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 ${\mathcal{O}}_{\ensuremath{\mu}}^{3}$ in the SU(3) second quantization formalism and use it to formulate an octupole-octupole residual interaction ${\mathcal{O}}^{3}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathcal{O}}^{3}$ 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 analyze the $B(E3;{3}_{1}^{\ensuremath{-}}\ensuremath{\rightarrow}{0}_{1}^{+})$ transition strength of the isotope $^{224}\mathrm{Th}$ using the semimicroscopic algebraic quartet model based on the proxy-SU(3) scheme.