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

Abstract

In this paper, four degree of freedom γ-rigid solutions of the Bohr Hamiltonian at γ0 = π/6 are
obtained. The relative variation between γ and the potential of the Hamiltonian is employed to retrieve X(4) from Z(4). γ is varied in the interval 0 ≤ γ0 ≤ π/6 while the potential minimum, β0, is varied in the interval 0 ≤ β0 ≤ ∞. Very small value of β0 yields Z(4) while a large value of β0 produces X(4) and SU(3) is realized at β0 ≈ ∞. The solutions at γ0 = 0 correspond to X(4) while the solutions at γ0 = π/6 yield Z(4): a dynamic link between Z(4) and X(4) critical point symmetries (CPSs) has been provided. The fact that γ and 
β0 of the potential play the same role in the dynamical link between Z(4) and X(4) shows that β0 also measures the departure from axial symmetry to other shapes as γ does. In the experimental realization of the model, the conformation of 192Pt and 194Pt isotopes to the present Z(4) model shows that the present model can be employed in the description of triaxial rotors and γ-soft isotopes. 194Pt is shown to be the best choice for triaxial rigid rotor candidacy. 130Xe, a critical point isotope, which belongs to the class of γ-soft nuclei, reproduces this present model very well in all the states. Consequently, this present model can serve as a critical point model. 40,0 - β0 distribution for Z(4) and X(4) candidate isotopes shows a significant interval between the two models where the T (4) CPS lies.