A Practical Method to Solve Cut-off Coulomb Problems in the Momentum Space -- Application to the Lippmann-Schwinger Resonating-Group Method and the pd Elastic Scattering --

TL;DR

This paper introduces a practical momentum-space method for solving cut-off Coulomb problems in two-cluster systems, enabling accurate phase shift calculations and applications to alpha-alpha and proton-deuteron scattering.

Contribution

The paper presents a new approach to handle cut-off Coulomb interactions in momentum space, improving the calculation of phase shifts in two-cluster nuclear systems.

Findings

Complete solutions achieved with finite cut-off radius 

Nuclear phase shifts stable within appropriate  range

Successful application to  alpha-alpha scattering and pd elastic scattering

Abstract

A practical method to solve cut-off Coulomb problems of two-cluster systems in the momentum space is given. When a sharply cut-off Coulomb force with a cut-off radius $\rho$ is introduced at the level of constituent particles, two-cluster direct potential of the Coulomb force becomes in general a local screened Coulomb potential. The asymptotic Hamiltonian yields two types of asymptotic waves; one is an approximate Coulomb wave with $\rho$ in the middle-range region, and the other a free (no-Coulomb) wave in the longest-range region. The constant Wronskians of this Hamiltonian can be calculated in either region. We can evaluate the Coulomb-modified nuclear phase shifts for the screened Coulomb problem, using the matching condition proposed by Vincent and Phatak for the sharply cut-off Coulomb problem. We apply this method first to an exactly solvable model of the $\alpha \alpha$ scattering with the Ali-Bodmer potential and confirm that a complete solution is obtained with a finite $\rho$. The stability of nuclear phase shifts with respect to the change of $\rho$ in some appropriate range is demonstrated in the $\alpha \alpha$ resonating-group method (RGM) using the Minnesota three-range force. An application to the pd elastic scattering is also discussed.