1 | |
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2 | \section[Bremsstrahlung]{Bremsstrahlung} \label{sec:em.ebrem} |
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3 | |
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4 | The class $G4eBremsstrahlung$ provides the energy loss of electrons and |
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5 | positrons due to the radiation of photons in the field of a nucleus |
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6 | according to the approach described in Section \ref{en_loss}. |
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7 | Above a given threshold energy the energy loss is simulated by the explicit |
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8 | production of photons. Below the threshold the emission of soft photons is |
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9 | treated as a continuous energy loss. |
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10 | |
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11 | %{\color{red} |
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12 | Below electron/positron energies of 1 GeV, the cross |
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13 | section evaluation is based on a dedicated parameterization, see |
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14 | Sec.\ \ref{sec:em.ebrem.param}. Above this limit an analytic cross |
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15 | section is used, cf.\ Sec.\ \ref{sec:em.ebrem.lpm}. |
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16 | %} |
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17 | In GEANT4 the Landau-Pomeranchuk-Migdal effect has also been implemented. |
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18 | |
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19 | \subsection{Cross Section and Energy Loss}\label{sec:em.ebrem.param} |
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20 | |
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21 | $d\sigma(Z,T,k)/dk$ is the differential cross section for the production of a |
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22 | photon of energy $k$ by an electron of kinetic energy $T$ in the field of an |
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23 | atom of charge $Z$. If $k_c$ is the energy cut-off below which the soft |
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24 | photons are treated as continuous energy loss, then the mean value of the |
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25 | energy lost by the electron is |
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26 | \begin{equation}\label{eq:ebrem.eloss} |
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27 | E_{Loss}^{brem} (Z,T,k_c ) = |
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28 | \int_{0}^{k_ c}k\frac{d \sigma (Z,T,k)}{dk}dk . |
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29 | \end{equation} |
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30 | The total cross section for the emission of a photon of energy larger than |
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31 | $k_c$ is |
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32 | \begin{equation}\label{eq:ebrem.discrete} |
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33 | \sigma_{brem} (Z,T,k_c ) = \int_{k_c}^{T}\frac{d \sigma (Z,T,k)}{dk} dk . |
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34 | \end{equation} |
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35 | \\ |
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36 | |
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37 | \subsubsection{Parameterization of the Energy Loss and Total Cross Section} |
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38 | |
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39 | The cross section and energy loss due to bremsstrahlung have been |
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40 | parameterized using the EEDL (Evaluated Electrons Data Library) data set |
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41 | \cite{eedl} as input. |
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42 | |
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43 | \noindent |
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44 | The following parameterization was chosen for the electron bremsstrahlung |
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45 | cross section : |
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46 | \begin{equation} |
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47 | \label{ebrem.a} |
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48 | \sigma (Z,T,k_c ) = Z(Z+\xi_{\sigma} ) (1-c_{sigh} Z^{1/4}) |
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49 | \left[ \ln \frac{T}{k_c} \right]^{\alpha} \frac{{f_s}}{N_{Avo}} |
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50 | \end{equation} |
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51 | where $f_s$ is a polynomial in $x = lg(T)$ with $Z$-dependent coefficients for |
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52 | $x < x_l$ , $f_s= 1 $ for $x \ge x_l$, $\xi_{\sigma}, c_{sigh}, \alpha$ are |
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53 | constants, $N_{Avo}$ is the Avogadro number. |
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54 | For the case of low energy electrons ($T \le T_{lim} = 10 MeV$) the above |
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55 | expression should be multiplied by |
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56 | |
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57 | \begin{equation} |
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58 | \left(\frac{T_{lim}}{T}\right)^{c_l} \cdot \left(1 + \frac {a_l}{\sqrt{Z} T}\right), |
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59 | \end{equation} |
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60 | with constant $c_l, a_l$ parameters. |
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61 | |
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62 | The energy loss parameterization is the following : |
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63 | |
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64 | \begin{equation} |
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65 | \label{ebrem.b} |
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66 | E_{Loss}^{brem} (Z,T,k_c ) =\frac{Z(Z+ \xi_l)(T+m)^2 } |
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67 | {(T+2m)}\left[\frac{k_c}{T}\right]^\beta (2-c_{lh} Z^{\frac{1}{4}} ) |
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68 | \frac{a + b \frac{T}{T_{lim}}}{1 + c \frac{T}{T_{lim}}} |
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69 | \frac{f_l}{N_{Avo}} |
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70 | \end{equation} |
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71 | where $m$ is the mass of the electron, $\xi_l, \beta, c_{lh}, a,b,c$ are |
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72 | constants, $f_l$ is a polynomial in $x = lg(T)$ with $Z$-dependent |
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73 | coefficients for $x < x_l$ , $f_l= 1 $ for $x \ge x_l$. |
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74 | For low energies this expression should be divided by |
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75 | |
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76 | \begin{equation} |
