Low-energy scattering parameters: A theoretical derivation of the effective range and scattering length for arbitrary angular momentum

TL;DR

This paper derives general formulas for low-energy scattering parameters like scattering length and effective range for any angular momentum, applicable to various model potentials and near Feshbach resonances, aiding understanding of scattering theory.

Contribution

It provides a unified theoretical framework for calculating scattering parameters for arbitrary angular momentum and different potentials, including resonance behavior.

Findings

Formulas for scattering length and effective range for any angular momentum.

Application of formulas to hard-sphere, soft-sphere, spherical well, and well-barrier potentials.

Analysis of scattering parameter behavior near Feshbach resonances.

Abstract

The most important parameters in the study of low-energy scattering are the s-wave and p-wave scattering lengths and the s-wave effective range. We solve the scattering problem and find two useful formulas for the scattering length and the effective range for any angular momentum, as long as the Wigner threshold law holds. Using that formalism, we obtain a set of useful formulas for the angular-momentum scattering parameters of four different model potentials: hard-sphere, soft-sphere, spherical well, and well-barrier potentials. The behavior of the scattering parameters close to Feshbach resonances is also analyzed. Our derivations can be useful as hands-on activities for learning scattering theory.