[1208] | 1 | \section{Correcting the cross section for energy variation } \label{integral} |
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| 2 | |
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| 3 | As described in Sections \ref{en_loss} and \ref{ip} the step size limitation |
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| 4 | is provided by energy loss processes in order to insure the precise |
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| 5 | calculation of the probability of particle interaction. It is generally |
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| 6 | assumed in Monte Carlo programs that the particle cross sections are |
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| 7 | approximately constant during a step, hence the reaction probability $p$ at |
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| 8 | the end of the step can be expressed as |
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| 9 | \begin{equation} |
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| 10 | \label{int_a} |
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| 11 | p = 1 - \exp \left ( -n s \sigma(E_i) \right ), |
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| 12 | \end{equation} |
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| 13 | where $n$ is the density of atoms in the medium, $s$ is the step length, |
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| 14 | $E_i$ is the energy of the incident particle at the beginning of the step, |
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| 15 | and $\sigma(E_i)$ is the reaction cross section at the beginning of the step. |
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| 16 | |
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| 17 | However, it is possible to sample the reaction probability from the exact |
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| 18 | expression |
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| 19 | \begin{equation} |
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| 20 | \label{int_b} |
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| 21 | p = 1 - \exp \left ( -\int_{E_i}^{E_f}{n \sigma(E) ds} \right ), |
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| 22 | \end{equation} |
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| 23 | where $E_f$ is the energy of the incident particle at the end of the step, |
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| 24 | by using the integral approach to particle transport. This approach is |
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| 25 | available for processes implemented via the $G4VEnergyLossProcess$ |
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| 26 | and $G4VEmProcess$ interfaces. |
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| 27 | |
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| 28 | The Monte Carlo method of integration is used for sampling the reaction |
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| 29 | probability \cite{int.unimod}. It is assumed that during the step |
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| 30 | the reaction cross section |
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| 31 | smaller, than some value $\sigma(E) < \sigma_m$. The mean free path |
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| 32 | for the given step is computed using $\sigma_m$. If the process is chosen |
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| 33 | as the process happens at the step, the sampling of the final state is performed only |
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| 34 | with the probability $p=\sigma(E_f)/\sigma_m$, alternatively no interaction |
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| 35 | happen and tracking of the particle is continued. |
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| 36 | To estimate the maximum value |
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| 37 | $\sigma_m$ for the given tracking step at Geant4 initialisation the energy |
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| 38 | $E_m$ of absoluted maximum $\sigma_{max}$ |
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| 39 | of the cross section for given material is determined and stored. |
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| 40 | If at the tracking time |
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| 41 | particle energy $E < E_m$, then $\sigma_m=\sigma(E)$. For higher initial energies |
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| 42 | if $0.8E>E_m$ then $\sigma_m=min(\sigma(E),\sigma(0.8E))$. In the opposit |
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| 43 | case $\sigma_m=\sigma_{max}$. Thus described method is precise if the |
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| 44 | cross section has only one maximum. |
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| 45 | Note, that it is a typical case for electromagnetic processes. |
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| 46 | |
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| 47 | |
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| 48 | The integral variant of step limitation is the default for the |
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| 49 | $G4eIonisation$, $G4eBremsstrahlung$ and some otehr process |
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| 50 | but is not automatically activated for others. |
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| 51 | To do so the boolean UI command ``/process/eLoss/integral true'' can be used. |
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| 52 | The integral variant of the energy loss sampling process is less dependent on |
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| 53 | values of the production cuts \cite{int.g403} and allows |
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| 54 | to have less step limitation, however it should be applied |
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| 55 | on a case-by-case basis because may require extra CPU. |
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| 56 | |
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| 57 | \subsection{Status of this document} |
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| 58 | 01.12.03 integral method subsection added by V. Ivanchenko \\ |
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| 59 | 17.08.04 moved to common to all charged particles by M. Maire \\ |
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| 60 | 25.11.06 revision by V. Ivanchenko \\ |
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| 61 | |
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| 62 | \begin{latexonly} |
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| 63 | |
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| 64 | \begin{thebibliography}{99} |
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| 65 | \bibitem{int.unimod} |
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| 66 | V.N.Ivanchenko et al., |
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| 67 | Proc. of Int. Conf. MC91: Detector and event simulation in high |
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| 68 | energy physics, Amsterdam 1991, pp. 79-85. (HEP INDEX 30 (1992) No. 3237). |
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| 69 | \bibitem{int.g403} |
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| 70 | V.N.Ivanchenko. |
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| 71 | Geant4 Workshop (TRIUMF, Canada, 2003) |
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| 72 | http://www.triumf.ca/\\ |
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| 73 | geant4-03/talks/04-Thursday-AM-1/02-V.Ivanchenko/eloss03.ppt |
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| 74 | |
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| 75 | \end{thebibliography} |
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| 76 | |
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| 77 | \end{latexonly} |
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| 78 | |
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| 79 | \begin{htmlonly} |
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| 80 | |
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| 81 | \subsection{Bibliography} |
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| 82 | |
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| 83 | \begin{enumerate} |
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| 84 | \item V.N.Ivanchenko et al., |
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| 85 | Proc. of Int. Conf. MC91: Detector and event simulation in high |
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| 86 | energy physics, Amsterdam 1991, pp. 79-85. (HEP INDEX 30 (1992) No. 3237). |
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| 87 | \item V.N.Ivanchenko. |
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| 88 | Geant4 Workshop (TRIUMF, Canada, 2003) |
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| 89 | http://www.triumf.ca/\\ |
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| 90 | geant4-03/talks/04-Thursday-AM-1/02-V.Ivanchenko/eloss03.ppt |
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| 91 | \end{enumerate} |
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| 92 | |
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| 93 | \end{htmlonly} |
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