Sextic potential for $\gamma$-rigid prolate nuclei

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.

21 pages, 4 figures

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• Collective states of even-even nuclei in gamma-rigid quadrupole Hamiltonian with Minimal Length under the sextic potential
• Axially Symmetric Quadrupole-Octupole Model incorporating Sextic Potential