Analytic description of critical-point actinides in a transition from octupole deformation to octupole vibrations

An analytic collective model in which the relative presence of the quadrupole and octupole deformations is determined by a parameter (φ₀), while axial symmetry is obeyed, is developed. The model [to be called the analytic quadrupole octupole axially symmetric model (AQOA)] involves an infinite well potential, provides predictions for energy and B(EL) ratios, which depend only on φ₀, draws the border between the regions of octupole deformation and octupole vibrations in an essentially parameter-independent way, and describes well ²²⁶Th and ²²⁶Ra, for which experimental energy data are shown to suggest that they lie close to this border. The similarity of the AQOA results with φ₀=45^^∘ for ground-state band spectra and $B(E2)$ transition rates to the predictions of the X(5) model is pointed out. Analytic solutions are also obtained for Davidson potentials of the form β²+β₀⁴/β², leading to the AQOA spectrum through a variational procedure.