On the accuracy of the Fokker-Planck and Fermi pencil beam equations for charged particle transport.
Larsen, Edward W.
- 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 Fokke... read morer-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. This is the peer reviewed version of the following article: C. Börgers and E. W. Larsen, "On the accuracy of the Fokker-Planck and Fermi pencil beam equations for charged particle transport," Medical Physics, vol. 23, no. 10, pp. 1749-1759, Oct. 1996, which has been published in final form at doi: 10.1118/1.597832. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.read less
- C. Börgers and E. W. Larsen, "On the accuracy of the Fokker-Planck and Fermi pencil beam equations for charged particle transport," Medical Physics, vol. 23, no. 10, pp. 1749-1759, Oct. 1996. doi: 10.1118/1.597832.