been observed that the isotopes $^{224}$Ra and $^{224}$Th have a positive–parity ground–state band with a sequence of level energies very close to the X(5) predictions [@bizzetierice; @bizzetirila]. Here, however, the presence of a very low lying negative parity band, soon merging with the positive–parity one for $J>5$, proves that the octupole mode of deformation plays an important role and should not be ignored in discussing the behavior of the phase transition.

In a previous paper [@bizzeti04] (henceforth referred to as I) a simple model has been introduced to describe the phase transitions in nuclear shape involving the octupole mode[^1]. To this purpose, a new parametrization of the collective coordinates describing the nuclear quadrupole and octupole deformation has been introduced and discussed. The nuclear shape is represented in the intrinsic frame defined by the principal axes of the overall tensor of inertia, in situations close to (but not necessarily coincident with) the axial–symmetry limit. In the same paper, a specific model is developed to describe the critical point of the phase transition in the octupole mode, between harmonic oscillations and permanent asymmetric deformation, in nuclei which already possess a stable quadrupole deformation. The Thorium isotopic chain was investigated and the experimental 