Approximate Analytic Approach to the Classical Scattering Problem

Abstract

An approximate analytic approach to the problem of determining differential scattering cross sections for classical central-field repulsive forces is described. It was shown that the impact parameter, b, could be approximated by b = R cos(B/2), where R is approximately the distance of closest approach and theta is the scattering angle in the center-of-mass system. A simple approximation gave the potential energy of interaction between two atoms as V (R) = 2E sin( theta /2), where E is the energy in the centerof-mass system. Simple analytic expressions for the differential scattering cross section, SIGMA , were derived from the above two relationships for three special cases of a two- parameter screened Coulomb potential energy, V (R)= Z/sub 1/Z/sub 2/e/sup 2/A expl--pAR) STA1 -- exp(-AR)!/sup -1/, where Z/sub 1/e is the charge on the ith atom, A/sup -1/ is a screening radius, and p is an adjustable parameter that is restricted to 1/2, 1, and 2 in this paper. A new and improved method for calculating SIGMA exactly is also discussed and was used to compute the exact behavior of SIGMA for p = 1. A table is presented which allows one to compare the exact and approximate SIGMA 's formore » p = 1 over a wide range of energy and scattering angles. The agreement is particularly good for large energy transfer. (auth)« less