1 | |
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2 | \section[Compton scattering]{Compton scattering}\label{sec:em.compton} |
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3 | \subsection{Cross Section and Mean Free Path} |
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4 | \subsubsection{Cross Section per Atom} |
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5 | When simulating the Compton scattering of a photon from an atomic electron, an |
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6 | empirical cross section formula is used, which reproduces the cross section |
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7 | data down to 10 keV: |
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8 | \begin{equation} |
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9 | \sigma(Z,E_{\gamma}) = \left [ P_{1}(Z) \ \frac{\log(1+2X)}{X} + |
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10 | \frac{P_{2}(Z)+P_{3}(Z) X + P_{4}(Z) X^{2}}{1+aX+bX^{2}+cX^{3}} \right ] . |
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11 | \end{equation} |
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12 | Here, |
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13 | \begin{eqnarray*} |
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14 | Z & = & \mbox{atomic number of the medium} \, \\ |
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15 | E_{\gamma} & = & \mbox{energy of the photon} \\ |
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16 | X & = & E_{\gamma}/mc^2 \\ |
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17 | m & = & \mbox{electron mass} \\ |
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18 | P_{i}(Z) & = & Z (d_{i} + e_{i}Z + f_{i}Z^{2}) . \, |
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19 | \end{eqnarray*} |
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20 | \\ |
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21 | The values of the parameters can be found within the method which computes |
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22 | the cross section per atom. A fit of the parameters was made to over 511 data |
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23 | points \cite{hubbell.comp,storm.comp} chosen from the intervals |
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24 | \begin{eqnarray*} |
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25 | 1 \leq Z \leq 100 |
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26 | \end{eqnarray*} |
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27 | and |
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28 | \begin{eqnarray*} |
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29 | E_{\gamma} \in [10 \mbox{ keV} , 100 \mbox{ GeV}] . |
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30 | \end{eqnarray*} |
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31 | The accuracy of the fit was estimated to be |
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32 | |
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33 | \vspace{.3cm} |
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34 | $$ |
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35 | \frac{\Delta\sigma}{\sigma} = \left\{ |
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36 | \begin{array}{lcl} |
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37 | \approx 10\% & \makebox[2cm][r]{\rm for } E_{\gamma} & \simeq 10 \mbox{ keV} |
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38 | -20 \mbox{ keV} \\ |
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39 | \leq 5-6\% & \makebox[2cm][r]{\rm for } E_{\gamma} & > 20 \mbox{ keV} |
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40 | \end{array} \right. |
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41 | $$ |
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42 | \subsubsection{Mean Free Path} |
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43 | In a given material the mean free path, $\lambda$, for a photon to interact |
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44 | via Compton scattering is given by |
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45 | \begin{equation} |
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46 | \lambda(E_{\gamma}) = |
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47 | \left( \sum_i n_{ati} \cdot \sigma_i (E_{\gamma}) \right)^{-1} |
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48 | \end{equation} |
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49 | where $n_{ati}$ is the number of atoms per volume of the $i^{th}$ element of |
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50 | the material. |
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51 | |
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52 | \subsection {Sampling the Final State} |
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53 | The quantum mechanical Klein-Nishina differential cross section per atom |
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54 | is \cite{klein.comp} : |
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55 | \begin{equation} |
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56 | \frac{d\sigma}{d\epsilon} =\pi r_e^2 \ |
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57 | \frac{m_e c^2}{E_0} \ Z |
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58 | \left[\frac{1}{\epsilon}+\epsilon\right] |
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59 | \left[1 - \frac{\epsilon \sin^2 \theta}{1+\epsilon^2}\right] |
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60 | \end{equation} |
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61 | where \quad |
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62 | \begin{tabular}[t]{l@{\ = \ }l} |
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63 | $r_e$ & classical electron radius \\ |
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64 | $m_e c^2$ & electron mass \\ |
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65 | $E_0$ & energy of the incident photon \\ |
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66 | $E_1$ & energy of the scattered photon \\ |
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67 | $\epsilon$ & $E_1/E_0$ . |
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68 | \end{tabular} |
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69 | |