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77 | \left(\frac{T_{lim}}{T}\right)^{c_l} |
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78 | \end{equation} |
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79 | |
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80 | and if $T < k_c$ the expression should be multiplied by |
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81 | |
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82 | \begin{equation} |
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83 | \left(\frac{T}{k_c}\right)^{a_l} |
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84 | \end{equation} |
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85 | |
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86 | with some constants $c_l, a_l$. |
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87 | The numerical values of the parameters and the coefficients of the |
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88 | polynomyals $f_s$ and $f_l$ can be found in the class code. |
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89 | \\ |
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90 | |
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91 | \noindent |
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92 | The errors of the parameterizations (\ref{ebrem.a}) and (\ref{ebrem.b}) |
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93 | were estimated to be |
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94 | |
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95 | \begin{eqnarray*} |
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96 | \frac{\Delta\sigma} {\sigma} & = & \left \{ |
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97 | \begin{array}{llr} |
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98 | 6-8 \% & \mbox{for } & T \leq 1 MeV \\ |
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99 | \leq 4-5\% & \mbox{for } & 1 MeV < T |
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100 | \end{array} |
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101 | \right . \\[1cm] |
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102 | \frac{\Delta E_{Loss}^{brem}} |
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103 | {E_{Loss}^{brem}} & = & \left \{ |
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104 | \begin{array}{llr} |
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105 | 8 -10\% & \mbox{for } & T \leq1 MeV \\ |
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106 | 5-6\% & \mbox{for } & 1 MeV < T . |
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107 | \end{array} |
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108 | \right . |
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109 | \end{eqnarray*} |
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110 | |
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111 | |
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112 | \noindent |
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113 | When running GEANT4, the energy loss due to soft photon bremsstrahlung is |
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114 | tabulated at initialization time as a function of the medium and of the |
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115 | energy, as is the mean free path for discrete bremsstrahlung. |
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116 | |
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117 | \subsubsection{Corrections for $e^+ e^-$ Differences} |
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118 | |
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119 | The preceding section has dealt exclusively with electrons. One might expect |
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120 | that positrons could be treated the same way. According to reference |
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121 | \cite{ebrem.kim} however, the differences between the radiative loss of positrons |
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122 | and electrons are considerable and cannot be disregarded. |
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123 | The ratio of the radiative energy loss for positrons |
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124 | to that for electrons obeys a simple scaling law, is a |
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125 | function only of the quantity $T/Z^2$. \\ |
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126 | |
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127 | \noindent |
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128 | The radiative energy loss for electrons or positrons is given by |
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129 | \begin{eqnarray*} |
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130 | -\frac{1}{\rho} \left ( \frac{dE}{dx} \right )_{rad}^{\pm} & = & |
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131 | \frac{N_{Av} \alpha r_e^2}{A} (T+m) Z^2 \Phi_{rad}^{\pm}(Z,T) \\ |
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132 | \Phi^{\pm}_{rad}(Z,T) & = & \frac{1}{\alpha r_{e}^2 Z^2 (T+m)} |
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133 | \int^{T}_{0}{k\frac{d\sigma^{\pm}}{dk}dk} |
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134 | \end{eqnarray*} |
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135 | and it is the ratio |
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136 | \begin{eqnarray*} |
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137 | \eta & = & \frac{\Phi_{rad}^{+}(Z,T)}{\Phi_{rad}^{-}(Z,T)} = |
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138 | \eta \left (\frac{T}{Z^2}\right ) |
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139 | \end{eqnarray*} |
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140 | that obeys the scaling law. \\ |
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141 | |
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142 | \noindent |
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143 | The authors have calculated this function in the range $10^{-7} |
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144 | \leq \frac{T}{Z^2} \leq 0.5$, where the kinetic energy $T$ is expressed in |
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145 | MeV. Their {\it data} can be fairly accurately reproduced using a |
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146 | parametrization: |
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147 | |
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148 | \begin{eqnarray*} |
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149 | \eta & = & \left \{ |
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150 | \begin{array}{llr} |
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151 | 0 & \mbox{if } & x \leq -8 \\ |
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152 | \frac{1}{2} + \frac{1}{\pi} \arctan \left( a_1 x + a_3 x^3 |
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153 | + a_5 x^5 \right ) & \mbox{if } & -8 < x < 9 \\ |
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154 | 1 & \mbox{if } & x \geq 9 |
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155 | \end{array} |
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156 | \right . |
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157 | \end{eqnarray*} |
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158 | where |
