$\gamma$-rigid solution of the Bohr Hamiltonian for the critical point description of the spherical to $\gamma$-rigidly deformed shape phase transition

The $\gamma$-rigid solution of the Bohr Hamiltonian with the $\beta$-soft potential and $0^{\circ }\le \gamma \le {30}^{\circ }$ is worked out. The resulting model, called T(4), provides a natural dynamical connection between the X(4) and the Z(4) critical-point symmetries, which thus serves as the critical-point symmetry of the spherical to $\gamma$-rigidly deformed shape phase transition. This point is further justified through comparing the model dynamics with those of the interacting boson model. As a preliminary test, the low-lying structures of ${}^{158}\mathrm{Er}$ are taken to compare the theoretical calculations, and the results indicate that this nucleus could be considered as the candidate of the T(4) model with an intermediate $\gamma$ deformation.

Figure 1

Selected energy ratios and $B\left(E2\right)$ ratios of the T(4) model and IBM for $N=10$ as functions of $\gamma$ (in degrees) with the assumption $\gamma \left(\chi \right)=\frac{\pi }{3\sqrt{7}}\chi +\frac{\pi }{6}$ for the IBM.

Figure 2

Transitional behavior of typical energy ratios and $B\left(E2\right)$ ratios as functions of $\eta$ in the IBM for $N=10$ solved from the Hamiltonian (38) with $\chi =-\sqrt{7}/4$ and the corresponding results obtained from the T(4) model at $\gamma =\pi /12$ based on the assumption $\gamma \left(\chi \right)=\frac{\pi }{3\sqrt{7}}\chi +\frac{\pi }{6}$.

Figure 3

The energy ratios $E_{0_{\mathrm{n}}}/E_{2_1}$, $E_{8_{\mathrm{n}}}/E_{2_1}$, $E_{0_{\mathrm{n}}}/E_{0_2}$, and $E_{8_{\mathrm{n}}}/E_{8_1}$ calculated in the T(4) model for $\gamma =5^{\circ },{15}^{\circ },{25}^{\circ }$, where $E_{0_n}$ with $n=1,2,3,4,5$ are the bandhead energies, and $E_{8_n}$ with $n=1,2,3,4$ represent the energy values of the states with $L_{\xi ,s}=8_{n,1}$.

Figure 4

The low-lying level patten of ${}^{158}\mathrm{Er}$ [41] and those solved from the T(4) model and the IBM, where all level energies have been normalized to $E\left(2_1\right)$ in each case. In calculations, the $\gamma$ value in the T(4) model is set as $\gamma =14.7^{\circ }$, and the parameters in the IBM Hamiltonian (38) are taken as $N=11$, $\eta =0.63$, and $\chi =-0.61$, which are the same as those adopted in Ref. [45].