lOa x 105 x lo3 x lo4 x 109 x lo2 x lo* x lo4 x 10” x lo5 x 1Ol2 x 109 x 104 x 10’ x lo6 x lo4 x IO” x 108
(28 3~ 2) 5 2.0) x 10’ A 0.80) x lo2
A Q 50
57+ 12.3 (. - 1.21 x lo4
(6.0 (2.68
with
Half-Life (se4
Decay
Partial
Beta
f
3 4860 i 40’ 2947 + 20’ 4913 i- 5h 3438 31 18 565 i 5 7410 + 150 5289 i 150 4518 + 150 44.1 f 1.0” 2491.9 31 1.1 412 f 8 598 i 41 3519 f 4h 1691 & 5 1817 + 10 4195 f 13 239.6 i 2.0 1213.8 f 2.7
4709
10422 f 3.5 4369 & 4 8679 + 15
tkW
E, (max)
Refs.
678
TOWNER,
WARBURTON,
AND
GARVEY
been fully described previously [3, 121. We shall only give a brief review here. The first step is the evaluation of the Fermi function fn = ,:” 4-T
W> W,
W P W+‘o -
W)’ d W,
(1)
where Wis the /? energy and W, the disintegration energy both in units of the electron rest mass (and including the rest mass), F(Z, W) is the ratio of the electron density at the nucleus to the density at infinity, and u,(Z, W) is the shape factor. The latter can be written in the form a,(Z, W) = (2n + l)! i: C~,“~2~w~“E;:,n’“~l
l;jz(n-“)
3
(2)
V=O
where, in our normalization, G,o(Z, w> = G*,(O, w> = Cn,,(Z a> = 1, and C,,,(Z, W) was evaluated from results summarized by Konopinski accuracy of the calculation has been discussed [12]. Ref.
to Table
[13]. The
I.
o From the atomic mass table of J. H. E. MATTAUCH, W. THIELE, AND A. H. WAPSTRA, Nucf. Phys. 67 (1965), 1, unless otherwise specified. b E. K. WARBURTON, W. R. HARRIS, AND D. E. ALBURGER, Phys. Rev. 175 (1968), 1275; J. K. BIENLEIN AND E. KALSCH, Nucl. Phys. 50 (1964), 202. c F. AJZENBERG~ELOVE AND T. LAURITSEN, Nucl. Phys. 11 (1959), 1. d M. G. SILBERT AND J. C. HOPKINS, Phys. Rev. B 134 (1964), 16. e P. M. ENDT AND C. VAN DER LEUN, Nucl. Phys. A 105 (1967), 1. IS. WIRJOAMIDJOJO AND B. D. KERN, Phys. Rev. 163 (1967), 1094; B. D. KERN, Phys. Rev. C 3 (1971), 413. g G. A. P. ENGELBER~NK AND J. W. OLNESS, Bull. Amer. Phys. Sot. 15 (1970), 566; and private communication. n D. S. KAPPE, Ph.D. Thesis, Pennsylvania State University, 1965, unpublished. *J. VAN KLINKEN, F. PLEITER, AND H. T. DIJKSTRA, Nucl. Phys. A 112 (1968), 372. j B. D. KERN, R. W. WINTERS, AND M. E. JERRELL, to be published; E. L. ROBINSON, M. S. Thesis, Purdue University, 1958, unpublished. b J. B. MARION, Nucl. Data Sect. A 4 (1968), 301. z No uncertainty quoted in literature; the value given here is assumed. m H. W. TAYLOR, J. D. KING, H. ING AND R. J. Cox, Can. J. Phys. 47 (1969), 1539. n H. MORINAGA AND G. WOLZAK, Phys. Lett. 11 (1964), 148. 0 L. G. MULTHAUF AND W. W. PRAY, Nucl. Phys. A 114 (1968), 476. f H. T. FISCHBECK, Bull. Amer. Phys. Sot. 13 (1968), 697. g S. T. HXJE, M. U. KIM, L. M. LANGER, E. H. SPEJLWSKI AND J. B. WILLETT, Nucl. Phys. A 101 (1967), 688. r J. C. COOPER AND B. CRASEMANN, Phys. Rev. C 2 (1970), 451. 8 K. I&WADE, H. Y~OTO, K. YOSHIKAWA, K. IIZAWA, I. KITAMLJRA, S. AMEMIYA, T. KATOH, AND Y. YOSHIZAWA, J. Phys. Sot. Japan 29 (1970), 43, give results from which (0.36 j, 0.03) % is extracted for this branch.