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159 | \begin{eqnarray*} |
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160 | x & = & \log \left ( C \frac{T}{Z^2} \right ) \mbox{(T in GeV)} \\ |
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161 | C & = & 7.5221 \times 10^{6} \\ |
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162 | a_1 & = & 0.415 \\ |
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163 | a_3 & = & 0.0021 \\ |
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164 | a_5 & = & 0.00054 . |
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165 | \end{eqnarray*} |
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166 | |
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167 | |
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168 | \noindent |
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169 | The $e^+ e^-$ energy loss difference is not purely a low-energy phenomenon |
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170 | (at least for high $Z$), as shown in Table~\ref{ebrem.c}. |
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171 | |
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172 | \begin{table}[hbt] |
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173 | \begin{centering} |
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174 | \begin{tabular}{rr|r|r} \hline |
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175 | \multicolumn{1}{c}{$\frac{T}{Z^2} (GeV)$} |
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176 | & \multicolumn{1}{c|}{T} |
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177 | & \multicolumn{1}{c|}{$\eta$} |
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178 | & \multicolumn{1}{c}{$\left ( \frac{rad. \ loss}{total \ loss} |
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179 | \right )_{e^-}$} \\[3mm] \hline |
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180 | $10^{-9}$ & $\sim 7 keV$ & $\sim 0.1$ & $\sim 0\%$ \\ |
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181 | $10^{-8}$ & $67 keV $ & $\sim 0.2$ & $\sim 1\%$ \\ |
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182 | $2 \times 10^{-7}$ & $1.35 MeV$ & $\sim 0.5$ & $\sim 15\%$ \\ |
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183 | $2 \times 10^{-6}$ & $13.5 MeV$ & $\sim 0.8$ & $\sim 60\%$ \\ |
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184 | $2 \times 10^{-5}$ & $135. MeV$ & $\sim 0.95$ & $> 90\%$ \\ \hline |
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185 | \end{tabular} |
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186 | \caption{ratio of the $e^+ e^-$ radiative energy loss in lead |
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187 | (Z=82).} |
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188 | \label{ebrem.c} |
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189 | \end{centering} |
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190 | \end{table} |
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191 | |
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192 | |
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193 | \noindent |
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194 | The scaling property will be used to obtain the positron energy loss and |
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195 | discrete bremsstrahlung from the corresponding electron values. However, |
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196 | while scaling holds for the ratio of the total radiative energy losses, it |
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197 | is significantly broken for the photon spectrum in the screened case. That is, |
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198 | \begin{eqnarray*} |
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199 | \frac{\Phi^+}{\Phi^-} = \eta \left ( \frac{T}{Z^2} \right ) |
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200 | & \hspace{3cm} & |
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201 | \frac{\frac{d\sigma^+}{dk}}{\frac{d\sigma^-}{dk}} = |
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202 | \mbox{does not scale .} |
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203 | \end{eqnarray*} |
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204 | For the case of a point Coulomb charge, scaling would be restored for the |
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205 | photon spectrum. In order to correct for non-scaling, it is useful to note |
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206 | that in the photon spectrum from bremsstrahlung reported in \cite{ebrem.kim}: |
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207 | \begin{eqnarray*} |
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208 | \frac{d\sigma^{\pm}}{dk} = S^{\pm} \left( \frac{k}{T} \right ) |
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209 | \hspace{2cm} |
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210 | \frac{S^{+}(k)}{S^{-}(k)} \leq 1 & \hspace{1cm} & S^{+}(1) = 0 |
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211 | \hspace{2cm} S^{-}(1) > 0 |
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212 | \end{eqnarray*} |
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213 | One can further assume that |
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214 | \begin{eqnarray} |
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215 | \frac{d\sigma^+}{dk} = f(\epsilon) \frac{d\sigma^-}{dk} , |
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216 | & \hspace{2cm} & |
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217 | \epsilon = \frac{k}{T} |
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218 | \label{ebrem.d} |
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219 | \end{eqnarray} |
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220 | and require |
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221 | \begin{eqnarray} |
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222 | \int^{1}_{0}{f(\epsilon)d\epsilon} & = & \eta |
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223 | \label{ebrem.e} |
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224 | \end{eqnarray} |
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225 | in order to approximately satisfy the scaling law for the ratio of the total |
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226 | radiative energy loss. From the photon spectra the boundary conditions |
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227 | \begin{eqnarray} |
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228 | \left . |
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229 | \begin{array}{l} |
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230 | f(0) = 1 \\ |
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231 | f(1) = 0 |
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232 | \end{array} |
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233 | \right \} \hspace{2cm} \mbox{for all $Z,T$} |
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234 | \label{ebrem.f} |
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235 | \end{eqnarray} |
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236 | may be inferred. Choosing a simple function for $f$ |
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237 | \begin{eqnarray} |
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238 | f(\epsilon) & = & C (1-\epsilon)^{\alpha} \hspace{3cm} C,\alpha > 0 , |
