17 October 1996

PHYSICS LElTERS B

Physics Letters B 387 ( 1996) 227-232

Gamow-Teller strengths in fp shell

S. Sarkar, K. Kar Saha Institute of Nuclear Physics, Theory Group, Block-AE Sector I, Bidhun Nagar: Calcutta 700 064, India

Received 19 January 1996; revised manuscript received 4 June 1996 Editor: W. Haxton

Abstract

We develop a simple parametrization based on the spectral distribution theory for the total Gamow-Teller (GT+ ) strength in the fp shell, which reproduces the sum of the observed B(GT) of (n,p) experiments well and is useful for nuclear stmctnre and astrophysical applications.

PACS: 23.4OHc; 2545Kv; 97.60Bw Keywords: Beta decays; Nuclear structure; Charge-exchange reactions; Supernovae

Recent studies of the Gamow-Teller (GT) strength in the GT+ direction in the fP shell are motivated pri- marily by two objectives. Firstly, experimental mea- surements of GT+ strengths for quite a few nuclei in the fP shell are now available and there is consider- able advance made [ 1,2] in the shell model calcula- tions involving the full f, space. This may help in obtaining a better understanding for the long-standing problem of quenching of GT strength and in general the spin-isospin response of nuclei in the intermediate energy.

Secondly, the importance of weak interactions at the late stages of Si burning of pre-supernova (preSN) in the form of electron capture (EC) as well as beta de- cay is now well-established [ 31. The rates of these in- teractions determine the electron-to-nucleon ratio, Y,, of the system and during the early stages of gravita- tional collapse of type II supernova (SN) the electron capture on nuclei heavier than “Fe reduces this ratio further until the nuclei have neutron number greater than 40 and allowed captures get neutron-shell blocked [ 4Sl. As the hydrodynamic shock, launched at a later

stage of the evolution and responsible for the eventual explosion, has its energy sensitively dependent on Y, a careful evaluation of the weak interaction rates is essential for the SN problem. The nuclei involved in this preSN and SN stages are the fP shell nuclei with 20 5 Z 5 40 and 20 5 N 5 40.

The allowed beta strength is dominated by the Gamow-Teller component, and the laboratory p- de- cay or /3’ decay/electron capture on these nuclei can normally reach only the tail of the giant resonance due to energy constraints. On the other hand reaction studies through intermediate energy (p, n) and (n, p) reactions give information about the full resonance [6,7] for the /?-(GT_) and p+/EC(GT+) direc- tions respectively. The CHARGEX Collaboration at TRIUMF measured the total strengths (denoted as C B( GT+) ) up to excitation energy about 8-9 MeV in the daughter. Experimental strengths beyond 8- 9 MeV are also available now. A recent (n,p) mea- surement [ 81 on 60*62*64Ni isotopes give total strengths (C B( GT+) > summed up to 30 MeV excitation en- ergy in daughter nuclei. But though this energy range

0370-2693/%/$12.00 Copyright 8 1996 Elsevier Science B.V. All rights reserved. PII SO370-2693(96)00966-5

228 S. Sarkur, K. Kar/Physics Letters B 387 (1996) 227-232

is large, the estimates above 8 MeV are subjected to systematic uncertainties which may be quite large. Earlier in the absence of full fP shell calculations the theoretical estimates commonly used were different truncated space shell model results and the Ikeda form [9] S,- - S,J+ = 3(N - Z) where Sp* are the GT p* total strength or sum rules. This relation is model independent in the nucleonic level but can only be used in the GT_ direction for nuclei for which ,fS+ transition is completely blocked. The recent shell model studies in the full fp space have started prob- ing how much of the quenching is due to truncation of space [ 10.11 . On the other hand there are efforts I II] to develop approximate forms for the GT sum rule strength which with very few parameters can give observed quenched strength for many nuclei in the fp shell. When these simple parametrized forms for the total GT+ strength have good predictability they are very useful for astrophysical applications as well as for estimates in nuclear physics without making detailed time-consuming computations. The parame- ters are determined by fitting the forms to available observed total strengths of nuclei.

