1 | \section[Transition radiation]{Transition radiation} |
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2 | |
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3 | \subsection{The Relationship of Transition Radiation to X-ray Cherenkov |
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4 | Radiation} |
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5 | X-ray transition radiation (XTR ) occurs when a relativistic charged |
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6 | particle passes from one medium to another of a different dielectric |
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7 | permittivity. In order to describe this process it is useful to begin |
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8 | with an explanation of X-ray Cherenkov radiation, which is closely related. |
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9 | |
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10 | The mean number of X-ray Cherenkov radiation (XCR) photons of frequency |
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11 | $\omega$ emitted into an angle $\theta$ per unit distance along a particle |
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12 | trajectory is ~\cite{griCR} |
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13 | % |
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14 | \begin{equation} |
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15 | \label{Nxcr} |
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16 | \frac{d^3 \bar{N}_{xcr}}{\hbar d\omega\,dx\,d\theta^2}= |
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17 | \frac{\alpha}{\pi\hbar c}\frac{\omega}{c}\theta^2 |
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18 | \textrm{Im}\left\{Z\right\}. |
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19 | \end{equation} |
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20 | % |
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21 | Here the quantity $Z$ is introduced as the {\em complex formation zone} of |
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22 | XCR in the medium: |
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23 | % |
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24 | \begin{equation} |
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25 | \label{Zj} |
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26 | Z=\frac{L}{1-i\displaystyle\frac{L}{l}},\quad L=\frac{c}{\omega} |
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27 | \left[\gamma^{-2}+\displaystyle\frac{\omega^2_p}{\omega^2}+\theta^2\right]^{-1}, |
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28 | \quad \gamma^{-2}=1-\beta^2. |
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29 | \end{equation} |
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30 | % |
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31 | with $l$ and $\omega_p$ the photon absorption length and the plasma |
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32 | frequency, respectively, in the medium. For the case of a transparent |
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33 | medium, $l \rightarrow \infty$ and the complex formation zone reduces to |
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34 | the {\em coherence length} $L$ of XCR. The coherence length roughly |
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35 | corresponds to that part of the trajectory in which an XCR photon can be |
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36 | created. |
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37 | |
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38 | Introducing a complex quantity $Z$ with its imaginary part proportional to |
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39 | the absorption cross-section ($\sim l^{-1}$) is required in order to account |
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40 | for absorption in the medium. Usually, |
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41 | $\omega_p^2/\omega^2 \gg c/\omega l$. Then it can be seen from Eqs. |
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42 | \ref{Nxcr} and \ref{Zj} that the number of emitted XCR photons is |
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43 | considerably suppressed and disappears in the limit of a transparent |
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44 | medium. This is caused by the destructive interference between the photons |
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45 | emitted from different parts of the particle trajectory. |
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46 | |
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47 | The destructive interference of X-ray Cherenkov radiation is removed if |
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48 | the particle crosses a boundary between two media with different |
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49 | dielectric permittivities, $\epsilon$, where |
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50 | % |
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51 | \begin{equation} |
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52 | \label{eps} |
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53 | \epsilon=1-\frac{\omega^2_p}{\omega^2}+ |
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54 | i\frac{c}{\omega l}. |
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55 | \end{equation} |
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56 | % |
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57 | Here the standard high-frequency approximation for the dielectric |
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58 | permittivity has been used. This is valid for energy transfers larger than |
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59 | the $K$-shell excitation potential. |
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60 | |
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61 | If layers of media are alternated with spacings of order $L$, the X-ray |
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62 | radiation yield from a trajectory of unit length can be increased by roughly |
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63 | $l/L$ times. The radiation produced in this case is called X-ray transition |
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64 | radiation (XTR). |
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65 | |
