Analytical expression for theα-decay half-life and understanding the data including very long life-times and superheavy nuclei

Abstract

An analytically solvable composite potential that can closely reproduce the combined potential of an $\ensuremath{\alpha}+\mathrm{nucleus}$ system consisting of attractive nuclear and repulsive electrostatic potentials is developed. The exact $s$-wave solution of the Schr\"odinger equation with this potential in the interior region and the outside Coulomb wave function are used to give a heuristic expression for the width or half-life of the quasibound state at the accurately determined resonance energy, called the $Q$ value of the decaying system. By using the fact that for a relatively low resonance energy, the quasibound state wave function is quite similar to the bound state wave function where the amplitude of the wave function in the interaction region is very large as compared to the amplitude outside, the resonance energy could easily be calculated from the variation of relative probability densities of inside and outside waves as a function of energy. By considering recent $\ensuremath{\alpha}$-decay systems, the applicability of the model is demonstrated with excellent explanations being found for the experimental data of $Q$ values and half-lives of a vast range of masses including superheavy nuclei and nuclei with very long lifetimes (of order ${10}^{22}$ s). Throughout the application, by simply varying the value of a single potential parameter describing the flatness of the barrier, we obtain successful results in cases with as many as 70 pairs of $\ensuremath{\alpha}+\mathrm{daughter}$ nucleus systems.