Correcting the Cross Section for Energy Variation
As described in Mean Energy Loss and Determination of the Interaction Point the step size limitation is provided by energy loss processes in order to insure the precise calculation of the probability of particle interaction. It is generally assumed in Monte Carlo programs that the particle cross sections are approximately constant during a step, hence the reaction probability \(p\) at the end of the step can be expressed as
where \(n\) is the density of atoms in the medium, \(s\) is the step length, \(E_i\) is the energy of the incident particle at the beginning of the step, and \(\sigma(E_i)\) is the reaction cross section at the beginning of the step.
However, it is possible to sample the reaction probability from the exact expression
where \(E_f\) is the energy of the incident particle at the end of the step, by using the integral approach to particle transport. This approach is available for processes implemented via the G4VEnergyLossProcess and G4VEmProcess interfaces.
The Monte Carlo method of integration is used for sampling the reaction probability [eal92]. It is assumed that during the step the reaction cross section smaller, than some value \(\sigma(E) < \sigma_m\). The mean free path for the given step is computed using \(\sigma_m\). If the process is chosen as the process happens at the step, the sampling of the final state is performed only with the probability \(p=\sigma(E_f)/\sigma_m\), alternatively no interaction happen and tracking of the particle is continued. To estimate the maximum value \(\sigma_m\) for the given tracking step at Geant4 initialisation the energy \(E_m\) of absolute maximum \(\sigma_{max}\) of the cross section for given material is determined and stored. If at the tracking time particle energy \(E < E_m\), then \(\sigma_m=\sigma(E)\). For higher initial energies if \(\xi E>E_m\) then \(\sigma_m=\max(\sigma(E),\sigma(\xi E))\), in the opposite case, \(\sigma_m=\sigma_{max}\). Here \(\xi\) is a parameter of the algorithm. Its optimal value is connected with the value of the dRoverRange parameter (see Mean Energy Loss), by default \(\xi = 1 - \alpha_R = 0.8\). Note, that described method is precise if the cross section has only one maximum, which is a typical case for electromagnetic processes.
The integral variant of step limitation is the default for the G4eIonisation, G4hIonisation, G4eBremsstrahlung and some other processes for charged particles. To enable/disable this option the Boolean UI command can be used:
/process/eLoss/integral true/false
The integral variant of the energy loss sampling process is less dependent on values of the production cuts [eal09] and allows to have less step limitation, however it should be applied on a case-by-case basis because may require extra CPU.
Bibliography
- eal09
J. Apostolakis et al. 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.
- eal92
V.N.Ivanchenko et al. In Proc. of Int. Conf. MC91: Detector and event simulation in high energy physics, 1991, HEP INDEX 30 (1992) No. 3237, 79–85. Amsterdam, 1992.