1 | \section{Bremsstrahlung}\label{lowebrems} |
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2 | |
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3 | The class G4LowEnergyBremsstrahlung calculates the continuous energy loss due to low energy gamma emission and |
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4 | simulates the gamma production by electrons. |
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5 | The gamma production threshold for a given material $\omega_c$ is used to separate the continuous and the |
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6 | discrete parts of the process. The energy loss of an electron with the incident energy $T$ are expressed |
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7 | via the integrand over energy of the gammas: |
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8 | |
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9 | \begin{equation} |
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10 | {dE\over dx}=\sigma(T){{\int^{\omega_c}_{0.1eV}t{d\sigma\over d\omega}d\omega} \over{\int^{T}_{0.1eV} |
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11 | {d\sigma\over d\omega}d\omega}}, |
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12 | \end{equation} |
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13 | |
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14 | where $\sigma(T)$ is the total cross-section at a given incident kinetic energy, $T$, $0.1eV$ is the low energy limit |
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15 | of the EEDL data. The production cross-section is a complimentary function: |
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16 | |
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17 | \begin{equation} |
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18 | \sigma=\sigma(T){{\int^{T}_{\omega_c}{d\sigma\over d\omega}d\omega}\over {\int^{T}_{0.1eV}{d\sigma\over d\omega}d\omega}}. |
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19 | \end{equation} |
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20 | |
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21 | The total cross-section, $\sigma_s$, is obtained from an interpolation of the evaluated cross-section data in the EEDL |
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22 | library~\cite{io-EEDL}, according to the formula (\ref{eqloglog}) in Section~\ref{subsubsigmatot}. |
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23 | |
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24 | The EEDL data~\cite{br-leg4} of total cross-sections are parametrised~\cite{br-EEDL} according to (\ref{eqloglog}). |
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25 | The probability of the emission of a photon with energy, $\omega$, considering an electron of incident kinetic energy, |
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26 | $T$, is generated according to the formula: |
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27 | |
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28 | \begin{equation} |
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29 | \label{eqbrem} |
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30 | {d\sigma \over d\omega} = {F(x) \over x}, \;\; \mbox{with} x = {\omega \over T}. |
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31 | \end{equation} |
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32 | |
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33 | The function, $F(x)$, describing energy spectra of the outcoming photons is taken from the EEDL library. For each |
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34 | element 15 points in $x$ from 0.01 to 1 are used for the linear interpolation of this function. The function $F$ is |
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35 | normalised by the condition $F(0.01) = 1$. The energy distributions of the emitted photons available in the EEDL |
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36 | library are for only a few incident electron energies (about 10 energy points between 10 eV and 100 GeV). For other |
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37 | energies a logarithmic interpolation formula (\ref{eqloglog}) is used to obtain values for the function, $F(x)$. |
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38 | For high energies, the spectral function is very close to: |
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39 | |
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40 | \begin{equation} |
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41 | F(x) = 1 - x + 0.75x^2. |
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42 | \end{equation} |
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43 | |
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44 | \subsection{Bremsstrahlung angular distributions} |
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45 | |
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46 | The angular distribution of the emitted photons with respect to the incident |
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47 | electron can be sampled according to three alternative generators described below. |
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48 | The direction of the outcoming electron is determined from the energy-momentum balance. |
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49 | This generators are currently implemented in G4ModifiedTsai, G4Generator2BS and |
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50 | G4Generator2BN classes. |
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51 | |
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52 | \subsubsection*{G4ModifiedTsai} |
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53 | |
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54 | \noindent |
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55 | The angular distribution of the emitted photons is obtained from a |
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56 | simplified \cite{br-g3} formula based on the Tsai cross-section \cite{br-tsai}, |
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57 | which is expected to become isotropic in the low energy limit. |
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58 | |
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59 | \subsubsection*{G4Generator2BS} |
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60 | |
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61 | In G4Generator2BS generator, the angular distribution of the emitted photons is obtained |
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62 | from the 2BS Koch and Motz bremsstrahlung double differential cross-section \cite{br-KandM}: |
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63 | |
