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 $\xi E>E_m$ then $\sigma_m=min(\sigma(E),\sigma(\xi E))$. In the opposit |
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43 | case $\sigma_m=\sigma_{max}$. Here $\xi$ is a |
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44 | parameter of the algorithm. Its optimal value is connected with the value |
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45 | of the {\it dRoverRange} parameter (see sub-chapter \ref{en_loss}), |
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46 | by default $\xi = 1 - \alpha_R = 0.8$. |
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47 | Note, that described method is precise if the |
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48 | cross section has only one maximum, which is a |
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49 | typical case for electromagnetic processes. |
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50 | |
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51 | |
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52 | The integral variant of step limitation is the default for the |
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53 | $G4eIonisation$, $G4eBremsstrahlung$ and some otehr process |
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54 | but is not automatically activated for others. |
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55 | To do so the boolean UI command can be used:\\ |
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56 | \\ |
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57 | {\it /process/eLoss/integral true} |
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58 | \\ |
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59 | \\ |
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60 | The integral variant of the energy loss sampling process is less dependent on |
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61 | values of the production cuts \cite{int.g403} and allows |
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62 | to have less step limitation, however it should be applied |
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63 | on a case-by-case basis because may require extra CPU. |
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64 | |
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65 | \subsection{Status of this document} |
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66 | 01.12.03 integral method subsection added by V. Ivanchenko \\ |
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67 | 17.08.04 moved to common to all charged particles by M. Maire \\ |
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68 | 25.11.06 revision by V. Ivanchenko \\ |
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69 | |
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70 | \begin{latexonly} |
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71 | |
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72 | \begin{thebibliography}{99} |
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73 | \bibitem{int.unimod} |
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74 | V.N.Ivanchenko et al., |
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75 | Proc. of Int. Conf. MC91: Detector and event simulation in high |
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76 | energy physics, Amsterdam 1991, pp. 79-85. (HEP INDEX 30 (1992) No. 3237). |
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77 | \bibitem{int.g403} |
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78 | V.N.Ivanchenko. |
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79 | Geant4 Workshop (TRIUMF, Canada, 2003) |
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80 | http://www.triumf.ca/\\ |
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81 | geant4-03/talks/04-Thursday-AM-1/02-V.Ivanchenko/eloss03.ppt |
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82 | |
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83 | \end{thebibliography} |
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84 | |
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85 | \end{latexonly} |
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86 | |
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87 | \begin{htmlonly} |
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88 | |
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89 | \subsection{Bibliography} |
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90 | |
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91 | \begin{enumerate} |
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92 | \item V.N.Ivanchenko et al., |
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93 | Proc. of Int. Conf. MC91: Detector and event simulation in high |
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94 | energy physics, Amsterdam 1991, pp. 79-85. (HEP INDEX 30 (1992) No. 3237). |
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95 | \item V.N.Ivanchenko. |
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96 | Geant4 Workshop (TRIUMF, Canada, 2003) |
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97 | http://www.triumf.ca/\\ |
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98 | geant4-03/talks/04-Thursday-AM-1/02-V.Ivanchenko/eloss03.ppt |
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99 | \end{enumerate} |
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100 | |
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101 | \end{htmlonly} |
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