In this work we construct a form for the total GT+ strength based on theoretical methods, known as the spectral distribution theory, which reproduces the av- eraged shell model values and relies on an asymptotic form for the density of energy levels in the large finite dimensional shell model space. We suggest improve- ments on a previous attempt [ 111 to develop empiri- cal forms for the total GT+ strength (C B(GT+) and show how the predictive power of these forms can be improved considerably.

The single particle estimate for the sum rule which assumes pure shell model configurations for the initial Ii) and final If) states and no neutron-proton correla- tions is given by,

where n$’ (v) is the number of proton particles in the orbit r of spin j, of the parent ground state and ng (v) is the number of neutron holes in the daughter final orbit s of spin jS, and the sum is over all the proton and neutron orbits. UT+ is the one-body Gamow-Teller operator.

In spectral distribution theory one observes that with

large shell model spaces where the number of valence particles is not too small (2 4), the density of states as a function of energy goes asymptotically towards a Gaussian. The expectation value of an operator K in the state IE) of energy E can be expanded in terms of a polynomial in energy [ 12,131,

i

X C( KP,(fO)“‘f’v(E)

&+(K~)“~+,_

= (K)"'+&,_KuK(E- 6)/u+... (2)

Here d(m) is the dimensionality of the space of m va-

lence particles in N single particle states (d(m) EC,,,), (K)m stands for the average of the operator in the

space ((K)"' = C(i(Kli)/d(m)),P,(x) are polyno- mials with the density of states as the weight function

(PO(X) = 1, PI(X) = (X - E)/(+, . . .). In the above

expression E and u are the centroid and width of the Hamiltonian, LTI( is the width of the operator K in the

m-particle space. One also observes that if under the transformation H + H + aK the shape of the den- sity of states p(E) does not change for LY 4 0, then K(E) is linear in energy, i.e., in the Eq. (2) only

the first two terms survive [ 121). In the final form of RHS in Eq. (2) [K-H is the correlation coefficient of

the operator K with H. If the correlation coefficient is very small the second term has small contribution

compared to the first. When one uses K = O~,O~T where OGT =

i C:, a( i)~i (i) for the Gamow-Teller (GT% ) op-

erator for the /I* decay, it is found that its correlation coefficient with most (1 + 2)-body realistic Hamil- tonians in the fp shell are indeed very small. Thus an expansion with only the first two terms in Eq. (2) is a good approximation to the shell model values [ 141. But in the sd shell the correlation coefficient of K with ( 1 + 2)-body realistic Hamiltonians is large (about 0.5 or greater) and there the terms beyond the second in the expansion is needed [ 151. This is one of the reasons for which the simple l- or 2-parameter forms considered by Koonin and Langanke [ 111 fail in the sd shell.

S. Sarkar, K. Kar/ Physics Letters B 387 (19%) 227-232 229

In spectral distribution theory, the result in the m-particle space is called a scalar result. To make contact with real nuclei one should work in (m,T) spaces where T is the isospin of the states in the space considered or one works in (m,, mp ) spaces where mn/mp stand for the valence neutron/proton number. These can also be extended to configuration-isospin (F%, T) and configuration proton-neutron number iiz”, EP) spaces. The notation (fin, tip) stands for m,m2,m,...,m, (neutron, proton) particles in orbits numbered as 1,2,3,. . . , r.

We choose the scalar neutron-proton space (m,, mp) and write the expression for the GT sum rule strength as a function of the energy E of the initial state. For our purposes E = Eg.$., the ground state energy of the target nucleus. So with K = OGT++OGT, EQ. (2) takes the form

K(E,,.) =a] mp(Nn - 4

N N

P n

+ bl w,(N, - mn)mp(Np - m,) (E,.,. - ~1

N,(N,, - l)NP(NP - 1) ’ u

(3)

where al is the average of the product of two one- body matrix elements of the GT operator and bl is the average of the product of these with the Hamiltonian two-body matrix element in the scalar neutron-proton space. N, and N,, are the total number of protons and neutrons that can be accommodated in the shell. So for fp shell Nn = NP = 20. It is seen that (E,,, - l ) /a in spectral distribution ,theory lies mostly in the range -3.5 to -5.0 [ 13,141 for the nuclei in the mid- fp shell and for our purposes we assume it to be a constant. Then Eq. (3) can be written as

