1 | \section[Ionization]{Ionization} |
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2 | |
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3 | \subsection{Method} |
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4 | |
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5 | The class $G4MuIonisation$ provides the continuous energy loss due to |
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6 | ionization and simulates the 'discrete' part of the ionization, that is delta |
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7 | rays produced by muons. The approach described in Section \ref{en_loss} is used. |
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8 | The value of the maximum energy transferable to a free electron $T_{max}$ |
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9 | is given by the following relation: |
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10 | \begin{equation} |
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11 | \label{muion.c} |
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12 | T_{max} =\frac{2mc^2(\gamma^2 -1)}{1+2\gamma (m/M)+(m/M)^2 } . |
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13 | \end{equation} |
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14 | Here $m$ is the electron mass and $M$ the muon mass. The method of |
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15 | calculation of the continuous energy loss and the total cross section are |
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16 | explained below. |
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17 | |
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18 | \subsection{Continuous Energy Loss} |
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19 | |
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20 | The integration of \ref{comion.a} leads to the Bethe-Bloch restricted energy |
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21 | loss formula \cite{muion.pdg} : |
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22 | \begin{equation} |
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23 | \label{muion1} |
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24 | \left. \frac{dE}{dx} \right]_{T < T_{cut}} = |
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25 | 2 \pi r_e^2 mc^2 n_{el} \frac{(z_p)^2}{\beta^2} |
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26 | \left [\ln \left(\frac{2mc^2 \beta^2 \gamma^2 T_{up}} {I^2} \right) |
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27 | - \beta^2 \left( 1 + \frac{T_{up}}{T_{max}} \right) |
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28 | - \delta - \frac{2C_e}{Z} \right ] |
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29 | \end{equation} |
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30 | where |
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31 | \[ |
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32 | \begin{array}{ll} |
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33 | r_e & \mbox{classical electron radius:} |
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34 | \quad e^2/(4 \pi \epsilon_0 mc^2 ) \\ |
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35 | mc^2 & \mbox{mass-energy of the electron} \\ |
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36 | n_{el} & \mbox{electrons density in the material} \\ |
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37 | I & \mbox{mean excitation energy in the material}\\ |
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38 | \gamma & \mbox{$E/mc^2$} \\ |
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39 | \beta^2 & 1-(1/\gamma^2) \\ |
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40 | T_{up} & \min(T_{cut},T_{max}) \\ |
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41 | \delta & \mbox{density effect function} \\ |
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42 | C_e & \mbox{shell correction function} |
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43 | \end{array} |
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44 | \] |
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45 | In a single element the electron density is |
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46 | $$ n_{el} = Z \: n_{at} = Z \: \frac{\mathcal{N}_{av} \rho}{A} $$ |
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47 | ($\mathcal{N}_{av}$: Avogadro number, $\rho$: density of the material, |
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48 | $A$: mass of a mole). In a compound material |
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49 | $$ |
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50 | n_{el} = \sum_i Z_i \: n_{ati} |
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51 | = \sum_i Z_i \: \frac{\mathcal{N}_{av} w_i \rho}{A_i} . |
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52 | $$ |
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53 | $w_i$ is the proportion by mass of the $i^{th}$ element, with molar mass |
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54 | $A_i$. |
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55 | |
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56 | |
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57 | The mean excitation energy, $I$, for all elements is tabulated according to the |
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58 | ICRU recommended values \cite{muion.icru1}. |
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59 | |
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60 | \subsubsection{Density Correction} |
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61 | |
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62 | $\delta$ is a correction term which takes into account the reduction |
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63 | in energy loss due to the so-called {\it density effect}. This becomes |
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64 | important at high energy because media have a tendency to become |
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65 | polarised as the incident particle velocity increases. As a consequence, |
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66 | the atoms in a medium can no longer be considered as isolated. To correct |
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67 | for this effect the formulation of Sternheimer~\cite{muion.sternheimer} |
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68 | is used: |
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69 | \input{electromagnetic/utils/densityeffect} |
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70 | |
