1 | |
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2 | \section[Gamma Conversion into an Electron - Positron Pair] |
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3 | {Gamma Conversion into an Electron - Positron Pair}\label{sec:em.conv} |
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4 | |
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5 | \subsection{Cross Section and Mean Free Path} |
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6 | |
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7 | \subsubsection{Cross Section per Atom} |
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8 | |
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9 | The total cross-section per atom for the conversion of a gamma into an |
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10 | $(e^+,e^-)$ pair has been parameterized as |
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11 | \begin{equation} \label{conv1} |
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12 | \sigma(Z,E_\gamma) = Z(Z+1) \: \left[ F_1(X) + F_2(X) \: Z + \frac{F_3(X)}{Z} |
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13 | \right], |
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14 | \end{equation} |
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15 | where $E_{\gamma}$ is the incident gamma energy and $X = \ln (E_{\gamma}/m_{e}c^2)$ . The functions $F_n$ are given by |
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16 | \begin{eqnarray} |
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17 | F_1(X) & = & a_0 + a_1 X + a_2 X^2 + a_3 X^3 + a_4 X^4 + a_5 X^5 \\ |
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18 | F_2(X) & = & b_0 + b_1 X + b_2 X^2 + b_3 X^3 + b_4 X^4 + b_5 X^5 \nonumber \\ |
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19 | F_3(X) & = & c_0 + c_1 X + c_2 X^2 + c_3 X^3 + c_4 X^4 + c_5 X^5 , \nonumber |
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20 | \end{eqnarray} |
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21 | with the parameters $a_i, b_i, c_i$ taken from a least-squares fit to the |
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22 | data \cite{conv.hubb}. Their values can be found in the function which |
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23 | computes formula \ref{conv1}. |
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24 | \\ |
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25 | This parameterization describes the data in the range |
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26 | \begin{eqnarray*} |
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27 | 1\leq Z\leq 100 |
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28 | \end{eqnarray*} |
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29 | and |
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30 | \begin{eqnarray*} |
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31 | E_\gamma \in [1.5 \mbox{ MeV} , 100 \mbox{ GeV}] . |
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32 | \end{eqnarray*} |
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33 | The accuracy of the fit was estimated to be |
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34 | $\frac{\Delta\ \sigma}{\sigma}\leq 5\% $ with a mean value of |
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35 | $\approx 2.2\%$. Above 100 GeV the cross section is constant. Below |
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36 | $E_{low} = 1.5 \mbox{ MeV}$ the extrapolation |
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37 | \begin{equation} |
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38 | \sigma(E) = \sigma(E_{low}) \cdot \left( \frac{E-2m_e c^2}{E_{low}-2m_e c^2} |
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39 | \right)^2 |
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40 | \end{equation} |
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41 | is used. |
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42 | |
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43 | \subsubsection{Mean Free Path} |
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44 | In a given material the mean free path, $\lambda$, for a photon to convert into |
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45 | an $(e^+,e^-)$ pair is |
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46 | \begin{equation} |
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47 | \lambda(E_{\gamma}) = |
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48 | \left( \sum_i n_{ati} \cdot \sigma (Z_i,E_{\gamma}) \right)^{-1} |
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49 | \end{equation} |
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50 | where $n_{ati}$ is the number of atoms per volume of the $i^{th}$ element of |
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51 | the material. |
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52 | |
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53 | \subsection{Final State} |
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54 | \subsubsection{Choosing an Element} |
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55 | The differential cross section depends on the atomic number $Z$ of the |
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56 | material in which the interaction occurs. In a compound material the element |
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57 | $i$ in which the interaction occurs is chosen randomly according to the |
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58 | probability |
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59 | \begin{equation} |
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60 | Prob(Z_i,E_{\gamma}) = |
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61 | \frac{n_{ati} \sigma(Z_i,E_{\gamma})} |