HINDRANCE
679
PHENOMENA
The p-moment, (G,Jz, is then given in terms offat, where t is the partial half-life for the transition in question, (G,)‘J
= (‘n 2) (--$$)
[(:n++1)l;!12&,)2n(fnt>-‘,
(3)
where A,, is the Compton wavelength of the electron. Giving fnt and (In 2)(2rr3/g2CA2) the units of time, (G,)2 has the units of A:: . We shall use (In 2)(2.rr3/g2CA2)= (4.04 & 0.07) x lo3 set [14, 151 and X,, = 386.M with the result that (G1)2 = 1.81 x log (fit)-‘F2, (G2)2 = 4.04 x 1015Cfit)-’ F4,
(4)
(G3)2 = 2.11 x 1O22(f3t)-l F6. TABLE Extraction
of the fit
Transition
and
from
11 the Experimental
Data
h&2)
fit
(1.26 + 0.10) x lo9 (9.2 j, 3.1) x lo8 (3.6 & 1.1) x log (1.5
“‘s(p-)“‘Cl ““S(j?)““Cl 38C1(/+)38Ar
;I;)
x 10’0
9.10 + 0.04 9.96 & 0.15 9.56 5 0.13
(2.13 (1.18 (1.64
+ 0.26) * 0.17) & 0.04)
x lo9 x 109 x lOa
9.33 9.07 9.22
+ & ?E &
o’51 0.10 0.05 0.06 0.01
(2.11
& 0.61)
x lo8
9.32
f
0.12
* f + + & * i k * + * i
0.02 0.15 0.10 0.11 0.07 0.04 0.07 0.16 0.005 0.07 0.05 0.10
(1.25 (3.90 (9.42 (2.57 (3.46 (4.97 (2.28 (2.05 (2.98 (1.60 (5.46 >(2.29
:
10.16
10.10 9.59 9.97 9.41 11.54 9.70 10.36 9.31 9.474 10.20 9.74 29.36
I
(se4 16N(p-)160 ‘“N(p-)‘“O* “N(/F)“O
of Table
I-t & & f & i f i * + + *
0.06) x 10”’ 1.3) x 108 2.1) x lo9 0.7) x lo9 0.55) x 10” 0.48) x 10s 0.36) x 10”’ 0.80) x lOa 0.03) x lo9 0.27) x 1O’O 0.60) x lOa 0.55) x LO9
(1.90
*
0.67)
x 10”’
10.28
* 0.15
(4.29
+
1.43)
x 1O’O
10.63
& 0.14
1.43 0.20 0.50 0 13 ’ 0.85 1.53 1.10 0 84 ’ 0.145 0.46 0.19 0.70 0.0052 0.37 0.079 0.88 0.607 0.11 0.33 GO.79 o 1o . o o4 .
* 5 i + rt i + + f * * + & & If * + 5 * & + + -
0.11 0.07 0.15 0.03 0.09 0.10 0.22 0.03 0.34 0.20 0.007 0.15 0.04 0.20 0.0008 0.03 0.013 0.34 0.014 0.02 0.035 0.19 0.05 0.02
0.02 0.01
680
TOWNER, WARBURTON,
AND GARVEY
Table II summarizes our evaluations of fit, log(f, t), and (G1)2 for the data shown in Table I. The evaluation for the two cases of electron capture and the two cases of IZ = 3 decay were discussed previously [3]. For our shell-model calculations, we shall use an isospin formalism so that the p-moment has the form [3] (G,) = (- l)Tt-=3t ( -‘--ff
;
;,,
(J,T{;~~;;!,~T~)
,
(5)
where the definition of the Wigner-Eckhart theorem follows de-Shalit and Talmi [l] and the matrix element is reduced with respect to both J and T. GF+l’ is assumed to be a sum over all A nucleons of a one-body operator. For a single-particle ;;i;3jTansition, the isospin dependence gives unity and evaluation of Eq. (5)
(jf 11P[P) X c~](~+l)11ji) (2ji + l)lj2
= 61j2(2n + 3)li2 (2jf + 1)1/2 (n,l, 1rn 1nJi)(l~ 1)Ccn) 11li)
2
(6)
where a 9j-coefficient appears on the far right; (lf II Cfn) 11Ii) = (-)“’
[(2lf + 1)(2/l + l)]“”
(,”
z
k);
(7)
our convention being to couple I + s = j.
III. SIMPLE SHELL MODEL
ESTIMATES
The beta decays in question are of the following type lb, 4-M (p,fY;
T) + lb, W-)
(~,f>~-‘;
T - I>,
09
that is, a state of isospin T and a configuration of N particles-M holes beta decays to a state of isospin T - 1 and a configuration of (N - 1) particles444 - 1) holes. The magnitude of the isospin T is &(M + N). The situation for positron emission or electron capture is given by reversing the direction of the arrow in Eq. (8). The spins of the initial and final nuclei for the unique first-forbidden decays which we shall consider are given in Table III. An exception to this scheme is the electron
681
HINDRANCE PHENOMENA
capture of the 40K 4-, T = 1 ground state to the 40Ar 2+, T = 2 ground state.l Furthermore, 40K has two branches for unique third-forbidden decay which we also consider below. TABLE III The Spins of the N Particle-A4 Hole Initial State and the (N - 1) Particle-@4 - 1) Hole Final State in Unique First-Forbidden p- Decay M
N
Initial Spin
Final Spin
even odd even odd
even odd odd even
o+ 27/23/2+
20+ 3/2+ 7/2-
We define the extreme single-particle estimate to be the transition rate obtained in the following calculation. Consider all M holes of the initial state to be in the l&,-orbit in a seniority zero (one) configuration for M even (odd). Further, consider all N particles to be in the If,,,-orbit again, in the lowest seniority configuration. With similar assumptions for the final nuclei, algebraic expressions for the p-moment are easily derivable for the transition l&/P
Tl , ‘3112)~ T, ; T) + I@s/P’-~)
Tl - 4; (h/F’
T, - ;; T -
I>,
(9)
where T = Tl + T, , M = 2T, and N = 2T, . They are obtained by standard techniques of fractional parentage expansions. Using algebraic formulae for the cfps as given in de-Shalit and Talmi [ 1] one finds there are four cases: (i)
M = even, N = odd, i.e., l/2- -+ 3/2f (G1)& = $9
(ii)
M = odd, N = even, i.e., 3/2+ + 7/2(G,):,
(iii)
- 2TJ T,(G, ; (7/2- + 3/2+))2;
= Q(5 - 2T,) T,(G, ; (7/2- -+ 3/2+))2;
M = even, N = even, i.e., O++ 2(G)&
= T,T,
1 We do not consider the decays of ““Cl and 4aK to excited states of 4oAr and %a. The experimental evidence for the former was reported after this work was completed while the latter, like the mass 4.5 and 47 decays of Tables I and II (also not considered further), is forbidden in first order and thus is not of direct interest to the present work.