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71 | \subsubsection{Shell Correction} |
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72 | |
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73 | $2C_e/Z$ is the so-called {\it shell correction term} which accounts for the |
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74 | fact that, at low energies for light elements and at all energies for heavy |
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75 | ones, the probability of collision with the electrons of the inner atomic |
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76 | shells (K, L, etc.) is negligible. The semi-empirical formula used in |
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77 | {\sc Geant4}, applicable to all materials, is due to |
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78 | Barkas \cite{muion.barkas}: |
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79 | \begin{equation} |
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80 | \label{muion2} |
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81 | C_e(I, \beta\gamma) = \frac{a(I)}{(\beta\gamma)^2} |
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82 | +\frac{b(I)}{(\beta\gamma)^4} |
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83 | +\frac{c(I)}{(\beta\gamma)^6} . |
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84 | \end{equation} |
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85 | The functions a(I), b(I), c(I) can be found in the source code. This formula |
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86 | breaks down at low energies, and is valid only when |
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87 | $\beta\gamma > 0.13$ ($T > 7.9$ MeV for a proton). For $\beta\gamma \leq |
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88 | 0.13$ the shell correction term is calculated as: |
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89 | \begin{equation} |
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90 | \label{muion3} |
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91 | \left . C_{e}(I,\beta\gamma) \rule{0mm}{5mm} \right |_{\beta\gamma \leq 0.13} |
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92 | = C_{e}(I,\beta\gamma=0.13)\frac{\ln(T/T_{2l})}{\ln(7.9 \: \rm MeV/T_{2l})} |
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93 | \end{equation} |
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94 | i.e. the correction is switched off logarithmically from $T=7.9$ MeV |
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95 | to $T=T_{2l}=2$ MeV. |
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96 | |
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97 | |
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98 | \subsubsection{Parameterization} |
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99 | |
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100 | The mean energy loss can be described by the Bethe-Bloch formula |
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101 | (\ref{muion1}) only if the projectile velocity is larger than that of the |
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102 | orbital electrons. In the low energy region this is not the case, and the |
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103 | parameterization from the ICRU'49 report \cite{muion.ICRU49} |
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104 | is used in the $G4BraggModel$ class. The Bethe-Bloch model is applied to |
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105 | muons of higher kinetic energies |
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106 | \begin{equation} |
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107 | \label{muion.1} |
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108 | T > 2 * M_{\mu}/M_{proton} MeV. |
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109 | \end{equation} |
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110 | The details of the low energy parameterization are described in |
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111 | Section \ref{le_had_ion}. |
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112 | |
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113 | \subsection{Total Cross Section per Atom and Mean Free Path} |
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114 | |
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115 | For $T \gg I $ the differential cross section can be written as |
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116 | \cite{muion.pdg} |
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117 | \begin{equation} |
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118 | \label{muion.i} |
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119 | \frac{d\sigma }{dT} = 2\pi r_e^2 mc^2 Z \frac{z_p^2}{\beta^2} \frac{1}{T^2} |
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120 | \left[ 1 - \beta^2 \frac{T}{T_{max}} + \frac{T^2}{2E^2} \right] . |
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121 | \end{equation} |
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122 | In {\sc Geant4} $T_{cut} \geq 1$ keV. Integrating from $T_{cut}$ to |
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123 | $T_{max}$ gives the total cross-section per atom : |
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124 | \begin{eqnarray} |
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125 | \sigma (Z,E,T_{cut}) & = & \frac {2\pi r_e^2 Z z_p^2}{\beta^2}mc^2 \times |
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126 | \\ & & \left[ \left( \frac{1}{T_{cut}} - \frac{1}{T_{max}} \right) |
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127 | - \frac{\beta^2}{T_{max}} \ln \frac{T_{max}}{T_{cut}} |
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128 | + \frac{T_{max} - T_{cut}}{2E^2} |
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129 | \right] . \nonumber |
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130 | \end{eqnarray} |
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131 | In a given material the mean free path is |
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132 | \begin{equation} |
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133 | \begin{array}{lll} |
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134 | \lambda = (n_{at} \cdot \sigma)^{-1} & or & |
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135 | \lambda = \left( \sum_i n_{ati} \cdot \sigma_i \right)^{-1} . |
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136 | \end{array} |
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137 | \end{equation} |
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138 | The mean free path is tabulated during initialization as a function of the |
