Shell-model description of collective deformation variables $β$ and $γ$ in $^{164-188}$Hf nuclei

Abstract

Symmetry-based models, such as the pseudo-SU(3) scheme, have proved to be highly useful tools for describing the low-energy structure of nuclei in the rare-earth region. This model is employed to study the evolution of the collective deformation variables beta and gamma in the ground state (0+) along the isotopic chain 164-188Hf. The truncation is carried out by considering the irreducible representations (irreps) with high spatial symmetry and pseudo-spin 0 and 1, so that both the most and less symmetric relevant irreps are included, the latter being of particular importance for the description of low-energy properties, especially the shape transition. The Hamiltonian includes spherical Nilsson single-particle energies, the quadrupole-quadrupole interaction, and pairing correlations. It is parameterised in a systematic manner, so that the description is obtained without free parameters. In agreement with experimental data, the maximum deformation beta occurs at N = 100, although the model predicts values that are about 20% smaller. As in other theoretical studies, a prolate deformation (gamma = 8o) is predicted for almost all nuclei in the shell, with a clear transition towards triaxial shapes for N > 110. An analysis of the effects of the pairing and Nilsson terms on the values of $β$ and gamma indicates that it is the latter that determine the prolate-oblate transition, with the pairing term playing only a marginal role. The absence of nucleons in the intruder sector is the most significant limitation of the model.