1 | \section[Ionization]{Ionization} \label{sec:em.eion} |
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2 | |
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3 | \subsection{Method} |
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4 | |
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5 | The $G4eIonisation$ class provides the continuous and discrete |
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6 | energy losses of electrons and positrons due to ionization in a material |
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7 | according to the approach described in Section \ref{en_loss}. |
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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{eion.c} |
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12 | T_{max} = \left\{ \begin{array}{ll} |
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13 | E-mc^2 & {for \hspace{.2cm} e^+} \\ |
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14 | (E-mc^2)/2 & {for \hspace{.2cm} e^- } \\ |
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15 | \end{array} \right . |
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16 | \end{equation} |
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17 | where $mc^2$ is the electron mass. |
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18 | Above a given threshold energy the energy loss is simulated by the |
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19 | explicit production of delta rays by M\"{o}ller scattering ($e^- e^-$), or |
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20 | Bhabha scattering ($e^+ e^-$). Below the threshold the soft electrons |
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21 | ejected are simulated as continuous energy loss by the incident |
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22 | ${e^{\pm}}$. |
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23 | |
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24 | \subsection{Continuous Energy Loss} \label{seceloss} |
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25 | |
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26 | The integration of \ref{comion.a} leads to the Berger-Seltzer |
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27 | formula \cite{eion.messel}: |
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28 | \begin{equation} |
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29 | \label{eion.d e} |
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30 | \left. \frac{dE}{dx} \right]_{T < T_{cut}} = |
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31 | 2 \pi r_e^2 mc^2 n_{el} \frac{1}{\beta^2} |
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32 | \left [\ln \frac{2(\gamma + 1)} {(I/mc^2)^2}+ F^{\pm} (\tau , \tau_{up}) |
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33 | - \delta \right ] |
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34 | \end{equation} |
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35 | with |
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36 | \[ |
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37 | \begin{array}{ll} |
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38 | r_e & \mbox{classical electron radius:} |
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39 | \quad e^2/(4 \pi \epsilon_0 mc^2 ) \\ |
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40 | mc^2 & \mbox{mass energy of the electron} \\ |
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41 | n_{el} & \mbox{electron density in the material} \\ |
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42 | I & \mbox{mean excitation energy in the material}\\ |
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43 | \gamma & \mbox{$E/mc^2$} \\ |
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44 | \beta^2 & 1-(1/\gamma^2) \\ |
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45 | \tau & \gamma-1 \\ |
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46 | T_{cut} & \mbox{minimum energy cut for $\delta$ -ray production} \\ |
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47 | \tau_c & \mbox{$T_{cut}/mc^2$} \\ |
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48 | \tau_{max} & \mbox{maximum energy transfer: |
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49 | $\tau$ for $e^+$, $\tau/2$ for $e^-$} \\ |
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50 | \tau_{up} & \min(\tau_c,\tau_{max}) \\ |
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51 | \delta & \mbox{density effect function} . |
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52 | \end{array} |
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53 | \] |
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54 | In an elemental material the electron density is |
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55 | $$ n_{el} = Z \: n_{at} = Z \: \frac{\mathcal{N}_{av} \rho}{A} . $$ |
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56 | $\mathcal{N}_{av}$ is Avogadro's number, $\rho$ is the material density, |
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57 | and $A$ is the mass of a mole. In a compound material |
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58 | $$ |
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59 | n_{el} = \sum_i Z_i \: n_{ati} |
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60 | = \sum_i Z_i \: \frac{\mathcal{N}_{av} w_i \rho}{A_i} , |
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61 | $$ |
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62 | where $w_i$ is the proportion by mass of the $i^{th}$ element, with molar |
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63 | mass $A_i$ . |
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64 | \par |
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65 | \noindent |
