1 Bohr Hamiltonian for 0   with Davidson Potential I. YIGITOGLU and M. GOKBULUT Department of Physics, Faculty of Arts and Sciences , Gaziosmanpas ̧a University, 60250, Tokat, Turkey ibrahim.yigitoglu@gop.edu.tr, melek.kgb@gmail.com A  -rigid solution of the Bohr Hamiltonian is derived for 0   utilizing the Davidson potential in the  variable. This solution is going to be called X(3)-D. The energy eigenvalues and wave functions are obtained by using an analytic method which has been developed by Nikiforov and Uvarov. BE(2) transition rates are calculated. A variational procedure is applied to energy ratios to determine whether or not the X(3) model is located at the critical point between spherical and deformed nuclei. I. Introduction The studies describing analytically the critical point at the shape-phase transitions between different dynamical symmetries and enlightening structural properties in atomic nuclei with experimental evidence have been started with the introduction of two new critical point symmetries, called E(5) [1] (between U(5) vibrational and O(6)  -unstable nuclei) and X(5) [2] (between U(5) vibrational and SU(3) axially deformed nuclei). The E(5) symmetry is a  independent exact solution of the Bohr Hamiltonian [3], while the X(5) symmetry is an approximate solution for 0   . The method is based on constructing the Bohr Hamiltonian, choosing different types of potentials such as Morse [4], Kratzer [5-7], Coulomb [5,6], Davidson [5,8], Eckart [9], Manning-Rosen [10], Killingbeck [11] and solving the eigenfunction-eigenvalue problem in search for the quadropole collective dynamics of nuclei. The geometric potentials admitting analytical solutions for the Schrödinger equation belong into two groups. The potentials in the first group depend on both  and  and can be written in the form ( , ) ( ) () V V V      [2,12], where the separation of variables can be done