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70 | \noindent |
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71 | Assuming an elastic collision, the scattering angle $\theta$ is defined by the |
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72 | Compton formula: |
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73 | \begin{equation} |
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74 | E_1 = E_0 \ \frac{m_{\rm e}c^2}{ m_{\rm e}c^2 + E_0(1-\cos\theta )} . |
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75 | \end{equation} |
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76 | |
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77 | \subsubsection{Sampling the Photon Energy} |
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78 | The value of $\epsilon$ corresponding to the minimum photon energy (backward |
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79 | scattering) is given by |
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80 | \begin{equation} |
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81 | \epsilon_0 = \frac{m_{\rm e}c^2}{m_{\rm e}c^2+2E_0} , |
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82 | \end{equation} |
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83 | hence $\epsilon \in [\epsilon_0, 1]$. Using the combined composition and |
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84 | rejection Monte Carlo methods described in |
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85 | \cite{butch.comp,messel.comp,ford.comp} one may set |
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86 | \begin{equation} |
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87 | \Phi(\epsilon) |
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88 | \simeq \left[ \frac{1}{\epsilon}+\epsilon \right] |
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89 | \left[ 1 - \frac{\epsilon \sin^2 \theta}{1+\epsilon^2} \right] |
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90 | = f(\epsilon) \cdot g(\epsilon) |
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91 | = \left[ \alpha_1 f_1(\epsilon) + \alpha_2 f_2(\epsilon) \right] |
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92 | \cdot g(\epsilon) , |
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93 | \end{equation} |
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94 | where |
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95 | $$ |
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96 | \begin{array}{lcl} |
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97 | \alpha_1 = \ln (1/\epsilon_0) & ; & |
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98 | f_1(\epsilon) = 1/(\alpha_1\epsilon) \\ |
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99 | \alpha_2 = (1-\epsilon_0^2)/2 & ; & |
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100 | f_2(\epsilon) = \epsilon/\alpha_2 . |
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101 | \end{array} |
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102 | $$ |
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103 | $f_1$ and $f_2$ are probability density functions defined on the interval |
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104 | $\lbrack\epsilon_0, 1\rbrack$, and |
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105 | $$ |
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106 | g(\epsilon) = \left[ 1 - \frac{\epsilon}{1+\epsilon^2} \sin^2\theta \right] |
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107 | $$ |
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108 | is the rejection function |
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109 | $\forall \epsilon \in [\epsilon_0, 1] \Longrightarrow 0 < g(\epsilon) \leq 1$.\\ |
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110 | |
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111 | \noindent |
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112 | Given a set of 3 random numbers $r, r', r''$ uniformly distributed on the |
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113 | interval [0,1], the sampling procedure for $\epsilon$ is the following: |
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114 | \begin{enumerate} |
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115 | \item |
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116 | decide whether to sample from $f_1(\epsilon)$ or $f_2(\epsilon)$: \\ |
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117 | if $ r < \alpha_1/(\alpha_1+\alpha_2)$ select $f_1(\epsilon)$, otherwise |
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118 | select $f_2(\epsilon)$ |
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119 | \item |
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120 | sample $\epsilon$ from the distributions corresponding to $f_1$ or $f_2$: \\ |
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121 | for $f_1 : \epsilon = \epsilon_0^{r'} \qquad (\equiv \exp(-r' \alpha_1))$ \\ |
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122 | for $f_2 : \epsilon^2 = \epsilon_0^2 + (1-\epsilon_0^2)r'$ |
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123 | |
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124 | \item |
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125 | calculate $\sin^2\theta = t(2-t)$ where |
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126 | $t \equiv (1-\cos\theta) = m_e c^2 (1-\epsilon)/(E_0 \epsilon)$ |
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127 | |
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128 | \item |
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129 | test the rejection function: \\ |
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130 | if $g(\epsilon) \geq r''$ accept $\epsilon$, otherwise go to step 1. |
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131 | \end{enumerate} |
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132 | |
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133 | \subsubsection{Compute the Final State Kinematics} |
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134 | After the successful sampling of $\epsilon$, the polar angles of the |
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135 | scattered photon with respect to the direction of the parent photon |
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136 | are generated. The azimuthal angle, $\phi$, is generated isotropically and |
