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

Abstract

In the applications of proxy-SU(3) model in the context of determining $({\hat{I}}^2,{\hat{I}}^3)$ 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) $(\hat{I}{»}_H,\hat{I}{¼}_H)$ but also the lower $SU(3)$ irreps $(\hat{I}»,\hat{I}¼)$ such that $2\hat{I}»+\hat{I}¼=2\hat{I}{»}_H+\hat{I}{¼}_H-3r$ 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 $(\hat{I}»,\hat{I}¼)$ with $2\hat{I}»+\hat{I}¼=2\hat{I}{»}_H+\hat{I}{¼}_H-3r$, $r=0,1,2$ 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.

24 pages, 10 tables