Correcting the cross section for energy variation

As described in Sections 7.1 and 5.2 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

$$p = 1 - \exp \left ( -n s \sigma(E_i) \right ),$$ (7.28)

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

$$p = 1 - \exp \left ( -\int_{E_i}^{E_f}{n \sigma(E) ds} \right ),$$ (7.29)

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 $G4VEnergyLossProcess$ interface.

The Monte Carlo method of integration is used for sampling the reaction probability [1]. It is assumed that the reaction cross section increases with energy, so that the cross section at the end of the step is always smaller, $\sigma(E_f) < \sigma(E_i)$. This assumption is correct for electromagnetic physics.

The integral variant of step limitation is the default for the $G4eIonisation$ process but is not automatically activated for others. To do so the boolean UI command ``/process/eLoss/integral true'' can be used. The integral variant of the energy loss sampling process is less dependent on values of the production cuts [2], however it should be applied on a case-by-case basis because heavy particles taking large steps in an absorber can cause inaccurate sampling of energy loss fluctuations.

Status of this document

01.12.03 integral method subsection added by V. Ivanchenko
17.08.04 moved to common to all charged particles (mma)

Bibliography

1. V.N.Ivanchenko et al., Proc. of Int. Conf. MC91: Detector and event simulation in high energy physics, Amsterdam 1991, pp. 79-85. (HEP INDEX 30 (1992) No. 3237).

2. V.N.Ivanchenko. Geant4 Workshop (TRIUMF, Canada, 2003) http://www.triumf.ca/
   geant4-03/talks/04-Thursday-AM-1/02-V.Ivanchenko/eloss03.ppt