A Computational Phase Function Approach for Obtaining $αα$ Wavefunctions

TL;DR

This paper addresses the problem of obtaining explicit radial scattering wavefunctions for the $\alpha\alpha$ system from phase-shift data. It applies the Phase Function Method (PFM) with a Morse short-range potential and a finite-size Coulomb term to derive first-order equations for the running phase $\delta_\ell(k,r)$ and radial amplitude $A_\ell(r)$, enabling direct construction of $u_\ell(k,r)$ for $\ell=0,2,4$ without solving the second-order Schrödinger equation. The results show well-behaved wavefunctions that asymptotically match physical expectations and agree with the resonating-group method of Hiura et al., validating the approach. This establishes PFM as a robust, efficient tool for cluster scattering problems and opens avenues for applying the method to more complex nuclear systems with inverse-data constraints.

Abstract

In this work, the phase function method (PFM) is applied for the first time to explicitly construct scattering wavefunctions for the $αα$ system using interaction parameters optimized from earlier phase shift analysis. While previous investigations employing PFM and related approaches have primarily focused on the reproduction of scattering phase shifts or cross sections, the present study advances the method to directly reconstruct physically meaningful radial scattering wavefunctions for the $\ell = 0$, 2, and 4 partial waves without solving the Schrödinger equation. The computed wavefunctions exhibit well-behaved near-origin and asymptotic characteristics and show excellent agreement with the established resonating-group method results of Hiura \textit{et al.}. This agreement confirms the numerical robustness and accuracy of the proposed framework. The present results establish PFM as an efficient, unified, and computationally attractive tool for scattering wavefunction reconstruction in cluster--cluster systems, opening new possibilities for its application to more complex nuclear scattering problems.

Figures (3)

• Figure 1: Detailed flowchart to obtain scattering phase shift $\delta(E)$, amplitude $A(r)$ and wavefunction $u(r)$ for $\ell=0$, 2 & 4 waves.
• Figure 2: Wavefunction for s, p & d-waves at resonance energies in comparison with results of Hiura et. al.
• Figure 3: Wavefunction for s-wave at 19.45 MeV in comparison with results of Yoshiharu et. al.