Quasi-particle effective interactions

Abstract

The study of effective interactions in a model space constitutes a basic ingredient of shell theory. One tries to avoid the intractable problem posed by the true nuclear Hamiltonian H, acting in an infinite-dimensional Hilbert space, by performing an act of truncation and considering only a modified or effective Hamiltonian H~, acting in a finite model space. BLOCH and HoowITz (~) have given the basic equations which allow one to determine the effective-interaction Hamiltonian once a finite model space has been chosen. The problem, however, is severely complicated due to the fact that one must deal with valence and core particles (2). The purpose of the present work is to formulate a similar treatment, but applied to Bogoliubov-Valatin quasi- particles (q.p.) (a). In this case no difficulties arise associated with a core, because one has to deal only with (~ valence ~) q.p. The only additional problem which must be faced is that of eliminating the spurious state resulting as a consequence of the violation of the conservation law for the number of particles induced by the q.p. transformation (a). We will restrict ourselves to a 2 q.p. space and consider a set D of 2 q.p. states lab}, where a and b denote all the quantum numbers characterizing the members of a given pair of q.p. The basis is ordered in such a way that these pairs are counted only once. We call W(abcd) the matrix element which connects the 2 q.p. states lab} and led}. Our problem is the following one: Having chosen a subset d of D around the Fermi surface, to determine the matrix dements of the effective q.p. interaction which acts in d from the one acting on D. The effective Hamiltonian must be of such a form that, in an eigenvalue problem within the model space, it reproduces some of the lowest eigenvectors of the complete Hamiltonian of the system. More precisely, if (~} and i~} denote the true and model eigenvectors, and P the operator which projects onto d, our requirement is that to