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

Abstract

In an event-by-event simulation of the trajectory of a particle moving in matter it is usually assumed that the probability for the particle to travel a distance s without interaction is exp(−s/λ), where λ=(n·σ)−1 is the total mean free path, n the number of scatterers per unit volume and σ the total cross section per scatterer. 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. It is here argued that this “conventional” sampling method, which basically assumes that the scattering medium may be regarded as a homogeneous continuum, may be erroneous unless λ is much larger than the average distance dnn between nearest neighbour scatterers, estimated by dnn=n−1/3. An alternative sampling method (“M” sampling) is proposed with a fixed step length D=dnn and a finite probability I=1−exp(−D/λ) of a single elastic or inelastic scattering event at the end of each step. According to this method, conventional sampling may exaggerate the number of events per unit path length; the corrected mean free path between events is found to be λc=D/(1−exp(−D/λ)). The correction is substantial when λ is comparable to or smaller than D, in practice for very low energy particles in liquids and solids. Consequently, quantities like stopping power may then be overestimated, while transport mean free path may be underestimated. 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.