Analytical solution for the Davydov-Chaban Hamiltonian with 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 considered potential shapes 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 $B(E2)$ 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 $B(E2)$ transition ratios are parameter independent. The energy spectra of the ground and first $beta$ and $gamma$ bands as well as the corresponding $B(E2)$ 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 $^{128,130,132}$Xe and $^{192,194,196}$Pt isotopes, revealing a qualitative agreement with experiment and a phase transition in Xe isotopes.

A solvable model which has X(5) as a limiting symmetry and removes some inherent drawbacks

Solvable Hamiltonians for the $beta$ and $gamma$ intrinsic shape coordinates are proposed. The eigenfunctions of the $gamma$ Hamiltonian are spheroidal periodic functions, while the Hamiltonian for the $beta$ degree of freedom involves the Davidsons potential and admits eigenfunctions which can be expressed in terms of the generalized Legendre polynomials. The proposed model goes to X(5) in the limit of $|gamma|$-small. Some drawbacks of the X(5) model, as are the eigenfunction periodicity and the $gamma$ Hamiltonian hermiticity, are absent in the present approach. Results of numerical applications to $^{150}$Nd, $^{154}$Gd and $^{192}$Os are in good agreement to the experimental data. Comparison with X(5) calculations suggests that the present approach provides a quantitative better description of the data. This is especially true for the excitation energies in the gamma band.