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239 | \label{ebrem.g} |
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240 | \end{eqnarray} |
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241 | the conditions (\ref{ebrem.e}), (\ref{ebrem.f}) lead to: |
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242 | \begin{eqnarray*} |
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243 | C & = & 1 \\ |
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244 | \alpha & = & \frac{1}{\eta} - 1 \hspace{2cm} |
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245 | \mbox{($\alpha > 0$ because $\eta < 1$)} \\ |
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246 | f(\epsilon) & = & (1-\epsilon)^{\frac{1}{\eta}-1} . |
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247 | \end{eqnarray*} |
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248 | |
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249 | \noindent |
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250 | Now the weight factors $F_{l}$ and $F_{\sigma}$ for the positron continuous |
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251 | energy loss and the discrete bremsstrahlung cross section can be defined: |
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252 | |
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253 | \begin{eqnarray} |
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254 | F_{l} = \frac{1}{\epsilon_{0}} \int^{\epsilon_{0}}_{0} |
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255 | {f(\epsilon)d\epsilon} & \hspace{3cm} & |
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256 | F_{\sigma} = \frac{1}{1-\epsilon_{0}} \int^{1}_{\epsilon_{0}} |
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257 | {f(\epsilon)d\epsilon} |
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258 | \label{ebrem.h} |
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259 | \end{eqnarray} |
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260 | where $\epsilon_{0} = \frac{k_c}{T}$ and $k_c$ is the photon cut. In this |
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261 | scheme the positron energy loss and discrete bremsstrahlung can be calculated |
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262 | as: |
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263 | \begin{eqnarray*} |
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264 | \left ( - \frac{dE}{dx} \right )^{+} = F_{l} |
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265 | \left ( - \frac{dE}{dx} \right )^{-} & \hspace{2cm} & |
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266 | \sigma^{+}_{brems} = F_{\sigma} \sigma^{-}_{brems} |
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267 | \end{eqnarray*} |
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268 | |
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269 | \noindent |
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270 | In this approximation the photon spectra are identical, therefore the same |
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271 | sampling is used for generating $e^-$ or $e^+$ bremsstrahlung. The following |
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272 | relations hold: |
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273 | \begin{eqnarray*} |
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274 | F_{\sigma} & = & \eta (1-\epsilon_{0})^{\frac{1}{\eta}-1} |
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275 | < \eta \\ |
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276 | \epsilon_{0} F_{l} + (1-\epsilon_{0}) F_{\sigma} & = & \eta |
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277 | \hspace{6cm} \mbox{from the def (\ref{ebrem.h})} \\ |
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278 | \Rightarrow F_{l} & = & \eta \frac{1-(1-\epsilon_{0})^{\frac{1} |
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279 | {\eta}})}{\epsilon_{0}} > \eta \frac{1-(1-\epsilon_{0})} |
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280 | {\epsilon_{0}} = \eta \hspace{1cm} |
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281 | \Rightarrow \left \{ |
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282 | \begin{array}{l} |
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283 | F_{l} > \eta \\ |
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284 | F_{\sigma} < \eta |
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285 | \end{array} \right . |
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286 | \end{eqnarray*} |
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287 | which is consistent with the spectra. The effect of the difference in $e^-$ |
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288 | and $e^+$ bremsstrahlung can also be seen in electromagnetic shower |
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289 | development when the primary energy is not too high. |
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290 | |
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291 | |
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292 | %{\color{red} |
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293 | The dielectric suppression of the photon spectrum (density effect or |
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294 | TerMikaelian effect \cite{ebrem.migdal}) is governed by the suppression function |
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295 | \begin{equation} |
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296 | \label{eq:ebrem.density} |
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297 | S_p = \frac{k^2}{k^2 + {k_p}^2} , |
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298 | \end{equation} |
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299 | where the charakteristic energy $k_p$ is defined by |
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300 | \begin{equation} |
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301 | \label{eq:ebrem.o} |
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302 | k_p^2 = \frac{r_0 \lambda^2_e n}{\pi} \cdot E^2 . |
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303 | \end{equation} |
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304 | and the parameters are defined by |
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305 | \[ |
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306 | \begin{array}{ll} |
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307 | r_0 & \mbox{classical electron radius} \\ |
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308 | \lambda_e & \mbox{electron Compton wavelength} \\ |
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309 | n & \mbox{electron density in the material.} |
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310 | \end{array} |
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311 | \] |
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312 | %} |
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313 | |
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314 | \subsection{Simulation of Discrete Bremsstrahlung} |
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315 | |
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316 | The energy of the final state photons is sampled according to the spectrum |
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317 | \cite{ebrem.seltzer} of Seltzer and Berger. They have calculated the |
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318 | bremsstrahlung spectra for materials with atomic numbers Z = 6, 13, 29, 47, |
