On the accuracy of the Fokker-Planck and Fermi pencil beam equations for charged particle transport.

Electron beam dose calculations are often based on pencil beam formulas such as the Fermi-Eyges formula. The Fermi-Eyges formula gives an exact solution of the Fermi equation. The Fermi equation can be derived from a more fundamental mathematical model, the linear Boltzmann equation, in two steps. First, the linear Boltzmann equation is approximated by the Fokker-Planck equation. Second, the Fokker-Planck equation is approximated by the Fermi equation. In this paper, we study these approximations. We use a simplified model problem, but choose parameter values closely resembling those relevant in electron beam therapy. Our main conclusions are: (1) The inaccuracy of the Fokker-Planck approximation is primarily due to neglect of large-angle scattering. (2) When computing an approximate solution to the Fokker-Planck equation by Monte Carlo simulation of a transport process, one should let the polar scattering angle be deterministic. (3) At shallow depths, the discrepancy between the linear Boltzmann and Fokker-Planck equations is far more important than that between the Fokker-Planck and Fermi equations. The first of these conclusions is certainly not new, but we state and justify it more rigorously than in previous work.