Approximate Analytic Approach to the Classical Scattering Problem

An approximate analytic approach to the problem of determining differential scattering cross sections for classical centralfield repulsive forces is described. It is shown that the impact parameter, $b$, can be approximated by $b=Rcos\left(\frac{\theta }{2}\right)$, where $R$ is approximately the distance of closest approach and $\theta$ is the scattering angle in the center-of-mass system. A simple approximation gives the potential energy of interaction between two atoms as $V\left(R\right)=2Esin\left(\frac{\theta }{2}\right)$, where $E$ is the energy in the center-of-mass system. Simple analytic expressions for the differential scattering cross section, $\sigma$, are derived from the above two relationships for three special cases of a two-parameter screened Coulomb potential energy, $V\left(R\right)=Z_1{Z}_2{e}^{2}A\exp \mathrm{}(-pAR)\mathrm{}{[1-\exp \mathrm{}(-AR\left)\right]}^{-1\mathrm{}},$ where $Z_{i}e$ is the charge on the $i\mathrm{th}$ atom, $A^{-1\mathrm{}}$ is a screening radius, and $p$ is an adjustable parameter which is restricted to ½, 1, and 2 in this paper.