682
TOWNER,
(iv)
WARBURTON,
AND
GARVEY
M = odd, N = odd, i.e., 2- + O+
&(5 -
2T,)(9 - 2T,)(G,
; (7/2- + 3/2+))2;
where (G, ; (7/2- -+ 3/2+))2 is the single-particle (sp) p-moment which from Eq. (6) is (G; (7/2- + 3/2+))” = j+ (r)2 = 8F2. Similar expressions for the two cases of unique third-forbidden (i)
M = even, N = even, i.e., Of + 4
(ii)
decay are as follows:
; (7/2- -
3/2+)>“;
M = odd, N = odd, i.e., 4- + 0+ (G3)&, = &5
- 2T,)(9 - 2T,)(G,
; (7/2- + 3/2+))“;
where (G3 ; (7/2- --f 3/2+))” = &- (r3)” w 300F6. All the above formulae hold for electron emission. Expressions for positron emission are obtained by multiplying the appropriate formula by [(2JY + 1)/(2J, + l)], where Ji and Jf are the initial and final nuclear spins in the positron emission process. In Table IV we list these single-particle estimates together with a hindrance factor, h, defined as
It is immediately obvious from Table IV that the experimental p-moments,
HlNDRANCE
683
PHENOMENA
TABLE
IV
Single Particle Estimates of the Beta Moment Initial P;
Decay
T
Final Jm; T
h”
Unique First Forbidden “‘s(p-)“‘Cl 3qp--)w1 38Cl(,P)38Ar
7/2-; 512 Of;3 2-;2
3]2+ ; 312 2-; 2 Of; 1
0.85 & 0.10 1.53 f 0.22 1.10 f 0.03
39Cl(,+)39Ar
3/2+;
512
7/2-;
312
0.84
7/2-; 2-; 4-; 7/2-; 7/2-; o+; 22; 312’;
312 3 1 512 l/2 3 2 5/2
3/2+; 0+;2 2+;2 3/2+; 3/2+; 2-;2 Of; 7/2-;
l/2
0.145 0.46 0.0052 0.37 0.079 0.88 0.607 0.33
3/2 3/2 1 3/2
+ * i i &
o’34 0.20 0.007 0.15 0.0008 0.03
8.0 16.0 6.4
9.4 i 1.1 10.5 * 1.7 5.8 i 0.2
2.0
2.4 A 0.7
4.0 4.8
28 & 1 10.4 & 3.4
3.0 2.0 16.0 9.6 8.0
l 0.013 * 0.34 + 0.014 * 0.035
8.1 25 18 15.8 24
& + * 3~ i
0.6 4 6 0.4 2
Unique Third
Forbidden
40K(p +)aoAr “OK(fl-)‘OCa ah =
4-; 4-;
~:G,)&/
Of;2 o+;o
1 1
10.6 & 1.3 51.6 k 1.3
33.0 266.0
3.1 5 0.4 5.1 * 0.1
.
between the orbits fT12and ds12. This is estimated from the binding energies (BE) of the calcium isotopes, viz., AE = 2BE(40Ca) - BE(3gCa) - BE(41Ca) = 7.2 MeV. The diagonalizations were performed using the Oxford shell-model program [17], and the resulting p-moments are given in Table V. In several cases a very small /?-moment results from the calculation. Consider, for example, the 40K(EC) 40Ar decay. The 40K wavefunction must be a pure If,,,-Id,,, particle-hole state in this model, whereas the 40Ar wavefunction may have as many as four terms: I 40Ar; 2+T
=
2)
=
~o(d3di=fo,~=l +
44/&,~
(f7~2)~=2,~=1 (f&1
+
+
4(d3&
4(4&2,
.
