We propose a Quantization-Constrained Parameter Method (QCPM) for determining the Woods-Saxon (WS) potential depth $V_0$ and diffuseness $a$ in studies of $\alpha$ and cluster decay half-lives. In this approach, the parameters are derived analytically by imposing the Bohr-Sommerfeld (BS) quantization condition, thereby eliminating the need for parameter fitting. The calculated diffuseness values exhibit a strong dependence on nuclear shell structure. The reliability of the QCPM-derived WS potential is validated through optical model analyses of elastic $\alpha$ scattering data. The resulting $\alpha$ and cluster decay half-lives show good agreement with experimental values. Our results highlight the significant role of daughter-nucleus deformation in improving the consistency between theoretical predictions and experimental data. In addition, we introduce a modified Woods-Saxon (mWS) potential that effectively approximates the surface behavior of folding potentials incorporating the nuclear medium effect. This modification leads to closer agreement with the measured half-lives. Free of adjustable parameters such as $V_0$ and $a$, the QCPM enhances the predictive power of phenomenological potentials for $\alpha$ and cluster decay studies.

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• Alpha decay
• Nuclear decay
• Nuclear structure & decays