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

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)exp(−s/λ), where λ=(n·σ)−1λ=(n·σ)−1 is the total mean free path, n   the number of scatterers per unit volume and σσ 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

► Conventional sampling inconsistent when mean free path λ≤averageλ≤average distance D between scatterers. ► Proposed alternative: step length=D   with probability 1−exp(−D/λ)1−exp(−D/λ) of interaction at end of step. ► Consequence: corrected mean free path λc=D/(1−exp(−D/λ))λc=D/(1−exp(−D/λ)); less scattering. ► Agreement with conventional sampling for λ⪢Dλ⪢D. ► Conventional sampling using λcλc appears to be a good approximation.