(11)
The p-moment amplitude for this decay is (G,) = 3 dz b[A, - 7 d$A1
- 6 d$4,
+ &
A3],
(12)
Forbidden
4oK@+)40Ar 4oKQ3-)40Ca
Third
4-i 4-;
l/2- ; 2-;3 4-; l/2-; l/2-; o+; 2-;2 3/2+;
39Ar(j-)38K 4~C1@-)4~Ar 4oK(EC)40Ar* 41Ar@-)41K *Q(EC)“K 4aAr(p-)42K 42K(/?)42Ca 4sKt’J-)4SCa
1 1
512
1 512 l/2 3
312
512
3/2+;
Wl@-)=Ar
Forbidden
l/2-; 512 Of; 3 2-;2
First
0+;2 o+;o
1 312
312 312
l/2
; 312
3/2+; 0+;2 2+;2 3/2+ ; 3/2+; 2-;2 o+; l/2-;
l/2-
3/2+; 312 2-;2 Of; 1
Final
J”; T
Jn; T
Moment
Initial
of the Beta
Yqp-y’cl ~~s(p-)=Cl 38C1(/-)48Ar
a Ref. [16]. *Ref. [5]. ’ htheo =
Unique
Unique
Decay
Calculations
using
TABLE
i f f + + 6 i * f f * &
Interaction
V
0.10 0.22 0.03 0.34 0.20 0.007 0.15 0.0008 0.03 0.013 0.34 0.014 0.035
10.6 & 1.3 51.6 * 1.3
0.85 1.53 1.10 o 84 * 0.145 0.46 0.0052 0.37 0.079 0.88 0.607 0.33
the Residual
61. 266.
0.2 3.4 0.01 2.8 0.04 6.9 4.9 1.4
8.0
8.1 16.3 6.5
of Kuo
1.2 1.7 0.2
3.5 0.2 0.4
0.6
2.4
5.8 & 0.8 5.1 & 0.1
7.9 f 8.1 f 4.3 *
7.4 f 7.6 f -
-
9.5 + 3.0
9.5 f 10.6 i 5.9 *
and Browna
19. 266.
4 x IO-5 2.9 0.015 2.4 0.004 6.4 4.3 1.0
8.2
8.1 16.3 6.5
and of Perez”
7.4 f 5.1 *
0.8 0.1
7.4 + 3.5 7.1 + 0.2 3.3 ,. 0.4
6.5 * 0.6
6.3 h 2.1 -
-
9.8 i= 3.0
9.5 * 1.2 10.6 & 1.7 5.9 f 0.2
HINDRANCE
685
PHENOMENA
where the radial integral (Y) has been evaluated using oscillator wavefunctions of length parameter b. The computed 40Ar wavefunction, using the residual interaction of Perez [5], gives A, = 0.748, A, = 0.512, A, = -0.300 and A, = 0.299. It is immediately obvious that there is a large degree of cancellation in Eq. (12), which is responsible for the small p-moment. Similar arguments explain all the very small ((G$ < 0.15F2) /3-moments and thus we see that the three most inhibited transitions observed yield very small p-moments with wavefunctions diagonalized within the l~71p-ld3,2 space. A small variation of the residual interaction could probably give complete agreement with experiment for these three cases. The remaining cases, however, still show significant hindrances which are not yet explained by the theory. For these the hindrance factor, hKuo or h,,,,, , is much more uniform than the corresponding values obtained from the extreme singleparticle estimates given in Table IV. For the unique first-forbidden decays the hindrance factor appears to be 7 f 1, and for the two cases of unique thirdforbidden decays it appears to be a little smaller at 6 + 1. The near constancy of the hindrance factor for first-forbidden decay in this region of the periodic table has been noticed before in the work of Oquidam and Jancovici [6]. It gives weight to the procedure of obtaining p-moments from simple shell-model calculations, but employing renormalized coupling constants. In this case the axial vector coupling constant C, has to be divided by approximately 2.5 f 0.3 (i.e., something in the range 45-d@. This is exactly analogous to the common practice of introducing an effective charge into shell-model calculations of EL y-ray transitions. The remainder of this paper will be concerned in investigating the origin of these hindrance factors.
IV. FIRST ORDER PERTURBATIONTHEORY The calculations of Section III are extended in the following way. To the initial nuclear wavefunction terms are added which connect via a one-body operator with the dominant terms in the final state wavefunction; and similarly to the final state wavefunction terms are added which connect with the dominant terms in the initial wavefunction. Figure 1 shows a picture of typical configurations involved. Schematically the calculation can be viewed in the following way: l(h.)-M
(f7dN;
T) + c c&ae+ah l(d~J(~-~)
(f,,JNel;
T -
1)
Ph
- WW~)-(~-~) (.f,~~)~-~; T- 1)+ cDh db,+ahl(4d-“(f7dN;T).
(13)
686
TOWNER,
WARBURTON,
INITIAL CONFIGURATIONS
AND
GARVEY
FINAL CONFIGURATIONS
leading term
leading term
final state
FIG. 1. Diagram of some of the relevant configurations entering for first forbidden tions. Arrows indicate the configurations linked by the beta decay.
