Systematic study of the α decay preformation factors of the nuclei around the Z = 82, N = 126 shell closures within the generalized liquid drop model

In this study, we systematically investigate the \begin{document}$\alpha$\end{document} decay preformation factors, \begin{document}$P_{\alpha}$\end{document} , and the \begin{document}$\alpha$\end{document} decay half-lives of 152 nuclei around Z = 82, N = 126 closed shells based on the generalized liquid drop model (GLDM) with \begin{document}$P_{\alpha}$\end{document} being extracted from the ratio of the calculated \begin{document}$\alpha$\end{document} decay half-life to the experimental one. The results show that there is a remarkable linear relationship between \begin{document}$P_{\alpha}$\end{document} and the product of valance protons (holes) \begin{document}$N_p$\end{document} and valance neutrons (holes) \begin{document}$N_n$\end{document} . At the same time, we extract the \begin{document}$\alpha$\end{document} decay preformation factor values of the even–even nuclei around the Z = 82, N = 126 closed shells from the study of Sun \begin{document}${et\ al.}$\end{document} [J. Phys. G: Nucl. Part. Phys., 45: 075106 (2018)], in which the \begin{document}$\alpha$\end{document} decay was calculated by two different microscopic formulas. We find that the \begin{document}$\alpha$\end{document} decay preformation factors are also related to \begin{document}$N_pN_n$\end{document} . Combining with our previous studies [Sun \begin{document}${et\ al.}$\end{document} , Phys. Rev. C, 94: 024338 (2016); Deng \begin{document}${et\ al.}$\end{document} , ibid. 96: 024318 (2017); Deng \begin{document}${et\ al.}$\end{document} , ibid. 97: 044322 (2018)] and that of Seif \begin{document}${et\ al.,}$\end{document} [Phys. Rev. C, 84: 064608 (2011)], we suspect that this phenomenon of linear relationship for the nuclei around the above closed shells is model-independent. This may be caused by the effect of the valence protons (holes) and valence neutrons (holes) around the shell closures. Finally, using the formula obtained by fitting the \begin{document}$\alpha$\end{document} decay preformation factor data calculated by the GLDM, we calculate the \begin{document}$\alpha$\end{document} decay half-lives of these nuclei. The calculated results agree with the experimental data well.