Next highest weight and other lower $SU(3)$ irreducible representations with proxy-$SU(4)$ symmetry for nuclei with $32 \le \mbox{Z,N} \le 46$

TL;DR

The paper extends proxy-SU(3) nuclear structure analysis by incorporating proxy-SU(4) symmetry for Ge–Pd isotopes, where valence nucleons occupy the same proxy shell. It introduces a simple, computable method to reduce $U({\cal N})$ irreps to $SU(3)$ irreps and implements it for two- and four-column cases, validating hw results and enabling calculation of next-higher weight irreps. The main results tabulate the hw, next-highest weight (nhw), and next-to-next highest weight (nnhw) irreps $(\lambda,\mu)$ for Ge–Pd isotopes under proxy-$SU(4)$ symmetry, including multiplicities that weight the contributions to deformation parameters. These tables provide a valuable resource for assessing triaxial shapes and shape coexistence in this nuclear region and complement prior work using pn-proxy-SU(3). The study thus advances the quantitative foundation for analyzing $(\beta,\gamma)$ shapes and fosters future exploration of additional symmetries in these nuclei.

Abstract

In the applications of proxy-SU(3) model in the context of determining $(β,γ)$ values for nuclei across the periodic table, for understanding the preponderance of triaxial shapes in nuclei with $Z \ge 30$, it is seen that one needs not only the highest weight (hw) or leading $SU(3)$ irreducible representation (irrep) $(λ_H, μ_H)$ but also the lower $SU(3)$ irreps $(λ,μ)$ such that $2λ+ μ=2λ_H + μ_H-3r$ with $r=0,1$ and $2$ [Bonatsos et al., Symmetry {\bf 16}, 1625 (2024)]. These give the next highest weight (nhw) irrep, next-to-next highest irrep (nnhw) and so on. Recently, it is shown that for nuclei with $32 \le \mbox{Z,N} \le 46$, there will be not only proxy-$SU(3)$ but also proxy-$SU(4)$ symmetry [Kota and Sahu, Physica Scripta {\bf 99}, 065306 (2024)]. Following these developments, presented in this paper are the $SU(3)$ irreps $(λ,μ)$ with $2λ+ μ=2λ_H + μ_H-3r$, $r=0,1,2$ for various isotopes of Ge, Se, Kr, Sr, Zr, Mo, Ru and Pd (with $32 \le \mbox{N} \le 46$) assuming good proxy-$SU(4)$ symmetry. A simple method for obtaining the SU(3) irreps is described and applied. The tabulations for proxy-$SU(3)$ irreps provided in this paper will be useful in further investigations of triaxial shapes in these nuclei.