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

Relative variation between γ coordinate and the potential of the Bohr Hamiltonian producing X(4) from Z(4)

“…corresponds to the Z(4) solutions obtained in [16,23]. Equation (32) and equation (33) suggest that these two critical point symmetries (CPSs) are the two limit cases of this model: the Z(4) corresponds to the upper limit while the X(4) corresponds to the lower limit. The solutions of a L,s for other values of γ between γ 0 = π/6 and γ 0 = 0, that is, 0 < γ 0 < π/6 region are obtained: this region is referred to as T(4) critical region [12].…”

“…Albeit, in the literature [16,24,25,[31][32][33][34], the variation of the potential parameter has provided some shape phase transitions at some constant or approximate values of γ, a simultaneous relative variation between γ and the potential has not been reported until in this present paper. These special shape phase solutions provide some geometric coordinates which link the potential and γ, as their relative variations link Z(4) and X(4) CPSs together.…”

“…The solutions of a L,s for other values of γ between γ 0 = π/6 and γ 0 = 0, that is, 0 < γ 0 < π/6 region are obtained: this region is referred to as T(4) critical region [12]. The solutions of equation (33) have been provided in [38] and their corresponding expansion coefficients, a b s L , ( ) are written as…”

“…The procedure for finding the eigenvalues are stated in [33,34]. If the first condition of the listed procedure is considered, then the acceptable expression for the spectra is written as:…”

“…The description of these CPSs, if this present method must be used, requires that a potential with a minimum or a free parameter potential such as Davidson potential used in [16], inverse square potential in [24,33,34], etc., must be used. In this present paper, the minimum of the potential, serves as an entity that can change the axial symmetry between the two CPSs.…”