geometrical model. It is usual to assume a constant [*electric polarizability*]{} of the nuclear matter[@bohr57; @bohr58] to obtain the E1 operator in the form $$\mathcal{M}{\rm (E1)}= C_1 \beta_2 \beta_3 Y_1\ .
\label{E:15}$$ This [*ansatz*]{} should be validated by proper microscopic calculations.

Actually, such a calculation has been performed by Tsvenkov [*et al.*]{} [@tsvenkov02] for a number of Radium, Thorium and Uranium isotopes, in the frame of the Skirme–Hartree–Foch model. The electric dipole moment turns out to be almost independent of the angular frequency in a given isotope, but can change drastically (even in the sign) along the isotopic chain. The small value of the electric dipole moments in $^{224}$Ra is correctly predicted by these calculations.

Values of the ratios of the E1 matrix elements, $R_J(E1)=(J|\!|\mathcal{M}(E1)|\!|J-1) /
(1^-|\!|\mathcal{M}(E1)|\!|0^+)$, obtained with the standard form (Eq. \[E:15\]) of the E1 operator, are shown in the fig. \[F:7\]$(b)$. They reach a maximum for $\beta_3^w/\beta_2^w$ somewhat below 1, [*i.e.*]{} just in a region including the values assumed for $^{224}$Ra and $^{224}$Th (0.81 and 0.85, respectively). The calculated values of $(J_i|\!|\mathcal{M}(EL)|\!|J_f)$ for E2 and E1 transitions in the ground state bands of $^{224}$Ra ($b=0.81$), $^{226}$Ra ($b=0.68$) and $^{224}$Th ($b=0.85$) are given in the upper part of 