Preformation probability and kinematics of cluster emission yielding Pb-daughters*

Abstract

In the present study, the newly established preformation formula is applied for the first time to study the kinematics of the cluster emission from various radioactive nuclei, especially those that decay to the double shell closure $^{208}$ the Pb nucleus and its neighbors as daughters. The recently proposed universal cluster preformation formula has been established based on the concepts that underscores the influence of the mass and charge asymmetry ($\eta_A$ and $\eta_Z$), cluster mass $A_c$ and the Q-value, paves the way to quantify the energy contribution during the preformation as well as the tunnelling process separately. The cluster-daughter interaction potential is obtained by folding the relativistic mean-field (RMF) densities with the recently developed microscopic R3Y using the NL$3^*$ and the phenomenological M3Y NN potentials to compare their adaptability. The penetration probabilities are calculated from the WKB approximation. With the inclusion of the new preformation probability $P_0$, the predicted half-lives of the R3Y and M3Y interactions are in good agreement with the experimental data. Furthermore, a careful inspection reflects slight differences in the decay half-lives, which arise from their respective barrier properties. The $P_0$ for systems with double magic shell closure $^{208}$Pb daughter are found to be relatively higher by an order of $\approx 10^2$ than those with neighbouring Pb daughter nuclei. By exploring the contributions of the decay energy, the recoil effect of the daughter nucleus is appraised, unlike several other conjectures. Thus, the centrality of the $Q-$value in the decay process is demonstrated and redefined within the preformed cluster-decay model. Besides, we have introduced a simple and intuitive set of criteria that governs the estimation of recoil energy in the cluster radioactivity.