Multi-shell effective interactions

Abstract

Background: Effective interactions, either derived from microscopic theories or based on fitting selected properties of nuclei in specific mass regions, are widely used inputs to shell-model studies of nuclei. The commonly used unperturbed basis functions are given by the harmonic oscillator. Until recently, most shell-model calculations have been confined to a single oscillator shell like the $sd$ shell or the $pf$ shell. Recent interest in nuclei away from the stability line requires, however, larger shell-model spaces. Because the derivation of microscopic effective interactions has been limited to degenerate models spaces, there are both conceptual and practical limits to present shell-model calculations that utilize such interactions.Purpose: The aim of this work is to present a novel microscopic method to calculate effective nucleon-nucleon interactions for the nuclear shell model. Its main difference from existing theories is that it can be applied not only to degenerate model spaces but also to nondegenerate model spaces. This has important consequences, in particular for intershell matrix elements of effective interactions.Methods: The formalism is presented in the form of a many-body perturbation theory based on the recently developed extended Kuo-Krenciglowa method. Our method enables us to microscopically construct effective interactions not only in one oscillator shell but also for several oscillator shells.Results: We present numerical results using effective interactions within (i) a single oscillator shell (a so-called degenerate model space) like the $sd$ shell or the $pf$ shell and (ii) two major shells (nondegenerate model space) like the $sd{f}_{7}{p}_{3}$ shell or the $pf{g}_{9}$ shell. We also present energy levels of several nuclei that have two valence nucleons on top of a given closed-shell core.Conclusions: Our results show that the present method works excellently in shell-model spaces that comprise several oscillator shells, as well as in a single oscillator shell. We show, in particular, that the microscopic intershell interactions are much more attractive than has been expected by degenerate perturbation theory. The consequences for shell-model studies are discussed.