We investigate the formation of a pocket structure in the $\alpha$-nucleus potential based on the dynamic double-folding potential model. We improve the model with the density-dependent Migdal nucleon-nucleon (NN) interaction and combine it with the nuclear medium effect. The improved potential naturally generates a pocket structure at the nuclear surface, which physically agrees with the characteristics suggested by the microscopic many-body calculations. The result reveals that the formation of the pocket structure is due to the strong Pauli repulsion caused by the variation of the NN interaction when the $\alpha$ cluster and the daughter nucleus have large density overlaps. The existence of the nuclear medium effect is essential to the physical self-consistency of the pocket position. These findings highlight the importance of the medium effect on the NN interaction and provide solid theoretical support for the understanding that $\alpha$ clustering occurs at the surface of heavy nuclei.

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• Alpha decay
• Lifetimes & widths
• Nuclear forces
• Nuclear structure & decays
• Cluster models