High-precision inverse potentials for neutron-proton scattering using piece-wise smooth Morse functions

Abstract：

The aim of this study is to construct inverse potentials for various ℓ-channels of neutron-proton scattering using a piece-wise smooth Morse function as a reference. The phase equations for single-channel states and the coupled equations of multi-channel scattering are solved numerically using the 5 ^thorder Runge-kutta method. We employ a piece-wise smooth reference potential comprising three Morse functions as the initial input. Leveraging a machine learning-based genetic algorithm, we optimize the model parameters to minimize the mean-squared error between simulated and anticipated phase shifts. Our approach yields inverse potentials for both single and multi-channel scattering, achieving convergence to a mean-squared error $ \leq 10^{-3} $ . The resulting scattering lengths " $ a_0 $ " and effective ranges " r"for $ ^3S_1 $ and $ ^1S_0 $ states, expressed as [ $ a_0 $ , r], are found to be [5.445(5.424), 1.770(1.760)] $\rm fm$ and [–23.741(–23.749), 2.63(2.81)] $\rm fm$ , respectively; these values are in excellent agreement with experimental ones. Furthermore, the calculated total scattering cross-sections are highly consistent with their experimental counterparts, having a percentage error of less than 1%. This computational approach can be easily extended to obtain interaction potentials for charged particle scattering.