Shell model for N = 49 nuclei

Volume 53B, number 3

PHYSICS LETTERS

9 December 1974

S H E L L M O D E L F O R N = 49 N U C L E I R. GROSS and A. FRENKEL The Weizmann Institute of Science, Rehovot, Israel Received 1 October 1974 The behavior o f N = 49 nuclei is examined in terms of a model based on a 88Sr core with a lgg/z neutronhole. Effective interactions between lg9/2 neutrons and lg9/2-2pi/2 protons are determined from the spectra of 88y, 89Zr' 9ONb and 91Mo. Close agreement is obtained also with experimental level spacings of 92Tc and 93Ru, as well as binding energies. Nuclei in the mass-90 region are of particular interest both from experimental and theoretical points of view, since they can be treated by the shell model. Nuclei with N = 50 are well described by 2Pl/2 and lgg/2 protons outside an inert core of 88Sr [ 1 - 6 ] . As a natural extension, one would hope to calculate cases where the lg9/2 and 2Pl/2 neutron sub-shells are also active. In this preliminary report the first step is done by using the N = 49 nuclei which are assumed to have a lgg/2 neutron hole. The similar particle-hole multiplets in 88y [7] and in 90Nb [ 8 - 9 ] seem to be in agreement with these assumptions [10]. Nevertheless the (Pl/2-Pl/2)proton-neutron multiplet in 88y sits quite low in energy [7], so the levels assumed to belong to the lgg/2 neutron hole multiplets should be chosen carefully. Recent!~,, new experimental data on the spectra of 89Zr, 91Mo, 92Tc and 93Ru have been published [10-12] thus enabling this extension. The agreement of the calculated and experimental level spacings is already evident even when only the 1g9/2 neutron sub-shell is active. There are 24 matrix elements of the effective interaction which completely define the shell model configuration space of active lg9/2-2Pl/2 proton and lgg/2 neutron sub-shells: 3 single particle energies and 21 two-body matrix elements. In order to fit 24 parameters a great number of energy levels should be fitted. We take eleven of these from refs [5, 6] where they fit data using N = 50 with an average deviation of 53 keV. The sets of effective interactions in reL [5, 6] are consistent and the r.m.s, deviation is the same; we choose one of them arbitrarily (ref. [5]). We made

use of 26 excited states in nuclei with N = 49 in order to obtain the remaining proton-neutron effective interactions. The Hamiltonian matrices in terms of the effective interactions were calculated using a convenient shell model computer code - WSMCC [13]. No binding energies were considered and thus all proton-neutron matrix elements used were determined up to a constant. We choose this constant so that the p-h interaction ¢gg/2(p)g~l(n)l Vlgg/2(p )g~l (n))j= 8 = O. Also the single particle energy of 1 g9/2 neutron-hole remains undetermined. The energy levels included in the fit should be chosen carefully, since admixtures of other neutron hole orbits can be significant. From the g9/2 (P) - gg/~ (n) multiplet in 88y we have taken only the J - 3 ÷ - 9 + states. The 0 + and 1+ were excluded because of possible contribution of the Pl/2 neutron hole, and the 2 + seems to be strongly mixed [7]. The calculated 2 + level of 88y comes out to be about 300 keV higher than the experimentally measured 2.225 MeV level, the deviation being much larger than that for other levels. The same consideration led to the choice of J = 2 + 9 + states in 90Nb. In addition, we have taken all the positive parity states of 89Zr (9/2 +, 13/2 +, 17/2 +, 21/2 +) and 91Mo (9/2 +, 13/2 +, 17/2 +, 21/2 +, 23/2 +, 25•2 +) *. The negative parity states of 91Mo ( 2 7 / 2 - , 29/2-, 3 1 / 2 - ) should be included in the fit since, due to angular momentum considerations, they are completely clean of 2pl/2 neutron contribution. -

* Calculations by J. Verrier of the positive parity states of 91Mo axe reported as private communication in ref. [4]. 227

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contribute to the energy of all the J=#0 states considered. This can be checked by varying the value of this parameter through the calculation. We did this and obtained that the low lying states considered are indeed independent of this J = 0 parameter. It is of special interest to test the consistency between the proton-neutron hole interaction obtained and the proton-proton interaction usedff The particlehole interactions are linear combinations of protonneutron effective interactions while the particle-particle interactions are proton-proton effective interactions which contain the Coulomb interaction. The Coulomb effect is largely eliminated by comparing differences of matrix elements and the J = 0 interaction of each set. In this way the contribution of the J= 0 + interaction of the proton and the neutron-hole is eliminated. Thus, we required that the difference between the Pandya transformation of the particlehole interactions be equal to the differences of proton-proton interactions. This requirement was satisfied to within 50 keV. The best values of the effective matrix elements are given in table 1 and the energy levels in figs. 1 and 2. (States included in the fit are marked by the letter f.) The large errors [14] on the effective interactions should be noted and especially that of J = 1+. This will probably enable the good agreement of the

Table 1 Effective interaction matrix elements Proton neutron-hole matrix elements Obtained by putting V(gg-t .1=8)=0

Statis- Actual values tieal obtained errors from binding energies

236 293 2658 929 677 304 223

±112 a 97 ±566 a178 ±117

V(gg/2(p)gg~(n) .1=6)

6

± 91

129

V(gg~(p)g~(n).1=7)

53

± 75

V(pl/2(p)gg~(n)J=4) V(Pt/2(P)g~(n).I=5) V(g9/2(p)gg~(n).1=1) V(gg/2(p)gg~(n)J=2) V(gg/~(p)gg~(n).1=3) V(gl,/2(p)gg~(n),/=4) V(gg/2(p)gg~(n)J=5)

368 417 2781 1053 801 427 346

0

0

177 123

836

± 56

960

-

11160

V(gg~(p)g~(n).1=8) V(gg/2(p)g~(n).1=9)

±113

±112

g~2 neutron single hole energy

-

All in all we have 30 states which furnish us with 26 energy spacings of excited states. These should be fitted by eleven relative effective interaction parameters. It should be noted that due to isospin considerations (g9/2(p)g~/~(n)[ V]g~l(p)g~-/~(n)b=0 does not

88 3;Y49

147~"

SWe would like to thank Prof. I. Talmi for making this suggestion.

