Exact Series Solution of Classical Deflection Angle for Repulsive Inverse Power Potential

Abstract

An exact series solution of the classical deflection angle is developed for the potential V=C/r8 (C > 0). The series converges at all impact parameters. In addition, upper and lower bounds for θ are constructed from the series as an aid to numerical computation. Finally, the classical differential cross section is evaluated from the deflection angle.

A general relationship between the scattering angle and the impact parameter in collisions driven by repulsive potentials

“…A general analytical expression for the integral (2) cannot be obtained. Various studies of the integral have yielded a number of series expansions (having different accuracies and different radii of convergence) and, in the case of repulsive interactions, approximate analytical formulae for the angle g in small-angle scatterings (Kihara 1953, Mott-Smith 1960, Lehmann and Leibfried 1963, Leibfried and Plesser 1965, Smith et al 1966, Gislason 1972, Pauly 1979, Kunc 1993.…”