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66 | \subsection{Calculating the X-ray Transition Radiation Yield} |
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67 | |
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68 | Using the methods developed in Ref. \cite{gri01} one can derive the relation |
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69 | describing the mean number of XTR photons generated per unit photon |
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70 | frequency and $\theta^2$ {\em inside} the radiator for a general XTR |
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71 | radiator consisting of $n$ different absorbing media with fluctuating |
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72 | thicknesses: |
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73 | % |
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74 | \begin{eqnarray} |
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75 | %\begin{equation} |
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76 | \label{Nin} |
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77 | &&\frac{d^2 \bar{N}_{in}}{\hbar d\omega\,d\theta^2}= |
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78 | \frac{\alpha}{\pi\hbar c^2}\omega\theta^2 |
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79 | \textrm{Re}\left\{\sum_{i=1}^{n-1}(Z_{i}-Z_{i+1})^2+ |
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80 | \right. \\ |
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81 | &+&\left. |
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82 | 2\sum_{k=1}^{n-1}\,\sum_{i=1}^{k-1}(Z_{i}-Z_{i+1})\left[\prod_{j=i+1}^{k}F_{j}\right](Z_{k}-Z_{k+1}) |
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83 | \right\},\,F_j=\exp\left[-\frac{t_j}{2Z_j}\right]. \nonumber |
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84 | %\end{equation} |
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85 | \end{eqnarray} |
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86 | % |
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87 | In the case of gamma distributed gap thicknesses (foam or fiber radiators) the values |
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88 | $F_j$, ($j=1,2$) can be estimated as: |
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89 | % |
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90 | \begin{equation} |
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91 | \label{Hj} |
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92 | F_j = \int_0^{\infty}dt_j\, |
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93 | \left(\frac{\nu_j}{\bar{t}_j}\right)^{\nu_j} |
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94 | \frac{t_j^{\nu_j - 1}}{\Gamma(\nu_j)} |
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95 | \exp\left[-\frac{\nu_j t_j}{\bar{t}_j}-\,i\frac{t_j}{2Z_j}\right]= \left[1 + |
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96 | \displaystyle i\frac{\bar{t}_j}{2Z_j\nu_j}\right]^{-\nu_j}, |
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97 | \end{equation} |
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98 | % |
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99 | where $Z_j$ is the complex formation zone of XTR |
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100 | (similar to relation \ref{Zj} for XCR) in the $j$-th medium |
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101 | \cite{gri01,g4xtr}. $\Gamma$ is the Euler gamma function, $\bar{t}_j$ is |
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102 | the mean thickness of the $j$-th medium in the radiator and $\nu_j > 0$ is |
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103 | the parameter roughly describing the relative fluctuations of $t_j$. In |
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104 | fact, the relative fluctuation is $\delta t_j/\bar{t}_j\sim 1/\sqrt{\nu_j}$. |
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105 | |
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106 | In the particular case of $n$ foils of the first medium ($Z_1, F_1$) |
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107 | interspersed with gas gaps of the second medium ($Z_2, F_2$), one obtains: |
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108 | % |
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109 | \begin{equation} |
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110 | \label{Nn1} |
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111 | \frac{d^2 \bar{N}_{in}}{\hbar d\omega\,d\theta^2} = |
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112 | \frac{2\alpha}{\pi\hbar c^2}\omega\theta^2 |
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113 | \textrm{Re}\left\{\langle R^{(n)}\rangle\right\},\quad F = F_1 F_2, |
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114 | \end{equation} |
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115 | % |
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116 | \begin{equation} |
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117 | \label{Rn} |
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118 | \langle R^{(n)}\rangle=(Z_1-Z_2)^2\left\{n\frac{(1-F_1)(1-F_2)}{1-F}+ |
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119 | \frac{(1-F_1)^2F_2[1-F^n]}{(1-F)^2}\right\}. |
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120 | \end{equation} |
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121 | % |
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122 | Here $\langle R^{(n)}\rangle$ is the stack factor reflecting the radiator geometry. |
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123 | The integration of (\ref{Nn1}) with respect to $\theta^2$ can be simplified for the |
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124 | case of a regular radiator ($\nu_{1,2}\rightarrow\infty$), transparent in terms |
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125 | of XTR generation media, and $n\gg 1$~\cite{gar71}. The frequency spectrum of |
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126 | emitted XTR photons is given by: |
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127 | % |
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128 | \begin{eqnarray} |
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129 | \label{Nntr} |
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130 | &&\frac{d \bar{N}_{in}}{\hbar d\omega}= |