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139 | material and of the energy of the incident muon. |
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140 | |
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141 | \subsection{Simulating Delta-ray Production} |
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142 | |
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143 | A short overview of the sampling method is given in Chapter \ref{secmessel}. |
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144 | Apart from the normalization, the cross section \ref{muion.i} can be |
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145 | factorized : |
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146 | \begin{eqnarray} |
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147 | \frac{d\sigma}{dT}=f(T) g(T) &with& T \in [T_{cut}, \ T_{max}] |
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148 | \end{eqnarray} |
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149 | where |
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150 | \begin{eqnarray} |
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151 | f(T) &=& \left(\frac{1}{T_{cut}} - \frac{1}{T_{max}} \right) \frac{1}{T^2} \\ |
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152 | g(T) &=& 1 - \beta^2 \frac{T}{T_{max}} + \frac{T^2}{2E^2} . |
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153 | \end{eqnarray} |
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154 | The energy $T$ is chosen by |
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155 | \begin{enumerate} |
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156 | \item sampling $T$ from $f(T)$ |
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157 | \item calculating the rejection function $g(T)$ and accepting the |
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158 | sampled $T$ with a probability of $g(T)$. |
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159 | \end{enumerate} |
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160 | After successful sampling of the energy, the direction of the scattered |
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161 | electron is generated with respect to the direction of the incident muon. |
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162 | The azimuthal angle $\phi$ is generated isotropically. The polar angle |
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163 | $\theta$ is calculated from energy-momentum conservation. This information |
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164 | is used to calculate the energy and momentum of both scattered |
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165 | particles and to transform them into the {\em global} coordinate system. |
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166 | |
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167 | \subsection{Status of this document} |
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168 | 09.10.98 created by L. Urb\'an. \\ |
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169 | 14.12.01 revised by M.Maire \\ |
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170 | 30.11.02 re-worded by D.H. Wright \\ |
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171 | 01.12.03 revised by V. Ivanchenko \\ |
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172 | |
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173 | \begin{latexonly} |
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174 | |
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175 | \begin{thebibliography}{99} |
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176 | |
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177 | \bibitem{muion.pdg} |
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178 | Particle Data Group. Rev. of Particle Properties. |
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179 | Eur. Phys. J. C15. (2000) 1. http://pdg.lbl.gov |
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180 | \bibitem{muion.icru1} |
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181 | ICRU Report No. 37 (1984) |
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182 | \bibitem{muion.sternheimer} |
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183 | R.M.Sternheimer. Phys.Rev. B3 (1971) 3681. |
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184 | \bibitem{muion.barkas} |
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185 | W. H. Barkas. Technical Report 10292,UCRL, August 1962. |
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186 | \bibitem{muion.ICRU49}ICRU (A.~Allisy et al), |
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187 | Stopping Powers and Ranges for Protons and Alpha |
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188 | Particles, |
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189 | ICRU Report 49, 1993. |
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190 | \end{thebibliography} |
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191 | |
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192 | \end{latexonly} |
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193 | |
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194 | \begin{htmlonly} |
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195 | |
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196 | \subsection{Bibliography} |
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197 | |
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198 | \begin{enumerate} |
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199 | \item Particle Data Group. Rev. of Particle Properties. |
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200 | Eur. Phys. J. C15. (2000) 1. http://pdg.lbl.gov |
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201 | \item ICRU Report No. 37 (1984) |
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202 | \item R.M.Sternheimer. Phys.Rev. B3 (1971) 3681. |
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203 | \item W.H. Barkas. Technical Report 10292,UCRL, August 1962. |
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204 | \item ICRU (A.~Allisy et al), |
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205 | Stopping Powers and Ranges for Protons and Alpha |
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206 | Particles, ICRU Report 49, 1993. |
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207 | \end{enumerate} |
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208 | |
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209 | \end{htmlonly} |
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