Symmetric-Core Collective Model for Odd-Odd Nuclei with Applications in the2s−1dShell

Abstract

A model for odd-odd nuclei is described in which the odd neutron and proton are coupled to a symmetric, rotating core. The Hamiltonian consists of four parts: ${H}_{R}$, the Hamiltonian for a rotating core of fixed shape; ${H}_{p}$ and ${H}_{n}$, the Hamiltonians for a proton and a neutron moving in a symmetric oscillator potential with 1\ifmmode\cdot\else\textperiodcentered\fi{}s and ${1}^{2}$ terms; and ${V}_{\mathrm{pn}}$, the residual neutron-proton interaction. This latter was taken to have a Gaussian radial dependence with a Serber exchange mixture with parameters picked to reproduce the low-energy singlet and triplet scattering lengths. The energy eigenvalues were obtained by an exact diagonalization of the total Hamiltonian using a core-particle basis to which a truncation procedure is applied to account for the filling of the shells. The model has been applied to $^{28}\mathrm{Al}$, $^{30}\mathrm{P}$, $^{32}\mathrm{P}$, and $^{36}\mathrm{Cl}$, determining the core-strength parameter $P$, the deformation parameter $\ensuremath{\beta}$, and the well-depth parameter $\ensuremath{\lambda}$ by fitting to the known energy-level sequence. The state functions so obtained were used to calculate static magnetic dipole and electric quadrupole moments, and for $^{30}\mathrm{P}$ the mixing ratio ${\ensuremath{\delta}}^{2}$ for two ${2}^{+}$ to ground transitions. The fit to the measured energy levels is quite successful; however, only in $^{28}\mathrm{Al}$ and $^{30}\mathrm{P}$ is the residual neutron-proton interaction needed. The values obtained for the static moments are rather poor. The results are compared, where possible, to a shell-model calculation.