Search for critical-point nuclei in terms of the sextic oscillator

Abstract

The spherical to deformed $\ensuremath{\gamma}$-unstable shape transition in nuclei is discussed in terms of the sextic oscillator as a $\ensuremath{\gamma}$-independent potential in the Bohr Hamiltonian. The wave functions, energy eigenvalues, and electric quadrupole and monopole transition rates are calculated in closed analytical form for the lowest-lying energy levels. It is shown that the locus of critical points for the spherical to deformed $\ensuremath{\gamma}$-unstable shape phase transition corresponds to a parabola in the parameter space of the model. The ratios of energy eigenvalues and electromagnetic transition probabilities are constant along this parabola. It is thus possible to associate parameter-free benchmark values to the ratios of relevant observables at the critical point of the transition that can be compared to experimental data. In addition, systematic studies of the shape evolution in isotope chains can be performed within the model. As an application, the model parameters are fitted to the energy spectra of the chains of even-even Ru, Pd, and Cd isotopes and the electric quadrupole transition probabilities are calculated. It is found that $^{104}\mathrm{Ru}$, $^{102}\mathrm{Pd}$, and $^{106,108}\mathrm{Cd}$ nuclei, which are usually considered to be good candidates for the $E$(5) symmetry, lie rather close to the critical parabola that separates the spherical and deformed $\ensuremath{\gamma}$-unstable domains. The isotope $^{116}\mathrm{Cd}$ is proposed as a new candidate for a similar critical-point nucleus.