Seniority-two valence-shell building blocks of the octupole phonon

Abstract

The relative ordering of $J^{-}$ levels of multiplets resulting from two-body excitations, which include a $J^{\pi }=3^{-}$ state that can contribute to the octupole phonon, are investigated in a simplistic shell-model approach. To calculate the relative level ordering, harmonic oscillator wavefunctions and a residual $\delta$ interaction are used. The simplistic approach confirms for the particle-particle channel, the often stated preference of the $\Delta j=3,\Delta l=3$ subshell combination over the $\Delta j=3,\Delta l=1$ subshell combination through an enhanced energy gain. Furthermore, it is shown that, in the particle-hole channel, the gain is less pronounced for the $\Delta j=3,\Delta l=3$ subshell configuration. In combination with the overall structure of an oscillator shell, these results explain the comparatively low excitation energy for 3${}_1^{-}$ excitations observed for the octupole-soft proton and neutron numbers predicted by various models.