The effects of the spin-orbit coupling strength and pairing correlation on the $\alpha$ decay of ${}^{210}\mathrm{Po}$ are explored in the framework of the spherical Skyrme-Hartree-Fock Bogoliubov theory and mean-field cluster potential approach with the SLy4 interaction. It is shown that the localization of the $\alpha$ cluster is enhanced by the increasing spin-obit coupling strength. However, the localization of the $\alpha$ cluster is getting weaker and weaker with the growing pairing field. Therefore, the roles of the spin-orbit coupling strength and pairing correlation in the $\alpha$-cluster preformation are competitive. Besides, it is found that the effect on the half-life from the $\alpha$-decay energy is much stronger than that from the $\alpha$-cluster localization although they all are influenced by the spin-obit coupling strength and pairing field.

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• Alpha decay
• Nuclear forces
• 190 ≤ A ≤ 219
• Nuclear density functional theory