The coefficients aD,hare found from first-order perturbation
cd;;= <(4/P’
(fv,dN; T I ~G,+G WNP-~)
transi-
theory
(f,,zF;
T - 1) ,
(14)
AE;;
with a similar expression for cy. ‘,’ . We shall take the energy denominator, AE,, , to be approximately given by the single-particle energies, E, of the orbits involved, viz., AE(~) = Ef - Ed + Eh - ED, Db (15)
A&
= Ed -
Ef +
ch
-
Ep
*
The contribution to the p-moment from these admixtures depends on %h
(16)
where N, represents the number of oscillator quanta in the orbital j, and similarly for Nh . Thus, for unique first-forbidden beta decay, j, and jh may differ by only one oscillator quantum, hence in Eq. (13) the sum over p and h will only involve particles in the (If, 2p)-shell and holes in the (2s, Id)-shell. For unique third-
HINDRANCE
687
PHENOMENA
forbidden decays we shall also consider particles in the (3s, 2d, Ig)-shell and holes in the (lp)-shell. The admixtures in the initial wavefunction o$; are induced through the particlehole force. That this force will give rise to a hindrance in the beta decay can be seen most simply from schematic model calculations [3] and from the calculations of Bertsch and Molinari [8]. The admixtures in the final-state wavefunction, ala;, that are connected to the dominant term in the initial state via a one-body operator have N + 1 particles and M + 1 holes. They are often termed ground-state correlations. Such correlations are known to be important when treating electromagnetic transitions in doubly magic nuclei in so far as they increase the transition probability to “collective” partile-hole states by a factor of two or so. At the outset of this study it was not evident to us that these ground-state correlations would have an important effect on retarded transitions. As will be shown in what follows their effect is appreciable, in some cases as large as the effects arising from the extended diagonalization of the initial state. This large effect, due to the final-state correlation, comes about even though the energy denominator, d Eih , is larger than the corresponding quantity in the initial state, because there are more configurations possible which taken in total give rise to an appreciable correlation. Numerical examples are presented in Section V. It is instructive to evaluate the retardation of the decay due to configuration mixing under certain simplifying assumptions. We calculate the ratio of the contribution to the transition matrix element from a particular particle-hole admixture in the initial wavefunction, viz., A:; = a$
G, ; I aa+ah / (d3,2)-(M-1) (f,,#‘-l; ( + I(&&(-) (f7/dN-? JtW)>
relative to the contribution
J,T,); JiTi) (17)
from the dominant terms, viz.,
D = ( G, ; l(4,P
(f,dN; JiTi) - IQ&,& (“-lYf,,zF1; J,W).
(18)
Here Ji , Ti are the spin and isospin of the initial state and Jf, Tf that for the final state. Summing (A$ + A$/0 over all particle and hole orbits, p and h, and squaring will give a measure of the hindrance. After some Racah algebra we find that A(i) -
uh
D
zzz
1
2.L
Jq-
x
+
2/g
1
(G
; (j,
+
jh)>
; (n + 1) 1 I VI d;if7,2 ; (n + 1) 1).
(19)
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Note that the only particle-hole matrix elements involved are those with T = 1 and spin equal to the multipolarity of the transition. In the derivation we have assumed j(d3,2)-(M-1) (f,,2)N-1; JfTf) that j, # hi2 and jh f G2 , i.e., the configuration can be taken as an inert core of spin Jf and isospin Tf, and the creation (a,+) and annihilation (ah) operators have their usual commutation relations. For the case j, = fCj2 or jh = daj2 the above expression is obtained on neglecting antisymmetrization. In the numerical results given in the next section, antisymmetrization is correctly taken into account using the methods of fractional parentage coefficients. Notice that the hindrance in the p-moment is independent of Ji , Ti , Jf , Tt , N and A4 and depends only on the orbits involved and the multipolarity of the transition. The hindrance can further be shown to depend only on orbital angular momenta. This is achieved by assuming that the energy denominators AEfi and all radial integrals depend only on orbital angular momenta and summing over particle and hole orbits j, and j, , assuming that both members of a spin-orbit pair j = 1 f l/2 are present in the summation over p and h. First the particle-hole matrix element is expressed in L-S coupling:
x ((l,-l1,); LST = 1 1 I’ j (d-y);
LST = 1).
(20)
Then substituting this expression into Eq. (19), evaluating the single-particle /?-moments from Eq. (6), and summing over the two possible j values, 1 =t l/2, it is seen that the orthonormality conditions on the 9j-symbols selects L = n and S = 1, in which case the sum over all particle-hole admixtures becomes
x ((&‘Z,); L = n, S = 1, T = 1 1 V 1(d-lf);
L = n, S = 1, T = 1). (21)
This expression can be further simplified on evaluating these particle-hole matrix elements using a zero-range residual interaction, and assuming that the radial integrals can be replaced by a constant average value, 1, independent of the orbits.
HINDRANCE
PHENOMENA
689
In which case
(22) Here I’,, is the strength of the residual particle-particle interaction (negative for attractive particle-particle forces) and (B + H) is the proportion of spin and isospin exchange forces in the interaction. Repeating the entire procedure for admixtures in the final nucleus wavefunction, we find, for an analogously defined quantity Afph, the identical expression:
For attractive particle-particle forces the energy denominators dEPh defined in Eq. (15) are negative as well as V,, . Furthermore, the value of B + H is certainly between 0 and 1 from what we understand about the nuclear force. The assumption that the radial integrals, Z, are approximately constant and are all positive implies a choice of sign convention for the radial wavefunctions, namely that the functions are positive asymptotically. This convention also makes the integrals (P) all positive, thus we can assert that for attractive particle-particle forces Eqs. (22) and (23) are negative definite and hence the admixtures shown schematically in Eq. (13) will produce a hindrance in the beta-decay rate. Although a number of drastic approximations have been made in deriving Eqs. (22) and (23) they enable a number of conclusions to be drawn which should hold for more realistic calculations. We summarize these results as follows: (i) The hindrance residual particle-particle
in the beta-decay rate increases as the strength of the interaction, V, , increases for attractive forces.
