1 | \chapter{Lepton Hadron Interactions} |
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2 | |
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3 | The photonuclear interaction of muons is currently the only process treated |
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4 | in this category. |
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5 | |
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6 | \section{\it G4MuonNucleusProcess} |
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7 | |
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8 | This class simulates the photonuclear interaction of muons in a material. |
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9 | The muon interacts electromagnetically with a nucleus, exchanging a virtual |
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10 | photon. At energies above a few GeV, the photon interacts hadronically with |
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11 | the nucleus, producing hadronic secondaries. |
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12 | |
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13 | The outcome of the simulation depends heavily upon the interaction model |
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14 | chosen. Hence the model-dependent part of the process is implemented in |
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15 | the {\it G4MuonNucleusInteractionModel} class, which can be easily replaced by |
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16 | another model. |
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17 | |
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18 | {\it G4MuonNucleusInteractionModel} calculates the cross section and final |
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19 | states of the muon and hadronic secondaries. The final muon momentum is |
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20 | given by a double-differential cross section which depends on the |
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21 | photoabsorption cross sections for longitudinally and transversely polarized |
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22 | photons. The final hadronic state is determined by replacing the virtual |
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23 | photon with a charged pion of the same $Q^2$ and then allowing the pion to |
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24 | interact with the nucleus. The charge of the pion is chosen at random. The |
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25 | pion interactions with the nucleus are modeled by processes derived from the |
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26 | GHEISHA \cite{GHEISHA} package. These processes are: \\ |
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27 | |
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28 | \begin{tabular}[t]{ll} |
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29 | {\it G4LEPionPlusInelastic}, {\it G4LEPionMinusInelastic} & $E \leq 25$ GeV \\ |
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30 | {\it G4HEPionPlusInelastic}, {\it G4HEPionMinusInelastic} & $E > 25$ GeV \\ |
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31 | \end{tabular} |
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32 | \\ |
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33 | |
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34 | \subsection{Cross Section Calculation} |
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35 | |
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36 | The cross section for the above process in a material is given roughly by |
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37 | |
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38 | \begin{eqnarray*} |
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39 | \sigma_{\mu A} = A \sigma_{\mu N} |
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40 | \end{eqnarray*} |
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41 | |
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42 | \noindent where $A$ is the atomic mass number of the material and |
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43 | $\sigma_{\mu N}$ is the cross section for the process on a single nucleon: |
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44 | |
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45 | \[ |
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46 | \sigma_{\mu N} = |
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47 | \left\{ \begin{array}{ll} |
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48 | 0.3 & (E \leq 30GeV) \\ |
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49 | 0.3 (E/30)^{0.25} & (E > 30GeV) \\ |
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50 | \end{array} \right. [\mu b] . |
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51 | \] |
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52 | |
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53 | \noindent |
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54 | The differential cross section, in terms of muon energy $E$ and emission solid |
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55 | angle $\Omega$, can be expressed as: |
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56 | |
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57 | \begin{eqnarray*} |
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58 | \frac{d\sigma}{d\Omega dE} =\Gamma\,(\sigma_T + \epsilon \sigma_L) |
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59 | \end{eqnarray*} |
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60 | where $\sigma_L$ and $\sigma_T$ are the photoabsorption cross sections for |
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61 | longitudinal and transverse photons, respectively. $\Gamma$ is the transverse |
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62 | photon flux and $\epsilon$ is the polarization of the intermediate photon. |
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63 | The photoabsorption cross sections are parameterized as: |
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64 | \begin{eqnarray*} |
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65 | \sigma_L &=& 0.3\,\left( 1 - \frac{1}{1.868} Q^2 \nu \right)\,\sigma_T \\ |
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66 | \sigma_T &\sim& const = 0.12 mb \\ |
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67 | \end{eqnarray*} |
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68 | |
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69 | \noindent while the flux and polarization are given by |
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70 | \begin{eqnarray*} |
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71 | \Gamma &=& \frac{K \alpha}{2\pi} \frac{E^\prime}{E} \frac{1}{1-\epsilon} \\ |
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72 | \epsilon &=& \left[ 1 + 2 \frac{Q^2 + \nu^2}{Q^2} tan^2 \frac{\theta}{2} \right]^{-1} . \\ |
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73 | \end{eqnarray*} |
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74 | |
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75 | \noindent |
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76 | $E$ and $E^{\prime}$ are the initial and final muon energies, $Q^2$ and $\nu$ |
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77 | are the scaling variables |
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78 | \begin{eqnarray*} |
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79 | Q^2 &=& -q^2 = 2 (EE^{\prime} - PP^{\prime} cos \theta - m_\mu^2) \\ |
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80 | \nu &=& E - E^{\prime} , \\ |
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81 | \end{eqnarray*} |
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82 | and $K$ is given using the Gilman convention |
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83 | \begin{eqnarray*} |
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84 | K = \nu + \frac{Q^2}{2\nu} . |
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85 | \end{eqnarray*} |
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86 | |
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87 | |
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88 | \section{Status of this document} |
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89 | 20.04.02 re-written by D.H. Wright \\ |
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90 | 23.10.98 created by M.Takahata \\ |
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91 | |
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92 | \begin{latexonly} |
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93 | |
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94 | \begin{thebibliography}{99} |
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95 | \bibitem{GHEISHA} H.Fesefeldt |
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96 | {\em GHEISHA The Simulation of Hadronic Showers} 149 |
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97 | {\em RWTH/PITHA 85/02} (1985) |
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98 | \end{thebibliography} |
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99 | |
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100 | \end{latexonly} |
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101 | |
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102 | \begin{htmlonly} |
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103 | |
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104 | \section{Bibliography} |
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105 | |
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106 | \begin{enumerate} |
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107 | \item H.Fesefeldt |
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108 | {\em GHEISHA The Simulation of Hadronic Showers} 149 |
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109 | {\em RWTH/PITHA 85/02} (1985) |
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110 | \end{enumerate} |
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111 | |
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112 | \end{htmlonly} |
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113 | |
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