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64 | \begin{eqnarray*} |
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65 | d\sigma_k,_\theta & = & \frac{4Z^2 r_0^2}{137} \frac{dk}{k} ydy \left\{ |
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66 | \frac{16y^2E}{(y^2+1)^4E_0}-\right.{} \nonumber \\ |
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67 | & & \left.\frac{(E_0+E)^2}{(y^2+1)^2E_0^2} + |
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68 | \left[ \frac{E_0^2+E^2}{(y^2+1)^2E_0^2}- \frac{4y^2E}{(y^2+1)^4E_0}\right]ln M(y)\right\} |
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69 | \end{eqnarray*} |
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70 | |
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71 | \noindent |
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72 | where $k$ the photon energy, $\theta$ the emission angle, $E_0$ and $E$ are the |
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73 | initial and final electron energy in units of $m_e c^2$, $r_0$ is the classical |
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74 | electron radius and $Z$ the atomic number of the material. $y$ and $M(y)$ are |
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75 | defined as: |
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76 | \begin{eqnarray*} |
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77 | y&=&E_0\theta \nonumber \\ |
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78 | \frac{1}{M(y)}&=&\left(\frac{k}{2E_0E}\right)^2+\left(\frac{Z^{1/3}}{111(y^2+1)}\right)^2 |
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79 | \end{eqnarray*} |
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80 | |
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81 | The adopted sampling algorithm is based on the sampling scheme developed by |
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82 | A. F. Bielajew et al. \cite{br-pirs}, and latter implemented in EGS4. In this sampling algorithm |
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83 | only the angular part of 2BS is used, with the emitted photon energy, $k$, determined by |
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84 | GEANT4 $\left(\frac{d\sigma}{dk}\right)$ differential cross-section. |
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85 | |
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86 | |
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87 | \subsubsection*{G4Generator2BN} |
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88 | |
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89 | \noindent |
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90 | The angular distribution of the emitted photons is obtained from the 2BN Koch and Motz bremsstrahlung |
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91 | double differential cross-section \cite{br-KandM} that can be written as: |
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92 | |
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93 | \begin{eqnarray*} |
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94 | d\sigma_k,_\theta & = & \frac{Z^2 r_0^2}{8\pi 137}\frac{dk}{k} \frac{p}{p_0} d\Omega_k |
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95 | \left \{ \frac{8\sin^2\theta (2E_0^2-1)}{p_0^2\Delta_0^4}- \right.{} \nonumber \\ |
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96 | & & \left.\frac{2(5E_0^2+2EE_0+3)}{p_0^2\Delta_0^2} |
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97 | - \frac{2(p_0^2-k^2)}{Q^2\Delta_0}+\frac{4E}{p_2^2\Delta_0}+\frac{L}{pp_0} \right.{} \nonumber \\ |
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98 | & & \left. \left[ \frac{4E_0\sin^2\theta(3k-p_0^2E)}{p_0^2\Delta^4} + \frac{4E_0^2(E_0^2+E^2)} |
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99 | {p_0^2\Delta_0^2}+ \right.\right.{} \nonumber \\ |
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100 | & & \left.\left. \frac{2-2(E_0^2-3EE_0+E^2)}{p_0^2\Delta_0^2}+\frac{2k(E_0^2+EE_0-1)} |
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101 | {p_0^2\Delta_0}\right] \right.{} \nonumber \\ |
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102 | & & \left. -\left(\frac{4\epsilon}{p\Delta0}\right) + \left(\frac{\epsilon^Q}{pQ}\right) |
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103 | \left[\frac{4}{\Delta^2_0}-\frac{6k}{\Delta_0}-\frac{2k(p_0^2-k^2)}{Q^2\Delta_0}\right]\right \} |
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104 | \end{eqnarray*} |
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105 | \noindent in which: |
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106 | \begin{eqnarray*} |
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107 | L&=&\ln\left[\frac{EE_0-1+pp_0}{EE_0-1-pp_0}\right] \nonumber \\ |
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108 | \Delta_0&=&E_0-p_0\cos\theta \nonumber \\ |
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109 | Q^2&=&p_0^2+k^2-2p_0k\cos\theta \nonumber \\ |
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110 | \epsilon&=&\ln\left[\frac{E+p}{E-p}\right] \qquad \epsilon^Q=\ln\left[\frac{Q+p}{Q-p}\right] |
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111 | \end{eqnarray*} |
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112 | |
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113 | \noindent |
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114 | where $k$ is the photon energy, $\theta$ the emission angle and $(E_0,p_0)$ and $(E,p)$ are the total |
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115 | (energy, momentum) of the electron before and after the radiative emission, all in units of $m_e c^2$.\\ |
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116 | Since the 2BN cross--section is a 2-dimensional non-factorized distribution an |
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117 | acceptance-rejection technique was the adopted. For the 2BN distribution, two functions |
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118 | $g_1(k)$ and $g_2(\theta)$ were defined: |
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119 | |
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120 | \begin{equation} |
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121 | g_1(k) = k^{-b} \qquad\qquad g_2(\theta)=\frac{\theta}{1+c\theta^2} |
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122 | \end{equation} |
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123 | |
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124 | \noindent |
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125 | such that: |
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126 | |
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127 | \begin{equation} |
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128 | Ag_1(k)g_2(\theta) \ge \frac{d\sigma}{dkd\theta} |