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319 | 74 and 92 in the electron kinetic energy range 1 keV - 10 GeV. Their tabulated |
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320 | results have been used as input in a fit of the parameterized function |
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321 | |
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322 | \[ |
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323 | S(x) = C k \frac{d \sigma}{d k} , |
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324 | \] |
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325 | which will be used to form the rejection function for the sampling process. |
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326 | The parameterization can be written as |
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327 | \begin{equation} |
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328 | \label{eq:phys341-1} |
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329 | S(x) = \left \{ |
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330 | \begin{array}{ll} |
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331 | (1-a_{h} \epsilon )F_{1}(\delta) + b_{h} \epsilon^{2} F_{2} (\delta) |
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332 | & T \geq 1 MeV \\ |
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333 | 1 + a_{l} x + b_{l} x^{2} & T < 1 MeV |
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334 | \end{array} \right . |
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335 | \end{equation} |
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336 | where |
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337 | \[ |
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338 | \begin{array}{lcl} |
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339 | C & & \mbox{normalization constant} \\ |
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340 | k & & \mbox{photon energy} \\ [1mm] |
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341 | T, E & & \mbox{kinetic and total energy of the primary electron} \\ |
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342 | x & = & \frac{k}{T} \\ [2mm] |
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343 | \epsilon & = & \frac{k}{E} = x \frac{T}{E} \\ |
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344 | \end{array} |
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345 | \] |
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346 | and $a_{h,l}$ and $b_{h,l}$ are the parameters to be fitted. The |
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347 | $F_{i}(\delta)$ screening functions depend on the screening variable |
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348 | \[ |
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349 | \begin{array}{lcll} |
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350 | \delta & = & \frac{136 m_{e}}{Z^{1/3} E} \frac{\epsilon}{1-\epsilon} \\ |
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351 | F_{1}(\delta) & = & F_{0} (42.392 - 7.796 \delta +1.961 \delta^{2} - F) |
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352 | & \delta \leq 1 \\ |
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353 | F_{2}(\delta) & = & F_{0} (41.734 - 6.484 \delta +1.250 \delta^{2} - F) |
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354 | & \delta \leq 1 \\ |
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355 | F_{1}(\delta) & = & F_{2}(\delta) = |
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356 | F_{0} (42.24 - 8.368 \ln(\delta + 0.952) -F) & \delta > 1 \\ |
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357 | F_{0} & = & \frac{1}{42.392-F} \\ |
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358 | F & = & 4 \ln Z - 0.55 (\ln Z)^{2} . |
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359 | \end{array} |
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360 | \] |
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361 | |
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362 | \noindent |
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363 | The ``high energy'' ($T >$ 1 MeV) formula is essentially the |
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364 | Coulomb-corrected, screened Bethe-Heitler formula (see e.g. |
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365 | \cite{ebrem.williams,ebrem.butcher,ebrem.egs4}). However, |
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366 | Eq.~(\ref{eq:phys341-1}) differs from Bethe-Heitler in two ways: |
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367 | \begin{enumerate} |
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368 | \item $a_{h}, b_{h}$ depend on $T$ and on the atomic number $Z$, whereas in |
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369 | the Bethe-Heitler spectrum they are fixed ($a_{h} = 1$, $b_{h} =0.75$); |
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370 | \item the function $F$ is not the same as that in the Bethe-Heitler |
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371 | cross-section; the present function gives a better behavior in the |
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372 | high frequency limit, i.e. when $k \rightarrow T$ ($x \rightarrow 1$). |
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373 | \end{enumerate} |
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374 | |
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375 | \noindent |
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376 | The $T$ and $Z$ dependence of the parameters are described by the equations: |
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377 | |
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378 | \begin{eqnarray*} |
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379 | a_{h} & = & 1 + \frac{a_{h1}}{u}+\frac{a_{h2}}{u^{2}}+\frac{a_{h3}}{u^{3}} \\ |
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380 | b_{h} & = & 0.75+\frac{b_{h1}}{u}+\frac{b_{h2}}{u^{2}}+\frac{b_{h3}}{u^{3}} \\ |
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381 | a_{l} & = & a_{l0} + a_{l1} u + a_{l2} u^{2} \\ |
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382 | b_{l} & = & b_{l0} + b_{l1} u + b_{l2} u^{2} \\ |
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383 | \mbox{with} \\ |
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384 | u & = & \ln \left ( \frac{T}{m_{e}} \right ) |
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385 | \end{eqnarray*} |
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386 | The parameters $a_{hi}, b_{hi}, a_{li}, b_{li}$ are polynomials of second order |
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387 | in the variable: |
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388 | |
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389 | \[ |
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390 | v = [Z (Z+1)]^{1/3} . |
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391 | \] |
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392 | In the limiting case $T \rightarrow |
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393 | \infty$, $a_{h} \rightarrow 1, b_{h} \rightarrow 0.75$, |
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394 | Eq.~(\ref{eq:phys341-1}) gives the Bethe-Heitler cross section. \\ |
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395 | |