K(Es.,) = am,(20 - m,)

+ bm,(20-m,)m,(20-m,) (4)

or to put it in the form of Koonin-Langanke (KL) [ 111, who use Zvar and Nvar instead of mP and m,, one gets,

CB(GT+) i;a1(20 - &I)

= a + b(20 - Zval)Nval (5)

If we put b = 0 then we get back the KL form exactly. Thus the spectral distributions give a theory which in its simplest form recovers the KL expression,

Table 1 Total Gamow-Teller strength c B(GT+) extracted from (n,p) data for 12 nuclei along with the corresponding single particle estimate (Eq. (I)). Column (a) gives the c B(GT+) used in

Ref. [II]. Both column (a) and (b) give c B(GT+) below 9 MeV except for the nuclei (s’V, 59C~) and 4sTi for which column (a) gives strengths summed up to 12 and 15 MeV re- spectively. Slightly different values for the

& B (GT+ ) in column

(b) than that in (a) for the 4 nuclei 51V, xs6Fe and 58Ni are taken from most recent references as quoted in column (b) (see text also).

No. Nucleus c B(GT+) Single

(b) particle

(a) estimate,

Eq. (1)

1

2

3

4

5

6

I

8

9

10

11

12

2.1

1.48f0.16

1.72 f 0.20

2.39 f 0.25

1.31 *to.20

3.lOf0.60

2.85 f 0.30

3.76 f 0.40

0.84 f 0.13

0.46f0.05 [17] 33114

1.201tO.16 [18] 3617

1.70 f 0.20 [ 191 6017

1.90f0.21 [ 181 8417

0.91 zt 0.22 [ 201 57/14

3.30-f 0.51 [21] 72/7

2.90 f 0.30 [ 191 7217

3.80 f 0.40 [ 191 9617

3.11 f0.32 [8] 9617

2.53 f 0.26 [ 81 6417

1.72f0.19 [8] 3217

0.84 f 0.13 [22] 1613

and is also able to provide a correction term involving one extra parameter. We shall see that this correction term is indeed necessary in order to improve the pre- dictive power throughout the fp shell.

In Table 1 we give C B(GT+) values for a set of fp shell nuclei along with the single particle estimates using Eq. ( 1) . Column (a) of Table 1 is identical to the C B (GT+ > values used by Koonin and Langanke [ 111 (given by their Table 1, column 3) except for the nuclei 5’V and 59Co. For 5*V and 59Co they give two values for each, but we quote only one value for each, the ones which were actually used for their fits. But we feel that going up to 12 and 15 MeV excitation energy for only three nuclei (% 59Co and 48Ti respectively) is not proper. A better procedure should be adding up to the same excitation energy for all nuclei and so looking up the references of the experimental papers we construct c B (GT+ ) up to excitation of 8-9 MeV and display the values in column (b) of Table 1. For

230 s. Sarkar, K. Km/Physics Letters B 387 (1996) 227-232

Table 2 Description of fitting to E!q. (5) and KLl form (i.e. with only the first term of Eq. (5)) with the set of c B(GT+) vaiues (column (a) I(b) of Table I ) corresponding to various sets of nuclei constructed out of the 12 nuclei given in Table 1.

