Constructing inverse scattering potentials for charged particles using a reference potential approach

An accurate way to incorporate long-range Coulomb interaction alongside short-range nuclear interaction has been a challenge for theoretical physicists. In this paper, we propose a methodology based on the reference potential approach for constructing inverse potentials for charged particle scattering. The central idea is to obtain the inverse potential directly from the expected scattering phase shifts by comparing them with those obtained by solving the phase equation for a chosen reference potential. The design of the reference potential is key to incorporating the Coulomb interaction successfully. Here, a combination of two smoothly joined Morse functions, one regular followed by an inverted one, is considered. While the former takes care of short-range nuclear and Coulomb interactions, the latter accounts for expected barrier height due to the long-range Coulomb part that dominates once nuclear interaction subsides. The final step is to incorporate the phase equation within an iterative loop of an optimization algorithm to obtain the model parameters for the reference potential by minimizing the mean absolute percentage error between the obtained and expected scattering phase shifts. We have applied the methodology to the $\alpha -\alpha$ system and constructed the inverse potentials for its $S, D$, and $G$ states with mean absolute percentage errors of 0.9, 0.5, and 0.4 respectively. Their respective resonances (experimental), in MeV, are found to be at 0.1240 (0.0918), 2.95 (3.03), and 11.89 (11.35). One can conclude that the reference potential approach using a combination of smoothly joined Morse functions is successful in accurately accounting simultaneously for the short-range nuclear and the long-range Coulomb interactions between charged particles in nuclear scattering studies.