A new asymmetric parabolic effective fusion barrier model for heavy ion fusion is developed.

In a one-dimensional quantal solution of Schroedinger equation, the general expressions for reflection and transmission coefficients are derived for a potential constituting n number of rectangular wells and barriers. These expressions are readily used for the estimation of eigenvalues of a smooth potential which is simulated by a multi-step potential. The applicability of this method is demonstrated with success in potentials with different forms including the most versatile Ginocchio potential where the widely used numerical method like Runge–Kutta integration algorithm fails to yield the result. Accurate evaluation of eigenvalues free from numerical problem for any form of potentials, whether analytically solvable or not, is the highlight of the present multi-step approximation method in the theory of potential scattering.

A general decay formula for the emission of charged particles from metastable nuclei is developed based on the basic phenomenon of resonances occurring in quantum scattering process under Coulomb-nuclear potential. It relates the half-lives of radioactive decays with the 𝑄 values of the outgoing elements with masses and charges of the nuclei involved in the decay. The relation is found to be a generalization of the Geiger–Nuttall law in 𝛼 radioactivity and explains well all the known emissions of charged particles including clusters, alpha and proton.

A nucleus, initially in a bound state akin to a Dirac δ well potential, under severe action of reflection from a repulsive electrostatic interaction at very small separation between the newly born α cluster and the daughter nucleus goes to the quasistationary state from which the decay of α sets in. The energy derivative of the phase-shift of the $S$-matrix of transition scattering from an isolated quasibound state to a scattering state is related to the width (${\Gamma}$)of the resonance and hence the half-life T$_{1/2}$= ln 2$\bar{h}$/${\Gamma}$of α-decay. Using exact solutions of the delta well driven Coulomb potential, we derive a compact expression for log10 T$_{1/2}$ in terms of Q-value and the mass and charge numbers of the α emitter with a radius parameter specifying the sum of the charge radii of α and daughter nuclei or the radius of the parent emitter. The computed results of half-lives successfully explain the experimental values of α-decay half-lives ranging from 10$^{−7}$s to 10$^{25}$s in many light, heavy and super heavy radioactive nuclei with charge number Z = 52–120 and mass number A = 106–299. Showing uniformity in the applications of formulation to the decay of positive ions, namely α and cluster ions, the universal nature of the decay rule is established.