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 (${\ensuremath{\phi}}_{0}$), 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 $B(\mathit{EL})$ ratios, which depend only on ${\ensuremath{\phi}}_{0}$, draws the border between the regions of octupole deformation and octupole vibrations in an essentially parameter-independent way, and describes well $^{226}\mathrm{Th}$ and $^{226}\mathrm{Ra}$, for which experimental energy data are shown to suggest that they lie close to this border. The similarity of the AQOA results with ${\ensuremath{\phi}}_{0}={45}^{\ifmmode^\circ\else\textdegree\fi{}}$ 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 ${\ensuremath{\beta}}^{2}+{\ensuremath{\beta}}_{0}^{4}/{\ensuremath{\beta}}^{2}$, leading to the AQOA spectrum through a variational procedure.

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