E(5) and X(5) critical point symmetries obtained from Davidson potentialsthrough a variational procedure

Davidson potentials of the form β²+β₀⁴∕β², when used in the E(5) framework bridge the U(5) and O(6) symmetries, while they bridge the U(5) and SU(3) symmetries when used in the X(5) framework. Using a variational procedure, we determine for each value of angular momentum L the value of β₀ at which the rate of change of various physical quantities [energy ratios, intraband B(E2) ratios, quadrupole moment ratios] has a maximum, the collection of the values of the physical quantity formed in this way being a candidate for describing its behavior at the relevant critical point. Energy ratios lead to the E(5) and X(5) results, while intraband B(E2) ratios and quadrupole moments lead to the E(5)-β⁴ and X(5)-β⁴ models. A new derivation of the Holmberg-Lipas formula for nuclear energy spectra is obtained as a by-product.