Searching of critical point symmetries in atomic nuclei with Bohr Hamiltonian γ-rigid solutions

In this work, the special solutions of the Bohr Hamiltonian which describe the collective motion of the atomic nucleus are presented. Analytical solutions of the Bohr Hamiltonian are constructed using the Davidson potential for the β-part in the cases of Z(4) and X(3) models, corresponding to a γ-rigid solutions of the Bohr Hamiltonian for γ=30o and γ=0o, respectively. These new solutions are called as Z(4)-D and X(3)-D. The eigenvalues and eigenvectors problem of Schrodinger equation for both models are solved through the Nikiforov-Uvarov analytical method developed by Nikiforov and Uvarov. Separately for both models, intraband and interband B(E2) electric quadrupole transitions rates are calculated and the obtained results for energies and B(E2) transition rates are compared with existing experimental data and the theoretical results in the literature. The agreement with the experimental data is achieved. In addition, variation procedure, proposed in order to determine the behavior of physical quantity at the point of shape-phase transition is applied by taking a Davidson potential containing a free parameter. The relevant results are compared with Z(4) and X(3) models predictions. it is shown that Z(4) model, corresponding to the transition from spherical to rigid triaxial rotor, and X(3) model, corresponding to the transition from spherical to prolate axially deformed structure are the solutions at the shape phase transition points.