Stochastic interpretation of the fermi pencil beam problem

Abstract

The problem of the transverse spatial spreading and angular dispersion of a pencil beam of charged particles with depth in a highly forward scattering medium, as originally posed by Fermi, is given a stochastic interpretation. It is shown that the spatial and angular relaxation of the beam is equivalently described by the random walk, or Brownian motion, of a particle in phase space. The joint probability distribution function of the particle's lateral position and associated direction is shown to satisfy a Fokker-Planck equation, which reduces to Fermi's equation for the angular flux if the particle diffusion coefficient is equated to the transport cross section. This stochastic analogy enables explicit solutions to be constructed with ease using well known results from the theory of Gaussian stochastic processes.