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129 | \end{equation} |
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130 | |
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131 | \noindent |
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132 | where A is a global constant to be completed. Both functions have an analytical |
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133 | integral $G$ and an analytical inverse $G^{-1}$. The $b$ parameter of $g_1(k)$ was |
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134 | empirically tuned and set to $1.2$. For positive $\theta$ values, $g_2(\theta)$ has a maximum |
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135 | at $\frac{1}{\sqrt(c)}$. $c$ parameter controls the function global shape and it was |
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136 | used to tune $g_2(\theta)$ according to the electron kinetic energy.\\ |
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137 | To generate photon energy $k$ according to $g_1$ and $\theta$ according to $g_2$ the |
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138 | inverse-transform method was used. The integration of these functions gives |
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139 | |
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140 | \begin{equation} |
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141 | G_1 = C_1 \int_{k_{min}}^{k_{max}} k'^{-b}dk' = C_1 \frac{k^{1-b}-k^{1-b}_{min}}{1-b} |
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142 | \end{equation} |
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143 | |
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144 | \begin{equation} |
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145 | G_2 = C_2 \int_{0}^{\theta} \frac{\theta'}{1+c\theta'^2}d\theta'=C_2 \frac{\log(1+c\theta^2)}{2c} |
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146 | \end{equation} |
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147 | |
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148 | \noindent |
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149 | where $C_1$ and $C_2$ are two global constants chosen to normalize the integral in the overall range |
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150 | to the unit. The photon momentum $k$ will range from a minimum cut value $k_{min}$ (required to avoid |
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151 | infrared divergence) to a maximum value equal to the electron kinetic energy $E_k$, while the polar |
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152 | angle ranges from 0 to $\pi$, resulting for $C_1$ and $C_2$: |
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153 | |
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154 | \begin{equation} |
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155 | C_1 = \frac{1-b}{E_k^{1-b}} \qquad\qquad C_2 = \frac{2c}{\log(1+c\pi^2)} |
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156 | \end{equation} |
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157 | |
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158 | \noindent |
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159 | $k$ and $\theta$ are then sampled according to: |
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160 | |
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161 | \begin{equation} |
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162 | k = \left[ \frac{1-b}{C_1}\xi_1 + k_{min}^{1-b} \right] \qquad\qquad \theta = \sqrt{\frac{\exp\left(\frac{2c\xi_2}{C_1}\right)}{2c}} |
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163 | \end{equation} |
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164 | |
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165 | \noindent |
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166 | where $\xi_1$ and $\xi_2$ are uniformly sampled in the interval (0,1). The event is accepted if: |
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167 | |
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168 | \begin{equation} |
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169 | uAg_1(k)g_2(\theta) \le \frac{d\sigma}{dkd\theta} |
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170 | \end{equation} |
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171 | |
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172 | \noindent |
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173 | where $u$ is a random number with uniform distribution in (0,1). The $A$ and $c$ parameters were computed |
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174 | in a logarithmic grid, ranging from 1 keV to 1.5 MeV with 100 points per decade. |
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175 | Since the $g_2(\theta)$ function has a maximum at $\theta = \frac{1}{\sqrt{c}}$, |
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176 | the $c$ parameter was computed using the relation $c=\frac{1}{\theta_{max}}$. At the point ($k_{min},\theta_{max}$) |
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177 | where $k_{min}$ is the $k$ cut value, the double differential cross-section has its maximum value, since it is |
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178 | monotonically decreasing in $k$ and thus the global normalization parameter $A$ is estimated from the relation: |
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179 | |
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180 | \begin{equation} |
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181 | A g_1(k_{min})g_2({\theta_{max}})= \left(\frac{d^2\sigma}{dkd\theta}\right)_{max} |
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182 | \end{equation} |
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183 | |
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184 | \noindent |
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185 | where $g_1(k_{min})g_2({\theta_{max}}) = \frac{k_{min}^{-b}}{2\sqrt{c}}$. |
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186 | Since $A$ and $c$ can only be retrieved for a fixed number of electron kinetic energies there exists the possibility that |
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187 | $A g_1(k_{min})g_2({\theta_{max}})\le\left(\frac{d^2\sigma}{dkd\theta}\right)_{max}$ for a given $E_k$. This is a small |
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188 | violation that can be corrected introducing an additional multiplicative factor to the $A$ parameter, which was |
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189 | empirically determined to be 1.04 for the entire energy range.\\ |
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190 | |
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191 | \subsubsection*{Comparisons between Tsai, 2BS and 2BN generators} |
