In the present work, the cluster radioactivity preformation probability $P_c$ in the scheme of $N_p{N}_n$ for the effective number of the valence particles (holes) in trans-lead nuclei has been systematically investigated. This quantity has been explored in the simplified parametrization of $N_p{N}_n$ as well as the multiplication $N_p{N}_{n}I$ of this product with the isospin asymmetry $I$. The calculations for $P_c$ are both performed in microscopic and model-dependent way. Within the microscopic approach, based on our previous work [Chin. Phys. C 47, 014101 (2023)], $P_c$ is calculated in cluster formation model combined with the exponential relationship of $P_c$ to the $\alpha$ decay preformation probability $P_{\alpha }$ when the mass number of the emitted cluster $A_c\le$ 28. While $A_c\ge$ 28, $P_c$ is obtained through the charge-number dependence of $P_c$ on the decay products proposed by Ren et al. [Phys. Rev. C 70, 034304 (2004)]. In the model-dependent approach, $P_c$ is extracted through the ratios from calculated cluster radioactivity half-lives in the framework of unified fission model proposed by Dong et al. [Eur. Phys. J. A 41, 197 (2009)] to experimental ones. Both of the results show $P_c$ in logarithmic form are linear to $N_p{N}_n$ as well as $N_p{N}_{n}I$. For comparison, the parent-mass-number dependence analytical formula as well as the model proposed by Wei and Zhang [Phys. Rev. C 96, 021601(R) (2017)] are also used. Furthermore, the preformation mechanic for cluster radioactivity has also been discussed.

• Received 21 April 2023
• Revised 18 June 2023
• Accepted 11 July 2023

DOI:https://doi.org/10.1103/PhysRevC.108.014325