True Step Length

Geant4 simulation of particle transport is performed step by step [SA03]. A true step length for a next physics interaction is randomly sampled using the mean free path of the interaction or by various step limitations established by different Geant4 components. The smallest step limit defines the new true step length.

The Interaction Length or Mean Free Path

Computation of mean free path of a particle in a media is performed in Geant4 using cross section of a particular physics process and density of atoms. In a simple material the number of atoms per volume is:

\[n = \frac{\mathcal{N}\rho}{A}\]

where:

\[\begin{split}\mathcal{N} &= \mbox{Avogadro's number} \\ \rho &= \mbox{density of the medium} \\ A &= \mbox{mass of a mole}\end{split}\]

In a compound material the number of atoms per volume of the $i^{th}$ element is:

\[n_{i} = \frac{\mathcal{N}\rho w_{i}}{A_{i}}\]

where:

\[\begin{split}w_{i} &= \mbox{proportion by mass of the $i^{th}$ element}\\ A_{i} &= \mbox{mass of a mole of the $i^{th}$ element}\end{split}\]

The mean free path of a process, $\lambda$, also called the interaction length, can be given in terms of the total cross section:

\[\lambda(E) = \left( \sum_i \lbrack n_i \cdot \sigma(Z_i,E) \rbrack \right)^{-1}\]

where $\sigma(Z,E)$ is the total cross section per atom of the process and $\sum_{i}$ runs over all elements composing the material.

$\sum\limits_{i}{\lbrack n_{i} \sigma(Z_{i},E)\rbrack}$ is also called the macroscopic cross section. The mean free path is the inverse of the macroscopic cross section.

Cross sections per atom and mean free path values may be tabulated during initialisation.

Determination of the Interaction Point

The mean free path, $\lambda$, of a particle for a given process 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:

\[n_\lambda =\int_{x_1}^{x_2} \frac{dx}{\lambda(x)} ,\]

which is independent of the material traversed. If $n_r$ is a random variable denoting the number of mean free paths from a given point to the point of interaction, it can be shown that $n_r$ has the distribution function:

\[P( n_r < n_\lambda ) = 1-e^{-n_\lambda}\]

The total number of mean free paths the particle travels before reaching the interaction point, $n_\lambda$, is sampled at the beginning of the trajectory as:

\[n_\lambda = -\log \left ( \eta \right )\]

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

\[n'_\lambda=n_\lambda -\frac{\Delta x }{\lambda(x)}\]

until the step originating from $s(x) = n_\lambda \cdot \lambda(x)$ is the shortest and this triggers the specific process.

Step Limitations

The short description given above is the differential approach to particle transport, which is used in many other simulation codes. In this approach besides the other (discrete) processes the continuous energy loss and multiple scattering imposes a limit on the step-size too [JA09][eal16], because the cross section of different processes depend of the energy of the particle. Then it is assumed that the step is small enough so that the particle cross sections remain approximately constant during the step. In principle, one must use very small steps in order to insure an accurate simulation, but computing time increases as the step-size decreases and the default model of energy loss fluctuations will not be accurate for extremly small steps. A good compromise depends on required accuracy of a concrete simulation. The problem is reduced using integral approach, which is described below in Correcting the Cross Section for Energy Variation. However, this only provides effectively correct cross sections but step limitation is needed also for more precise tracking. Thus, in Geant4 any process may establish additional step limitation, the most important limits come from ionisation and multiple scattering (see details in sub-chapters Step-size Limit and Step Limitation Algorithm correspondingly).

Updating the Particle Time

The laboratory time of a particle should be updated after each step:

\[\Delta t_{lab} = 0.5 \Delta x \left( \frac{1}{v_1} + \frac{1}{v_2} \right),\]

where $\Delta x$ is a true step length traveled by the particle, $v_1$ and $v_2$ are particle velocities at the beginning and at the end of the step correspondingly.

Bibliography

eal16: J. Allison et al. Recent developments in geant4. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 835:186–225, nov 2016. URL: https://doi.org/10.1016/j.nima.2016.06.125, doi:10.1016/j.nima.2016.06.125.
JA09: et al. J. Apostolakis. Geometry and physics of the geant4 toolkit for high and medium energy applications. Radiation Physics and Chemistry, 78(10):859–873, oct 2009. URL: https://doi.org/10.1016/j.radphyschem.2009.04.026, doi:10.1016/j.radphyschem.2009.04.026.
SA03: et al. S. Agostinelli. Geant4—a simulation toolkit. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 506(3):250–303, jul 2003. URL: https://doi.org/10.1016/S0168-9002(03)01368-8, doi:10.1016/s0168-9002(03)01368-8.