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396 | \noindent |
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397 | There are altogether 36 linear parameters in the formulae and their values are |
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398 | given in the code. This parameterization reproduces the Seltzer-Berger tables |
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399 | to within 2-3 \% on average, with the maximum error being less than 10-12 \%. |
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400 | The original tables, on the other hand, agree well with the experimental data |
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401 | and theoretical (low- and high-energy) results ($<$ 10 \% below 50 MeV and |
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402 | $<$ 5 \% above 50 MeV). \\ |
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403 | |
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404 | \noindent |
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405 | Apart from the normalization the cross section differential in photon |
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406 | energy can be written as |
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407 | \[ |
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408 | \frac{d \sigma}{d k} = -\frac{g(x)}{x\ln x_{c}} = -\frac{S(x)}{S_{max}x\ln x_{c}}, |
---|
409 | \] |
---|
410 | where $x_{c} = k_{c}/T$ and $k_{c}$ is the photon cut-off energy below |
---|
411 | which the bremsstrahlung is treated as a continuous energy loss. Using this |
---|
412 | decomposition of the cross section and two random numbers $r_{1}$, $r_{2}$ |
---|
413 | uniformly distributed in $[0,1]$, the sampling of $x$ is done as follows: |
---|
414 | \begin{enumerate} |
---|
415 | \item |
---|
416 | sample $x$ from $-\frac{1}{x\ln x_{c}}$ setting $x = e^{r_{1} \ln x_{c}}$ |
---|
417 | \item calculate the rejection function $g(x)$ and: |
---|
418 | \begin{itemize} |
---|
419 | \item if $r_{2} > g(x)$ reject $x$ and go back to 1; |
---|
420 | \item if $r_{2} \leq g(x)$ accept $x$. |
---|
421 | \end{itemize} |
---|
422 | \end{enumerate} |
---|
423 | |
---|
424 | \noindent |
---|
425 | The application of the dielectric suppression \cite{ebrem.migdal} %and |
---|
426 | %the LPM effect |
---|
427 | requires that $\epsilon$ also be sampled. First, the rejection |
---|
428 | function must be multiplied by a suppression factor |
---|
429 | \[ |
---|
430 | C_M (\epsilon) =\frac{1 + C_0 / \epsilon_c^2} |
---|
431 | {1 + C_0 / \epsilon^2} |
---|
432 | \] |
---|
433 | where |
---|
434 | \[ |
---|
435 | C_0 =\frac{nr_0 \lambda^2 }{\pi}, \hspace{1cm} \epsilon_c = \frac{k_{c}}{E} |
---|
436 | \] |
---|
437 | \begin{itemize} |
---|
438 | \item[$n$] electron density in the medium |
---|
439 | \item[$r_0$] classical electron radius |
---|
440 | \item[$\lambda$] reduced Compton wavelength of the electron. |
---|
441 | \end{itemize} |
---|
442 | Apart from the Migdal correction factor, this is simply expression |
---|
443 | \ref{eq:ebrem.density} . This correction decreases the cross-section for low photon |
---|
444 | energies. \\ |
---|
445 | |
---|
446 | %\noindent |
---|
447 | %While sampling $\epsilon$, the suppression factor $f_{LPM}=\frac{S}{S_p}$ is |
---|
448 | %also used as a rejection function in order to take into account the LPM effect. |
---|
449 | %Here the supression factor is compared to a random number $r$ uniformly |
---|
450 | %distributed in the interval $[0,1]$. If $f_{LPM} \geq r$ the simulation |
---|
451 | %continues, otherwise the bremsstrahlung process concludes {\em without photon |
---|
452 | %production}. It can be seen that this procedure performs the LPM suppression |
---|
453 | %correctly. \\ |
---|
454 | |
---|
455 | \noindent |
---|
456 | After the successful sampling of $\epsilon$, the polar angles of the radiated |
---|
457 | photon are generated with respect to the parent electron's momentum. It is |
---|
458 | difficult to find simple formulae for this angle in the literature. For |
---|
459 | example the double differential cross section reported by |
---|
460 | Tsai~\cite{ebrem.tsai1,ebrem.tsai2} is |
---|
461 | \begin{eqnarray*} |
---|
462 | \frac{d \sigma}{dkd \Omega} |
---|
463 | & = & \frac{2 \alpha^{2}e^{2}}{\pi k m^{4}} |
---|
464 | \left\{ \left[ \frac{2\epsilon-2}{(1+u^2)^2}+ |
---|
465 | \frac{12u^2(1-\epsilon)}{(1+u^2)^4}\right] |
---|
466 | Z(Z+1) \right. \\ |
---|
467 | & & \mbox{} + \left. \left[ \frac{2-2\epsilon-\epsilon^{2}}{(1+u^2)^2}- |
---|
468 | \frac{4u^2(1-\epsilon)}{(1+u^2)^4} |
---|
469 | \right] |
---|
470 | \left[ X-2Z^{2}f_{c}((\alpha Z)^{2})\right] |
---|
471 | \right\} \\ |
---|
472 | u & = & \frac{E \theta}{m} \\ |
---|
473 | X & = & \int_{t_{min}}^{m^{2}(1+u^{2})^{2}} |
---|
474 | {\left [ G_{Z}^{el}(t) + G_{Z}^{in}(t) \right ] \frac{t-t_{min}} |
---|
475 | {t^{2}} dt} \\ |
---|
476 | G_{Z}^{el, in}(t) & & \mbox{atomic form factors} \\ |
---|
477 | t_{min} & = & \left [ \frac{k m^{2} (1+u^{2})}{2 E (E-k)} \right ] ^{2} |
---|
478 | = \left [ \frac{\epsilon m^{2} (1+u^{2})}{2 E (1-\epsilon)} \right ] ^{2} . |
---|
479 | \end{eqnarray*} |
---|
480 | The sampling of this distribution is complicated. It is also only an |
---|
481 | approximation to within a few percent, due at least to the presence of the |
---|
482 | atomic form factors. The angular dependence is contained in the variable |
---|
483 | $u = E \theta m^{-1}$. For a given value of $u$ the dependence of the shape |
---|
484 | of the function on $Z$, $E$ and $\epsilon = k/E$ is very weak. Thus, the |
---|
485 | distribution can be approximated by a function |
---|
486 | \begin{equation} |
---|
487 | f(u) = C \left( u e^{-au} + d u e^{-3au} \right) |
---|
488 | \end{equation} |
---|
489 | where |
---|
490 | \[ |
---|
491 | C = \frac{9a^{2}}{9 + d} \hspace{1cm} a = 0.625 \hspace{1cm} |
---|
492 | d = 27 |
---|
493 | \] |
---|
494 | where $E$ is in GeV. While this approximation is good at high energies, |
---|
495 | it becomes less accurate around a few MeV. However in that region the |
---|
496 | ionization losses dominate over the radiative losses. \\ |
---|
497 | |
---|
498 | \noindent |
---|
499 | The sampling of the function $f(u)$ can be done with three random numbers |
---|
500 | $r_i$, uniformly distributed on the interval [0,1]: |
---|
501 | \begin{enumerate} |
---|
502 | \item choose between $u e^{-au}$ and $d u e^{-3au}$: |
---|
503 | \[ |
---|
504 | b = \left \{ \begin{array}{ll} |
---|
505 | a & \mbox{if\hspace{0.5cm}}r_{1} < 9/(9+d) \\ |
---|
506 | 3a & \mbox{if\hspace{0.5cm}}r_{1} \geq 9/(9+d) |
---|
507 | \end{array} \right . |
---|
508 | \] |
---|
509 | \item sample $u e^{-bu}$: |
---|
510 | \[ |
---|
511 | u=-\frac{\log ( r_{2} r_{3}) }{b} |
---|
512 | \] |
---|
513 | \item check that: |
---|
514 | \[ |
---|
515 | u \leq u_{max} = \frac{E \pi}{m} |
---|
516 | \] |
---|
517 | otherwise go back to 1. |
---|
518 | \end{enumerate} |
---|
519 | The probability of failing the last test is reported in |
---|
520 | table~\ref{tb:phys341-1}. \\ |
---|
521 | |
---|
522 | \begin{table} |
---|
523 | \begin{centering} |
---|
524 | \begin{tabular}{|l|l|} |
---|
525 | \multicolumn{2}{c}{$\displaystyle |
---|
526 | P = \int^{\infty}_{u_{max}}{f(u) \: du} \hfill $} \\ [0.5cm] |
---|
527 | \hline |
---|
528 | E (MeV) & P(\%) \\ \hline |
---|
529 | 0.511 & 3.4 \\ |
---|
530 | 0.6 & 2.2 \\ |
---|
531 | 0.8 & 1.2 \\ |
---|
532 | 1.0 & 0.7 \\ |
---|
533 | 2.0 & $<$ 0.1 \\ \hline |
---|
534 | \end{tabular} |
---|