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62 | {\sum_i [ n_{ati} \cdot \sigma_i (E_{\gamma})]} . |
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63 | \end{equation} |
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64 | |
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65 | \subsubsection{Corrected Bethe-Heitler Cross Section} |
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66 | As written in \cite{conv.heit}, the Bethe-Heitler formula corrected for various |
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67 | effects is |
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68 | \begin{eqnarray} \label{conv.eq1} |
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69 | \frac{d \sigma(Z,\epsilon)}{d \epsilon} & = & \alpha r_e^2 Z [Z + \xi(Z)] |
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70 | \left \{ [\epsilon^2 + ( 1 -\epsilon)^2] |
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71 | \left[ \Phi_1(\delta(\epsilon)) - \frac{F(Z)}{2} \right] \right. \nonumber \\ |
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72 | & & + \left. \frac{2}{3}\epsilon (1-\epsilon) |
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73 | \left[ \Phi_2(\delta(\epsilon)) - \frac{F(Z)}{2} \right] \right \} |
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74 | \end{eqnarray} |
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75 | where $\alpha$ is the fine-structure constant and $r_e$ the classical electron |
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76 | radius. Here $\epsilon = E/E_\gamma$, $E_\gamma$ is the energy of the photon |
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77 | and $E$ is the total energy carried by one particle of the $(e^+,e^-)$ pair. |
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78 | The kinematical limits of $\epsilon$ are therefore |
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79 | \begin{equation} |
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80 | \frac{m_e c^2}{E_\gamma} = \epsilon_0 \leq \epsilon \leq 1-\epsilon_0 . |
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81 | \end{equation} |
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82 | |
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83 | \paragraph*{Screening Effect} |
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84 | The {\it screening variable}, $\delta$, is a function of $\epsilon$ |
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85 | \begin{equation} |
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86 | \delta(\epsilon)=\frac{136}{Z^{1/3}} \ \frac{\epsilon_0}{\epsilon(1-\epsilon)}, |
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87 | \end{equation} |
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88 | and measures the 'impact parameter' of the projectile. Two screening |
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89 | functions are introduced in the Bethe-Heitler formula : |
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90 | \begin{eqnarray} |
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91 | \mbox{for } \delta \leq 1 & \Phi_1 (\delta) = |
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92 | & 20.867 - 3.242 \delta + 0.625 \delta^2 \\ |
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93 | & \Phi_2 (\delta) = |
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94 | & 20.209 - 1.930 \delta - 0.086 \delta^2 \nonumber \\ |
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95 | \mbox{for } \delta > 1 & \Phi_1 (\delta) = |
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96 | & \Phi_2 (\delta) = 21.12 - 4.184 \ln(\delta+0.952). |
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97 | \nonumber |
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98 | \end{eqnarray} |
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99 | Because the formula \ref{conv.eq1} is symmetric under the exchange $\epsilon |
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100 | \leftrightarrow (1-\epsilon)$, the range of $\epsilon$ can be restricted to |
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101 | \begin{equation} |
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102 | \epsilon \in [ \epsilon_0 , 1/2] . |
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103 | \end{equation} |
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104 | |
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105 | \paragraph*{Born Approximation} |
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106 | The Bethe-Heitler formula is calculated with plane waves, but Coulomb waves |
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107 | should be used instead. To correct for this, a |
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108 | {\it Coulomb correction function} is introduced in the Bethe-Heitler formula : |
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109 | \begin{eqnarray} |
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110 | \mbox{for $E_\gamma$ } < 50 \mbox{ MeV :} & F(z) = & 8/3 \ln Z \\ |
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111 | \mbox{for $E_\gamma$ } \geq 50 \mbox{ MeV :} & F(z) = & 8/3 \ln Z + 8 f_c (Z) |
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112 | \nonumber |
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113 | \end{eqnarray} |
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114 | with |
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115 | \begin{eqnarray} |
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116 | f_c(Z) &=& (\alpha Z)^2 \left[ \frac{1}{1+(\alpha Z)^2} \right. |
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117 | \\ & & \left. |
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118 | + 0.20206 - 0.0369 (\alpha Z)^2 + 0.0083 (\alpha Z)^4 - 0.0020 (\alpha Z)^6 |
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119 | + \cdots \right]. \nonumber |