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137 | $\theta$ is as defined in the previous section. The momentum vector of the |
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138 | scattered photon, $\overrightarrow{P_{\gamma1}}$, is then transformed into |
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139 | the {\tt World} coordinate system. The kinetic energy and momentum of the |
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140 | recoil electron are then |
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141 | \begin{eqnarray*} |
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142 | T_{el} & = & E_0 - E_1 \\ |
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143 | \overrightarrow{P_{el}} & = & |
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144 | \overrightarrow{P_{\gamma0}} - \overrightarrow{P_{\gamma1}} . |
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145 | \end{eqnarray*} |
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146 | |
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147 | \subsection{Validity} |
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148 | |
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149 | The differential cross-section is valid only for those collisions in which the |
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150 | energy of the recoil electron is large compared to its binding energy (which |
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151 | is ignored). However, as pointed out by Rossi \cite{rossi.comp}, this has a |
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152 | negligible effect because of the small number of recoil electrons produced at |
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153 | very low energies. |
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154 | |
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155 | \subsection{Status of this document} |
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156 | 09.10.98 created by M.Maire. \\ |
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157 | 14.01.02 minor revision (mma) \\ |
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158 | 22.04.02 reworded by D.H. Wright \\ |
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159 | 18.03.04 include references for total cross section (mma) \\ |
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160 | |
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161 | \begin{latexonly} |
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162 | |
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163 | \begin{thebibliography}{99} |
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164 | \bibitem{hubbell.comp} Hubbell, Gimm and Overbo. |
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165 | {\em J. Phys. Chem. Ref. Data 9} 1023 (1980) |
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166 | \bibitem{storm.comp} H. Storm and H.I. Israel |
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167 | {\em Nucl. Data Tables A7} 565 (1970) |
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168 | \bibitem{klein.comp} O. Klein and Y. Nishina. |
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169 | {\em Z. Physik 52} 853 (1929) |
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170 | \bibitem{butch.comp} J.C. Butcher and H. Messel. |
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171 | {\em Nucl. Phys. 20} 15 (1960) |
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172 | \bibitem{messel.comp} H. Messel and D. Crawford. |
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173 | {\em Electron-Photon shower distribution, Pergamon Press} (1970) |
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174 | \bibitem{ford.comp} R. Ford and W. Nelson. |
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175 | {\em SLAC-265, UC-32} (1985) |
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176 | \bibitem{rossi.comp} B. Rossi. |
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177 | {\em High energy particles, Prentice-Hall} 77-79 (1952) |
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178 | \end{thebibliography} |
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179 | |
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180 | \end{latexonly} |
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181 | |
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182 | \begin{htmlonly} |
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183 | |
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184 | \subsection{Bibliography} |
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185 | |
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186 | \begin{enumerate} |
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187 | \item Hubbell, Gimm and Overbo. |
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188 | {\em J. Phys. Chem. Ref. Data 9} 1023 (1980) |
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189 | \item H. Storm and H.I. Israel |
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190 | {\em Nucl. Data Tables A7} 565 (1970) |
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191 | \item O. Klein and Y. Nishina. |
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192 | {\em Z. Physik 52} 853 (1929) |
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193 | \item J.C. Butcher and H. Messel. |
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194 | {\em Nucl. Phys. 20} 15 (1960) |
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195 | \item H. Messel and D. Crawford. |
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196 | {\em Electron-Photon shower distribution, Pergamon Press} (1970) |
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197 | \item R. Ford and W. Nelson. |
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198 | {\em SLAC-265, UC-32} (1985) |
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199 | \item B. Rossi. |
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200 | {\em High energy particles, Prentice-Hall} 77-79 (1952) |
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201 | \end{enumerate} |
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202 | |
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203 | \end{htmlonly} |
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