Constructing inverse potentials for resonant states of <i>α</i>-³H and <i>α</i>-³He scattering directly from phase shifts

Abstract In this paper, the inverse potentials for the resonant f states of α - 3 H and α - 3 He are constructed using the phase function method by utilizing an reference potential approach. Central Idea: A combination of three Morse functions are chosen over three regions of interaction and are joined smoothly at the intervening boundaries 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. Methodology: The phase equation is solved using the Runge-Kutta 5 th 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. Results: Our approach successfully constructed the inverse potentials for the resonant f states of the α - 3 H and α - 3 He systems, achieving convergence with a minimized mean square error. The resonance energies and widths for the α - 3 H system for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>5</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>7</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>5</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>7</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> states of the α - 3 He 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. Conclusion: Our fitting procedure of the available experimental data 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.