Analytic scattering parameters at low energies

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.