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120 | \end{eqnarray} |
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121 | It should be mentioned that, after these additions, the cross section |
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122 | becomes negative if |
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123 | \begin{equation} |
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124 | \delta > \delta_{max} (\epsilon_1) = \exp \left[ \frac{42.24 - F(Z)}{8.368} |
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125 | \right] - 0.952 . |
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126 | \end{equation} |
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127 | This gives an additional constraint on $\epsilon$ : |
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128 | \begin{equation} |
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129 | \delta \leq \delta_{max} \Longrightarrow |
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130 | \epsilon \geq \epsilon_1 = \frac{1}{2} - |
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131 | \frac{1}{2} \sqrt{1-\frac{\delta_{min}}{\delta_{max}}} |
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132 | \end{equation} |
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133 | where |
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134 | \begin{equation} |
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135 | \delta_{min} = \delta \left(\epsilon = \frac{1}{2} \right) |
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136 | = \frac{136}{Z^{1/3}} \ 4 \epsilon_0 |
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137 | \end{equation} |
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138 | has been introduced. Finally the range of $\epsilon$ becomes |
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139 | \begin{equation} |
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140 | \epsilon \in [ \epsilon_{min}=\max (\epsilon_0,\epsilon_1),\ 1/2] . |
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141 | \end{equation} |
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142 | \vspace{0mm} |
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143 | \begin{center} |
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144 | \includegraphics* |
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145 | [width=\textwidth,height=0.4\textheight,draft=false] |
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146 | {electromagnetic/standard/conv.eps} |
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147 | \end{center} |
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148 | \paragraph*{Gamma Conversion in the Electron Field} |
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149 | The electron cloud gives an additional contribution to pair creation, |
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150 | proportional to $Z$ (instead of $Z^2$). This is taken into account through |
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151 | the expression |
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152 | \begin{equation} |
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153 | \xi(Z) = \frac{\ln(1440/Z^{2/3})}{\ln(183/Z^{1/3}) - f_c(Z)} . |
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154 | \end{equation} |
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155 | |
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156 | \paragraph*{Factorization of the Cross Section} |
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157 | $\epsilon$ is sampled using the techniques of 'composition+rejection', as |
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158 | treated in \cite{conv.ford,conv.butch,conv.messel}. First, two auxiliary |
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159 | screening functions should be introduced: |
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160 | \begin{eqnarray} |
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161 | F_1(\delta) &=& 3 \Phi_1(\delta) - \Phi_2(\delta) - F(Z) \nonumber \\ |
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162 | F_2(\delta) &=& \frac{3}{2} \Phi_1(\delta) - \frac{1}{2} \Phi_2(\delta) - F(Z) |
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163 | \end{eqnarray} |
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164 | It can be seen that $F_1(\delta)$ and $F_2(\delta)$ are decreasing |
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165 | functions of $\delta$, $\forall \delta \in [\delta_{min} , \delta_{max}]$. |
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166 | They reach their maximum for $\delta_{min} = \delta(\epsilon = 1/2)$ : |
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167 | \begin{eqnarray} |
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168 | F_{10} = \max F_1(\delta) = F_1(\delta_{min}) \nonumber \\ |
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169 | F_{20} = \max F_2(\delta) = F_2(\delta_{min}) . \, |
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170 | \end{eqnarray} |
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171 | After some algebraic manipulations the formula \ref{conv.eq1} can be written : |
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172 | \begin{eqnarray} \label{conv.eq2} |
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173 | \frac{d \sigma(Z,\epsilon)}{d \epsilon} &=& \alpha r_e^2 Z [Z + \xi(Z)] |
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174 | \frac{2}{9} \left[ \frac{1}{2} - \epsilon_{min} \right] \nonumber \\ |
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175 | & & \times \left[ |
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176 | N_1 \ f_1(\epsilon) \ g_1(\epsilon) + N_2 \ f_2(\epsilon) \ g_2(\epsilon) |
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177 | \right] , |
