A moment-preserving single-event Monte Carlo model of electron and positron energy-loss straggling

Analog simulation of energy straggling of electrons and positrons is computationally impractical because of long-range Coulomb forces resulting in highly peaked cross sections about small energy-losses and extremely small collision mean free paths. The resulting transport process is dominated by very frequent small energy transfer collisions but a significant contribution to the overall energy-loss distribution comes from the infrequent high energy-losses. Sufficient resolution in energy-loss spectra and dose profiles using single-event Monte Carlo methods would then require a large number of particle samples. In this thesis, we demonstrate that a pseudo-differential cross section designed to approximately yet accurately preserve energy-loss moments is capable of yielding accurate energy-loss spectra and dose distributions in a single-event Monte Carlo formulation. A benchmark solution for the analog problem for incident electrons and positrons is developed in order to provide an exact solution in which our approximation is evaluated against. A random walk sequence is used to randomly sample a distance to collision followed by a sampled energy-loss at the distance traveled by the particle. This process was completed until specific boundaries or cutoffs were met. Due to the non-linearity of the probability distribution functions for the electron and positron energy-loss differential cross sections, analog energy-loss sampling is simulated using the rejection method. The Landau straggling distribution is examined in detail and its accuracy is quantitatively assessed. Under the constraints in the formulation of Landau, we show that the number of energy-loss moments preserved is equal to the number of energy-flux moments preserved. More specifically, when the energy-losses within a given distance are sufficiently small so that the mean free path can be considered constant, the number of energy-loss moments preserved up to order $N$ is equal to the number of energy-flux moments preserved up to order $N$. Energy-flux moments of Landau and the analog solution are compared. This moment-preserving theory provides the foundation in which a pseudo-transport model based on the Landau energy-loss distribution is then constructed. Next, the Landau distribution is used in formulating a pseudo-transport model. The energy-dependent mean free paths of this model are exponentially sampled while the energy-loss will be sampled using the Landau energy-loss distribution in terms of the respective mean free path, incident energy and mean energy-loss of the particle. The Landau Pseudo-Transport(LPT) model allows longer mean free paths and smoother distributions increasing the efficiency of electron and positron transport. Extensive numerical comparisons of the LPT model against the benchmark are conducted for energy-loss spectra and depth-dose profiles. It is shown that while high fidelity dose distributions can be obtained at a fraction of the cost of the analog calculation, energy spectra are difficult to resolve because of the presence of artifacts associated with the Landau distribution itself. Although these artifacts are reduced in thicker materials, a loss of computational efficiency results in step-sizes that are relatively close to the analog mean free path when attempting to resolve these deficiencies. The step-size of the simulations must be chosen to balance the greatest efficiency with the highest accuracy. In summation, a pseudo-transport model with longer mean free paths and smoother cross sections has been developed to increase computational time producing results comparable to the analog solution