This balance of neutral particles is cast as the linear Boltzmann Transport Equation (BTE).
The BTE is an integro-differential equation which describes the behavior of neutral particles in
terms of spatial, angular, and energy domains as they interact in a system; the steady-state
transport equation is given in Equation (1-5):
dE 'WrR dn' o-, (,',E 'a) Yr, ', ')qE,0E (15
0 4xr
where y(r7, OE)d'rdEdai is the angular flux with energy between E an~d E dE, at position r
within the volume element dF~, and in direction aZ within the solid angle daZ. The scattering
term, defined with the double integral term, represents the sum of the particles scattered into
dEdaZ from all the dE 'dnZ' after scattering collisions represented by the double differential
cross-secti on Gs (F,ER 4 E, iZ' 4 6) q (r-, 6,E ) represent the external source.
The left side of Equation (1-5) represents streaming and collision terms (loss), and the right
side represents scattering and independent sources (gain). Since it describes the flow of radiation
in a 3-D geometry with angular and energy dependence, this is one of the most challenging
equations to solve in terms of complexity and model size. For rendering a deterministic
computational solution for a large problem in a reasonable time, it is necessary to utilize a robust
parallel transport algorithm and a high performance computing system. All the methods
described below are different approaches to solving this equation. For cases with charged particle
transport, the Boltzmann and Fokker-Plank formalisms (Boltzmann-FP) 10are generally used. A
simplified form of this equation is usually used, which is: