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In present work, the unfavored $\mathcal{\alpha}$ decay half-lives and$\mathcal{\alpha}$ preformation probabilities of closed shell nuclei related toground and isomeric states around $Z=82$, $N=82$ and 126 shell closures areinvestigated by adopting two-potential approach from the perspective of valencenucleon (hole) and isospin asymmetry of the parent nucleus. The resultsindicate that $\mathcal{\alpha}$ preformation probability is linear dependenceon $N_p N_n$ or $N_p N_n I$, the same as the case of favored $\mathcal{\alpha}$decay in our previous work [X.-D. Sun \textit{et al.}, Phys. Rev. C 94, 024338(2016)]. $N_p$, $N_n$, and $I$ represent the number of valence protons (holes),the number of valence neutrons (holes), and the isospin of the parent nucleus,respectively. Fitting the $\mathrm{\alpha}$ preformation probabilities dataextracted from the differences between experimental data and calculatedhalf-lives without a shell correction, we give two linear formulas of the$\mathcal{\alpha}$ preformation probabilities and the values of correspondingparameters. Based on the formulas and corresponding parameters, we calculatethe $\mathrm{\alpha}$ decay half-lives for those nuclei. The calculated resultscan well reproduce the experimental data.

Systematic study of unfavored $\mathcal{\alpha}$ decay half-lives of closed shell nuclei related to ground and isomeric states