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2 | \section[Positron - Electron Annihilation]{Positron - Electron Annihilation}\label{sec:em.annil} |
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3 | |
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4 | \subsection{Introduction} |
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5 | The process {\tt G4eplusAnnihilation} simulates the in-flight annihilation |
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6 | of a positron with an atomic electron. As is usually done in shower |
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7 | programs \cite{egs4}, it is assumed here that the atomic electron is |
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8 | initially free and at rest. Also, annihilation processes producing one, or |
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9 | three or more, photons are ignored because these processes are negligible |
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10 | compared to the annihilation into two photons \cite{egs4,messel}. |
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11 | |
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12 | \subsection{Cross Section and Mean Free Path} |
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13 | \subsubsection{Cross Section per Atom} |
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14 | The annihilation in flight of a positron and electron is described by |
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15 | the cross section formula of Heitler \cite{heitler,egs4}: |
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16 | |
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17 | \begin{eqnarray} |
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18 | \sigma(Z,E) & = & \frac{Z \pi r_e^2}{\gamma +1} |
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19 | \left[\frac{\gamma^2 + 4 \gamma +1}{\gamma^2 -1} |
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20 | \ln \left(\gamma +\sqrt{\gamma^2 -1} \right) -\frac |
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21 | {\gamma +3}{\sqrt{\gamma^2 -1}} \right] \\ |
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22 | {\rm where}\nonumber\\ |
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23 | E & = & \mbox{total energy of the incident positron} \nonumber \\ |
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24 | \gamma & = & E/m c^2 \nonumber \\ |
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25 | r_e & = & \mbox{classical electron radius} \nonumber |
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26 | \end{eqnarray} |
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27 | |
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28 | \subsubsection{Mean Free Path} |
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29 | In a given material the mean free path, $\lambda$, for a positron to be |
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30 | annihilated with an electron is given by |
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31 | \begin{equation} |
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32 | \lambda(E) = |
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33 | \left( \sum_i n_{ati} \cdot \sigma(Z_i,E) \right)^{-1} |
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34 | \end{equation} |
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35 | where $n_{ati}$ is the number of atoms per volume of the $i^{th}$ element |
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36 | composing the material. |
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37 | |
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38 | \subsection {Sampling the final state} |
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39 | |
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40 | The final state of the $e+e-$ annihilation process |
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41 | \[e^+ \; e^- \to \gamma_a \; \gamma_b \] |
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42 | is simulated by first determining the kinematic limits of the photon energy |
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43 | and then sampling the photon energy within those limits using the |
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44 | differential cross section. Conservation of energy-momentum is then used to |
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45 | determine the directions of the final state photons. |
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46 | |
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47 | \subsubsection{Kinematic Limits} |
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48 | |
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49 | If the incident $e^+$ has a kinetic energy $T$, then the total energy is |
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50 | $E_e = T + mc^2$ and the momentum is $Pc = \sqrt{T(T+2mc^2)}$. |
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51 | The total available energy is $E_{tot} = E_e + mc^2 = E_a + E_b$ and |
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52 | momentum conservation requires $ \vec{P} = \vec{P}_{\gamma_a} + \vec{P}_{\gamma_b}$ . The fraction of the total energy transferred to one photon |
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53 | (say $\gamma_a$) is |
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54 | \begin{equation} |
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55 | \epsilon = \frac{E_a}{E_{tot}} \equiv \frac{E_a}{T+2mc^2} . |
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56 | \end{equation} |
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57 | The energy transfered to $\gamma_a$ is largest when $\gamma_a$ is emitted in |
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58 | the direction of the incident $e^+$. In that case |
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59 | $E_{a max} = (E_{tot}+Pc)/2$ . The energy transfered to $\gamma_a$ is |
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60 | smallest when $\gamma_a$ is emitted in the opposite direction of the |
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61 | incident $e^+$. Then $E_{a min} = (E_{tot}-Pc)/2$ . Hence, |
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62 | \begin {eqnarray} |
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63 | \epsilon_{max} &=& \frac{E_a \mathsf{max}}{E_{tot}} = |
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64 | \frac{1}{2} \left\lbrack 1+ \sqrt{\frac{\gamma - 1}{\gamma+1}} \right\rbrack \\ |
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65 | \epsilon_{min} &=& \frac{E_a \mathsf{min}}{E_{tot}} = |
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66 | \frac{1}{2} \left\lbrack 1- \sqrt{\frac{\gamma - 1}{\gamma+1}} |
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67 | \right\rbrack |
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68 | \end {eqnarray} |
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69 | \\ |
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70 | where $\qquad \gamma = (T + mc^2)/mc^2$ . Therefore the range of $\epsilon$ |
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71 | is |
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72 | $\quad \lbrack \epsilon_{min} \; ; \; \epsilon_{max} \rbrack $ |
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73 | $\qquad ( \equiv \lbrack \epsilon_{min} \; ; \; 1-\epsilon_{min} \rbrack) $. |
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74 | |
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75 | \subsubsection{Sampling the Gamma Energy} |
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76 | |
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77 | A short overview of the sampling method is given in Chapter \ref{secmessel}. |
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78 | |
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79 | The differential cross section of the two-photon positron-electron |
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80 | annihilation can be written as |
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81 | \cite{heitler,egs4}: |
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82 | \begin{equation} |
