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

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 ${\beta }_{\lambda \mu }$ with even $\mu$ 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 ${\beta }_{32}$ deformation is included. Although the inclusion of the ${\beta }_{30}$ deformation can also reduce the energy around ${\beta }_{20}=0$ and lead to minima with pear-like shapes for nuclei around ${}^{110}\mathrm{Zr}$, these minima are unstable due to their shallowness.

Figure 1

The potential energy curve of ${}^{110}\mathrm{Zr}$ from MDC-RHB calculations with different symmetries imposed (a) and with different truncations of the ADHO basis [(b), (c), and (d)]. The calculations are preformed with axial symmetry and reflection symmetry (AS & RS), axial symmetry and reflection asymmetry (AS & RA), and triaxial symmetry and reflection asymmetry (TA & RA) imposed, respectively. In (b), (c), and (d), the results calculated with $N_f=12$, 14, and 16 are depicted by dotted, dashed, and solid curves, respectively. The corresponding amplified figures (e), (f), and (g) show the detailed structure of the PES's near ${\beta }_{20}=0$; the width of each subfigure is 0.06 and the height is 1 MeV.

Figure 2

The binding energy (a) and ${\beta }_{32}$ deformation (b) of ${}^{110}\mathrm{Zr}$ from MDC-RHB calculations with different truncations made in the Fourier expansion of densities and potentials. The calculations are performed with $N_f=14$. The functional DD-PC1 is used in the MDC-RHB calculations.

Figure 3

Potential energy curves ($E\sim {\beta }_{20}$) for Zr isotopes. The energy is normalized with respect to the energy minimum with ${\beta }_{20}<0$ for each nucleus. The dotted line denotes results with axial symmetry (AS) and reflection symmetry (RS) imposed, the dash-dotted line denotes results restricted to AS and reflection asymmetry (RA), while the solid line represents results with both AS and RS broken, i.e., with triaxial (TA) deformation and RA. The functional DD-PC1 is used in MDC-RHB calculations.

Figure 4

The single-particle levels near the Fermi surface for protons (a) and neutrons (b) of ${}^{110}\mathrm{Zr}$ as functions of ${\beta }_{32}$ with ${\beta }_{20}$ fixed at zero. The functional DD-PC1 is used in the MDC-RHB calculations.

Figure 5

The proton and neutron pairing energies of ${}^{110}\mathrm{Zr}$ as functions of ${\beta }_{20}$. The dotted line denotes the results with the axial symmetry (AS) and reflection symmetry (RS) imposed, the dash-dotted line denotes the results restricted to the AS and reflection asymmetry (RA), while the solid line represents the results with both triaxial (TA) and RA shapes allowed. The functional DD-PC1 is used in the MDC-RHB calculations.

Figure 6

The potential energy surfaces of ${}^{106–114}\mathrm{Zr}$ in the $({\beta }_{30},{\beta }_{32})$ plane with ${\beta }_{20}$ fixed at zero. The contour interval is 0.1 MeV. The functional DD-PC1 is used in the MDC-RHB calculations.