Forward and inverse functional variations in elastic scattering

This paper considers the response of various types of elastic collision cross sections to functional variations in the intermolecular potential. The following cross sections are considered differential, total, effective diffusion, and effective viscosity. A very simple expression results for the diffusion and viscosity cross sections at high energy relating the variations to the classical deflection function. Attention is first given to the forward sensitivity densities δσ(E)/δV(R) [i.e., the functional derivative of cross sections σ(E) with respect to the potential surface F(R)]. In addition inverse sensitivity densities δV(R)/δσ(E) are obtained. These inverse sensitivity densities are of interest since they are the exact solution to the infinitesimal inverse scattering problem. Although the inverse densities do not in themselves form an inversion algorithm, they do give a quantitative measure of the importance of performing particular measurements for the ultimate purpose of inversion. In addition, the degree to which different regions of a potential surface are correlated to a given set of cross sections are calculated by means of the densities {δV(R)/ δV(R′)}. The overall numerical results contain elements which are physically intuitive as well as perplexing. This latter interesting and unexpected behavior is a direct result of allowing for unconstrained cross section ↔ potential response, as well as the presence of quantum interference processes. The present focus on elastic scattering is simply for the purpose of illustration of the functional variation technique which has broad applicability in all types of scattering processes.