In the present work, we systematically study the $\alpha$ decay preformation factors $P_{\alpha }$ within the cluster-formation model and $\alpha$ decay half-lives by the proximity potential 1977 formalism for nuclei around $Z=82,N=126$ closed shells. The calculations show that the realistic $P_{\alpha }$ is linearly dependent on the product of valance protons (holes) and valance neutrons (holes) $N_p{N}_n$. It is consistent with our previous works [Sun et al., Phys. Rev. C 94, 024338 (2016); Deng et al., Phys. Rev. C 96, 024318 (2017)], in which $P_{\alpha }$ are model dependent and extracted from the ratios of calculated $\alpha$ half-lives to experimental data. Combining with our previous works, we confirm that the valance proton-neutron interaction plays a key role in the $\alpha$ preformation for nuclei around $Z=82,N=126$ shell closures whether the $P_{\alpha }$ is model dependent or microcosmic. In addition, our calculated $\alpha$ decay half-lives by using the proximity potential 1977 formalism taking $P_{\alpha }$ evaluated by the cluster-formation model can well reproduce the experimental data and significantly reduce the errors.

• Received 2 February 2018

DOI:https://doi.org/10.1103/PhysRevC.97.044322