A generalized Fokker-Planck theory for electron transport problems.
Leakeas, Christopher Louis
1999
Abstract
We consider the problem of including large-angle scattering effects in electron pencil beam problems. A model scattering operator is devised based on the angular diffusion operator used in Fokker-Planck (FP) theory. This model operator preserves higher angular moments of the exact linear Boltzmann scattering operator, and therefore allows the inclusion of large-angle scattering effects that are not included in the small-angle scattering FP theory. The resulting Generalized Fokker-Planck (GFP) equation preserves higher moments of the exact transport equation. The GFP<italic>n</italic> equation preserves <italic> n</italic> angular moments of the true Boltzmann equation. We focus on the GFP3 and GFP5 scattering operators and show that these operators may be used to accurately solve electron pencil beam problems. In the first part of this thesis, we solve the energy-independent GFP3 and GFP5 equations in slab geometry with varying amounts of large-angle scattering using deterministic and Monte Carlo methods. We show that the GFP3 and GFP5 equations may be written as integro-differential Boltzmann equations and solved using Monte Carlo, or as a system of partial differential equations that may be solved using finite-difference or finite-element techniques. We solve the 1-D (transverse-integrated) GFP3 and GFP5 equations using a deterministic sweeping method similar to source iteration, and introduce an efficient angular coupling that significantly accelerates these calculations. We also solve the GFP3 and GFP5 equations in 1-D and 3-D using Monte Carlo. Finally, we present numerical results for both deterministic and Monte Carlo GFP3 and GFP5 calculations. We show that 3-D GFP5 calculations are very accurate, but that both GFP3 and GFP5 calculations yield flux peaks near the central axis of the beam. In 1-D, both GFP3 and GFP5 yield very accurate transverse-integrated flux distributions. In the second part of this thesis, we determine an approximate closed-form solution of the energy-dependent 3-D GFP3/CSD equation for electron pencil beam problems in a homogeneous medium by performing an asymptotic expansion. This expansion results in a leading-order equation identical to the well-known Fermi-Eyges (FE) equation used in electron radiotherapy treatment planning. A correction equation is obtained that includes large-angle scattering effects. Together, these solutions form Corrected Fermi-Eyges theory with large-angle scattering (CFE/LA). We obtain a closed-form solution for the CFE/LA scalar flux and show that CFE/LA yields more accurate radial flux distributions and spatial electron distributions than FE for energy-independent problems, relative to Monte Carlo simulations of the exact Boltzmann equation. The scalar fluxes become less accurate as the amount of large-angle scattering increases, but the CFE/LA solution remains more accurate than the FE solution for a wide range of physically relevant screening parameters.Subjects
Electron Transport Fermi Eyges Theory Fokker-planck Theory Generalized Large Angle Pencil Beams Problems Scattering
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