An estimate of Alpha decay half-life from the poles of S-matrix of an exactly solvable potential

Abstract

We develop a versatile and analytically solvable potential which fairly reproduces the combined potential of an $\alpha+$nucleus system resulting from both the attractive nuclear and repulsive electrostatic potentials. The potential is expressed in terms of the radial position, mass and proton number of the $\alpha$-particle and the daughter nucleus and certain parameters governing the depth, height and steepness of the barrier. The potential generated is typically a pocket near the origin and a barrier adjacent to it. The Schr\"odinger equation with the above mentioned potential is then solved for the wave function. This potential produces discrete positive energy quasibound state known as resonance state. By matching the wave function and its derivative with the regular Coulomb wave function $F_0$ and irregular Coulomb wave function $G_0$, we obtain the S-matrix analytically. The resonance is obtained from the pole in complex energy plane which gives the width of the corresponding time of decay through the imaginary part of the energy pole position. We make a comparative study of the measured half-lives of various nuclei with the calculated half-lives. The calculated values of half-lives closely match with the corresponding experimental results.