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131 | \int_{0}^{\sim 10\gamma^{-2}}d\theta^2\frac{d^2 \bar{N}_{in}}{\hbar d\omega\,d\theta^2}= |
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132 | \frac{4\alpha n}{\pi\hbar\omega}(C_1+C_2)^2 \nonumber \\ |
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133 | &&\cdot\sum_{k=k_{min}}^{k_{max}} |
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134 | \frac{(k-C_{min})}{(k-C_1)^2(k+C_2)^2} |
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135 | %\cdot \nonumber \\ |
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136 | \sin^2\left[\frac{\pi t_1}{t_1+t_2}(k+C_2)\right],\nonumber \\ |
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137 | \end{eqnarray} |
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138 | % |
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139 | % |
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140 | \[ |
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141 | C_{1,2}=\frac{t_{1,2}(\omega^2_1-\omega^2_2)}{4\pi c\omega},\quad |
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142 | C_{min}=\frac{1}{4\pi c}\left[\frac{\omega(t_1+t_2)}{\gamma^2}+ |
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143 | \frac{t_1\omega^2_1+t_2\omega^2_2}{\omega}\right]. |
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144 | \] |
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145 | The sum in (\ref{Nntr}) is defined by terms with $k\geq k_{min}$ corresponding |
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146 | to the region of $\theta\geq 0$. Therefore $k_{min}$ should be the nearest |
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147 | to $C_{min}$ integer $k_{min}\ge C_{min}$. The value of $k_{max}$ is defined by the |
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148 | maximum emission angle $\theta^2_{max}\sim 10\gamma^{-2}$. It can be evaluated as the |
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149 | integer part of |
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150 | \[ |
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151 | C_{max}=C_{min}+\frac{\omega(t_1+t_2)}{4\pi c}\frac{10}{\gamma^2}, \quad |
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152 | k_{max}-k_{min}\sim10^2\div 10^3\gg 1. |
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153 | \] |
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154 | Numerically, however, only a few tens of terms contribute substantially to the |
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155 | sum, that is, one can choose $k_{max}\sim k_{min}+20$. Equation (\ref{Nntr}) |
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156 | corresponds to the spectrum of the total number of photons emitted inside a |
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157 | regular transparent radiator. Therefore the mean interaction length, $\lambda_{XTR}$, |
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158 | of the XTR process in this kind of radiator can be introduced as: |
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159 | \[ |
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160 | \lambda_{XTR}=n(t_1+t_2)\left[\int_{\hbar\omega_{min}}^{\hbar\omega_{max}} |
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161 | \hbar d\omega\frac{d \bar{N}_{in}}{\hbar d\omega}\right]^{-1}, |
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162 | \] |
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163 | where $\hbar\omega_{min}\sim 1$ keV, and $\hbar\omega_{max}\sim 100$ keV for the |
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164 | majority of high energy physics experiments. Its value is constant along |
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165 | the particle trajectory in the approximation of a transparent regular radiator. |
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166 | The spectrum of the total number of XTR photons {\em after} regular transparent |
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167 | radiator is defined by (\ref{Nntr}) with: |
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168 | \[ |
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169 | n\rightarrow n_{eff}=\sum_{k=0}^{n-1}\exp[-k(\sigma_1t_1+\sigma_2t_2)]= |
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170 | \frac{1-\exp[-n(\sigma_1t_1+\sigma_2t_2)]} |
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171 | {1-\exp[-(\sigma_1t_1+\sigma_2t_2)]}, |
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172 | \] |
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173 | where $\sigma_1$ and $\sigma_2$ are the photo-absorption cross-sections corresponding |
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174 | to the photon frequency $\omega$ in the first and the second medium, respectively. |
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175 | With this correction taken into account the XTR absorption in the |
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176 | radiator (\ref{Nntr}) corresponds to the results of \cite{fab75}. In the more |
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177 | general case of the flux of XTR photons {\em after} a radiator, the XTR absorption |
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178 | can be taken into account with a calculation based on the stack factor derived |
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179 | in \cite{gar74}: |
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180 | % |
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181 | \begin{eqnarray} |
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182 | \label{Rflux} |
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183 | \langle R^{(n)}_{flux}\rangle&=& (L_1-L_2)^2\left\{ |
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184 | \frac{1-Q^n}{1-Q}\frac{(1 + Q_1)(1 + F) - 2F_1 - 2 Q_1 F_2}{2(1-F)}\right.\nonumber \\ |
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185 | &+&\left.\frac{(1 - F_1 )(Q_1 - F_1)F_2 (Q^n -F^n)}{(1 - F)(Q - F)} |
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186 | \right\}, |
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187 | \end{eqnarray} |
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188 | % |
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189 | % |
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190 | \[ |
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191 | Q = Q_1\cdot Q_2, \quad Q_j=\exp\left[-t_j/l_j\right]=\exp\left[-\sigma_j t_j\right],\quad j=1,2. |
