Tetrahedral shapes of neutron-rich Zr isotopes from a multidimensionally constrained relativistic Hartree-Bogoliubov model

Abstract

We develop a multidimensionally constrained relativistic Hartree-Bogoliubov (MDC-RHB) model in which the pairing correlations are taken into account by making the Bogoliubov transformation. In this model, the nuclear shape is assumed to be invariant under the reversion of $x$ and $y$ axes; i.e., the intrinsic symmetry group is ${V}_{4}$ and all shape degrees of freedom ${\ensuremath{\beta}}_{\ensuremath{\lambda}\ensuremath{\mu}}$ with even $\ensuremath{\mu}$ are included self-consistently. The RHB equation is solved in an axially deformed harmonic oscillator basis. A separable pairing force of finite range is adopted in the MDC-RHB model. The potential energy curves of neutron-rich even-even Zr isotopes are calculated with relativistic functionals DD-PC1 and PC-PK1 and possible tetrahedral shapes in the ground and isomeric states are investigated. The ground state shape of $^{110}\mathrm{Zr}$ is predicted to be tetrahedral with both functionals and so is that of $^{112}\mathrm{Zr}$ with the functional DD-PC1. The tetrahedral ground states are caused by large energy gaps around $Z=40$ and $N=70$ when ${\ensuremath{\beta}}_{32}$ deformation is included. Although the inclusion of the ${\ensuremath{\beta}}_{30}$ deformation can also reduce the energy around ${\ensuremath{\beta}}_{20}=0$ and lead to minima with pear-like shapes for nuclei around $^{110}\mathrm{Zr}$, these minima are unstable due to their shallowness.