The 3− level in A40

Nuclear Physics 4 (1957) 469--471; ~

North-Holland Publishing Co., Amsterdam

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

T h e 3 - L E V E L I N A *° R. T H I E B E R G E R

The Weizmann Institute o/ Science, Rehovoth and The Israel Atomic Energy Commission, Tel-Aviv and A. D E - S H A L I T

The Weizmann Institute o] Science, Rehovoth Received 3 J u n e 1957 The calculation of the 3- level in A 4° is a t t e m p t e d . The calculation is based on c o m p a r i n g energy levels in n e i g h b o u r i n g nuclei. The result o b t a i n e d is 2.51 MeV.

Abstract:

Recent calculations by Talmi and coworkers 1-3) have revealed a very good agreement between the results obtained from the shell model and experimental data concerning binding energies of nuclei and the energies of some related configurations in different nuclei. It is not yet clear, however, how general this agreement is and to what sort of data it applies. It has been pointed out by Glaubman 4) that lowest excited states of odd parity in even nuclei have odd multiples of/~ for their total angular momentum. This experimental rule has been explained by Talmi 6) on the assumption that the odd parity state involves a new configuration, resulting from the breaking up of a nucleon pair. Thus the lowest odd parity state in an even nucleus is, in a sense, a "ground state", and it is interesting to check whether its binding energy could be reproduced by the shell model. The following is an example of the sort of calculation which m a y lead to the prediction of the location of odd parity states in even nuclei. We treat the nucleus of A 4° where some contradictory results have been reportede-S). A 3- state has been reported by Morinaga at an excitation of 4.2 MeV whereas Heitler et al. have observed a 2.4 MeV level and van Heerden and Prowse a 2.22 MeV or 2.66 MeV level which might be the 3state (see the results of the calculations below). The calculation is based on comparing energy levels in neighbouring nuclei. The levels used and their assumed configurations are given in table 1 t. A 4° probably has its last protons in the ds/~ shell and its last neutrons t We wish to t h a n k Prof. P. M. E n d t for i n f o r m a t i o n concerning the e x p e r i m e n t a l data. 469

470

R. THIEBERGER AND A. DE-SHALIT

in the f712shell. An odd parity state can be produced most easily by transferring a proton from its d819 state to the f~/2 shell; all other possibilities probably involve a much higher excitation. We therefore assume for the lowest 3- level in A *° a configuration rids/2 nf~/2 v~/z, where we use n, to denote a proton and a neutron respectively. We shall assume that we can work in a scheme where the two protons are coupled to J = 3, and the two neutrons to J = o. The assumption t h a t the lowest level of this configuration is 3 (and not 5) is based on comparison with the experimental data of A 88. The calculations mentioned above 1-*) showed that the potential well of the nucleus is constant, to a good approximation, at least fn nuclei where the same shell is being filled. Therefore, assuming that the two neutrons are coupled to zero, and assuming that the residual interaction between the two protons does not change, we can expect that if the lowest level of the configuration ~ds/o. ~f712 is 3 (as is probably the case in A 8s) then this is also true for A *°. The same argument of constant potential well will be used in the treatment of the experimental data of K 89 and K .1, as will be seen later. In the following calculations it is shown that under the assumption mentioned above, the difference between the excitations of the 3- level in A ~8 and A *° is the same as the diference between the excitations of the 7/2 level in K 39 and K *1. The levels used and their assumed configurations are given in table 1. TABLE 1 The experimental levels used in the calculation and their assumed configurations nucleus

level (MeV)

spin and parity

88 18A20

g.s. 3.75

0+ 3--

19K~

g.s. 2.5.3

3/2+ 7/2+ (assumed)

41 19K22

3/2+ g's' 1.29

I

712

configuration d2/2 d3/2 f712 d~/2 d2/2 (0) f7/2 3 "2 d3/2 f7/2(0) d212 (0) f7/2 f7212(0)

Let us write the energy of a given level in the form

E (il"(]~)7"~'~(J2)]) = E (?'i"(]~))+ E (i2~ (]2))--I-V(i1"(Jl)J2"(12)J), where/'~(Jt) stands for n nucleons ~'~coupled to a total angular momentum J~; E is the interaction between the group of nucleons with the closed shells and with one another, plus their kinetic energy; V is the interaction between the two groups of nuclei, when coupled to resultant angular momentum J. Writing the energy in this form we get for the difference between the excitations of the two 7/2 levels in K 89 and in K41:

THE

3- L~V~L IN

471

A t°

[Eztd~/~ (0)~f~12(7/2) 7/2) -- E (ztd~/~(3/2)) 3 - - [E (ztd~/9.(0)zdT/2 (7/2),v~12 (0)7/2) -- E (:tdaa/~(3/2), v~/9.(0)3/2)3

= V(~ds/2(3/2 ), v~12(O)3/2)--V(~f~72(7/2),

v~1~(0)7/2).

In a similar w a y we get for the difference between the excitations of the two 3- levels in A 38 and in A4°: [E (ztda/2(3/2):tf7/9. (7/2)3) -- E (~td~/2(0)) 3 -- EE (:tds/9. (3/2)ztf~/~. (7/2), V~l ~(0)3) -- E (:td~/~ (0), vff/2 (0)0)] ---- V (zdsl ~(3/2), vff/2(0)3/2) -- V (~f712(7/2), vff/2(0) 7/2). In these calculations we used the two relations V (ztdl/2 (0), Vffl 2 (0)0) = 2V (:td3/~(3/2), Vffl 2 (0)3/2), V (gd]/2 (3/2), vff/2 (0)3/2) = 3V (z~da/2(3/2), ?)~]2(0)3/2). These relations result from the fact that the two f7/2 particles are coupled to ] ---- 0, and therefore the interaction with t h e m depends only on the n u m b e r of particles 9). Therefore the energy difference between the ground state and the 3-level in A 4° is E (~dsi2 (3/2)zd7/2 (7/2), rift 2 (0) 3) -- E (~d~/2(0), rift 2(0)0)

= [E (~d3/2 (3/2):~fm (7/2)3) -- E (=d~2 (0))3 + [Z (=dl/2 (0)~fm (7/2), vff~2(0)7/2) -- E (=d~:~(3/2), vff/~(0)3/2)3-- [Z (~di/2 (0)~fm (7/2) 7/2)-- E (~d~/~(3/2))] 2.51 MeV. The three sets of differences in this expression are the excitation energies of the three nuclei given in table 1. With these values we obtain the result 2.51 MeV. We would like to stress that no assumptions were made concerning the interactions between nucleons or about the shape of the wave functions; therefore a fair fit with experiment is expected. As the experimental material is not yet conclusive, it would be desirable to obtain more information concerning the experimental levels in the vicinity of our calculated 3- level. References 1) 2) 3) 4) 5) 6) 7) 8) 9)

S. Goldstein a n d I. Talmi, Phys. Rev. 102 (1956) 589 I. Talmi and R. Thieberger, Phys. Rev. 103 (1956) 718 S. Goldstein a n d I. Talmi, P h y s . Rev. 105 (1957) 995 M. J. Glaubman, Phys. l~ev. 90 (1953) 1000 I. Talmi, Phys. Rev. 90 (1953) 1001 H. Heifler, A. N. May a n d C. F. Powell, Proc. Roy. Soc. A 190 (1947) 180 H. Morinaga, Phys. Rev. 103 (1956) 504 I. J. v a n Heerden and D. J. Prowse, Phil. Mag. 1 (1956) 967 N. Zeldes, Nuclear Physics 2 (1956) 1