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66 | The mean excitation energies $I$ for all elements are taken from |
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67 | \cite{ioni.icru1}. |
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68 | \par |
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69 | \noindent |
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70 | The functions $ F^{\pm}$ are given by : |
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71 | \begin{eqnarray} |
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72 | F^+ (\tau,\tau_{up}) & = &\ln(\tau\tau_{up} ) \\ |
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73 | & & -\frac{\tau_{up}^2}{\tau}\left[\tau + 2 \tau_{up} - |
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74 | \frac{3\tau_{up}^2 y } {2} -\left(\tau_{up} - \frac{\tau_{up}^3 }{3} \right) y^2 |
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75 | - \left (\frac{\tau_{up}^2}{2} - \tau |
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76 | \frac{\tau_{up}^3}{3} + \frac{\tau_{up}^4 } {4} \right) |
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77 | y^3 \right] \nonumber |
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78 | \end{eqnarray} |
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79 | |
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80 | \begin{eqnarray} |
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81 | F^- (\tau,\tau_{up} ) & = & -1 -\beta^2 \\ |
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82 | & & +\ln \left [(\tau - \tau_{up}) |
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83 | \tau_{up} \right ] + \frac{\tau}{\tau -\tau_{up}} |
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84 | + \frac{1}{\gamma^2} \left [ |
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85 | \frac{\tau_{up}^2}{2} + ( 2\tau +1) \ln |
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86 | \left (1- \frac{\tau_{up}}{\tau} \right ) \right ] \nonumber |
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87 | \end{eqnarray} |
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88 | where $y = 1/(\gamma+1)$. |
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89 | |
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90 | |
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91 | The density effect correction is calculated according to the formalism of |
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92 | Sternheimer \cite{eion.sternheimer}: |
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93 | \input{electromagnetic/utils/densityeffect} |
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94 | |
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95 | \subsection{Total Cross Section per Atom and Mean Free Path } \label{sectot} |
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96 | |
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97 | The total cross section per atom for M\"{o}ller scattering ($e^- e^-$) and |
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98 | Bhabha scattering ($e^+ e^-$) is obtained by integrating Eq.~\ref{comion.b}. |
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99 | In {\sc Geant4} $T_{cut}$ is always 1 keV or larger. For delta ray energies |
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100 | much larger than the excitation energy of the material ($T \gg I$), the |
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101 | total cross section becomes \cite{eion.messel} for M\"{o}ller scattering, |
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102 | \begin{eqnarray} |
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103 | \sigma ( Z,E,T_{cut} ) & = & \frac {2 \pi r_e^2 Z}{\beta^2(\gamma -1)} \times \\ |
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104 | & & \left[\frac{(\gamma-1)^2} {\gamma^2}\left(\frac{1}{2}-x\right) |
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105 | +\frac{1}{x}-\frac{1}{1-x}-\frac{2\gamma-1}{\gamma^2}\ln |
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106 | \frac{1-x}{x}\right] , \nonumber |
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107 | \end{eqnarray} |
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108 | and for Bhabha scattering ($e^+ e^-$), |
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109 | \begin{eqnarray} |
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110 | \sigma (Z,E,T_{cut}) & = & \frac{ 2 \pi r_e^2 Z }{(\gamma -1)} \times \\ |
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111 | & & \left [\frac {1 }{\beta^2} \left(\frac{1}{x}-1\right) |
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112 | + B_1 \ln x + B_2 (1-x) - |
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113 | \frac {B_3 } {2} ( 1-x^2 ) +\frac{B_4}{3}(1-x^3)\right] . \nonumber |
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114 | \end{eqnarray} |
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115 | Here |
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116 | \[ |
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117 | \begin{array}{lcllcl} |
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118 | \gamma & = & E/mc^2 & |
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119 | B_1 & = & 2-y^2 \\ |
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120 | \beta^2 & = & 1-(1/\gamma^2) & |
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121 | B_2 & = & (1-2y)(3+y^2 ) \\ |
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122 | x & = & T_{cut}/(E-mc^2) & |
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123 | B_3 & = & (1-2y)^2+(1-2y)^3 \\ |
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124 | y & = & 1/(\gamma + 1) & |
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125 | B_4 & = & (1-2y)^3 . |
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126 | \end{array} |
