Analytic scattering parameters at low energies

TL;DR

This paper derives and verifies analytic formulas for low-energy scattering parameters like scattering length and effective range across various potentials, aiding research in physics and chemistry.

Contribution

It introduces new analytic expressions and a calculation technique for scattering parameters for inverse-power, Woods-Saxon, and Yukawa potentials.

Findings

Analytic formulas for scattering length and effective range are derived.

Verification shows good agreement with numerical calculations.

Results are applicable in multiple physics and chemistry fields.

Abstract

The scattering of two and more particles at low energies is described by the so called effective-range expansion. The leading terms of this expansion are the scattering length and effective range. The analytic expressions for both of the aforementioned scattering parameters are presented for the inverse-power potential and the Woods-Saxon potential. A technique for calculating the approximate scattering parameters is proposed. Approximate analytic formulas representing the scattering length and effective range are obtained for the Yukawa potential. The corresponding figures demonstrate a few interesting features of the effective range. All analytic formulas, both exact and approximate, were verified by comparing with the corresponding results obtained by direct numerical calculations. Wolfram Mathematica is heavily used. The presented results can be used with advantage in the fields of nuclear physics, atomic and molecular physics, quantum chemistry and many others.