535 | \caption{Angular sampling efficiency} |
---|
536 | \label{tb:phys341-1} |
---|
537 | \end{centering} |
---|
538 | \end{table} |
---|
539 | |
---|
540 | |
---|
541 | \noindent |
---|
542 | The function $f(u)$ can also be used to describe the angular distribution of |
---|
543 | the photon in $\mu$ bremsstrahlung and to describe the angular distribution in |
---|
544 | photon pair production. \\ |
---|
545 | |
---|
546 | \noindent |
---|
547 | The azimuthal angle $\phi$ is generated isotropically. Along with $\theta$, |
---|
548 | this information is used to calculate the momentum vectors of the radiated |
---|
549 | photon and parent recoiled electron, and to transform them to the |
---|
550 | global coordinate system. |
---|
551 | The momentum transfer to the atomic nucleus is neglected. |
---|
552 | |
---|
553 | \subsection{Bremsstrahlung of high-energy electrons}\label{sec:em.ebrem.lpm} |
---|
554 | % |
---|
555 | %{\color{red} |
---|
556 | Above an electron energy of 1 GeV an analytic |
---|
557 | differential cross section representation is used |
---|
558 | \cite{ebrem.perl}, which was modified to account for the density |
---|
559 | effect and the Landau-Pomeranchuk-Migdal (LPM) effect |
---|
560 | \cite{ebrem.klein,ebrem.stanev}. |
---|
561 | %} |
---|
562 | |
---|
563 | \subsubsection{Relativistic Bremsstrahlung cross section} |
---|
564 | |
---|
565 | %{\color{red} |
---|
566 | The basis of the implementation is the well known high energy limit of |
---|
567 | the Bremsstrahlung process \cite{ebrem.perl}, |
---|
568 | \begin{multline} |
---|
569 | \frac{d\sigma}{dk} = \frac{4 \alpha r_e^2}{3k} \biggl[ \{ y^2 + 2 [ 1 |
---|
570 | + (1-y)^2 ] \} [ Z^2 ( F_{el} - f ) + Z F_{inel} ] \\+ (1-y) \frac{Z^2 |
---|
571 | + Z }{3} \biggr] |
---|
572 | \label{eq:perl} |
---|
573 | \end{multline} |
---|
574 | The {\em elastic from factor} $F_{el}$ and {\em inelastic form factor} |
---|
575 | $F_{inel}$, |
---|
576 | describe the scattering on the nucleus and on the shell electrons, |
---|
577 | respectively, and for $Z>4$ are given by \cite{ebrem.PDGreview} |
---|
578 | \begin{align*} |
---|
579 | F_{el} &= \log\biggl( \frac{184.15}{Z^{\frac{1}{3}} } |
---|
580 | \biggr) |
---|
581 | &\mbox{and}\quad |
---|
582 | F_{inel} &= \log\biggl( \frac{1194.}{Z^{\frac{2}{3}} } \biggr) \;. |
---|
583 | \end{align*} |
---|
584 | This corresponds to the complete screening approximation. |
---|
585 | The Coulomb correction is defined as \cite{ebrem.PDGreview} |
---|
586 | \begin{align*} |
---|
587 | f &= \alpha^2 Z^2 \sum_{n=1}^\infty |
---|
588 | \frac{1}{n ( n^2 + \alpha^2Z^2 )} |
---|
589 | \end{align*} |
---|
590 | % |
---|
591 | This approach provides an analytic differential cross section for an |
---|
592 | efficient evaluation in a Monte Carlo computer code. Note |
---|
593 | that in this approximation the differential cross section $d\sigma/dk$ |
---|
594 | is independent of the energy of the initial electron and is also valid |
---|
595 | for positrons. |
---|
596 | |
---|
597 | The total integrated cross section $\int d\sigma/dk \, dk$ is |
---|
598 | divergent, but the energy loss integral $\int k d\sigma/dk \, dk$ is finite. |
---|
599 | % |
---|
600 | This allows the usual separation into |
---|
601 | continuous enery loss, and discrete photon production according to |
---|
602 | Eqs.\ (\ref{eq:ebrem.eloss}) and (\ref{eq:ebrem.discrete}). |
---|
603 | |
---|
604 | |
---|
605 | |
---|
606 | %} |
---|
607 | |
---|
608 | \subsubsection{Landau Pomeranchuk Migdal (LPM) effect} |
---|
609 | |
---|
610 | %{\color{red} |
---|
611 | At higher energies matter effects become more and |
---|
612 | more important. In GEANT4 the two leading matter effects, the |
---|
613 | LPM effect and the dielectric suppresion (or Ter-Mikaelian effect), |
---|
614 | are considered. The analytic cross section representation, eq.\ |
---|
615 | \eqref{eq:perl}, provides the basis for the incorporation of these |
---|
616 | matter effects. |
---|
617 | %} |
---|
618 | |
---|
619 | The LPM effect (see for example \cite{ebrem.galitsky, ebrem.anthony, ebrem.hansen} ) is the |
---|
620 | suppression of photon production due to the multiple scattering of the |
---|
621 | electron. If an electron undergoes multiple scattering while traversing the |
---|
622 | so called ``formation zone'', the bremsstrahlung amplitudes from before and |
---|
623 | after the scattering can interfere, reducing the probability of bremsstrahlung |
---|
624 | photon emission (a similar suppression occurs for pair production). The |
---|
625 | suppression becomes significant for photon energies below a certain value, |
---|
626 | given by |
---|
627 | \begin{equation} |
---|
628 | \label{ebrem.k} |
---|
629 | \frac{k}{E} < \frac{E}{E_{LPM}} , |
---|
630 | \end{equation} |
---|
631 | where |
---|
632 | \[ |
---|
633 | \begin{array}{ll} |
---|
634 | k & \mbox{photon energy} \\ |
---|
635 | E & \mbox{electron energy} \\ |
---|
636 | E_{LPM} & \mbox{characteristic energy for LPM effect (depend on the medium).} |
---|
637 | \end{array} |
---|
638 | \] |
---|
639 | The value of the LPM characteristic energy can be written as |
---|
640 | \begin{equation} |
---|
641 | \label{eq:ebrem.elpm} |
---|
642 | E_{LPM} = \frac{\alpha m^2 X_0}{4 h c} , |
---|
643 | \end{equation} |
---|
644 | where |
---|
645 | \[ |
---|
646 | \begin{array}{ll} |
---|
647 | \alpha & \mbox{fine structure constant} \\ |
---|
648 | m & \mbox{electron mass} \\ |
---|
649 | X_0 & \mbox{radiation length in the material} \\ |
---|
650 | h & \mbox{Planck constant} \\ |
---|
651 | c & \mbox{velocity of light in vacuum.} |
---|
652 | \end{array} |
---|
653 | \] |
---|
654 | |
---|
655 | %{\color{red} |
---|
656 | \noindent |
---|
657 | At high energies (approximately above 1 GeV) the differential cross section including the |
---|
658 | Landau-Pomeranchuk-Migdal effect, can be expressed using an evaluation |
---|
659 | based on \cite{ebrem.migdal,ebrem.stanev,ebrem.klein} |
---|
660 | \begin{multline} |
---|
661 | \label{eq:ebrem.lpm} |
---|
662 | \frac{d\sigma}{dk} = \frac{4 \alpha r_e^2}{3k} \biggl[ \xi(s) |
---|
663 | \{ y^2 G(s) + 2 [ 1 + (1-y)^2 ] \phi(s) \} \\ |
---|
664 | \times [ Z^2 (F_{el} - f ) + Z F_{inel} ] + (1-y) \frac{Z^2 + Z }{3} \biggr] |
---|
665 | \end{multline} |
---|
666 | where LPM suppression functions are defined by \cite{ebrem.migdal} |
---|
667 | \begin{align} |
---|
668 | G(s) &= 24s^2 \biggl( \frac{\pi}{2} - \int_0^\infty |
---|
669 | e^{-st} \frac{\sin(st)}{\sinh(\frac{t}{2})} dt\biggr) |
---|
670 | \end{align} |
---|
671 | and |
---|
672 | \begin{align} |
---|
673 | \phi(s) &= 12s^2 \Biggl( -\frac{\pi}{2} + \int_0^\infty |
---|
674 | e^{-st} \sin(st) \sinh\Bigl(\frac{t}{2}\Bigr)\, dt\Biggr) |
---|
675 | \end{align} |
---|
676 | They can be piecewise approximated with simple analytic functions, see e.g.\ \cite{ebrem.stanev}. |
---|
677 | The suppression function $\xi(s)$ is recursively defined via |
---|
678 | \begin{align*} |
---|
679 | s = \sqrt{ \frac{k\,E_{\rm LPM} }{8 E(E-k) \xi(s)} } |
---|
680 | \end{align*} |
---|
681 | but can be well approximated using an algorithm introduced by \cite{ebrem.stanev}. |
---|
682 | % |
---|
683 | The material dependent characteristic energy $E_{\rm LPM}$ is defined |
---|
684 | in Eq.\ (\ref{eq:ebrem.elpm}) |
---|
685 | %\begin{align} |
---|
686 | % E_{\rm LPM} = X_0 \frac{\alpha\,m_e^2}{8\pi\hbar c} |
---|
687 | %\end{align} |
---|
688 | according to \cite{ebrem.anthony}. Note that this definition differs from |
---|
