On the Gaussian overlap approximation for the collective excitations of odd nuclei

The multi-dimensional generator coordinate method is applied to describe the quadrupole collective states of odd nuclei. The laboratory Cartesian components of the quadrupole deformation tensor are used as the generator coordinates. The two mean-field states of an odd nucleus constituting a Kramers' doublet are taken as the two intrinsic generating states which can rotate in the laboratory reference frame. The time-signature invariance of these intrinsic states is assumed. A version of the Gaussian overlap approximation (GOA) for the two generating states is constructed. A system of two coupled differential equations for the collective excitations of odd nuclei is derived from the variational principle using the GOA. The system forms a generalized eigenvalue differential equation in the two-dimensional 'alispin' space. Breaking the time-reversal symmetry and alignment of the angular momentum in the mean-field states modifies the rotational part of the Hamiltonian compared to that of the standard Bohr Hamiltonian, whereas the quadrupole vibrations are modified through the coupling of the Kramers states.