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Harmonic Oscillator Representation of Scattering Theory in the Presence of Coulomb Potential

Ustin M Yanikov, Vasily A Kulikov, Andrey M Shirokov

TL;DR

The paper tackles incorporating the long-range Coulomb interaction into the Harmonic Oscillator Representation of Scattering Equations (HORSE) for charged-particle scattering. It introduces a diagonal Coulomb contribution $V_{nn}^{ad,l}$ in the asymptotic three-term recurrence relation and uses a truncated, smoothly truncated nuclear potential alongside a sharply truncated Coulomb term to compute scattering observables. By solving for asymptotic coefficients $S_{nl}(k)$ and $C_{nl}(k)$ and employing a Green-function correction, it obtains phase shifts with good convergence, validated against exact Numerov results and prior methods. The results suggest this approach is accurate and efficient for ab initio, many-body, and multichannel scattering problems.

Abstract

Considering the problem of scattering of charged particles, we introduce a new approach of taking the Coulomb interaction into account within the HORSE formalism. Compared to the conventional HORSE approach for uncharged particles, we add a diagonal Coulomb term to} the three-term recurrent relation for expansion coefficients in the asymptotic region. The method simplifies calculations and demonstrates a good agreement with numerical solution.

Harmonic Oscillator Representation of Scattering Theory in the Presence of Coulomb Potential

TL;DR

The paper tackles incorporating the long-range Coulomb interaction into the Harmonic Oscillator Representation of Scattering Equations (HORSE) for charged-particle scattering. It introduces a diagonal Coulomb contribution in the asymptotic three-term recurrence relation and uses a truncated, smoothly truncated nuclear potential alongside a sharply truncated Coulomb term to compute scattering observables. By solving for asymptotic coefficients and and employing a Green-function correction, it obtains phase shifts with good convergence, validated against exact Numerov results and prior methods. The results suggest this approach is accurate and efficient for ab initio, many-body, and multichannel scattering problems.

Abstract

Considering the problem of scattering of charged particles, we introduce a new approach of taking the Coulomb interaction into account within the HORSE formalism. Compared to the conventional HORSE approach for uncharged particles, we add a diagonal Coulomb term to} the three-term recurrent relation for expansion coefficients in the asymptotic region. The method simplifies calculations and demonstrates a good agreement with numerical solution.

Paper Structure

This paper contains 6 sections, 21 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Coulomb interaction in HORSE formalism
  3. Analysis of TRR for coefficients $S_{nl}(k)$
  4. Phase shifts
  5. Results
  6. Conclusion

Figures (2)

  • Figure 1: Dependence of $S_{nl}(k)$ on $n$ in $p$-wave $p{ -} \alpha$ scattering ($\mu=626.4$ MeV) at $E=20$ MeV and $\hbar\omega=20$ MeV. Solid line: calculated by the integral \ref{['sn_int']}; dashed line: obtained by TRR \ref{['TRR']} starting from the asymptotic coefficients $S_{n_s+2,\,l}(k)$ and $S_{n_s+1,\,l}(k)$ with $n_s=40$; crosses: asymptotic values \ref{['sn_as']}.
  • Figure 2: Phase shift $\delta_l(k)$ dependence on relative motion energy $E$ for $p{-}{\rm ^{15}N}$ scattering ($\mu=881.2$ MeV) in the $s$ wave obtained by different methods. Solid line: numerical integration of Schrödinger equation by Numerov method; dots: method suggested in Ref. OKHRIMENKO1984121 with $M=70$; dashed line: method suggested in Ref. Bang1999PmatrixAJ with Coulomb interaction cut at $b=7.0$ fm; oblique crosses: shifts obtained by the approach proposed here with $n_s=200$. In all calculations the nuclear interaction is smoothly truncated at $N=10$ and $\hbar\omega=18$ MeV.