Ab-initio Approach for Constructing Inverse Potentials for Resonant States of α-3H and α-3He Scattering

Abstract: In this paper, the inverse potentials for the resonant f states of {\alpha}-3H and {\alpha}-3He are constructed using the phase function method by utilizing an ab-initio approach. A combination of three Morse functions are joined smoothly to prepare the reference potential. While the regular Morse function captures the nuclear and Coulomb interactions at short and medium ranges, an inverse Morse function is chosen to obtain the Coulomb barrier that arises because of the long-range Coulomb interaction. This reference potential is representative of a large family of curves consisting of eight distinct model parameters and two intermediate points that define the boundaries that exist between the three regions. The phase equation is solved using the Runge-Kutta 5th order method for the input reference potential to obtain the scattering phase shifts at various center of mass energies. The model parameters are then adjusted using the genetic algorithm in an iterative fashion to minimize the mean square error between the simulated and expected phase shift values. Our approach successfully constructed the inverse potentials for the resonant f states of the {\alpha}-3H and {\alpha}-3He systems, achieving convergence with a minimized mean square error. The resonance energies and widths for the {\alpha}-3H system for the f-5/2 and f-7/2 states are determined to be [4.19 (4.14), 1.225 (0.918)] MeV and [2.20 (2.18), 0.099 (0.069)] MeV, respectively. For the f-5/2 and f-7/2 states of the {\alpha}-3He system, the resonance energies and widths are [5.03 (5.14), 1.6 (1.2)] MeV and [2.99 (2.98), 0.182(0.175)] MeV, respectively. Our ab-initio approach to solve the phase equation utilizing a combination of smoothly joined Morse functions effectively captures both short-range nuclear and long-range Coulomb interactions, providing an accurate model for nuclear scattering involving charged particles.