(ii) The hindrance decreases as the proportion forces (B + H) increases in the residual interaction.
of spin and isospin exchange
(iii) For a transition of multipolarity, IZ + 1, predominantly described by the matrix element (G, ; (ji -+ j,)), the hindrance introduced by particle-hole admixtures is approximately independent of the particular nuclei involved (i.e., independent of Ji , Ti , Jf , T, , Nand 44). This justifies the procedure of renormalizing the one-body operator, G(“+l), the renormalization depending only on the multipolarity and the orbital angular momenta Ii and If . 595/w2-19
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(iv) The hindrance introduced by admixtures in the initial wavefunction are greater than those introduced by admixtures in the final wavefunction (correlations of the core) since the energy denominator AEiA is less than AE$ . However the core correlations are not negligible as will be seen by the computations in the next section.
V. NUMERICAL
RESULTS
We now compute A!;, A$ and D as defined in Eqs. (17) and (18) and hence obtain a theoretical estimate for the @moment (24) where Ni and Nf are normalization
constants,
(25) Nf2 =
1 + c / 01 nh
and the o19%are the admixture coefficients defined in Eq. (14). The particle-hole matrix elements were calculated using a simple phenomenological finite-range residual interaction Jqr) = ~Op+ro2 [w + BP, + MP, + HP,],
(26)
where W, B, M and H are the strengths of the Wigner, Bartlett, Majorama, and Heisenberg exchange forces, V,, is the overall strength of the interaction, PH = -P, and Y,,is the range. Their values were taken from the calculations of Perez [5] in which the parameters had been adjusted until a good fit to the experimental particle-hole energy levels in 40Ca was obtained. This procedure gave V, = -40.84 MeV, W = 0.13, B = 0.26, M = 0.57, and H = 0.04. Perez, however, used a Yukawa shape cur/(w) for the residual interaction with 01 = 0.65F-l. For simplicity we have used a Gaussian shape, the range r. was chosen by requiring that the Gaussian and Yukawa shapes should have the same volume integral, in which case r. = (+‘+P)W
a-1
HINDRANCE
691
PHENOMENA
giving rD = 2.018F. All radial integrals were evaluated using harmonic oscillator wavefunctions, the length parameter was chosen such that b = 1.006A1J6F, which for mass A = 40 gives b = 1.86F, this being the value used in our previous calculations of unique third-forbidden beta decay [3]. First-order perturbation theory was used to calculate A$ and AFL as discussed in the last section. The cases for which j, = fT12or j, = dsj2 are correctly evaluated using the methods of fractional parentage expansions [l]. The energy denominators AE,, were approximated by Eq. (15). Note, however, that the energy denominator AE$ can be quite small; in fact on neglecting spin-orbit splitting and using oscillator energies AE;; = 0. Thus the use of first-order perturbation theory for the initial wavefunction must be viewed with some suspicion. The procedure adopted was to make the full matrix diagonalization calculation for the initial wavefunction, and then to repeat the calculation in perturbation theory using the single-particle energies given in Table VI. A constant was then added to the energy denominator AE$ which would bring the perturbation theory calculation into agreement with the full diagonalization for the important admixtures. The required constant varied from case to case but ranged from zero to 3 MeV in the worst cases (these typically being Of states). We therefore conclude that the use of perturbation theory was not so inaccurate as might be suspected. There is no (f) this is typically of the order problem with the final state energy denominator A Eyh, of 2 hw. TABLE
VI
Single Particle Energies, E, used in the Evaluation the Energy Denominators AE,,
ORBIT
C&V)
IP,h IPI/? Ids/z
-31.0 -28.0 -20.74 -18.11 -15.64 -8.36 -6.30 -2.70 -4.22 -3.38 0.60 2.60 6.00 7.00
2% M/s lh13 2P3/2
If512 2P1/9 kd2 2&z 3.%/z 2&z I&/z
of
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The results of the computations are given in Table VII. For the transitions with very small experimental /I-moments, (G1)2 < 0.15Fs, the calculations also predict very small values due primarily to cancellations among various dominant terms in D. This was discussed in Section III. In these cases the results are very sensitive to the parameters of the calculation, and with the simple phenomenological residual interaction used here, we do not expect to obtain a good numerical agreement. We are content to note that while the hindrance phenomena under consideration are present in these cases, the small &moment can be well explained by a shell-model calculation in the highly truncated space involving only ds12and f,,2 orbits. Two of the decays considered by Bertsch and Molinari [8] are of this category, and their failure to obtain good quantitative agreement with experiment is the same as ours, namely that the computations predict a near perfect cancellation in the transition matrix element. TABLE Theoretical
Beta Moments Calculated A!; and A$ Which Interfere
VII
using Perturbation Theory for the Admixtures with the Dominant Terms, D
.Z,,,,A”’ (F$?
