Classical Scattering by Some Important Repulsive Potentials

Abstract

Classical scattering by repulsive potentials of the form V = Br−n exp (−r/a) is analyzed in terms of two energy-dependent parameters, the distance of closest approach in head-on collision c, and a slope parameter γ = −(d ln V/d ln r)r=c. Use of these brings out important relationships among values of cross sections for various potentials which are in the literature and also facilitates the derivation of certain asymptotic analytic relations, the most interesting applying when γ is large, and others when γ is near unity. For large γ, the differential cross section σ(θ) is given by 4σ(θ)/c2 = 1 + Q1/γ + Q2/γ2, where Q1 is a simple function of θ only, while Q2 is a more complicated function which depends also on n/γ. This asymptotic expression for σ(θ) and transport cross sections σl calculated from it are compared with some of the available data. For the exponential potential, the asymptotic analytic behavior of collision integrals I(l, s) for large values of α = ln (B/kT) is determined and compared with numerical results of Monchick. For the exponentially screened Coulomb potential, a simple expression describing the dependence of σ1/πc2 on γ is selected and used in treating the energy loss of a fast heavy atom being stopped by elastic collisions in a gas of light atoms. This discussion makes clearer the basis and scope of the Bohr-Nielsen range-energy equation.