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192 | |
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193 | The currently available generators can be used according to the user required |
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194 | precision and timing requirements. Regarding the energy range, validation results |
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195 | indicate that for lower energies ($\le$ 100 keV) there is a significant |
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196 | deviation on the most probable emission angle between Tsai/2BS generators |
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197 | and the 2BN generator - Figure \ref{br-dist}. The 2BN generator maintains however a good agreement |
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198 | with Kissel data \cite{Kissel}, derived from the work of Tseng and co-workers \cite{Pratt}, |
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199 | and it should be used for energies between 1 keV and 100 keV \cite{IEEE}. |
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200 | As the electron kinetic energy increases, the different distributions tend to overlap |
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201 | and all generators present a good agreement with Kissel data. |
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202 | |
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203 | \begin{figure}[hbtp] |
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204 | \begin{center} |
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205 | \setlength{\unitlength}{0.0105in}% |
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206 | \includegraphics[width=4.7cm]{electromagnetic/lowenergy/br-10kev.eps}% |
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207 | \includegraphics[width=4.7cm]{electromagnetic/lowenergy/br-100kev.eps}% |
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208 | \includegraphics[width=4.7cm]{electromagnetic/lowenergy/br-500kev.eps}% |
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209 | \end{center} |
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210 | \caption{Comparison of polar angle distribution of bremsstrahlung photons ($k/T=0.5$) for |
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211 | 10 keV ({\em left}) and 100 keV ({\em middle}) and 500 keV ({\em right}) electrons in silver, |
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212 | obtained with Tsai, 2BS and 2BN generator} |
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213 | \label{br-dist} |
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214 | \end{figure} |
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215 | |
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216 | \noindent |
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217 | In figure \ref{br-eff} the sampling efficiency for the different generators are presented. |
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218 | The sampling generation efficiency was defined as the ratio between the |
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219 | number of generated events and the total number of trials. As energies increases the sampling efficiency |
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220 | of the 2BN algorithm decreases from 0.65 at 1 keV electron kinetic energy down to almost 0.35 at 1 MeV. |
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221 | For energies up to 10 keV the 2BN sampling efficiency is superior or equivalent to the one of the |
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222 | 2BS generator. These results are an indication that precision simulation of low energy bremsstrahlung |
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223 | can be obtained with little performance degradation. For energies above 500 keV, Tsai generator can be |
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224 | used, retaining a good physics accuracy and a sampling efficiency superior to the 2BS generator. |
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225 | % |
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226 | \begin{figure}[hbtp] |
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227 | \begin{center} |
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228 | \setlength{\unitlength}{0.0105in}% |
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229 | \includegraphics[width=8cm]{electromagnetic/lowenergy/br-eff.eps} |
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230 | \end{center} |
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231 | \caption{Sampling efficiency for Tsai generator, 2BS and 2BN Koch and Motz generators.} |
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232 | \label{br-eff} |
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233 | \end{figure} |
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234 | |
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235 | \subsection{Status of the document} |
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236 | |
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237 | \noindent |
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238 | 30.09.1999 created by Alessandra Forti\\ |
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239 | 07.02.2000 modified by V\'eronique Lef\'ebure\\ |
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240 | 08.03.2000 reviewed by Petteri Nieminen and Maria Grazia Pia\\ |
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241 | 05.12.2001 modified by Vladimir Ivanchenko\\ |
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242 | 13.05.2002 modified by Vladimir Ivanchenko\\ |
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243 | 24.11.2003 modified by Andreia Trindade, Pedro Rodrigues and Luis Peralta\\ |
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244 | |
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245 | \begin{latexonly} |
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246 | |
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247 | \begin{thebibliography}{99} |
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248 | \bibitem{br-leg4} |
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249 | ``Geant4 Low Energy Electromagnetic Models for Electrons and Photons", |
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250 | J.Apostolakis et al., CERN-OPEN-99-034(1999), INFN/AE-99/18(1999) |
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251 | \bibitem{br-EEDL} |
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252 | %http://reddog1.llnl.gov/homepage.red/Electron.htm |
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253 | ``Tables and Graphs of Electron-Interaction Cross-Sections from 10~eV to 100~GeV Derived from |
