Figure 1

Upper row (a${}_1$ to a${}_4$): contour plots of the HFB energy surfaces $E(Q_{20},Q_{30})$ obtained with the Gogny D1M interaction are depicted for several nuclei considered in this work. The position of the ground state minimum is signalled by a bullet. In the middle row (b${}_1$ to b${}_4$) the ground state collective amplitudes $|g_{\sigma }(Q_{20},Q_{30}){|}^2$ are shown. In the bottom row (c${}_1$ to c${}_4$), the corresponding amplitudes for the first negative parity excited state are given. In this mass region a $Q_{20}$value of 1b corresponds to a ${\beta }_2=0.022$ and a $Q_{30}$ of 1 b${}^{3/2}$ to ${\beta }_3=0.035$. All quantities are symmetric under the $Q_{30}\to -Q_{30}$ exchange. For the HFB energies, contour lines (full and dashed) are plotted every 250 keV up to 4 MeV and from there on (dotted) every 1 MeV up to 8 MeV.Reuse & Permissions

Figure 2

Excitation energies of the $1^{-}$ state (panels a${}_1$ to $a_3$) and transition strengths to the ground state $B\left(E1\right)$ (panels b${}_1$ to $b_3$) and $B\left(E3\right)$ (panels c${}_1$ to $c_3$) (in Weisskopf units) are depicted for the radon, radium, and thorium isotopic chains. Experimental results are represented by bullets whereas the theoretical predictions obtained with the two-dimensional $Q_{20}$-$Q_{30}$ GCM calculations are plotted as full (dotted) black lines for Gogny D1S (D1M). Results of the one-dimensional GCM calculation with $Q_{30}$ as collective coordinate are represented by red (gray) lines. Experimental results are from [2] for ${}^{220}$Rn and ${}^{224}$Ra and from other sources [20, 21] for the other nuclei.Reuse & Permissions

Figure 3

The proton-neutron energy $V_{pn}$ defined in the text is plotted as a function of neutron number N for the radium (b) and thorium (a) isotopes.Reuse & Permissions