Generalized approximation to seniority shell model

Abstract

A generalized approximation to seniority shell model, based on the number conserving quasiparticle theory, is presented which includes the cases where both neutrons and protons are present in the valance shells. The numerical calculations are carried out for even Zr isotopes. The results obtained compare well with the corresponding exact shell model results, demonstrating thereby the validity of the present approximation for this nuclear region. It is well known that the shell model or even the seniority (v (4) shell model calculations with con- figuration mixing are permanently limited to -the cases where only a few' valence particles are pres- ent in a few valence levels. This is because with the increase in the number of valence particles (levels), the dimension of the Hamiltonian matrix increases tremendously which is formidable in practice. However, in such practical calculations, the dimensions of the Hamiltonian matrices can be reduced if the problm is reviewed with some ap- proximation based on phenomenology derived from the effective nucleon-nucleon interactions. One such practical approximation is the quasiparticle model based on the strong pairing correlations among nucleons. Unfortunately, the rehults of the quasiparticle calculation do not correspond to the specific nucleus due to the noncommutability of the actual number operator with the quasiparticle Hamiltonian. This number nonconservation also introduces spurious states. To remove these drawbacks and to.retain the basic advantages of the model, various methods' ' based on the number projection, have been proposed. One such method is the broken-pair approximation (BPA), ' the val- idity of which has been demonstrated~' in the 2plf- region as well as for N=50, even A. nuclei. So far these projection methods have been applied only for the identical particles (neutrons 'or protons) in the valence shells. In this report we generalize the broken-pair approximation (referred to as GBPA) to include the cases where both the neutrons and the protons are active in the valence shells. Approaching the problem in the same manner as in the identical nucleon case, we write the approx- imate ground state for even-even nuclei as a pro- duct of a proton paired (p pairs) state T, l 0), and a neutron paired (n pairs) state 7tl0), i.e.,