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254 | the LLNL Evaluated Electron Data Library (EEDL), Z=1-100" |
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255 | S.T.Perkins, D.E.Cullen, S.M.Seltzer, |
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256 | UCRL-50400 Vol.31 |
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257 | \bibitem{br-g3} |
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258 | ``GEANT, Detector Description and Simulation Tool", |
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259 | CERN Application Software Group, CERN Program Library Long Writeup W5013 |
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260 | \bibitem{br-tsai} |
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261 | ``Pair production and bremsstrahlung of charged leptons", |
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262 | Y. Tsai, Rev. Mod. Phys., Vol.46, 815(1974), Vol.49, 421(1977) |
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263 | \bibitem{br-KandM} |
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264 | ``Bremsstrahlung Cross-Section Formulas and Related Data", |
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265 | H. W. Koch and J. W. Motz, Rev. Mod. Phys., Vol.31, 920(1959) |
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266 | \bibitem{br-pirs} |
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267 | ``Improved bremsstrahlung photon angular sampling in the EGS4 code system'', |
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268 | A. F. Bielajew, R. Mohan and C.-S. Chui, Report NRCC/PIRS-0203 (1989) |
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269 | \bibitem{Kissel} |
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270 | ``Bremsstrahlung from electron collisions with neutral atoms'', |
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271 | L. Kissel, C. A. Quarls and R. H. Pratt, At. Data Nucl. Data Tables, Vol. 28, 382(1983) |
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272 | \bibitem{Pratt} |
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273 | ``Electron bremsstrahlung angular distributions in the 1-500 keV energy range'', |
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274 | H. K. Tseng, R. H. Pratt and C. M. Lee , Phys. Rev. A, Vol. 19, 187(1979) |
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275 | \bibitem{IEEE} |
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276 | ``GEANT4 Applications and Developments for Medical Physics Experiments'', |
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277 | P. Rodrigues et al. IEEE 2003 NSS/MIC Conference Record |
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278 | \end{thebibliography} |
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279 | |
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280 | \end{latexonly} |
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281 | |
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282 | \begin{htmlonly} |
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283 | |
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284 | \subsection{Bibliography} |
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285 | |
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286 | \begin{enumerate} |
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287 | \item |
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288 | ``Geant4 Low Energy Electromagnetic Models for Electrons and Photons", |
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289 | J.Apostolakis et al., CERN-OPEN-99-034(1999), INFN/AE-99/18(1999) |
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290 | \item |
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291 | %http://reddog1.llnl.gov/homepage.red/Electron.htm |
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292 | ``Tables and Graphs of Electron-Interaction Cross-Sections from 10~eV to 100~GeV Derived from |
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293 | the LLNL Evaluated Electron Data Library (EEDL), Z=1-100" |
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294 | S.T.Perkins, D.E.Cullen, S.M.Seltzer, |
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295 | UCRL-50400 Vol.31 |
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296 | \item |
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297 | ``GEANT, Detector Description and Simulation Tool", |
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298 | CERN Application Software Group, CERN Program Library Long Writeup W5013 |
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299 | \item |
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300 | ``Pair production and bremsstrahlung of charged leptons", |
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301 | Y. Tsai, Rev. Mod. Phys., Vol.46, 815(1974), Vol.49, 421(1977) |
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302 | \item |
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303 | ``Bremsstrahlung Cross-Section Formulas and Related Data", |
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304 | H. W. Koch and J. W. Motz, Rev. Mod. Phys., Vol.31, 920(1959) |
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305 | \item |
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306 | ``Improved bremsstrahlung photon angular sampling in the EGS4 code system'', |
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307 | A. F. Bielajew, R. Mohan and C.-S. Chui, Report NRCC/PIRS-0203 (1989) |
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308 | \item |
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309 | ``Bremsstrahlung from electron collisions with neutral atoms'', |
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310 | L. Kissel, C. A. Quarls and R. H. Pratt, At. Data Nucl. Data Tables, Vol. 28, 382(1983) |
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311 | \item |
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312 | ``Electron bremsstrahlung angular distributions in the 1-500 keV energy range'', |
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313 | H. K. Tseng, R. H. Pratt and C. M. Lee , Phys. Rev. A, Vol. 19, 187(1979) |
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314 | \item |
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315 | ``GEANT4 Applications and Developments for Medical Physics Experiments'', |
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316 | P. Rodrigues et al. IEEE 2003 NSS/MIC Conference Record |
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317 | \end{enumerate} |
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318 | |
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319 | \end{htmlonly} |
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320 | |
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321 | |
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