Cluster radioactivity preformation probability of trans-lead nuclei in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:msub><mml:mi>N</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math> scheme

Abstract

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 $\ensuremath{\alpha}$ decay preformation probability ${P}_{\ensuremath{\alpha}}$ when the mass number of the emitted cluster ${A}_{c}\ensuremath{\le}$ 28. While ${A}_{c}\ensuremath{\ge}$ 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.

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