Solution of an inverse problem in scattering theory

Abstract

Hoyt (1939) and Firsov devised methods in classical mechanics to deduce a central scattering potential from a measured differential effective cross-section in the nonrelativistic case. These methods are here extended to the relativistic case. A detailed analysis of the applicability of all methods has been undertaken for potentials of the form V(r) = ±αr^−k for sufficiently high energies of the colliding particles. It is found that Hoyt's method is inapplicable in the relativistic case only when the potential represents attraction. A relatively simple method is given for deducing the parameters α and k for a monotonic attraction potential that can be approximated by V(r) = −αr^−k. The method is based on simple arguments concerning the dimensions of the cross-section. It is sufficient to know only two values of the integral cross-section in the same range of angles but at different energies to determine the parameters.