A microscopic study of $\alpha$-decay half-lives is performed to examine the influence of the nuclear medium effect on the $\alpha$-daughter interaction. The conventional double-folding potential is improved by replacing the fixed free-$\alpha$ density with a density-dependent $\alpha$-cluster distribution that varies with the local density of the daughter nucleus. This modification accounts for the change of the $\alpha$-cluster size induced by Pauli blocking and the surrounding nuclear mean field during the decay process. Because $\alpha$ decay is highly sensitive to the surface region of the interaction potential, the medium-induced modification of the $\alpha$ density has a direct impact on the barrier penetrability and calculated half-lives. The calculations are carried out for a wide range of 256 nuclei, with special attention to isotopic chains around shell closures in order to clarify the role of magicity in $\alpha$-decay systematics. Nuclear deformation is included through deformed daughter-density distributions, and its effect on the orientation-dependent potential and half-life is investigated. The influence of the exchange part of the effective nucleon-nucleon interaction is also analyzed by comparing zero-range and finite-range exchange treatments of the M3Y-Paris interaction. The results show that the nuclear medium effect improves the description of experimental half-lives, while deformation and finite-range exchange provide additional refinements, especially for open-shell and deformed nuclei. The observed behavior around magic numbers confirms the strong connection between shell structure, $\alpha$-cluster formation, and decay probability. These findings indicate that the combined treatment of medium effects, deformation, and exchange interaction is important for a reliable microscopic description of $\alpha$ decay.

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• Alpha decay
• Lifetimes & widths
• Nuclear matter
• Nuclear reactions
• Nuclear structure & decays