Analytical solution for the Davydov-Chaban Hamiltonian with a sextic potential for $\gamma ={30}^{\circ }$

An analytical solution for the Davydov-Chaban Hamiltonian with a sextic oscillator potential for the variable $\beta$ and $\gamma$ fixed to ${30}^{\circ }$ is proposed. The model is conventionally called Z(4)-sextic. For the potential shapes considered, the solution is exact for the ground and $\beta$ bands, while for the $\gamma$ band an approximation is adopted. Due to the scaling property of the problem, the energy and $B\left(E2\right)$ transition ratios depend on a single parameter apart from an integer number which limits the number of allowed states. For certain constraints imposed on the free parameter, which lead to simpler special potentials, the energy and $B\left(E2\right)$ transition ratios are parameter independent. The energy spectra of the ground and first $\beta$ and $\gamma$ bands as well as the corresponding $B\left(E2\right)$ transitions, determined with Z(4)-sextic, are studied as function of the free parameter and presented in detail for the special cases. Numerical applications are done for the ${}^{128,130,132}\mathrm{Xe}$ and ${}^{192,194,196}\mathrm{Pt}$ isotopes, revealing a qualitative agreement with experiment and a phase transition in Xe isotopes.

Figure 1

The shapes of the energy potential for $\gamma$ fixed at $\pi /6$ and with $K=1$, corresponding to $\alpha =-10,\pm 2\sqrt{c_0}(\pm 3.22),0,10$, are plotted as function of $y$.

Figure 2

The energy spectrum given by Eq. (3.27) is shown as function of $\alpha$ in the interval $[-10,10]$ for $K=1,2,3,4$. In panels (a), (c), (e), and (g) are plotted the energy curves of the ground band and $\beta$ band which go to infinity when $\alpha \to -\infty$. While in panels (b), (d), (f), and (h) are those corresponding to the $\gamma$ band, with the continuous and dashed curves representing $L$-even and $L$-odd states, respectively.

Figure 3

The $B\left(E2\right)$ transitions (a) $4_g^{+}\to 2_g^{+},2_{\beta }^{+}\to 0_{\beta }^{+},0_{\beta }^{+}\to 2_g^{+}$ and (b) $3_{\gamma }^{+}\to 2_{\gamma }^{+},2_{\gamma }^{+}\to 2_g^{+},3_{\gamma }^{+}\to 4_g^{+},4_{\gamma }^{+}\to 2_{\gamma }^{+},2_{\beta }^{+}\to 2_{\gamma }^{+}$, normalized to $B(E2,2_g^{+}\to 0_g^{+})$ and plotted as functions of $\alpha$ in the same interval $[-10,10]$. The continuous, dashed, dot-dashed, and dotted lines correspond to $K=1,2,3,4$, respectively.

Figure 4

The plot of ${\varphi }_{00}^1\left(y\right)y^{3/2}$ as function of $y$ for $\alpha \in [0,10]$. The square of this quantity is the probability density for the ground-state deformation with respect to the $dy$ integration measure. The horizontal red line marks the the wave-function behavior at the critical value ${\alpha }_c$. The evolution of the peak's position with $\alpha$ is visualized by another red curve. The difference between two consecutive contour lines is 0.1.

Figure 5

Energy spectra and some $E2$ transitions, normalized to the energy of the state $2_g^{+}$ and to the transition probability $B(E2,2_g^{+}\to 0_g^{+})$, respectively, are visualized for $K=1,2,3,4$ when $\alpha =0$.

Figure 6

The same as in Fig. 5, but for $\alpha ={\alpha }_c$.

Figure 7

The staggering S(4) given by Z(4)-sextic for $\alpha \in [-10,10]$ is compared with the values yielded by the Z(4) and Z(4)-${\beta }^2$ models.

Figure 8

The theoretical energy spectra, given by Eq. (3.27), are compared with the experimental data [40, 41, 42] of the ${}^{128,130,132}\mathrm{Xe}$ isotopes and with the Z(4)-model predictions. The corresponding rms values for Z(4)-sextic are 0.516, 0.478, and 0.403, while for Z(4) one obtains 0.528, 0.389, and 0.611.

Figure 9

The same as Fig. 8, but for the experimental data [43, 44, 45] of the ${}^{192,194,196}\mathrm{Pt}$ isotopes. The corresponding rms values for Z(4)-sextic are 0.614, 0.543, and 0.682, while for Z(4) one obtains 0.662, 0.707, and 0.732.

Figure 10

Ground-state potential (3.20) with $K=4$ plotted as a function of $y$ for all considered nuclei with $\alpha$ resulting from the fits of Figs. 8 and 9. The critical-point potential $\left(\alpha ={\alpha }_c\right)$ as well as a potential close to the spherical limit $\left(\alpha =20\right)$ are given for reference.