Quadrupole-Octupole Residual Interaction in the Proxy-$SU(3)$ Scheme

The octupole deformation of atomic nuclei is a relevant research area given its implications in the nuclear structure and fundamental physics, however, inclusion of octupole degrees of freedom in the nuclear interaction has been explored little in $SU(3)$ symmetry-based models. In this article we expand the octupole operator $\mathcal{O}^3_\mu$ in the $SU(3)$ second quantization formalism and use it to formulate an octupole-octupole residual interaction $\mathcal{O}^3\cdot \mathcal{O}^3$ which is incorporated in Elliott's model Hamiltonian. We compute the matrix elements of this extended Hamiltonian and use them to calculate energy levels and analyse the $B(E3; 3^-_1\rightarrow 0^+_1)$ transition strength of the isotope $^{224}$Th using the semi-microscopic algebraic quartet model (SAQM) based on the proxy-$SU(3)$ scheme.