Numerical construction of parametrized potentials for nucleon-alpha inverse scattering using variational Monte Carlo and phase function method

Neutron and proton scattering with stable alpha particles can be represented as a two-particle system. In this work, we present a numerical algorithm developed to obtain parametric potentials of the inverse scattering problem by solving the Riccati-type non-linear differential equation, known as the phase equation, alongside the Variational Monte Carlo method. Local interaction potentials for the resonant $p_{1/2}$ and $p_{3/2}$ channels are constructed using the Morse potential, derived from scattering phase shift (SPS) data for neutron-alpha and proton-alpha scattering. The non-local interaction potential in the $p\alpha$ system is modeled using a screened Coulomb potential. The phase equation for the $p_{1/2}$ and $p_{3/2}$ channels is solved numerically using the fifth-order Runge–Kutta method. The parameters of the Morse function are optimized by minimizing the mean absolute percentage error (MAPE) between the calculated and experimental phase shifts, through an iterative process involving Monte Carlo sampling and variational techniques. This ensures the accurate reproduction of experimental SPS data. Resonance energies, obtained (experimental) from partial cross-section plots for the $p_{1/2}$ and $p_{3/2}$ states in the $n\alpha$ system, are 4.10 (4 $\pm$ 1) MeV and 0.93 (0.89) MeV, respectively, while for the system, they are 5.31 (5 $\pm$ 2) MeV and 1.97 (1.96) MeV. The total cross-section for the $n\alpha$ system is in agreement with values found in the literature. This study demonstrates the effectiveness of the proposed numerical algorithm for constructing parametric potentials and shows its ability to accurately reproduce phase shifts and cross-sections in line with experimental data.