Z(5): critical point symmetry for the prolate to oblate nuclear shape phase transition

Abstract

A critical point symmetry for the prolate to oblate shape phase transition is introduced, starting from the Bohr Hamiltonian and approximately separating variables for γ = 30 o . Parameter-free (up to overall scale factors) predictions for spectra and B(E2) transition rates are found to be in good agreement with experimental data for 194 Pt, which is supposed to be located very close to the prolate to oblate critical point, as well as for its neighbours ( 192 Pt, 196 (5) [3], related to the transition from axial to triaxial shapes. All these critical point symmetries have been constructed by considering the original Bohr equation [8], separating the collective β and γ variables, and making different assumpions about the u(β) and u(γ) potentials involved.Furthermore, it has been demonstrated [9] that experimental data in the Hf-Hg mass region indicate the presence of a prolate to oblate shape phase transition, the nucleus 194 Pt being the closest one to the critical point. No critical point symmetry for the prolate to oblate shape phase transition originating from the Bohr equation has been given so far, although it has been suggested [10,11] that the (parameter-dependent) O(6) limit of the Interacting Boson Model (IBM) [12] can serve as the critical point of this transition, since various physical quantities exhibit a drastic change of behaviour at O(6), as they should [13].In the present work a parameter-free (up to overall scale factors) critical point symmetry, to be called Z(5), is introduced for the prolate to oblate shape phase transition, leading to parameter-free predictions which compare very well with the experimental data for 194 Pt. The path followed for constructing the Z(5) critical point symmetry is described here: 1) Separation of variables in the Bohr equation [8] is achieved by assuming γ = 30 o . When considering the transition from γ = 0 o (prolate) to γ = 60 o (oblate), it is reasonable to expect that the triaxial region (0 o < γ < 60 o ) will be crossed, γ = 30 o lying in its middle. Indeed, there is experimental evidence supporting this assumption [14].2) For γ = 30 o the K quantum number (angular momentum projection on the bodyfixedẑ ′ -axis) is not a good quantum number any more, but α, the angular momentum projection on the body-fixedx ′ -axis is, as found [15] in the study of the triaxial rotator [16,17].3) Assuming an infinite well potential in the β-variable and a harmonic oscillator potential having a minimum at γ = 30 o in the γ-variable, the Z(5) model is obtained.On these choices, the following comments apply: 1) Taking γ = 30 o does not mean that rigid triaxial shapes are prefered. In fact, it has been pointed out [18] that a nucleus in a γ-flat potential [19] (as it should be expected for a prolate to oblate shape phase transition) oscillates uniformly over γ from γ = 0 o to γ = 60 o , having an average value of γ av = 30 o , and, therefore, the triaxial case to which it should be compared is the one with γ = 30 o . Furthermore, it is known [20] that many prediction...

Description of the isotope chain180−196Pt within several solvable approaches

“…[5] shows a good agreement between calculations and experimental data. In this context the application of the ISWMA and SMA to these isotopes will provide a sensible comparison of the formalisms on one hand and the Z(5) on the other hand.…”

Shape phase transition in γ-rigid prolate nuclei

“…The importance of E(5) and X(5) comes from the fact that they can approximately describe the critical points [13,14] of the spherical vibrator to γ-unstable shape phase transition and of the spherical vibrator to axial rotor shape phase transition, respectively. A similar influence on the development of the research field in discussion had the so called Y(5) [15] and Z(5) [16] solutions, describing the critical points of the shape phase transitions from an axial to a triaxial shape and from a prolate to an oblate shape, respectively. These simple solutions have the advantage of offering numerical results which are free of any parameter being ideal signatures in looking for experimental candidates for these critical points.…”

Solutions of the Bohr Hamiltonian, a compendium

“…A critical point symmetry for a shape phase transition from prolate (γ = 0 o ) to oblate (γ = 60 o ) shapes was introduced in [43] and called Z(5). The authors argue that in such a transition the triaxial region is crossed and the middle lies at γ = 30 o .…”