Multiple Scattering in the Gaussian Approximation: Systematic Improvement of the Small-Angle Treatment

In the theory of the elastic multiple scattering of a charged particle in a plane, it is usually assumed that all angles involved are small. Working in the Gaussian approximation, we use a power-series expansion to effect a systematic improvement of the results of the small-angle approximation, these being the zero-order approximation to our expansion. A method of integration in function space is used to determine the joint probability of lateral and angular displacements of the scattered particle, in principle to any order in the expansion parameter; this computation is carried out explicitly to first order in the parameter. This joint probability is employed to obtain first-order corrections to previous results concerning, first, the lateral displacement y(L/2) midway between two selected points a distance L apart on the track of the scattered particle, and second, the mean squared curvature 〈C²〉ₐᵥ of the track; in the former instance the result is 〈y²(L/2)〉ₐᵥ=(L³48λ)1+(23L96λ), where λ is the scattering parameter, and in the latter, 〈C²〉ₐᵥ=(43λL)1-(25L96λ). For tracks both in two and in three dimensions, first-order corrections are determined also to results of the small-angle approximation for the actual path length of particles passing through a foil, both when the particles emerge normally to the foil and when they emerge in any direction. The results indicate that the discrepancy reported between theoretical predictions and some experimental measurements of multiple scattering is not attributable to the use of the small-angle approximation in the theoretical description.