Decay
Unique First Forbidden 3’S@-)3’Cl ““s(p-)““Cl 38Cl(/+)38Ar
0.92 0.81 0.95
0.96 0.98 0.91
2.84 4.04 2.56
-0.99 -1.50 -0.84
-0.75 -1.06 -0.67
0.96 1.56 0.83
38Cl(g-)38Ar
0.94
0.98
2.86
-1.21
-0.75
0.69
3eAr(fi-)38K %1(/3-)4OAr 40K(EC)40Ar* 41Ar(,!-)41K “Ca(EC)“‘K 4zAr(p-)4aK 42K(fl-)42Ca 43K(,3-)43Ca
0.96 0.96 0.99 0.95 0.97 0.93 0.95 0.96
0.97 0.95 0.95 0.98 0.98 0.98 0.93 0.98
0.00 1.72 0.12 1.57 0.05 2.54 2.07 1.01
-0.08 -0.38 -0.16 -0.72 -0.12 -0.99 -0.72 -0.49
0.00 -0.45 -0.03 -0.42 -0.01 -0.66 -0.55 -0.27
0.0065 0.66 0.003 3 0.16 0.0054 0.62 0.51 0.056
Unique Thud Forbidden 4oK@f)40Ar 40K(j-)40Ca
0.96 0.99
0.99 0.89
8.90 16.32
-3.14 -4.55
-0.83 -1.54
21.7 81.0
<‘=&a (F2’9
0.85 1.53 1.10 o 84 . 0.145 0.46 0.0052 0.37 0.079 0.88 0.607 0.33
f * + + f f +
0.10 0.22 0.03 0.34 0.20 0.007 0.15 0.0008 l 0.03 i- 0.013 * 0.34 & 0.014 f 0.035
10.6 51.6
f f
1.3 1.3
Of more interest are the decays in which simple shell-model calculations fail. In these cases the hindrance introduced by perturbing the initial and final wavefunctions is sufficient to bring theory and experiment into almost complete agreement.
HINDRANCE
693
PHENOMENA
Note in particular that the contribution from &ADh (f) , this being due to correlations in the core, is sizable despite the fact that the energy denominator AELf,’ used in this case is larger than that used when perturbing the initial wavefunction. It is quite clear that these core correlations cannot be neglected. It is of interest to see which particle and hole orbits are responsible for the hindrance. In Table VIII, a breakdown of the contributions are given for the decay 38C1(p-) 3*Ar. For admixtures in the 38C1wavefunction the largest values of Ati occur for j,, = dsj2 , j, = fT12 and j, = 2~~,~, j, = 2p,,, and for both these casesj, = Zh + l/2 and j, = Z, + l/2 with the consequence that in Eq. (6) for the single-particle p-moment, (G, ; (j, --f j,)), the 9j symbol is stretched and correspondingly has a large value. For the 4 hole-2 particle admixtures in 38Ar almost the entire contribution to the hindrance comes from the configuration (d3,J4 (f,,2)2. Again this is to be expected since this configuration has a large seniority zero component which is energetically favored, whereas all other configurations considered break the pairing energy by having seniority four or six.
Values of the Individual
TABLE VIII A$! and A$ for the Decay 38C1(~-)38Ar 5
Hole Orbit
l&z
2w2 l&z
Particle Orbit
Db
-0.353 -0.387 -0.131
2P3/2
lf5la
2PliZ
$0.033 -0.004 -0.324 -0.094 -0.029 -0.013
-0.008 -0.023 -0.087 -0.041
-0.001 -0.0001 -0.033 -0.015
a The upper entry gives A:,! and the lower entry ALfl in F. b D is the dominant term, in this case D = 2.555 F.
VI. CONCLUSIONS The fact that all unique n-forbidden beta decays, with n odd, among light nuclei (A < 50) show a characteristic hindrance relative to truncated shell-model calculations has been traced to the repulsive nature of the T = 1 particle-hole interaction. This repulsive particle-hole interaction (attractive particle-particle) causes thef,,, particles and d3,2 holes to move further apart Cf1,2 particles and d3,2 particles to move closer together) than can be obtained with simplef& dGy wavefunctions and hence considerable configuration mixing is required. The fact that
694
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the particle and hole move further apart leads to a retardation of the decays in which the particle decays into the hole. In this paper we have examined these correlations using a microscopic shell-model description. The degree of success we have had can be gauged from Table VII where the computations are compared with the experimental p-moments. The results are exceptionally good considering that there has been no adjustment of parameters to fit the data, and that the hindrance of - 7 has been explained with terms making up about 10% in intensity of the wavefunctions. The calculations can however be criticized on a number of points. First, the spurious center-of-mass motion has not been removed. Second, we have used simple oscillator wavefunctions and phenomenological residual interactions rather than more realistic forces. Finally, our choice of energy denominator, AE,, , which has a crucial bearing on the perturbation theory calculations, was rather arbitrarily chosen. Related to this is the choice of single-particle energies given in Table VI. We have used values taken from the positions of the single-particle and single-hole states in 41Ca and 3gCa, in particular the gap between the d3,2 and fT12 orbits was taken to be 7.2 MeV. Gerace and Green [19] reason that the shell-model f,,2-d3,2 separation energy should be 5.4 MeV rather than 7.2 MeV. This is because the contributions to the binding energy due to nuclear deformations are different for 40Ca, 3gCa, and 41Ca. One alternative procedure which avoids some of these problems is to first diagonalize the l-particle-l-hole states in 40Ca. Then these correlated particle-hole vibrations are coupled to the states l(d3d-cM-1) (hdN-l
; T - 1) and l(d3d-hf (hdN;
0
to give the admixtures shown schematically in Eq. (13). The particle-hole energies required in the energy denominator for perturbation theory are now given in terms of the eigenvalues of the particle-hole diagonalization. This technique has been used by Shukla and Brown [20] in discussing transitions in the 2-hole-l-particle system at mass A = 15, and by Mavromatis and Singh [21] for the 2-particlel-hole system at mass A = 17. To improve our calculations in the light of some of these comments would increase the complexity of the computations by an order of magnitude. It is doubtful whether these improvements will cast any further light on the nature of the correlations involved in the hindrance phenomenon, but they may well improve on the numerical agreement with experiment. Finally, we wish to repeat some of the conclusions reached in Section IV from some simple arguments, namely that the unique first-forbidden beta-decay hindrance (i) increases as the strength of the particle-hole force is increased; (ii) decreases as the proportion of spin and isospin exchange forces are increased;
HINDRANCE
(iii)
is approximately
PHENOMENA
695
independent of the particular nuclei involved;
(iv) depends essentially on orbital angular momenta and the multipolarity of the transition only.
