1 | \section[Ionization]{Muon Ionization} |
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2 | |
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3 | |
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4 | The class {\it G4MuIonisation} provides the continuous energy loss due to |
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5 | ionization and simulates the 'discrete' part of the ionization, that is, |
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6 | delta rays produced by muons. |
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7 | Inside this class the following models are used: |
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8 | \begin{itemize} |
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9 | \item |
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10 | {\it G4BraggModel} (valid for protons with $T < 0.2 \;MeV$) |
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11 | \item |
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12 | {\it G4BetherBlochModel} (valid for protons with $0.2 \;MeV < T < 1\; GeV$) |
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13 | \item |
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14 | {\it G4MuBetherBlochModel} (valid for protons with $T > 1\; GeV$) |
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15 | \end{itemize} |
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16 | The limit energy $0.2\; MeV$ is equivalent to the proton limit energy $2 MeV$ because of |
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17 | scaling relation (\ref{enloss.sc}), which allows |
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18 | simulation for muons with energy below $1\; GeV$ |
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19 | in the same way as for point-like hadrons with spin 1/2 described in the section \ref{en_loss}. |
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20 | |
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21 | For higher energies the {\it G4MuBetherBlochModel} is applied, in which |
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22 | leading radiative corrections are taken into account \cite{muion.br}. |
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23 | Simple analytical formula for the cross section, derived with the logarithmic |
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24 | are used. Calculation results appreciably differ from usual elastic $\mu -e$ |
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25 | scattering in the region of high energy transfers $m_e << T < T_{max}$ |
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26 | and give non-negligible correction to the total average energy loss of |
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27 | high-energy muons. The total cross section is written as following: |
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28 | \begin{equation} |
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29 | \label{muion.c} |
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30 | \sigma (E,\epsilon ) = \sigma_{BB}(E, \epsilon ) \left[ 1 + \frac{\alpha}{2\pi} |
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31 | ln\left( 1 + \frac{2\epsilon}{m_e} \right) |
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32 | ln\left(\frac{4m_e E(E - \epsilon )}{m_{\mu}^2( 2\epsilon + m_e) }\right) \right] , |
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33 | \end{equation} |
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34 | here $\sigma(E,\epsilon)$ is the differential cross sections, |
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35 | $\sigma(E,\epsilon)_{BB}$ is the Bethe-Bloch cross section (\ref{hion.i}), |
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36 | $m_e$ is the electron mass, $m_{\mu}$ is the muon mass, |
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37 | $E$ is the muon energy, $\epsilon$ |
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38 | is the energy transfer, $\epsilon = \omega + T$, where |
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39 | T is the electron kinetic energy and $\omega$ is the energy of radiative gamma. |
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40 | |
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41 | For computation of the truncated mean energy loss (\ref{comion.a}) the partial integration |
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42 | of the expression (\ref{muion.c}) is performed |
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43 | \begin{equation} |
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44 | \label{muion.d} |
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45 | S(E,\epsilon_{up}) = S_{BB}(E,\epsilon_{up}) + S_{RC}(E,\epsilon_{up}), \;\; \epsilon_{up} = min(\epsilon_{max},\epsilon_{cut}), |
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46 | \end{equation} |
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47 | where term $S_{BB}$ is the Bethe-Bloch truncated energy loss (\ref{hion.d}) for the interval |
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48 | of energy transfer $(0 - \epsilon_{up})$ |
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49 | and term $S_{RC}$ is a correction due to radiative effects. |
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50 | The function become smooth after log-substitution and |
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51 | is computed by numerical integration |
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52 | \begin{equation} |
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53 | \label{muion.e} |
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54 | S_{RC}(E,\epsilon_{up}) = |
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55 | \int^{\ln\epsilon_{up}}_{\ln\epsilon_1}\epsilon^2(\sigma(E,\epsilon) - \sigma_{BB}(E, \epsilon))d(\ln\epsilon), |
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56 | \end{equation} |
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57 | where lower limit $\epsilon_1$ does not |
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58 | effect result of integration in first order and |
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59 | in the class {\it G4MuBetheBlochModel} the default value $\epsilon_1 = 100 keV$ is used. |
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60 | |
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61 | For computation of the discrete cross section (\ref{comion.b}) another substitution is used |
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62 | in order to perform numerical integration of a smooth function |
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63 | \begin{equation} |
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64 | \label{muion.f} |
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65 | \sigma(E) = \int^{1/\epsilon_{up}}_{1/\epsilon_{max}}\epsilon^2\sigma (E,\epsilon )d(1/\epsilon ). |
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66 | \end{equation} |
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67 | The sampling of energy transfer is performed between $1/\epsilon_{up}$ and $1/\epsilon_{max}$ using |
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68 | rejection constant for the function $\epsilon^2\sigma(E,\epsilon)$. |
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69 | After the successful sampling of the energy transfer, the direction |
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70 | of the scattered electron is generated with respect to the direction of the |
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71 | incident particle. The energy of radiative gamma is neglected. |
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72 | The azimuthal electron angle $\phi$ is generated isotropically. |
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73 | The polar angle $\theta$ is calculated from energy-momentum conservation. |
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74 | This information is used to calculate the energy and momentum of both |
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75 | scattered particles and to transform them into the {\em global} coordinate |
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76 | system. |
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77 | |
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78 | \subsection{Status of this document} |
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79 | 09.10.98 created by L. Urb\'an. \\ |
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80 | 14.12.01 revised by M.Maire \\ |
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81 | 30.11.02 re-worded by D.H. Wright \\ |
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82 | 01.12.03 revised by V. Ivanchenko \\ |
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83 | 22.06.07 rewritten by V. Ivanchenko \\ |
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84 | |
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85 | \begin{latexonly} |
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86 | |
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87 | \begin{thebibliography}{99} |
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88 | |
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89 | \bibitem{muion.br} |
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90 | S.R.~Kelner, R.P.~Kokoulin, A.A.~Petrukhin, |
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91 | Phys. Atomic Nuclei 60 (1997) 576. |
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92 | \end{thebibliography} |
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93 | |
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94 | \end{latexonly} |
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95 | |
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96 | \begin{htmlonly} |
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97 | |
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98 | \subsection{Bibliography} |
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99 | |
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100 | \begin{enumerate} |
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101 | \item |
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102 | S.R.~Kelner, R.P.~Kokoulin, A.A.~Petrukhin, |
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103 | Phys. Atomic Nuclei 60 (1997) 576. |
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104 | \end{enumerate} |
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105 | |
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106 | \end{htmlonly} |
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