689 | other definition (e.g. \cite{ebrem.klein}) by a factor $\frac{1}{2}$. |
---|
690 | |
---|
691 | An additional multiplicative factor governs the dielectric suppression |
---|
692 | effect (Ter-Mikaelian effect) \cite{ebrem.terMikaelian1}. |
---|
693 | \begin{align*} |
---|
694 | S(k) = \frac{k^2}{k^2+k_p^2} |
---|
695 | \end{align*} |
---|
696 | The characteristic photon energy scale $k_p$ is given by the |
---|
697 | plasma frequency of the media, defined as |
---|
698 | \begin{align*} |
---|
699 | k_p = \hbar \omega_p \frac{E_e}{m_e c^2} = \frac{\hbar E_e}{m_e |
---|
700 | c^2}\cdot \sqrt{\frac{n_e e^2}{\epsilon_0 m_e} } |
---|
701 | \;. |
---|
702 | \end{align*} |
---|
703 | |
---|
704 | % |
---|
705 | %\begin{align*} |
---|
706 | % S_{LPM} = \sqrt{ \frac{k\,E_{LPM} }{E^2 } } |
---|
707 | %\end{align*} |
---|
708 | % |
---|
709 | |
---|
710 | Both suppression effects, dielectric suppresion and LPM effect, reduce |
---|
711 | the effective formation length of the photon, |
---|
712 | so the suppressions {\em do not simply multiply.} |
---|
713 | %For the total suppression |
---|
714 | %$S$ the following equation holds (see \cite{ebrem.galitsky}) |
---|
715 | |
---|
716 | A consistent treatment of the overlap region, where both suppression |
---|
717 | mechanism, was suggested by \cite{ebrem.terMikaelian2}. |
---|
718 | The algorithm garanties that the LPM suppression is turned off |
---|
719 | as the density effect becomes important. This is achieved by defining |
---|
720 | a modified suppression variable $\hat{s}$ via |
---|
721 | \begin{align*} |
---|
722 | \hat{s} &= s \cdot \biggl( 1 + \frac{k_p^2}{k^2} \biggr) |
---|
723 | \end{align*} |
---|
724 | and using $\hat{s}$ in the LPM suppression functions $G(s)$ and |
---|
725 | $\phi(s)$ instead of $s$ in Eq.\ (\ref{eq:ebrem.lpm}). |
---|
726 | |
---|
727 | %The presented algorithms could be translated into Monte Carlo code, |
---|
728 | %which proved to perform efficiently for moderately high energies (1 |
---|
729 | %GeV to 100 TeV). |
---|
730 | %} |
---|
731 | |
---|
732 | \subsection{Status of this document} |
---|
733 | 09.10.98 created by L. Urb\'an. \\ |
---|
734 | 21.03.02 modif in angular distribution (M.Maire) \\ |
---|
735 | 27.05.02 re-written by D.H. Wright \\ |
---|
736 | 01.12.03 minor update by V. Ivanchenko \\ |
---|
737 | 20.05.04 updated by L.Urban \\ |
---|
738 | 09.12.05 minor update by V. Ivanchenko \\ |
---|
739 | 15.03.07 modify definition of Elpm (mma) \\ |
---|
740 | 12.12.08 update LPM effect and relativistic Model \\ |
---|
741 | 03.12.09 correct total cross section, formula 3 (mma) |
---|
742 | |
---|
743 | \begin{latexonly} |
---|
744 | |
---|
745 | \begin{thebibliography}{99} |
---|
746 | \bibitem{eedl} |
---|
747 | S.T.Perkins, D.E.Cullen, S.M.Seltzer, UCRL-50400 Vol.31 |
---|
748 | \bibitem{ebrem.geant3} |
---|
749 | GEANT3 manual ,CERN Program Library Long Writeup W5013 (October 1994). |
---|
750 | \bibitem{ebrem.galitsky} |
---|
751 | V.M.~Galitsky and I.I.~Gurevich, Nuovo Cimento 32 (1964) 1820. |
---|
752 | \bibitem{ebrem.anthony} |
---|
753 | P.L. Anthony et al., |
---|
754 | Phys.\ Rev.\ D {\bf 56} (1997) 1373, |
---|
755 | SLAC-PUB-7413/LBNL-40054 (February 1997). |
---|
756 | \bibitem{ebrem.seltzer} |
---|
757 | S.M.Seltzer and M.J.Berger, % Nucl.Inst.Meth. 80 (1985) 12. |
---|
758 | Nucl.\ Inst.\ Meth.\ {\bf B12} (1985) 95. |
---|
759 | \bibitem{ebrem.egs4} W.R. Nelson et al.:The EGS4 Code System. |
---|
760 | {\em SLAC-Report-265 , December 1985 } |
---|
761 | \bibitem{ebrem.messel} |
---|
762 | H.Messel and D.F.Crawford. Pergamon Press,Oxford,1970. |
---|
763 | \bibitem{ebrem.migdal} |
---|
764 | A.B. Migdal, Phys.\ Rev.\ 103 (1956) 1811. |
---|
765 | \bibitem{ebrem.kim} |
---|
766 | L.~Kim et al., Phys.\ Rev.\ A33 (1986) 3002. |
---|
767 | \bibitem{ebrem.williams} |
---|
768 | R.W.~Williams, Fundamental Formulas of Physics, vol.2., Dover Pubs. (1960). |
---|
769 | \bibitem{ebrem.butcher} |
---|
770 | J.C.~Butcher and H. Messel., Nucl.\ Phys.\ 20 (1960) 15. |
---|
771 | \bibitem{ebrem.tsai1} |
---|
772 | Y-S.~Tsai, Rev.\ Mod.\ Phys 46 (1974) 815. |
---|
773 | \bibitem{ebrem.tsai2} |
---|
774 | Y-S.~Tsai, Rev.\ Mod.\ Phys 49 (1977) 421. |
---|
775 | \bibitem{ebrem.PDGreview} |
---|
776 | C. Amsler et al., |
---|
777 | %{\em 2008 Review of Particle Physics}, |
---|
778 | Physics Letters B667, 1 (2008). |
---|
779 | \bibitem{ebrem.perl} |
---|
780 | M.L.~Perl, |
---|
781 | % {\em Notes on the Landau Pomeranchuk Migdal effect: experiment and |
---|
782 | % theory}, |
---|
783 | in Procede Les Rencontres de physique de la Valle D'Aoste, |
---|
784 | SLAC-PUB-6514. |
---|
785 | \bibitem{ebrem.klein} |
---|
786 | S.~Klein, |
---|
787 | % {\em Suppression of bremsstrahlung and pair production due to |
---|
788 | % environmental factors}, |
---|
789 | Rev.\ Mod.\ Phys.\ {\bf 71} (1999) 1501-1538. |
---|
790 | \bibitem{ebrem.stanev} |
---|
791 | T.~Stanev et.al., |
---|
792 | % {\em Development of ultrahigh-energy electromagnetic cascades in |
---|
793 | % water and lead including the Landau-Pomeranchuk-Migdal effect} |
---|
794 | Phys. Rev. D25 (1982) 1291. |
---|
795 | \bibitem{ebrem.hansen} |
---|
796 | H.D.~Hansen et al., |
---|
797 | % {\em Landau-Pomeranchuk-Migdal effect for multihundred GeV electrons}, |
---|
798 | Phys.\ Rev.\ D {\bf 69} (2004) 032001. |
---|
799 | \bibitem{ebrem.terMikaelian1} |
---|
800 | M.L.~Ter-Mikaelian, Dokl.\ Akad. Nauk SSSR 94 (1954) 1033. |
---|
801 | \bibitem{ebrem.terMikaelian2} |
---|
802 | M.L.~Ter-Mikaelian, |
---|
803 | {\em High-energy Electromagnetic Processes in Condensed Media}, |
---|
804 | Wiley, (1972). |
---|
805 | |
---|
806 | \end{thebibliography} |
---|
807 | |
---|
808 | \end{latexonly} |
---|
809 | |
---|
810 | \begin{htmlonly} |
---|
811 | |
---|
812 | \subsection{Bibliography} |
---|
813 | |
---|
814 | \begin{enumerate} |
---|
815 | \item S.T.Perkins, D.E.Cullen, S.M.Seltzer, UCRL-50400 Vol.31 |
---|
816 | \item GEANT3 manual ,CERN Program Library Long Writeup W5013 (October 1994). |
---|
817 | \item V.M.Galitsky and I.I.Gurevich. Nuovo Cimento 32 (1964) 1820. |
---|
818 | \item P.L. Anthony et al., Phys. Rev. D56 (1997) 1373, SLAC-PUB-7413/LBNL-40054 (February 1997) |
---|
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829 | \item |
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833 | \item |
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834 | M.L.~Perl, |
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835 | in Procede Les Rencontres de physique de la Valle D'Aoste, |
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837 | \item |
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841 | T.~Stanev et.al., |
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846 | \item |
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847 | M.L.~Ter-Mikaelian, Dokl.\ Akad. Nauk SSSR 94 (1954) 1033. |
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848 | \item |
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849 | M.L.~Ter-Mikaelian, |
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850 | {\em High-energy Electromagnetic Processes in Condensed Media}, |
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851 | Wiley, (1972). |
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852 | |
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853 | \end{enumerate} |
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854 | |
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855 | \end{htmlonly} |
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