Sextic potential for $\gamma$-rigid prolate nuclei

TL;DR

This paper introduces an exactly solvable sextic oscillator model for $eta$-rigid prolate nuclei, enabling analysis of shape phase transitions and assignment of experimental states with closed-form energies and wave functions.

Contribution

It develops a novel sextic potential-based model for $eta$-rigid nuclei, providing exact solutions and insights into shape phase transitions in nuclear structure.

Findings

Model accurately describes 39 nuclei with shape phase transition behavior.

Identifies $^{104}$Ru and $^{120,126}$Xe as critical point candidates.

Provides closed-form solutions for energies and wave functions.

Abstract

The equation of the Bohr-Mottelson Hamiltonian with a sextic oscillator potential is solved for $\gamma$-rigid prolate nuclei. The associated shape phase space is reduced to three variables which are exactly separated. The angular equation has the spherical harmonic functions as solutions, while the $\beta$ equation is brought to the quasi-exactly solvable case of the sextic oscillator potential with a centrifugal barrier. The energies and the corresponding wave functions are given in closed form and depend, up to a scaling factor, on a single parameter. The $0^{+}$ and $2^{+}$ states are exactly determined, having an important role in the assignment of some ambiguous states for the experimental $\beta$ bands. Due to the special properties of the sextic potential, the model can simulate, by varying the free parameter, a shape phase transition from a harmonic to an anharmonic prolate $\beta$-soft rotor crossing through a critical point. Numerical applications are performed for 39 nuclei: $^{98-108}$Ru, $^{100,102}$Mo, $^{116-130}$Xe, $^{132,134}$Ce, $^{146-150}$Nd, $^{150,152}$Sm, $^{152,154}$Gd, $^{154,156}$Dy, $^{172}$Os, $^{180-196}$Pt, $^{190}$Hg and $^{222}$Ra. The best candidates for the critical point are found to be $^{104}$Ru and $^{120,126}$Xe, followed closely by $^{128}$Xe, $^{172}$Os, $^{196}$Pt and $^{148}$Nd.