On the sampling of step length in Monte Carlo simulation of trajectories with very small mean free path

Abstract

In detailed (event-by-event) simulation of trajectories of particles (electrons, ions) in liquids and solids the step length sampling method is often based on a total mean free path λ=(n·σ)−1, where n is the number of scatterers per unit volume and σ the total cross section. Typically, the step length s between scattering events is then generated by means of a sampling formula s=−λln(1−R), where R a random number in the interval 0<R<1; the probability for the particle to travel a distanced s without interaction is then exp(−s/λ). It is here argued that this “conventional” sampling method, which implies that the average distance between successive events is equal to λ, is erroneous unless λ is much larger than the distance dnn=n−1/3 between nearest neighbour scatterers. An alternative sampling method, M (monolayer) sampling, is proposed, with a fixed step length D=dnn and a finite probability I=1−exp(−D/λ) of interaction at the end of each step. According to M sampling, conventional sampling exaggerates the number of events per unit path length unless λ⪢dnn. Consequently, quantities like stopping power are overestimated, while transport mean free path is underestimated. The true (corrected) mean free path λc between events is found to be λc=D/(1−exp(−D/λ). The correction is substantial when λ is comparable to or smaller than the distance between the scatterers, in practice for very low energy particles in liquids and solids. In the opposite limit λ⪢D, conventional and M sampling produce the same result. Present results further indicate that conventional sampling using the corrected total mean free path λc is a good approximation to M sampling.