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Application of the sextic oscillator with centrifugal barrier and the spheroidal equation for some X(5) candidate nuclei

The eigenvalue equation associated to the Bohr-Mottelson Hamiltonian is considered in the intrinsic reference frame and amended by replacing the harmonic oscillator potential in the $β$ variable with a sextic oscillator potential with centrifugal barrier plus a periodic potential for the $γ$ variable. After the separation of variables, the $β$ equation is quasi-exactly solved, while the solutions for the $γ$ equation are just the angular spheroidal functions. An anharmonic transition operator is used to determine the reduced E2 transition probabilities. The formalism is conventionally called the Sextic and Spheroidal Approach (SSA) and applied for several X(5) candidate nuclei: $^{176,178,180,188,190}$Os, $^{150}$Nd, $^{170}$W, $^{156}$Dy, $^{166,168}$Hf. The SSA predictions are in good agreement with the experimental data of the mentioned nuclei. The comparison of the SSA results with those yielded by other models, such as X(5) \cite{Iache9}, Infinite Square Well (ISW) \cite{Raduta}, and Davidson (D) like potential \cite{Raduta} for the $β$, otherwise keeping the spheroidal functions for the $γ$, and the Coherent State Model (CSM) \cite{Rad1,Rad2,Rad3,Rad4,RaSa,Rad5} respectively, suggests that SSA represents a good approach to describe nuclei achieving the critical point of the U(5)$\rightarrow$SU(3) shape phase transition.