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178 | \end{eqnarray} |
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179 | where |
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180 | \begin{eqnarray*} |
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181 | N_1 = \left[ \frac{1}{2} - \epsilon_{min} \right]^2 F_{10} \hspace{5mm} & |
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182 | f_1(\epsilon) = \frac{3}{\left[ \frac{1}{2} - \epsilon_{min} \right]^3} |
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183 | \left[ \frac{1}{2} - \epsilon \right]^2 & |
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184 | \hspace{5mm} g_1(\epsilon) = \frac{F_1(\epsilon)}{F_{10}} |
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185 | \\ |
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186 | N_2 = \frac{3}{2} F_{20} \hspace{5mm} & |
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187 | f_2(\epsilon) = \mbox{const} |
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188 | = \frac{1}{\left[ \frac{1}{2} - \epsilon_{min} \right]} & |
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189 | \hspace{5mm} g_2(\epsilon) = \frac{F_2(\epsilon)}{F_{20}} . |
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190 | \end{eqnarray*} |
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191 | $f_1(\epsilon)$ and $f_2(\epsilon)$ are probability density functions on the |
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192 | interval $\epsilon \in [\epsilon_{min} , 1/2]$ such that |
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193 | $$ \int_{\epsilon_{min}}^{1/2} f_i(\epsilon) \, d\epsilon = 1 $$ , |
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194 | and $g_1(\epsilon)$ and $g_2(\epsilon)$ are valid rejection functions: |
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195 | $0 < g_i (\epsilon) \leq 1$ . |
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196 | |
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197 | \paragraph*{Sampling the Energy} |
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198 | Given a triplet of uniformly distributed random numbers $(r_a, r_b, r_c)$ : |
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199 | \begin{enumerate} |
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200 | \item use $r_a$ to choose which decomposition term in \ref{conv.eq2} to use: |
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201 | \begin{equation} |
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202 | \mbox{if } r_a < N_1/(N_1+N_2) \rightarrow f_1(\epsilon)\ g_1(\epsilon) \\ |
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203 | \mbox{ otherwise } \rightarrow f_2(\epsilon)\ g_2(\epsilon) |
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204 | \end{equation} |
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205 | \item sample $\epsilon$ from $f_1(\epsilon)$ or $f_2(\epsilon)$ with $r_b$ : |
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206 | \begin{equation} |
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207 | \epsilon = \frac{1}{2} - \left(\frac{1}{2} - \epsilon_{min} \right) r_b^{1/3} |
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208 | \hspace{5mm} \mbox{or} \hspace{5mm} |
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209 | \epsilon = \epsilon_{min} + \left(\frac{1}{2} - \epsilon_{min} \right) r_b |
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210 | \end{equation} |
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211 | \item reject $\epsilon$ if $g_1(\epsilon) \mbox{or } g_2(\epsilon) < r_c$ |
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212 | \end{enumerate} |
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213 | {\sc note} : |
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214 | below $E_{\gamma} = 2$ MeV it is enough to sample $\epsilon$ uniformly on |
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215 | $[\epsilon_0,\ 1/2]$, without rejection. |
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216 | \paragraph*{Charge} |
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217 | The charge of each particle of the pair is fixed randomly. |
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218 | |
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219 | \subsubsection{Polar Angle of the Electron or Positron} |
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220 | The polar angle of the electron (or positron) is defined with respect to the |
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221 | direction of the parent photon. The energy-angle distribution given by |
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222 | Tsai \cite{conv.tsai} is quite complicated to sample and can be approximated |
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223 | by a density function suggested by Urban \cite{conv.urban} : |
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224 | \begin{equation} \label{conv.eq3} |
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225 | \forall u \in [0,\ \infty [ \hspace{3mm} |
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226 | f(u) = \frac{9a^2}{9+d} \left[ u \exp (-a u) + d\ u \exp (-3a u) \right] |
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227 | \end{equation} |
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228 | with |
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229 | \begin{equation} |
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230 | a=\frac{5}{8} \hspace{5mm} d=27 \hspace{1cm} \mbox{and } |
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231 | \theta_\pm = \frac{mc^2}{E_\pm} \ u . |
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232 | \end{equation} |
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233 | A sampling of the distribution \ref{conv.eq3} requires a triplet of random |
