# Multiple $SU(3)$ algebras in shell model and \\ interacting boson model

## TL;DR

This paper explores multiple $SU(3)$ algebras in nuclear shell and interacting boson models, revealing different shapes and quadrupole properties, supported by analytical and numerical results across models.

## Contribution

It demonstrates the existence of multiple $SU(3)$ algebras in shell and boson models and analyzes their structural and shape implications using various theoretical approaches.

## Key findings

- Different $SU(3)$ algebras generate prolate and oblate shapes.
- One $SU(3)$ algebra produces small quadrupole moments.
- Results are supported by shell model and interacting boson model analyses.

## Abstract

Rotational $SU(3)$ algebraic symmetry continues to generate new results in the shell model (SM). Interestingly, it is possible to have multiple $SU(3)$ algebras for nucleons occupying an oscillator shell $\eta$. Several different aspects of the multiple $SU(3)$ algebras are investigated using shell model and also deformed shell model based on Hartree-Fock single particle states with nucleons in $sdg$ orbits giving four $SU(3)$ algebras. Results show that one of the $SU(3)$ algebra generates prolate shapes, one oblate shape and the other two also generate prolate shape but one of them gives quiet small quadrupole moments for low-lying levels. These are inferred by using the standard form for the electric quadrupole transition operator and using quadrupole moments and $B(E2)$ values in the ground $K=0^+$ band in three different examples. Multiple $SU(3)$ algebras extend to interacting boson model and using $sdg$IBM, the structure of the four $SU(3)$ algebras in this model are studied by coherent state analysis and asymptotic formulas for $E2$ matrix elements. The results from $sdg$IBM further support the conclusions from the $sdg$ shell model examples.