Functional Expansion Technique for Monte Carlo Electron Transport Calculations

A new method for Monte Carlo electron transport calculations has previously been outlined. Briefly, the method is as follows: the quantity of interest (e.g., x-ray photoemission angular distribution) is expanded in a complete set of orthogonal functions; the individual trajectories of the Monte Carlo sample are used to calculate the expansion coefficients; the resulting coefficients are used to evaluate a (continuous) representation of the relevant distribution. In this paper we discuss this orthogonal function expansion method at some length. A precise mathematical formulation is given to two types of expansions: expansions of the probability density of a random variable; expansions of functions defined on a stochastic process. The method as formulated is then used for several problems of practical interest. The POEM Monte Carlo electron transport code has been modified to provide expansions for the following: one- and two-dimensional x-ray photoemission angular distributions; dose enhancement profiles near high-Z/low-Z interfaces, and energy distributions for finite beams incident on an interface. A discussion is given of the applicability of the method, particularly as it relates to the specific applications. Several pathological examples are cited which indicate the discretion necessary in applying the technique. These examples serve to counterpoint the desirable properties of convergence, smoothing, and variance reduction which arise from a properly applied expansion.