Modeling the X-Ray Deposition Region by Numerical Integration of the Boltzmann Equation

Most numerical solutions of the X-ray deposition problem have involved either cloud-in-cell macroparticle techniques or prescribed source techniques. Macroparticle approaches often require 104 - 105 particles to reduce the simulation noise to a reasonable level and therefore can be quite costly for a multidimensional problem. Prescribed source techniques are cheaper but do not achieve a self-consistent solution. More recently, attempts have been made to model the X-ray deposition region by a set of fluid equations. This approach offers large savings in computer time but requires assumptions on the form of the distribution function. An alternate approach is to integrate the distribution function directly. This approach has been discussed by Higgins, Longmire, and O´Dell and was used by Holland to investigate the effects of secondary electron emission. In this paper, the Boltzmann equation, including ionization sources, is numerically solved by weather differencing in two spacial dimensions. The algorithm is stable and allows object boundary conditions to be installed in a natural manner. The experimental response of a cylindrical cavity to electron injection sources at pressures of 0.2 and 300 mtorr is compared to numerical predictions and found to be in reasonable agreement. The effects of various ionization cross section models are described and the validity of the assumption of velocity independent acceleration is examined.