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In the present work, the cluster radioactivity preformation probability Pc in the scheme of NpNn 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 NpNn as well as the multiplication NpNnI of this product with the isospin asymmetry I. The calculations for Pc are both performed in microscopic and model-dependent way. Within the microscopic approach, based on our previous work [Chin. Phys. C 47,014101 (2023)], Pc is calculated in cluster formation model (CFM) combined with the exponential relationship of Pc to the alpha decay preformation probability P alpha when the mass number of the emitted cluster Ac less than 28. While Ac greater than 28, Pc is obtained through the charge-number dependence of Pc on the decay products proposed by Ren et al. [Phys. Rev. C 70,034304 (2004)]. In the model-dependent approach, Pc is extracted through the ratios from calculated cluster radioactivity half-lives in the framework of unified fission model (UFM) proposed by Dong et al. [Eur. Phys. J. A 41,197 (2009)] to experimental ones. Both of the results show Pc in logarithmic form are linear to NpNn as well as NpNnI. For comparison, the parent-mass-number dependence analytical formula as well as the model proposed by K. Wei and H. F. Zhang [Phys. Rev. C 96,021601(R)(2017)] are also used. Furthermore, the preformation mechanic for cluster radioactivity has also been discussed.

In the present study, proton emission half-lives have been investigated for the deformed proton emitters with $53\leq Z \leq 83$ in the deformed Gamow-like model, where the deformation effect has been included in the Coulomb potential. The experimental half-lives of proton emitters can be reproduced within a factor of 3.45. For comparison, other results from the universal decay law and the new Geiger-Nuttall law are presented as well. Furthermore, the relevance of the half-lives to the angular momentum $l$ for $^{117}$La, $^{121}$Pr, $^{135}$Tb and $^{141}$Ho has been analyzed, and corresponding possible values of $l$ has been put forward: $l=$3, 3, 4, 4.

In the present work, we systematically study the spectroscopic factor of proton radioactivity ($S_p$) with $A>100$ using the deformed two-potential approach (D-TPA). It is found that there is a link between the quadrupole deformation parameter of proton emitter and $S_p$. Based on this result, we propose a simple analytic formula for estimating the spectroscopic factor of proton radioactivity. With the help of this formula, the calculated half-lives of proton radioactivity can reproduce the experimental data successfully within a factor of 2.77. Furthermore, we extend the D-TPA with this formula for evaluating the spectroscopic factor to predict the proton radioactivity half-lives of 12 proton radioactivity candidates whose radioactivity is energetically allowed or observed but not yet quantified in NUBASE2020. For comparison, the universal decay law for proton radioactivity (UDLP) and the new Geiger-Nuttall law (NG-N) are also used. It turns out that all of the predicted results are basically consistent with each other.