– is now $N=n_2 \cdot n_3$, and we obtain a finite dimensional $N\times N$ Hamiltonian matrix. This Hamiltonian has been diagonalized with the Implicitly Restarted Arnoldi – Lanczos method, using the ARPACK package [@arpack].

In the fig. \[F:3.1\], calculated values of the excitation energies (in units of $E(2^+_1$) ) are depicted as a function of the ratio $b=\beta_3^w/\beta_2^w$. At the limit for $\beta_3^w\rightarrow 0$, the curves corresponding to even $J$ and $\pi$ tend to the X(5) values, as expected. With increasing $b$, at the beginning these curves deviate substantially from the X(5) limit, but they come closer to the initial values for $b\approx 1$. In this region it is possible to find a good fit of the ground–state band of $^{224}$Ra and of $^{224}$Th, for $b=0.81$ and for $b=0.85$, respectively (Fig. \[F:5\]).

Moreover, a rather good fit of the ground–state band of $^{226}$Ra is obtained with $b=0.68$, [*i.e.*]{} close to the maximum of the curves for even parity and spin.

We can observe that, with our choice of the parameter $b$, the calculated $1^-$ level is always somewhat lower than the experimental one (Fig. \[F:5\]). This fact can be related to the inclusion, in the potential well, of a region 