Boson Expansion Method and the Coupling of Different Degrees of Freedom

Abstract

A model for the quintuplet-collective levels 1^-, 2^-, 3^-, 4^-, 5^- observed in some spherical nuclei is proposed. The shell model Hamiltonian with pairing, quadrupole-quadrupole and octupole-octupole effective forces is expanded up to fourth order in quadrupole and octupole phonons by the Belyaev-Zelevinsky technique. In order to improve the expansion convergence the basis given by the random phase approximation (RPA) is changed by a canonical transformation determined by the condition of minimum for the ground state energy. To specify the diagonalization basis one needs the multiplicity of the R₃ irreducible representation into R₇ representation. This was calculated using Weyl's procedure. Using some specific tensor properties of fourth order terms, the collective part of the expanded Hamiltonian is written in a compact form. The contribution of the noncollective degrees of freedom on the above-mentioned levels is simulated by an effective, collective and energy-dependent Hamiltonian which is written explicitly. The eigenvalue problem of the total Hamiltonian can be solved selfconsistently. The matrix elements of the fourth order octupole-like terms as well as of the mixing terms are expressed as products of two factors, one of them bearing the entire dependence on the number of phonons.