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192 | \] |
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193 | Both XTR energy loss (\ref{Rn}) and flux (\ref{Rflux}) models can be implemented |
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194 | as a discrete electromagnetic process (see below). |
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195 | |
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196 | |
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197 | \subsection{Simulating X-ray Transition Radiation Production} |
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198 | |
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199 | A typical XTR radiator consits of many ($\sim 100$) boundaries between different |
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200 | materials. To improve the tracking performance in such a volume one can introduce |
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201 | an artificial material \cite{g4xtr}, which is the geometrical mixture of foil and |
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202 | gas contents. Here is an example: |
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203 | \begin{verbatim} |
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204 | // In DetectorConstruction of an application |
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205 | // Preparation of mixed radiator material |
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206 | foilGasRatio = fRadThickness/(fRadThickness+fGasGap); |
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207 | foilDensity = 1.39*g/cm3; // Mylar |
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208 | gasDensity = 1.2928*mg/cm3 ; // Air |
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209 | totDensity = foilDensity*foilGasRatio + |
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210 | gasDensity*(1.0-foilGasRatio); |
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211 | fractionFoil = foilDensity*foilGasRatio/totDensity; |
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212 | fractionGas = gasDensity*(1.0-foilGasRatio)/totDensity; |
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213 | G4Material* radiatorMat = new G4Material("radiatorMat", |
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214 | totDensity, |
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215 | ncomponents = 2 ); |
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216 | radiatorMat->AddMaterial( Mylar, fractionFoil ); |
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217 | radiatorMat->AddMaterial( Air, fractionGas ); |
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218 | G4cout << *(G4Material::GetMaterialTable()) << G4endl; |
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219 | // materials of the TR radiator |
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220 | fRadiatorMat = radiatorMat; // artificial for geometry |
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221 | fFoilMat = Mylar; |
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222 | fGasMat = Air; |
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223 | \end{verbatim} |
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224 | |
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225 | This artificial material will be assigned to the logical volume in which |
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226 | XTR will be generated: |
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227 | |
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228 | \begin{verbatim} |
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229 | solidRadiator = new G4Box("Radiator", |
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230 | 1.1*AbsorberRadius , |
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231 | 1.1*AbsorberRadius, |
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232 | 0.5*radThick ); |
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233 | logicRadiator = new G4LogicalVolume( solidRadiator, |
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234 | fRadiatorMat, // !!! |
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235 | "Radiator"); |
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236 | physiRadiator = new G4PVPlacement(0, |
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237 | G4ThreeVector(0,0,zRad), |
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238 | "Radiator", logicRadiator, |
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239 | physiWorld, false, 0 ); |
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240 | \end{verbatim} |
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241 | |
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242 | XTR photons generated by a relativistic charged particle intersecting a |
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243 | radiator with $2n$ interfaces between different media can be simulated by |
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244 | using the following algorithm. First the total number of XTR photons is |
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245 | estimated using a Poisson distribution about the mean number of photons |
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246 | given by the following expression: |
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247 | % |
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248 | %\begin{equation} |
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249 | %\label{Nn2} |
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250 | \[ |
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251 | \bar{N}^{(n)}=\int_{\omega_1}^{\omega_2}d\omega |
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252 | \int_{0}^{\theta_{max}^2}d\theta^2 |
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253 | \frac{d^2 \bar{N}^{(n)}}{d\omega\,d\theta^2}= |
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254 | %\nonumber\\&=& |
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255 | \frac{2\alpha}{\pi c^2}\int_{\omega_1}^{\omega_2}\omega d\omega |
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256 | \int_{0}^{\theta_{max}^2}\theta^2 d\theta^2 |
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257 | \textrm{Re}\left\{\langle R^{(n)}\rangle\right\}. |
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258 | \] |
