Determination of the interatomic potential from elastic differential cross sections at fixed energy: Functional sensitivity analysis approach

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Abstract
Elastic differential cross sections in atomic crossed beam experiments contain detailed information about the underlying interatomic potentials. The functional sensitivity density of the cross sections with respect to the potential δσ(θ)/δV(R) reveals such information and has been implemented in an iterative inversion procedure, analogous to that of the Newton–Raphson technique. The stability of the inversion is achieved with the use of the regularization method of Tikhonov and Miller. It is shown that given a set of well resolved and noise-free differential cross section data within a limited angular range and given a reasonable starting reference potential, the recovered potential accurately resembles the desired one in the important region, i.e., the region to which the scattering data are sensitive. The region of importance depends upon the collision energy relative to the well depth of the potential under study; usually a higher collision energy penetrates deeper into the repulsive part of the potential and thus accordingly yields a more accurate potential in that part. The inversion procedure produces also a quality function indicating the well determined radial region. Moreover, the extracted potential is quite independent of the functional form of the reference potential in contrast to curve fitting approaches. As illustrations, the model inert gas systems He–Ne and Ne–Ar have been considered. For collision energies within an order of magnitude of the associated potential well depth, the attractive part of the potential can be determined to high precision provided that scattering data at small enough angles are available. On the other hand, the repulsive part of the potential must be scrutinized by high collision energy data.