Determination of the interaction point

The mean free path of a particle for a given process, $\lambda$ , depends on the medium and cannot be used directly to sample the probability of an interaction in a heterogeneous detector. The number of mean free paths which a particle travels is:

$N$_λ=∫{dxλ(x)}

and it is independent of the material traversed. If $N$_R is a random variable denoting the number of mean free paths from a given point until the point of interaction, it can be shown that $N$_R has the distribution function

$P(\; N$_R< N_λ) = 1-e^-N_λ

The total number of mean free paths the particle travels before the interaction point, $N$_λ , is sampled at the beginning of the trajectory as:

$N$_λ= -log( η)

where $\eta$ is a random number uniformly distributed in the range $(0,1)$ . $N$_λ is updated after each step $\Delta x$ according the formula:

$N\text{'}$_λ=N_λ-{Δx λ(x)}

until the step originating from $s(x)\; =\; N$_λλ(x) is the shortest and this triggers the specific process.