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234 | numbers such that |
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235 | \begin{equation} |
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236 | \mbox{if } r_1 < \frac{9}{9+d} \rightarrow u = \frac{-\ln(r_2 r_3)}{a} |
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237 | \hspace{5mm} \mbox{otherwise } u = \frac{-\ln(r_2 r_3)}{3a} . |
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238 | \end{equation} |
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239 | The azimuthal angle $\phi$ is generated isotropically. |
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240 | \paragraph*{Final State} |
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241 | The $e^+$ and $e^-$ momenta are assumed to be coplanar with the parent photon. |
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242 | This information, together with energy conservation, is used to calculate the |
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243 | momentum vectors of the $(e^+,e^-)$ pair and to rotate them to the global |
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244 | reference system. |
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245 | |
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246 | \subsection{Status of this document} |
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247 | 12.01.02 created by M.Maire. \\ |
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248 | 21.03.02 corrections in angular distribution (mma) \\ |
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249 | 22.04.02 re-worded by D.H. Wright \\ |
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250 | |
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251 | \begin{latexonly} |
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252 | |
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253 | \begin{thebibliography}{99} |
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254 | \bibitem{conv.hubb} J.H.Hubbell, H.A.Gimm, I.Overbo |
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255 | {\it Jou. Phys. Chem. Ref. Data 9:1023} (1980) |
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256 | \bibitem{conv.heit} W. Heitler |
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257 | {\it The Quantum Theory of Radiation, Oxford University Press} (1957) |
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258 | \bibitem{conv.ford} R. Ford and W. Nelson. |
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259 | {\it SLAC-210, UC-32} (1978) |
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260 | \bibitem{conv.butch} J.C. Butcher and H. Messel. |
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261 | {\it Nucl. Phys. 20} 15 (1960) |
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262 | \bibitem{conv.messel} H. Messel and D. Crawford. |
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263 | {\it Electron-Photon shower distribution, Pergamon Press} (1970) |
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264 | \bibitem{conv.tsai} |
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265 | Y. S. Tsai, {\em Rev. Mod. Phys. 46} 815 (1974), |
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266 | Y. S. Tsai, {\em Rev. Mod. Phys. 49} 421 (1977) |
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267 | \bibitem{conv.urban} L.Urban in {\sc Geant3} writeup, section PHYS-211. |
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268 | {\it Cern Program Library} (1993) |
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269 | \end{thebibliography} |
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270 | |
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271 | \end{latexonly} |
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272 | |
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273 | \begin{htmlonly} |
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274 | |
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275 | \subsection{Bibliography} |
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276 | |
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277 | \begin{enumerate} |
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278 | \item J.H.Hubbell, H.A.Gimm, I.Overbo |
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279 | {\it Jou. Phys. Chem. Ref. Data 9:1023} (1980) |
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280 | \item W. Heitler |
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281 | {\it The Quantum Theory of Radiation, Oxford University Press} (1957) |
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282 | \item R. Ford and W. Nelson. |
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283 | {\it SLAC-210, UC-32} (1978) |
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284 | \item J.C. Butcher and H. Messel. |
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285 | {\it Nucl. Phys. 20} 15 (1960) |
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286 | \item H. Messel and D. Crawford. |
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287 | {\it Electron-Photon shower distribution, Pergamon Press} (1970) |
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288 | \item Y. S. Tsai, {\em Rev. Mod. Phys. 46} 815 (1974), |
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289 | Y. S. Tsai, {\em Rev. Mod. Phys. 49} 421 (1977) |
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290 | \item L.Urban in {\sc Geant3} writeup, section PHYS-211. |
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291 | {\it Cern Program Library} (1993) |
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292 | \end{enumerate} |
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293 | |
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294 | \end{htmlonly} |
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