larger values of $N$, Th isotopes maintain a stable quadrupole deformation, while the octupole mode evolves towards the vibrational behavior, as indicated by the large excitation energies of all negative–parity levels. At $N=130$ or less, the quadrupole mode has a vibrational (or non collective) character. It turns out, therefore, that the octupole phase transition proceeds in the direction opposite to the one of quadrupole. We also observe that the phase transition only involves the axial octupole mode. In fact, the energy of the $J^\pi=1^-$ band head of the $K^\pi=0^-$ octupole band shows a sharp decrease, both in its absolute value and in the ratio to the $E(2^+)$, when the neutron number decreases below $N=142$. Other octupole bands (with $K>0$) do not show a similar trend (fig. \[F:1\]$(c)$), and one can conclude that non-axial octupole excitations maintain a vibrational character. A similar trend is apparent also for Ra isotopes.

In order to describe Th and Ra isotopes with $A < 226$, we must allow also $\beta_2$ to vary and perform (non necessarily harmonic) oscillations. If we consider the value $E(4^+)/E(2^+)=2.91$ as a signature of the critical point with respect to the quadrupole deformation, this would correspond approximately to $^{224}$Ra and $^{224}$Th.