Description of fitted set of nuclei

Set Description NO. of the set

I 8 nuclei of Table 1, column (a), 45Sc excluded

II same 8 nuclei as in I 111 all 12 nuclei of Table I IV all odd-even nuclei V all even-even nuclei VI all even-even nuclei

excluding &Ti & ‘OGe

c B(GT+) values used

(a)

(b) (b) (h) (b) (b)

Fitted parameter of

KLl form Eq. (5)

a (Error) 0 (Error) h (Error) x10-2 x10-2 x 10-d

4.14 (0.18) 6.75 ( 1.70) -1.89 (1.22)

3.79 (0.17) 6.76 ( 1.63) -2.13 (1.16) 3.88 (0.14) I .84 (0.57) I .54 (0.42) 3.20 (0.19) 2.36 (0.67) 0.68 (0.53) 4.64 (0.20) 2.51 (1.19) 1.48 (0.82) 4.95 (0.22) 3.48 (1.31) I .oo (0.88)

example for 48Ti the quoted C B( GT+ ) value in col-

umn (a) is 1.31 f0.20 which actually is the sum up to an excitation energy of 15 MeV whereas when we re-

strict the sum up to 9 MeV the value is 0.91 kO.22. For

column (b) we add a systematic uncertainty of 10%

of total B(GT+) along with the statistical uncertainty

when only the later is quoted. C B( GT+ f value for

45Sc is misquoted i n Ref. [ 111 as 2.1. But this value is 0.70 when strength is summed up to 21 11 MeV and

is 0.46 & 0.05 up to 9 MeV of daughter excitation.

For 45Sc we take only the systematic uncertainty as no statistical uncertainty is quoted.

We first do a parameter fit of Eq. (5) for the same set of 8 nuclei as used by Koonin and Langanke [ 111 us- ing the same set of experimental values of C B( GT+)

(column (a) of Table 1). The best fit values of the parameters “a” and “6” using Eq. (5) and also IU’s one-parameter form (i.e. keeping only the first term of

Q. (5), denoted henceforth as KLl form) are given in the first row of Table 2 and is called set I by us. The

x2 per degree of freedom for the two fits are found to

be 0.92 and 1.13 respectively. The KL two-parameter form (denoted throughout as KL2 form) is given as

c BCGT,) = aZ,,i(b - N,,I) (6)

The parameter “a” for the KLl fit comes out as a = (4.14 f 0.18) x 10M2 whereas for KL2 the fit gives u = (4.39 f 0.27) x 10m2, b = 19.60 i 0.19 with

a x2 = 1.05. (Ref. [ 111 gives a = 0.0429 f 0.0015

with x2 = 1.1 per degree of freedom for KLl form.

Also for KL2 form it gives a = 0.0455 f 0.025 and b = 19.54 i 0.32 with ,y* = 1.0. The differences of

our results with the values given in Ref. [ 111 are due

to the fact that Ref. [ 111 used rounded values for the

quantity B(GT+)/Z,,l in their fits [ 161).

It is to be noted that Eq. (5) and the KLl form are

constrained to be within the fp space whereas in the KL2 form the number of single particle states itself is a

parameter and can, in principle, involve orbits beyond

the fp shell. That is why we do not include KL2 for the comparison in Table 2 but discuss it separately at

a later stage. Table 2 also gives best fit parameters of the spectral

distribution form (Eq. (5)) and KLl form using 8

nuclei from column (b) of Table 1 (set II, see Table 2) as well as using all 12 nuclei for which data is now available (set III) and then making a separation of odd Z-even N (o-e) (set IV) and even Z-even N (e-e) (sets V and VI). Fig. 1 gives the difference

c B(GT+)ficted - CB(GT+)~XP.

plotted for the six sets of Table 2 along with the cor- responding x2 for KLl form and the spectral distri-

bution form (Eq. (5)). Because of the effect of pairing in the low lying

region the strength distribution for the e-e and o-e nuclei are different and the systematics of e-e and o- e nuclei are not similar as observed by Koonin and

S. Sarkar, K. Kar/ Physics Letters B 387 (1996) 227-232 231

-A I I I

Tdn a-

45sC 51 - -3

5eCo 0.56 T -

V 0.01 - -

0 1.55 - 1.26 -

-1

Fig. 1. The difference between the fitted and experimental c B(GT+) is plotted against mass number A, showing the deviation of the predicted total strength using l?q. (5) (black dots) and KLl form (open triangles) from the measured value for different nuclei. Also given are the x2 (per degree of freedom) values and the corresponding parameters are given in Table 2.

