Analytical and numerical solutions of generalized fokker-planck equations

摘要

A novel approach for the spatial, angular and energy spreading of a collimated beam of charaged particles as it penetrates an amorphous medium is described. A Generalized Fokker-Planck (GFP) approach is presented which accounts for larger angle scattering than is possible with traditional ,strictly Fokker-Planck formalism. An aymptotic analysis demonstrates that, for broad beam incidence, the GFP model gives more accurate results for the scalar flux as a function of depth when mu0 is not too close to unity. We have also developed a scheme to relax the near collimation approximation inherent in the classical Fermi-Eyges analysis. The resulting modified Fermi-Eyges formula retains the elegance and simplicity of the original but shows radically improved accuracy for the radial spreading of the beam, especially at large radii. A computationally efficient approach to account for electronic energy-loss straggling is discussed. Our methodology preserves the correct mean and mean-square energy loss in a multigroup setting and is remarkably accurate for the energy spreading of an initially monoenergetic beam. A discontinuous finite element discretization, with arbitrary order polynomial trial functions, is implemented to test the straggling model and results are compared against exact Monte Carlo simulations.