U(5) and O(6) shape phase transitions via E(5) inverse square potential solutions

The inverse square potential is used to provide the ‘pictures’ of the transitions symmetry from U(5) to O(6) via the variational procedure of the E(5) solutions. The variation of the single-parameter $$\beta _{0}$$ of the inverse square potential shifts the E(5) solutions: the energy eigenvalues increase when $$\beta _{0}$$ increases. In the theoretical prediction of $$R_{L/2}$$ , $$\beta _{0}=0$$ yields solutions that correspond to the U(5) symmetry of the Bohr Hamiltonian with $$R_{4/2}=2.00$$ . As $$\beta _{0}$$ increases, the solutions of E(5) materialize with $$R_{4/2}=2.189$$ at $$\beta _{0,max}=3.986$$ . The solutions leave E(5) and approach O(6) with $$R_{4/2}=2.466$$ at $$\beta _{0}=25$$ . The solutions of $$^{132}Xe$$ and $$^{134}Xe$$ of the E(5) symmetry candidates are fitted and compared with the available experimental values. With the same variational method, the ratios of B(E2) transition rates within the ground state leave O(6) and approach E(5) solution as $$\beta _{0}$$ increases.