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The eigenvalue equation associated to the Bohr-Mottelson Hamiltonian isconsidered in the intrinsic reference frame and amended by replacing theharmonic oscillator potential in the $\beta$ variable with a sextic oscillatorpotential with centrifugal barrier plus a periodic potential for the $\gamma$variable. After the separation of variables, the $\beta$ equation isquasi-exactly solved, while the solutions for the $\gamma$ equation are justthe angular spheroidal functions. An anharmonic transition operator is used todetermine the reduced E2 transition probabilities. The formalism isconventionally called the Sextic and Spheroidal Approach (SSA) and applied forseveral X(5) candidate nuclei: $^{176,178,180,188,190}$Os, $^{150}$Nd,$^{170}$W, $^{156}$Dy, $^{166,168}$Hf. The SSA predictions are in goodagreement with the experimental data of the mentioned nuclei. The comparison ofthe 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 $\beta$, otherwise keeping the spheroidal functions forthe $\gamma$, and the Coherent State Model (CSM)\cite{Rad1,Rad2,Rad3,Rad4,RaSa,Rad5} respectively, suggests that SSA representsa good approach to describe nuclei achieving the critical point of theU(5)$\rightarrow$SU(3) shape phase transition.

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