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

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·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 $\beta$, and the well-depth parameter $\lambda$ 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 ${\delta }^2$ 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.