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259 | %\end{equation} |
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260 | % |
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261 | Here $\theta_{max}^2\sim 10\gamma^{-2}$, $\hbar\omega_1\sim 1$~keV, |
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262 | $\hbar\omega_2\sim 100$~keV, and $\langle R^{(n)}\rangle$ correspond to the |
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263 | geometry of the experiment. For events in which the number of XTR |
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264 | photons is not equal to zero, the energy and angle of each XTR quantum |
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265 | is sampled from the integral distributions obtained by the numerical |
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266 | integration of expression (\ref{Nn1}). For example, the integral |
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267 | energy spectrum of emitted XTR photons, $\bar{N}^{(n)}_{>\omega}$, is defined |
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268 | from the following integral distribution: |
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269 | % |
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270 | %\begin{equation} |
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271 | %\label{Nomega} |
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272 | \[ |
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273 | \bar{N}^{(n)}_{>\omega}=\frac{2\alpha}{\pi c^2} |
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274 | \int_{\omega}^{\omega_2}\omega d\omega |
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275 | \int_{0}^{\theta_{max}^2}\theta^2 d\theta^2 |
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276 | \textrm{Re}\left\{\langle R^{(n)}\rangle\right\}. |
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277 | \] |
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278 | %\end{equation} |
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279 | % |
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280 | In { \sc Geant4} XTR generation {\em inside} or {\em after} radiators is |
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281 | described as a discrete electromagnetic process. It is convenient for the |
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282 | description of tracks in magnetic fields and can be used for the cases when |
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283 | the radiating charge experiences a scattering inside the radiator. The base |
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284 | class {\tt G4VXTRenergyLoss} is responsible for the creation of tables with |
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285 | integral energy and angular distributions of XTR photons. It also contains |
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286 | the {\tt PostDoIt} function providing XTR photon generation and motion |
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287 | (if fExitFlux=true) through a XTR radiator to its boundary. Particular models |
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288 | like {\tt G4RegularXTRadiator} implement the pure virtual function |
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289 | {\tt GetStackFactor}, which calculates the response of the XTR radiator |
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290 | reflecting its geometry. Included below are some comments for the declaration |
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291 | of XTR in a user application. |
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292 | |
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293 | In the physics list one should pass to the XTR process additional |
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294 | details of the XTR radiator involved: |
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295 | |
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296 | \begin{verbatim} |
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297 | // In PhysicsList of an application |
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298 | else if (particleName == "e-") // Construct processes for electron with XTR |
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299 | { |
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300 | pmanager->AddProcess(new G4MultipleScattering, -1, 1,1 ); |
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301 | pmanager->AddProcess(new G4eBremsstrahlung(), -1,-1,1 ); |
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302 | pmanager->AddProcess(new Em10StepCut(), -1,-1,1 ); |
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303 | // in regular radiators: |
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304 | pmanager->AddDiscreteProcess( |
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305 | new G4RegularXTRadiator // XTR dEdx in general regular radiator |
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306 | // new G4XTRRegularRadModel - XTR flux after general regular radiator |
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307 | // new G4TransparentRegXTRadiator - XTR dEdx in transparent |
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308 | // regular radiator |
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309 | // new G4XTRTransparentRegRadModel - XTR flux after transparent |
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310 | // regular radiator |
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311 | (pDet->GetLogicalRadiator(), // XTR radiator |
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312 | |
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313 | pDet->GetFoilMaterial(), // real foil |
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314 | pDet->GetGasMaterial(), // real gas |
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315 | pDet->GetFoilThick(), // real geometry |
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316 | pDet->GetGasThick(), |
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317 | pDet->GetFoilNumber(), |
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318 | "RegularXTRadiator")); |
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319 | // or for foam/fiber radiators: |
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320 | pmanager->AddDiscreteProcess( |
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321 | new G4GammaXTRadiator - XTR dEdx in general foam/fiber radiator |
