Analytical comparison between X(3) and X(5) models of the Bohr Hamiltonian.

The 3-D Bohr-Mottelson Hamiltonian for γ-rigid prolate isotopes is solved via inverse square potential having only one free parameter, β0. The exact form of the wave functions and the energy spectra are obtained as a function of the free parameter that determines the changes in the spectra ratios and the B(E2). The solutions are compared with the X(5) model and some new set of equations that show the relationships between the two models are stated. The entire solutions from β0 = 0 to ∞ are compared with U(5), X(5) and SU(3). The solutions spread from the region around U(5) over X(5) and approach SU(3) at β0 =  ∞ . The exact solutions obtained via variational procedure are compared with some existing X(3) models in the literature. The strong agreement between the present model and X(3) via infinite square well potential is discussed. Twelve best critical point isotopes, (102)Mo, (104-108)Ru, (120-126)Xe, (148)Nd, (184-188)Pt are chosen for experimental realization of the model and moderate agreements are recorded. An excellent agreement which appears in the first β-excited state in the comparison of the present model with three N = 90 isotones: (150)Nd, (154)Gd, and (156)Dy, known to be X(5) candidates, suggests that the present model compensates the X(5) models whose predictions are excellent in the ground states but moderately bad in the first β-excited states.