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127 | \] |
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128 | The above formulas give the total cross section for scattering above the |
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129 | threshold energies |
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130 | |
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131 | \begin{equation} |
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132 | T_{\rm Moller}^{\rm thr} =2T_{cut} \mbox{\hspace{2cm}and\hspace{2cm}} |
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133 | T_{\rm Bhabha}^{\rm thr} = T_{cut} . |
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134 | \end{equation} |
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135 | |
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136 | \noindent |
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137 | In a given material the mean free path is then |
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138 | \begin{equation} |
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139 | \begin{array}{lll} |
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140 | \lambda = (n_{at} \cdot \sigma)^{-1} & or & |
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141 | \lambda = \left( \sum_i n_{ati} \cdot \sigma_i \right)^{-1} . |
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142 | \end{array} |
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143 | \end{equation} |
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144 | |
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145 | \subsection{Simulation of Delta-ray Production} |
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146 | \subsubsection{Differential Cross Section} |
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147 | |
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148 | For $T \gg I$ the differential cross section per atom becomes |
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149 | \cite{eion.messel} for M\"{o}ller scattering, |
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150 | \begin{eqnarray} |
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151 | \label{eion.i} |
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152 | \frac{d\sigma }{d \epsilon } &=& \frac{2 \pi r_e^2 Z}{\beta^2 (\gamma -1)} |
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153 | \times \\ |
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154 | & & \left[ \frac{(\gamma -1 )^2} {\gamma^2 }+\frac{1}{\epsilon} |
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155 | \left(\frac{1}{\epsilon}-\frac{2 \gamma -1 } {\gamma^2 } \right) + |
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156 | \frac{1}{1- \epsilon}\left(\frac{1} {1- \epsilon} - \frac{2 \gamma - 1} |
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157 | {\gamma^2 }\right) \right] \nonumber |
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158 | \end{eqnarray} |
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159 | and for Bhabha scattering, |
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160 | \begin{equation} |
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161 | \label{eion.j} |
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162 | \frac{d \sigma}{d \epsilon}=\frac{2 \pi r_e^2 Z}{(\gamma -1)}\left[ |
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163 | \frac{1} {\beta^2 \epsilon^2}-\frac{B_1}{\epsilon}+B_2 - B_3 \epsilon |
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164 | + B_4 \epsilon^2\right] . |
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165 | \end{equation} |
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166 | Here $\epsilon = T/(E-mc^2)$. The kinematical limits of $\epsilon$ are |
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167 | \[ |
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168 | \epsilon_0 = \frac{T_{cut}}{E-mc^2} \leq \epsilon \leq \frac{1}{2} |
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169 | \mbox{\hspace{.2cm} for $e^- e^-$} \hspace{2cm} |
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170 | \epsilon_0 = \frac{T_{cut}}{E-mc^2} \leq \epsilon \leq 1 |
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171 | \mbox{\hspace{.2cm} for $e^+ e^-$} . |
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172 | \] |
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173 | |
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174 | \subsubsection{Sampling} |
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175 | The delta ray energy is sampled according to methods discussed in |
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176 | Chapter \ref{secmessel}. Apart from normalization, the cross section can |
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177 | be factorized as |
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178 | \begin{equation} |
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179 | \frac{d\sigma}{d\epsilon}=f(\epsilon) g(\epsilon) . |
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180 | \end{equation} |
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181 | For $e^- e^-$ scattering |
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182 | \begin{eqnarray} |
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183 | f(\epsilon)&=&\frac{1}{\epsilon^2} \frac{\epsilon_0 }{1- 2\epsilon_0} \\ |
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184 | g(\epsilon)&=&\frac{4}{9\gamma^2 - 10 \gamma + 5}\left[(\gamma -1)^2 |
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185 | \epsilon^2 - (2 \gamma^2 +2\gamma -1) \frac{\epsilon} {1- \epsilon }+ |
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186 | \frac{\gamma^2}{(1- \epsilon )^2 }\right] |
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187 | \end{eqnarray} |
