Asymptotic Expansion for High-Energy Potential Scattering

Abstract

An asymptotic expansion of a previously derived approximate expression for high-energy potential scattering is obtained. This is used to compute numerical values for the large-angle differential scattering cross section of a Schrödinger particle scattered by a spherically symmetric parabolic potential well and of a relativistic electron scattered by a uniformly charged sphere. In both cases the parameters are such that the Born approximation cannot be used. These values are compared with the results of an exact partial wave calculation and (in the parabolic-well case) with the results of numerical integration of the full approximate expression. The object of this work is to gauge the reliability both of the full approximate expression, and of its asymptotic expansion. It is found that for both the Schrödinger equation and the Dirac equation the locations of the maxima and the minima of the scattering cross sections are reproduced with good accuracy. The magnitudes of the minima, however, are not given reliably and although the general shapes of the curves are correct, the magnitudes of the maxima differ from the correct values by 10 to 50%, depending on the value of $\mathrm{kR}$.