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