The Problem of Multiple Scattering

Multiple scattering can be regarded as a succession of elementary events. The distribution function for the particles which have gone through $n+1\mathrm{}$ elementary events is the convolution of two functions. The first of these expresses the scattering law; the second one is the distribution function for particles which have gone through $n$ events. It is well known that such convolutions can be calculated very easily by means of Fourier transforms if the elementary event is the traversal of a free path in an arbitrary direction. In this case, the Fourier transform of the convolution is the product of the Fourier transforms of the convolvents. In the case of more general scattering laws, integrals over products of the distribution function, and or representations of the group which leaves the scattering law invariant, play the same role which the Fourier transforms play in the aforementioned case. From the present point of view, the exponential in the Fourier transform is a representation of the displacement group. It is shown that one can solve several problems of multiple scattering on the basis of the above observation. These problems include the scattering of a point particle without change of energy but an arbitrary angular distribution, and several more involved problems.