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188 | and for $e^+ e^-$ scattering |
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189 | \begin{eqnarray} |
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190 | f(\epsilon)&=&\frac{1}{\epsilon^2} \frac{\epsilon_0}{1- \epsilon_0 } \\ |
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191 | g(\epsilon)&=&\frac{B_0 -B_1 \epsilon +B_2 \epsilon^2 |
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192 | -B_3 \epsilon^3 +B_4 \epsilon ^4}{B_ 0-B_1\epsilon_0 |
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193 | +B_2\epsilon_0^2 |
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194 | -B_3 \epsilon_0^3 +B_4 \epsilon_0^4} . |
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195 | \end{eqnarray} |
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196 | Here $ B_0=\gamma^2/(\gamma^2-1)$ and all other quantities have been defined |
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197 | above. |
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198 | |
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199 | |
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200 | To choose $\epsilon$, and hence the delta ray energy, |
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201 | \begin{enumerate} |
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202 | \item $\epsilon$ is sampled from $f(\epsilon)$ |
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203 | \item the rejection function $g(\epsilon)$ is calculated using the sampled |
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204 | value of $\epsilon$ |
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205 | \item $\epsilon$ is accepted with probability $g(\epsilon)$. |
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206 | \end{enumerate} |
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207 | After the successful sampling of $\epsilon$, the direction of the ejected |
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208 | electron is generated with respect to the direction of the incident |
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209 | particle. The azimuthal angle $\phi$ is generated isotropically and the |
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210 | polar angle $\theta$ is calculated from energy-momentum conservation. |
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211 | This information is used to calculate the energy and momentum of both the |
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212 | scattered incident particle and the ejected electron, and to transform them |
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213 | to the global coordinate system. |
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214 | |
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215 | \subsection{Status of this document} |
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216 | \ 9.10.98 created by L. Urb\'an. \\ |
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217 | 29.07.01 revised by M.Maire. \\ |
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218 | 13.12.01 minor cosmetic by M.Maire. \\ |
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219 | 24.05.02 re-written by D.H. Wright. \\ |
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220 | 01.12.03 revised by V. Ivanchenko. \\ |
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221 | |
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222 | \begin{latexonly} |
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223 | |
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224 | \begin{thebibliography}{99} |
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225 | |
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226 | \bibitem{eion.messel} |
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227 | H.~Messel and D.F.~Crawford. {\em Pergamon Press, Oxford (1970).} |
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228 | \bibitem{ioni.icru1} |
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229 | ICRU (A.~Allisy et al), Stopping Powers for Electrons and Positrons, |
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230 | {\em ICRU Report No.37 (1984).} |
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231 | \bibitem{eion.sternheimer} |
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232 | R.M.~Sternheimer. {\em Phys.Rev. B3 (1971) 3681.} |
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233 | \end{thebibliography} |
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234 | |
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235 | \end{latexonly} |
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236 | |
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237 | \begin{htmlonly} |
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238 | |
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239 | \subsection{Bibliography} |
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240 | |
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241 | \begin{enumerate} |
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242 | \item H.~Messel and D.F.~Crawford. {\em Pergamon Press, Oxford (1970).} |
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243 | \item ICRU (A.~Allisy et al), Stopping Powers for Electrons and Positrons, |
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244 | {\em ICRU Report No.37 (1984).} |
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245 | \item R.M.~Sternheimer. {\em Phys.Rev. B3 (1971) 3681.} |
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246 | \end{enumerate} |
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247 | |
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248 | \end{htmlonly} |
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249 | |
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