Simple formula for leading $SU(3)$ irreducible representation for nucleons in an oscillator shell

TL;DR

This paper derives a simple formula to determine the leading $SU(3)$ irreducible representation for nucleons in an oscillator shell, aiding nuclear structure models and revealing prolate shape dominance.

Contribution

It provides a new, straightforward formula for the leading $SU(3)$ irrep in any given $U( ext{dimension})$ irrep, applicable to nuclear shell models.

Findings

Prolate shapes dominate over oblate shapes in the $SU(3)$ shell model.

The formula applies to various oscillator shells and particle numbers.

Results are explicitly calculated for relevant nuclear shells.

Abstract

Applications of rotational $SU(3)$ symmetry in nuclei, using Elliott's $SU(3)$ or pseudo-$SU(3)$ or proxy-$SU(3)$ model, often need just the lowest or leading $SU(3)$ irreducible representation (irrep) $(\lambda_H, \mu_H)$. For nucleons in an oscillator shell $\eta$, with ${\cal N}=(\eta +1)(\eta +2)/2$, we have the algebra $U(r{\cal N}) \supset [U({\cal N}) \supset SU(3)] \otimes SU(r)$; $r=2$ when there are only valence protons or neutrons and $r=4$ for nucleons with isospin $T$. Presented in this paper is a simple general formula for the leading $SU(3)$ irrep $(\lambda_H, \mu_H)$ in any given irrep $\{f\}$ of $U({\cal N})$. Results are provided for $(\lambda_H, \mu_H)$ irreps for $\eta$ values of interest in nuclei and for this for all allowed particle numbers. These results clearly show that prolate shape dominates over oblate shape in the shell model $SU(3)$ description.