Approximation Methods in Nuclear Intermediate Coupling Applied to the1pShell

Abstract

The intermediate-coupling shell model for nuclei is considered in terms of the classification of states in a harmonic oscillator according to the irreducible representations of the unitary unimodular group in three dimensions, $S{U}_3$, introduced by Elliott. The properties of this group are used to produce the approximate spectrum of a quadrupole force, acting within an oscillator shell. When specialized to the $1p$ shell, a more general interaction, including exchange forces, is shown to be approximately diagonal in the chosen representation, and its approximate spectrum is computed. A method is developed for calculating the matrix elements of interactions not diagonal in the representation, in particular the single-particle spin-orbit potential, using the generating functions of the group, $S{U}_3$. The intermediate-coupling energy spectra of the nuclei of the $1p$ shell are then calculated to the first or second order in perturbation theory. The results are compared with experimental spectra, and with calculations of Kurath.