Langanke [ 111. So to make further improvement of the predictive power of the spectral distribution form one has to treat these two types of nuclei separately. Doing that one sees that the x2 values come down to 1.26 and 1.55 for Eq. (5) and KLl form with e-e nuclei whereas for o-e nuclei the x2 values are 0.01 and 0.56 respectively and it remains to be seen whether the fit stays so good when one encounters a larger set of nuclei in future. But if one reduces the set for fitting the e-e nuclei by excluding 4*Ti and 70Ge (set VI of Table 2)) the x2 values get reduced drastically and the prediction with this set for 48Ti and 70Ge also turn out to be good (see Fig. 1). So we suggest that for other

e-e isotopes of Fe and Ni one should use E!q. (5) with set VI to predict the strength sums. But the drawback of this separation at this stage is the small sample sixes which increases the error in the parameters (the error in “b” has values 60% to 90% of the values of ‘3”). But we feel that the fitting with more data in future will reduce the errors and establish this separation of e-e and o-e nuclei on a firmer footing.

We now discuss briefly the results of fitting the KL2 form and compare it with fits to KLl.

KL,2 form fits better than the KLl form for both the e-e and o-e sets of nuclei. For the o-e set IV, KL2 form gives x2 = 0.21 and the corresponding a = (2.433 f

232 S. Surkar, K. Kar / Physics Letters B 387 (I 996) 227-232

0.710) x 10e2 and b = 23.50f3.92. For the set VI of

six e-e nuclei the fit gives a x2 = 0.05 and a = (4.32&

0.6 1) x lo-* and b = 21.10 f I .02. This gives for the

c B(GT+; E, 5 9 MeV) values as 1.30, 3.40, 2.88,

3.84, 3.14, 2.45, 1.76, 1.61 for the 8 e-e nuclei given

in the same order as in Table I. Thus the KL2 form reproduces the C B( GT+) values very well within

the fitting set but fails for the remaining two nuclei, Ti

and Ge. One can note that the uncertainty associated

with the two parameters of the KL2 form is also large

similar to Eq. (5). It is seen in some cases that “b”

is somewhat larger than 20 for KL2 form, indicating the participation of single particle states from the next

higher orbits.

Finally for an overall comparison of the spectral dis-

tribution form of Eq. (5) with KLl one should con-

sider the largest sample of 12 nuclei (set III) which

has the smallest errors. Then we find that the addi-

tion of the correction term involving the parameter

“b” brings the x2 down substantially from the value

of 3.68 to 2.72. The inclusion of this term becomes

progressively important as one goes to more neutron- rich nuclei. The Gamow-Teller strength sum for such

neutron-rich nuclei are considered important for elec-

tron captures both in the presupernova and supernova

evolution. To conclude, we have constructed here a phe-

nomenological form for the /?‘/< n, p) Gamow-Teller strength sum in the fp shell which is:

(i) based on a formal theory of statistically averaged

shell model results, and

(ii) include correction term to the form used by

Koonin and Langanke. We assert that this corrected form based on the spec-

tral distribution theory should be consistently used

for all nuclei through the fp shell, in particular the neutron-rich ones. Applications of the spectral distri- bution theory with configuration averaged forms is be- ing looked into. Also the stellar weak interaction rates

depend sensitively on the centroid of the Gamow- Teller resonances and an evaluation of the centroid us- ing averaged strength distribution by the spectral dis- tribution theory is being pursued. The evaluation of sum rule strengths for the excited states of the par- ent nuclei using spectral distribution theory is done by Kar et al. in Ref. [ 31 and needs follow-up work.

The authors would like to thank Alak Ray and

F.K. Sutaria (TIFR, India) for useful discussions and

valuable criticisms throughout this work, and Soumya

Chakravarti (USA) for encouragement. One of the

authors (S.S.) would like to thank the Theoretical

Astrophysics Group of Tata Institute of Fundamental Research, Bombay, India, for their hospitality where

part of this work was done. We also thank M. Saha

Sarkar and A. Bhattacharya for critically reading the manuscript.

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