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

Abstract

Abstract In this paper, four degree of freedom γ -rigid solutions of the Bohr Hamiltonian at γ 0 = 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 ≤ γ 0 ≤ π /6 while the potential minimum, β 0 , is varied in the interval 0 ≤ β 0 ≤ ∞ . Very small value of β 0 yields Z (4) while a large value of β 0 produces X (4) and SU (3) is realized at β 0 ≈ ∞ . The solutions at γ 0 = 0 correspond to X (4) while the solutions at γ 0 = 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 β 0 of the potential play the same role in the dynamical link between Z (4) and X (4) shows that β 0 also measures the departure from axial symmetry to other shapes as γ does. In the experimental realization of the model, the conformation of 192 Pt and 194 Pt 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. 194 Pt is shown to be the best choice for triaxial rigid rotor candidacy. 130 Xe, 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. 4 0,0 − β 0 distribution for Z (4) and X (4) candidate isotopes shows a significant interval between the two models where the T (4) CPS lies.