The potential for $\alpha$ induced nuclear scattering, reaction and decay, and a resonance-pole-decay model with exact explicit analytical solutions

The decay of $\alpha$ particle from a nucleus is viewed as a quantum resonance state of a two-body scattering process of the $\alpha$+daughter nucleus pair governed by a novel nucleus-nucleus potential in squared Woods-Saxon form. By the application of the rigorous optical model (OM) potential scattering (S-matrix) theory the genuineness of the potential for the system is established by giving good explanation of the elastic scattering and reaction cross sections data of the $\alpha$+nucleus pair. From the pole position in the complex momentum (k) plane of the S-matrix defined above, the energy and width of the resonance state akin to the decaying state of emission of $\alpha$ particle are extracted and from this width, the result of $\alpha$-decay half-life is derived to account for the experimental result of half-life in the cases of large number of $\alpha$-emitters including heavy and super-heavy nuclei. The S-matrix of the full OM calculation above is replaced by an analytical function expressed in terms of exact Schr\"{o}dinger solutions of a global potential that closely represents the Coulomb-nuclear interaction in the interior and the pure Coulomb wave functions outside, and the resonant poles of this S-matrix in the complex momentum plane are used to give satisfactory results of decay half-lives of $\alpha$ coming out from varieties of nuclei.