$$f(\beta_2,
\beta_3)=\sqrt{\frac{(\beta_2^2+\beta_3^2)(\beta_2^2+2\beta_3^2)}
{\beta_2^2+5\beta_3^2}}\ ,
\label{E:2.4}$$ from which one obtains $$G_0(\beta_2,\beta_3)=
\frac{(\beta_2^2+\beta_3^2)^2(\beta_2^2+2\beta_3^2)^4}
{(\beta_2^2+5\beta_3^2)^2}\ ,
\label{E:2.4b}$$

The critical potential in two dimensions
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Possible landscapes of axial quadrupole–octupole deformation in the Thorium region are exemplified in Fig. \[F:2\], where the potential energy is depicted as a function of the deformation parameters $\beta_2$ and $\beta_3$. Reported values have been obtained by Nazarewicz [*et al.*]{} [@nazarew85] with a Wood–Saxon–Bogolyubov cranking calculation. We notice that fig \[F:2\]$(e)$ shows a potential minimum which is localized around a fixed value in the $\beta_2$ direction, while a flat minimum extends over a sizable interval in the direction $\beta_3$. This is just the “critical” potential for the shape transition between octupole oscillation and permanent octupole deformation (combined with a fixed quadrupole deformation), corresponding to the fig.s \[F:2\]$(d)$ and \[F:2\]$(f)$, respectively. The fig.s \[F:2\]$(a), \ref{F:2}(b)$ and \[F:2\]$(c)$, instead, show a different kind of shape transition, proceeding directly from a fixed, reflection–asymmetric deformation (fig.\[F:2\]$(c)$) to quadrupole–octupole vibrations around a spherical shape (fig.\[F:2\]$(a)$).

![(color on line) Potential–energy surfaces in the $\beta_2$ – $\beta_3$ plane for several Th isotopes, as given by Nazarewicz [*et al.*]{} [@nazarew85].[]{data-label="F:2"}](biz_f2.ps){width="8.4cm"}

The potential corresponding to the critical point is not shown. It should be somewhere between fig. \[F:2\]$(a)$ 