940lNb49

exp 9t

9 December 1974

colc 9+

exp

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calc

exp

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-

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~

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843

calc

968 ( 5 ) - 854 -600(f)

9' $,.

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0

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373

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(4°)

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0

Fig. 1. Experimental and theoretical energy levels (in keV) of SSy, 90Nb and 92Tc. The SSy data are from z~f. [15]. ref. [8] and for 92Tcref. [11]. 228

8".

665 ~lS3

:57t ~556

431 Z64 I86 171 0

For 9°N'o see

Volume 53B, number 3

9 December 1974

PHYSICS LETTERS 91 42M049

exp

3l* T

~

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colc 3f 2

5217(f)

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~"

o

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,$80

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°

Fig. 2. Experimental and theoretical energy levels (in keV) of 89 Zr, ~z Me and 93Ru. The data are from ref. [ 10].

calculated levels with the experimental ones to be preserved when the 2Pl/2 neutron hole will be included in the configuration space. The r.m.s, deviation of our fit is 100 keV. Using our results we calculated other low lying states of those nuclei which may belong to the configurations¢onsidered. In these states the admixtures of

other neutron-holes were not known a priori to be insignificant. These states are 88Y(4-, 5 - ) , 89Zr ( 1 3 / 2 - ) , 90Nb(4-, 5 - ) and 91Mo ( 1 7 / 2 - , 25/2-). The deviations found for these levels are of order of magnitude of the r.m.s, deviations, so the agreement is quite good. It seems that the contribution of the 2Pl/2 neutron hole to these states is small. We note 229

Volume 53B, number 3

PHYSICS LETTERS

that the calculated 1+ level in 88y is high in energy which is consistent with the classification of the 0.393 MeV level as belonging mainly to the (Pl/2 Pl/2) multiplet I7]. We went further to calculate the spectra of 92Tc and 93Ru (9/2 +, 13/2 +, 17/2+). As to the latter the deviations are less than the calculated r.m.s. The analysis of the experimental spectrum of 92Tc is not yet completed [11 ]. The density of low lying states of odd-odd nuclei is relatively high and so it is for 92Tc. The calculated states are below 700 keV, much like the experimental spectrum. Our calculations seem to be consistent with the order of the positive parity levels [11-12] and the deviations in energy being of the order of the r.m.s. In particular the ground state of 92Tc came out to be 8 +. Binding energies of the nuclei considered minus that of 88Sr can be calculated by determining the value of V(gg-1 J = 8 ) p-h interaction and the energy of the lgg/2 neutron hole (denoted by e, which would be equal to the binding energy difference of 88Sr and 87Sr). The interaction between the neutron hole and the proton shell is proportional to the number of active protons (np), and so a term npV(gg-1 J = 8 ) + e should be added to the ground state energies, calculated with the normalized set in column 2 of table 1. The binding energies of 87Sr, 88y, 89Zr ' 90Nb ' 91Mo and 92Tc agree within 50 keV with the calculated values by using V(gg-1 J = 8) = 123 keV and e = + 11.16 MeV. The matrix elements in column two, table 1 can be now used to obtain the actual values of the effective matrix elements. These are listed in column 4, table 1.

230

9 December 1974

We plan to include the 2Pl/2 neutron sub-shell in the configuration space considered which may enable us to treat also other nuclei in this region. The authors would like to thank Prof. I. Talmi for his continued advice through the course of this work.

References [1] I. Talmi and I. Unna, Nucl. Phys. 19 (1960) 225. [2] N. Auerbach and I. Talmi, Nucl. Phys. 64 (1965) 458. [3] S. Cohen, R.D. Lawson, M.H. Macfarlane and M. Soga, Phys. Lett. 10 (1964) 195. [4] J. Vender, Nucl. Phys. 75 (1965) 17. [5] J.B. Ball, J.B. McGrory and J.S. Larsen, Phys. Lett. 41B (1972) 581. [6] D.H. Gloeckner and F.J.D. Serduke, Nucl. Phys. A220 (1974) 477. [7] J.R. Comfort and LP. Schiffer, Phys. Rev. C4 (1971) 803.

[8] R.C. Brarse, J.R. Comfort, J.P. Sohiffer, M.M. Stautherg and J.C. Stoltzfus, Phys. Rev. Lett. 23 (1969) 864. [9] Y. Yosiiida, M. Ogawa,T. Hattori and H. TaketanL Nucl. Phys. A187 (1972) 161. [10] A. Nilsson and M. Grecescu, Nucl. Phys. A212 (1973) 448. [11] S.I. Hayakawa et al., NucL Phys. A199 (1973) 560. [12] LC. de Lange, H. Verheul and W.B. Ewbank, Intern. Conf. on Nuclear structure and spectroscopy, Amsterdam, Sept. 1974. [13] R. Gross and Y. Accad, to appear in Computer Physics Communication 7 (1974). [ 14 ] H. Cramer, The elements of probability theory (John Wiley & Sons, 1955) p. 235. [15] M. Ishihara, K. Forssten, P. Monsen and A. Nilsson, annual report (1972) Research Institute for Physics, Stockholm, Sweden.