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83 | \frac{d \sigma (Z, \epsilon)} {d \epsilon} = |
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84 | \frac{Z \pi r_e^2}{\gamma - 1} \ \frac{1}{\epsilon} \ |
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85 | \left[ |
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86 | 1+\frac{2\gamma}{(\gamma+1)^2}-\epsilon-\frac{1}{(\gamma+1)^2}\frac{1}{\epsilon} |
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87 | \right] |
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88 | \end{equation} |
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89 | where |
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90 | $Z$ is the atomic number of the material, $r_e$ the classical electron |
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91 | radius, and $\epsilon \in [ \epsilon_{min} \; ; \; \epsilon_{max} ]$ . |
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92 | The differential cross section can be decomposed as |
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93 | \begin{equation} |
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94 | \frac{d \sigma (Z, \epsilon)} {d \epsilon} = |
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95 | \frac{Z \pi r_e^2}{\gamma - 1} |
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96 | \alpha f(\epsilon) g(\epsilon) |
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97 | \end{equation} |
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98 | where |
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99 | \begin{eqnarray} |
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100 | \alpha &=& \ln (\epsilon_{max}/\epsilon_{min}) \nonumber \\ |
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101 | f(\epsilon) &=& \frac{1}{\alpha \epsilon} \\ |
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102 | g(\epsilon) &=& \left[ |
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103 | 1+\frac{2\gamma}{(\gamma+1)^2}-\epsilon-\frac{1}{(\gamma+1)^2}\frac{1}{\epsilon} |
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104 | \right] \equiv |
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105 | 1-\epsilon+\frac{2 \gamma \epsilon -1}{\epsilon (\gamma +1)^2} |
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106 | \end{eqnarray} |
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107 | Given two random numbers $r, r' \in [0,1]$, the photon energies are chosen |
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108 | as follows: |
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109 | \begin{enumerate} |
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110 | \item |
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111 | sample $\epsilon$ from $f(\epsilon): |
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112 | \epsilon =\epsilon_{min} \left( \frac{\epsilon_{max}}{\epsilon_{min}} \right)^r$ |
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113 | \item |
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114 | test the rejection function: if $g(\epsilon) \geq r'$ accept |
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115 | $\epsilon$, otherwise return to step 1. |
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116 | \end{enumerate} |
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117 | Then the photon energies are |
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118 | $E_a = \epsilon E_{tot} \qquad E_b = (1-\epsilon) E_{tot}$ . |
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119 | |
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120 | \subsubsection{Computing the Final State Kinematics} |
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121 | If $\theta$ is the angle between the incident $e^+$ and $\gamma_a$, then |
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122 | from energy-momentum conservation, |
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123 | \begin{equation} |
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124 | \cos \theta = \frac{1}{Pc} \left[ T+mc^2 \frac{2\epsilon -1}{\epsilon} \right] |
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125 | = \frac{\epsilon(\gamma +1) - 1}{\epsilon \sqrt{\gamma^2 -1}} . |
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126 | \end{equation} \\ |
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127 | The azimuthal angle, $\phi$, is generated isotropically and the photon |
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128 | momentum vectors, $\vec{P_{\gamma_a}}$ and $\vec{P_{\gamma_b}}$, |
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129 | are computed from energy-momentum conservation and transformed into the lab |
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130 | coordinate system. |
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131 | |
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132 | \subsubsection{Annihilation at Rest} |
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133 | The method {\tt AtRestDoIt} treats the special case when a positron comes |
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134 | to rest before annihilating. It generates two photons, each with energy |
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135 | $k=mc^2$ and an isotropic angular distribution. |
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136 | |
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137 | \subsection{Status of this document} |
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138 | 09.10.98 created by M.Maire. \\ |
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139 | 01.08.01 minor corrections (mma) \\ |
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140 | 09.01.02 MeanFreePath (mma) \\ |
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141 | 01.12.02 Re-written by D.H. Wright \\ |
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142 | |
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143 | \begin{latexonly} |
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144 | |
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145 | \begin{thebibliography}{99} |
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146 | \bibitem{egs4} R. Ford and W. Nelson. |
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147 | {\em SLAC-265, UC-32} (1985) |
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148 | \bibitem{messel} H. Messel and D. Crawford. |
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149 | {\em Electron-Photon shower distribution, Pergamon Press} (1970) |
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150 | \bibitem{heitler} W. Heitler. |
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151 | {\em The Quantum Theory of Radiation, Clarendon Press, Oxford} (1954) |
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152 | \end{thebibliography} |
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153 | |
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154 | \end{latexonly} |
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155 | |
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156 | \begin{htmlonly} |
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157 | |
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158 | \subsection{Bibliography} |
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159 | |
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160 | \begin{enumerate} |
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161 | \item R. Ford and W. Nelson. |
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162 | {\em SLAC-265, UC-32} (1985) |
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163 | \item H. Messel and D. Crawford. |
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164 | {\em Electron-Photon shower distribution, Pergamon Press} (1970) |
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165 | \item W. Heitler. |
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166 | {\em The Quantum Theory of Radiation, Clarendon Press, Oxford} (1954) |
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167 | \end{enumerate} |
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168 | |
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169 | \end{htmlonly} |
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170 | |
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