Abstract.The equation of the Bohr-Mottelson Hamiltonian with a sextic oscillator potential is solved for γ-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 β 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 β 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 β-soft rotor crossing through a critical point. Numerical applications are performed for

An analytical solution for the Davydov-Chaban Hamiltonian with a sextic oscillator potential for the variable β and γ fixed to 30 • , is proposed. The model is conventionally called Z(4)-Sextic.For the considered potential shapes the solution is exact for the ground and β bands, while for the γ band an approximation is adopted. Due to the scaling property of the problem the energy and B(E2) transition ratios depend on a single parameter apart from an integer number which limits the number of allowed states. For certain constraints imposed on the free parameter, which lead to simpler special potentials, the energy and B(E2) transition ratios are parameter independent. The energy spectra of the ground and first β and γ bands as well as the corresponding B(E2) transitions, determined with Z(4)-Sextic, are studied as function of the free parameter and presented in detail for the special cases. Numerical applications are done for the 128,130,132 Xe and 192,194,196 Pt isotopes, revealing a qualitative agreement with experiment and a phase transition in Xe isotopes.

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Based on the competition between γ-stable and γ-rigid collective motions mediated by a rigidity parameter, a two-parameter exactly separable version of the Bohr Hamiltonian is proposed. The γ-stable part of the Hamiltonian is restricted to stiff oscillations around the γ value of the rigid motion. The separated potential for β and γ shape variables is chosen such that in the lower limit of this parameter, the model recovers exactly the ES-X(5) model, while in the upper limit it tends to the prolate γ-rigid solution X(3). The combined effect of the rigidity and stiffness parameters on the energy spectrum and wave function is duly investigated. Numerical results are given for few nuclei showing such ambiguous behaviour.

The experimentally available data on the α decay half lives and Q α values for 96 superheavy nuclei are used to fix the parameters for a modified version of the Brown empirical formula through two fitting procedures which enables its comparison with similar fits using Viola-Seaborg and Royer formulas. The new expressions provide very good agreement with experimental data having fewer or the same number of parameters.All formulas with the obtained parameters are then extrapolated to generate half lives predictions for 125 unknown superheavy α emitters. The nuclei where the employed empirical formulas maximally or minimally diverge are pointed out and a selection of 36 nuclei with exceptional superposition of predictions was made for experimental reference.

A prolate γ-rigid version of the Bohr-Mottelson Hamiltonian with a quartic anharmonic oscillator potential in β collective shape variable is used to describe the spectra for a variety of vibrationallike nuclei. Speculating the exact separation between the two Euler angles and the β variable, one arrives to a differential Schrödinger equation with a quartic anharmonic oscillator potential and a centrifugal-like barrier. The corresponding eigenvalue is approximated by an analytical formula depending only on a single parameter up to an overall scaling factor. The applicability of the model is discussed in connection to the existence interval of the free parameter which is limited by the accuracy of the approximation and by comparison to the predictions of the related X(3) and X(3)β 2 models. The model is applied to qualitatively describe the spectra for nine nuclei which exhibit near vibrational features.

An analytical expression for the energy spectrum of the ground and β bands was obtained through the JWKB approximation in the axially symmetric γ -rigid regime of the Bohr-Mottelson Hamiltonian with an oscillator potential and a sextic anharmonicity in the β shape variable. Due to the scaling property of the problem, the resulting energy depends up to an overall multiplicative constant on a single parameter. Studying the behavior of the energy spectrum as a function of the free parameter, one establishes the present model's place among other prolate γ -rigid models and in the more general extent of collective solutions. The agreement with experiment is achieved through model fits for few nuclei.

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