ACKNOWLEDGMENTS We are indebted to Dr. H. G. Benson for many fruitful discussions. This work was initiated while two of us (G.T.G. and E.K.W.) were guests at the Weizmann Institute in the winter of 1969. While there, stimulating discussions with Professors Igal Talmi and Amos de-Shalit were very important in focusing our attention on the long-standing problem of these first-forbidden unique decays. The hospitality extended to us at the institute during our short visit allowed us to become better acquainted with Prof. de-Shalit and thereby to feel most keenly the loss of this able physicist and generous person.
REFERENCES 1. A. DE-SHALIT AND I. TALMI, “Nuclear Shell Theory,” Academic Press, Inc., New York, 1963. 2. S. SIEGEL AND L. ZAMICK, Nrrcf. Phys. A 145 (1970), 89. 3. E. K. WARBURTON, G. T. GARVEY, AND I. S. TOWNER, Ann. Phys. New York 57 (1970), 174; I. S. TOWNER, Nucl. Phys. A 151 (1970), 97. 4. V. GILLET AND E. A. SANDERSON, Nucl. Phys. A 91 (1967), 292. 5. S. M. PEREZ, Nucl. Phys. A 133 (1969), 599. 6. B. OQUIDAM AND B. JANCOVICI, Nuouo Cimenfo 11 (1959), 578. 7. S. COHEN AND R. D. LAWSON, Phys. Lett. 17 (1965), 299. 8. G. BERTSCH AND A. MOLINARI, Nucl. Phys. A 148 (1970), 87. 9. H. EJIRI, K. IKEDA, AND J. FUJITA, Phys. Rev. 176 (1968), 1277; J. FUJITA AND K. IKEDA, Nucl. Phys. 67 (1965), 145; J. FUJITA, S. Funr, AND K. IKEDA, Phys. Rev. B 133 (1964), 549. 10. G. E. BROWN AND M. BOLSTERLI, Phys. Rev. Left. 3 (1959), 472. 11. G. E. BROWN, “Unified Theory of Nuclear Models and Forces,” North-Holland Publ. Co., Amsterdam, 1967. 12. E. K. WARB~TON, W. R. HARRIS, AND D. E. ALBURGER, Phys. Rev. 175 (1969), 1275. 13. E. J. KONOPINSKI, “The Theory of Beta Radioactivity,” Oxford University Press, Oxford, England, 1966. 14. C. J. CHRISTENSEN, A. NIELSEN, A. BAHNSEN, W. K. BROWN, AND B. M. RUSTAD, Phys. Lett. B 26 (1967), 11; C. J. CHRISTENSEN, V. E. KROHN, AND G. R. RINGO, Phys. Left. B 28 (1969), 411. 15. J. M. FREEMAN, J. G. JENKINS, G. MURRAY, AND W. E. BURCHAM, Phys. Rev. Left. 16 (1966), 959; J. M. FREEMAN, J. G. JENKINS, D. C. ROBINSON, G. MURRAY, AND W. R. BURCHAM, Phys. Left. B 27 (1968), 156. 16. T. T. S. Kuo AND G. E. BROWN, Nucl. Phys. A 114 (1968), 241; and private communication from T. T. S. Kuo. 17. I. S. TOWNER AND W. G. DAVIES, Oxford Nuclear Physics Theoretical Group Report No. 43, unpublished.
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18. S. G. NILSSON, Kgl. Danske Videnskab. Selskab, Mat. Fys. Medd. 29, No. 16 (1955). 19. W. J. GERACE AND A. M. GREEN, Nucl. Phys. A 113 (1958), 641. 20. A. P. SHUKL.A AND G. E. BROWN, Nucl. Phys. A 112 (1968), 296; A. P. SHUKLA, Thesis, Princeton University, 1967. 21. H. A. MAVROMATIS AND B. SINGH, Nucl. Phys. A 139 (1969), 451.