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322 | // new G4XTRGammaRadModel - XTR flux after general foam/fiber radiator |
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323 | ( pDet->GetLogicalRadiator(), |
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324 | 1000., |
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325 | 100., |
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326 | pDet->GetFoilMaterial(), |
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327 | pDet->GetGasMaterial(), |
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328 | pDet->GetFoilThick(), |
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329 | pDet->GetGasThick(), |
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330 | pDet->GetFoilNumber(), |
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331 | "GammaXTRadiator")); |
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332 | } |
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333 | \end{verbatim} |
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334 | Here for the foam/fiber radiators the values 1000 and 100 are the $\nu$ parameters |
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335 | (which can be varied) of the Gamma distribution for the foil and gas gaps, |
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336 | respectively. Classes G4TransparentRegXTRadiator and G4XTRTransparentRegRadModel |
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337 | correspond (\ref{Nntr}) to $n$ and $n_{eff}$, respectively. |
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338 | |
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339 | \subsection{Status of this document} |
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340 | 18.11.05 modified by V.Grichine \\ |
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341 | 29.11.02 re-written by D.H. Wright \\ |
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342 | 29.05.02 created by V.Grichine \\ |
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343 | |
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344 | \begin{latexonly} |
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345 | |
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346 | \begin{thebibliography}{99} |
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347 | |
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348 | \bibitem{griCR} V.M. Grichine, |
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349 | {\em Nucl. Instr. and Meth.}, {\bf A482} (2002) 629. |
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350 | |
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351 | \bibitem{gri01} V.M. Grichine, {\em Physics Letters}, {\bf B525} (2002) 225-239 |
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352 | |
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353 | \bibitem{gar71} G.M. Garibyan, |
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354 | {\em Sov. Phys. JETP} {\bf 32} (1971) 23. |
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355 | |
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356 | \bibitem{fab75} C.W. Fabian and W. Struczinski |
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357 | {\em Physics Letters}, {\bf B57 } (1975) 483. |
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358 | |
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359 | \bibitem{gar74} G.M. Garibian, L.A. Gevorgian, and C. Yang, |
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360 | {\em Sov. Phys.- JETP, 39 (1975) 265.} |
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361 | |
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362 | \bibitem{g4xtr} J. Apostolakis, S. Giani, V. Grichine et al., |
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363 | {\em Comput. Phys. Commun.} {\bf 132} (2000) 241. |
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364 | |
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365 | \end{thebibliography} |
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366 | |
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367 | \end{latexonly} |
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368 | |
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369 | \begin{htmlonly} |
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370 | |
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371 | \subsection{Bibliography} |
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372 | |
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373 | \begin{enumerate} |
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374 | \item V.M. Grichine, |
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375 | {\em Nucl. Instr. and Meth.}, {\bf A482} (2002) 629. |
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376 | |
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377 | \item V.M. Grichine, {\em Physics Letters}, {\bf B525} (2002) 225-239 |
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378 | |
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379 | \item G.M. Garibyan, |
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380 | {\em Sov. Phys. JETP} {\bf 32} (1971) 23. |
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381 | |
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382 | \item C.W. Fabian and W. Struczinski |
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383 | {\em Physics Letters}, {\bf B57 } (1975) 483. |
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384 | |
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385 | \item G.M. Garibian, L.A. Gevorgian, and C. Yang, |
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386 | {\em Sov. Phys.- JETP, 39 (1975) 265.} |
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387 | |
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388 | \item J. Apostolakis, S. Giani, V. Grichine et al., |
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389 | {\em Comput. Phys. Commun.} {\bf 132} (2000) 241. |
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390 | |
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391 | \end{enumerate} |
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392 | |
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393 | \end{htmlonly} |
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394 | |
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395 | |
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