Reexamining cluster radioactivity in trans-lead nuclei with consideration of specific density distributions in daughter nuclei and clusters

We further investigate the cluster emission from heavy nuclei beyond the lead region in the framework of the preformed cluster model. The refined cluster-core potential is constructed by the double-folding integral of the density distributions of the daughter nucleus and the emitted cluster, where the radius or the diffuseness parameter in the Fermi density distribution formula is determined according to the available experimental data on the charge radii and the neutron skin thickness. The Schrödinger equation of the cluster-daughter relative motion is then solved within the outgoing Coulomb wave-function boundary conditions to obtain the decay width. It is found that the present decay width of cluster emitters is clearly enhanced as compared to that in the previous case, which involved the fixed parametrization for the density distributions of daughter nuclei and clusters. Among the whole procedure, the nuclear deformation of clusters is also introduced into the calculations, and the degree of its influence on the final decay half-life is checked to some extent. Moreover, the effect from the bubble density distribution of clusters on the final decay width is carefully discussed by using the central depressed distribution.

Figure 1

Schematic diagram of the total interaction potential $V\left(r\right)=\lambda V_N\left(r\right)+V_C\left(r\right)$ between the daughter ${}^{208}\mathrm{Pb}$ and the cluster ${}^{14}\mathrm{C}$ for the cluster emitter ${}^{222}\mathrm{Ra}$. The black line denotes the case calc 1 with $\mathrm{\Delta }{R}_{np}=0$, the red dashed line presents the neutron skin type distribution with $\mathrm{\Delta }{R}_{np}=0.160$ fm, the blue dotted line gives the halo type case, while the magenta dash-dotted line indicates the halo daughter with the deduced density distribution of clusters (calc 3). The magnified version of the potential in the range of $r=8$–12 fm is plotted in the right panel for clarity.

Figure 2

Variation of the extracted $P_c$ value with the quantity $\sqrt{\mu }{\left(Z_c{Z}_d\right)}^{1/2}$ including the results in the three cases and the correspondingly fitted line, which are separated as even-even nuclei and odd-$A$ nuclei.

Figure 3

The ratio of the calculated half-life to that in the spherical case, denoted as $T_{1/2}^{{\beta }_2}/T_{1/2}^0$, varies with the change of deformation parameters ${\beta }_2$ of ${}^{20}\mathrm{O}$ in the cluster emission of ${}^{228}\mathrm{Th}$.

Figure 4

Variety of calculated half-lives with $P_c=1$ versus the increasing of the depression parameter $\omega$ for cluster decay ${}^{228}\mathrm{Th}\to {}^{208}\mathrm{Pb}+{}^{20}\mathrm{O}$, which is plotted in the left panel. Correspondingly, the central depressed degree, defined as the ratio of the deviation between the maximum and central densities to the maximum of the density, is plotted in the right panel to present a comprehensive view.