On the Theory of Multiple Scattering, Particularly of Charged Particles

Abstract

The general theory of the elastic multiple scattering of particles with a strongly anisotropic scattering function is investigated without making the small-angle approximation. The rigorous transport equation is used and approximations are introduced at a later stage. The paper consists of four parts. In the first part the general formulation of the problem is given. The approximations involved in the existing theories of small-angle forward scattering are discussed in some detail. In the second part the spherical harmonic method is formulated in a manner so as to permit an explicit expression for the general nth approximation. There is an ambiguity both in (a) the way of defining successive approximations and in (b) the way of introducing approximate boundary conditions. We have chosen (b) to give the best approximation to the exact solution of the Schwarzschild-Milne problem. In the third part it is shown that our choice of (a) for the spherical harmonic method leads to the same final formulas as the gaussian quadrature method. The relation of these two methods is discussed in detail. In the fourth part the problem of anisotropic multiple scattering is reduced to a quasi-isotropic one by using a generalized Goudsmit-Saunderson type distribution function (defined also for back scattering) as a first approximation. Three different methods are given for forward scattering (including large angles). The first method is a perturbation treatment. The second method is based on the approximate delta-function character of the scattering function and employs a Fokker-Planck type development for the peaked part of the scattering function. The third method is a Liouville-Neumann type of iteration applied directly to the transport equation. For back scattering the second and third of these methods also apply. In addition a special method is developed, based on the smallness of the back single scattering cross section. The generalized Goudsmit-Saunderson distribution function is developed in powers of the thickness of the scatterer and it is shown that all three methods lead to the same single scattering tail. The three methods can be applied for any single scattering law. The screened Born-Rutherford law is introduced in some cases as an example. The relation of the present work to previous theories is discussed.