1 | \section[Photoabsorption ionization model]{Photoabsorption Ionization Model} |
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2 | \label{secpai} |
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3 | |
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4 | \subsection{Cross Section for Ionizing Collisions} |
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5 | |
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6 | The Photoabsorption Ionization (PAI) model describes the ionization energy |
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7 | loss of a relativistic charged particle in matter. For such a particle, the |
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8 | differential cross section $d\sigma_i/d\omega$ for ionizing collisions with |
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9 | energy transfer $\omega$ can be expressed most generally by the following |
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10 | equations \cite{pai.asosk}: |
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11 | |
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12 | \begin{eqnarray} |
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13 | \label{PAI1} |
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14 | \frac{d\sigma_i}{d\omega} |
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15 | & = & \frac{2\pi Ze^4}{mv^2} |
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16 | \left\{ |
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17 | \frac{f(\omega)}{\omega\left|\varepsilon(\omega)\right|^2} |
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18 | \left[ |
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19 | \ln\frac{2mv^2}{\omega\left|1-\beta^2\varepsilon\right|} - |
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20 | \right. \right. \nonumber \\ |
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21 | && \left. \left. |
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22 | - \frac{\varepsilon_1 - \beta^2\left|\varepsilon\right|^2}{\varepsilon_2} |
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23 | \arg(1-\beta^2\varepsilon^*) |
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24 | \right] + |
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25 | \frac{\tilde{F}(\omega)}{\omega^2} |
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26 | \right\} , |
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27 | \end{eqnarray} |
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28 | |
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29 | \[ |
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30 | \tilde{F}(\omega) = \int_{0}^{\omega}\frac{f(\omega')} |
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31 | {\left|\varepsilon(\omega')\right|^2}d\omega' , |
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32 | \] |
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33 | |
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34 | \[ |
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35 | f(\omega) = \frac{m\omega\varepsilon_2(\omega)}{2\pi^2ZN\hbar^2} . |
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36 | \] |
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37 | Here $m$ and $e$ are the electron mass and charge, $\hbar$ is Planck's |
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38 | constant, $\beta = v/c$ is the ratio of the particle's velocity $v$ to |
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39 | the speed of light $c$, $Z$ is the effective atomic number, $N$ is the |
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40 | number of atoms (or molecules) per unit volume, and |
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41 | $\varepsilon = \varepsilon_1 + i\varepsilon_2$ is the complex dielectric |
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42 | constant of the medium. In an isotropic non-magnetic medium the dielectric |
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43 | constant can be expressed in terms of a complex index of refraction, |
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44 | $n(\omega) = n_1 + in_2$, $\varepsilon(\omega) = n^2(\omega)$. In the |
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45 | energy range above the first ionization potential $I_1$ for all cases of |
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46 | practical interest, and in particular for all gases, $n_1 \sim 1$. |
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47 | Therefore the imaginary part of the dielectric constant can be expressed in |
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48 | terms of the photoabsorption cross section $\sigma_{\gamma}(\omega)$: |
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49 | |
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50 | \[ |
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51 | \varepsilon_2(\omega) = 2n_1n_2 \sim 2n_2 = \frac{N\hbar c}{\omega} |
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52 | \sigma_{\gamma}(\omega) . |
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53 | \] |
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54 | The real part of the dielectric constant is calculated in turn from the |
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55 | dispersion relation |
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56 | |
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57 | \[ |
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58 | \varepsilon_1(\omega) - 1 = \frac{2N\hbar c}{\pi}V.p.\int_{0}^{\infty} |
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59 | \frac{\sigma_{\gamma}(\omega')}{\omega'^2 - \omega^2}d\omega' , |
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60 | \] |
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61 | where the integral of the pole expression is considered in terms of the |
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62 | principal value. In practice it is convenient to calculate the contribution |
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63 | from the continuous part of the spectrum only. In this case the normalized |
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64 | photoabsorption cross section |
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65 | |
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66 | \[ |
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67 | \tilde{\sigma}_{\gamma}(\omega) = \frac{2\pi^2\hbar e^2Z}{mc} |
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68 | \sigma_{\gamma}(\omega) |
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69 | \left[ |
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70 | \int_{I_1}^{\omega_{max}}\sigma_{\gamma}(\omega')d\omega' |
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71 | \right]^{-1}, \ \omega_{max} \sim 100 \ keV |
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72 | \] |
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73 | is used, which satisfies the quantum mechanical sum rule \cite{pai.fano}: |
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74 | |
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75 | \[ |
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76 | \int_{I_1}^{\omega_{max}}\tilde{\sigma}_{\gamma}(\omega')d\omega' = |
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77 | \frac{2\pi^2\hbar e^2Z}{mc} . |
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78 | \] |
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79 | |
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80 | \noindent |
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81 | The differential cross section for ionizing collisions is expressed by the |
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82 | photoabsorption cross section in the continuous spectrum region: |
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83 | |
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84 | \begin{eqnarray} |
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85 | \frac{d\sigma_i}{d\omega} |
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86 | & = & \frac{\alpha}{\pi\beta^2} |
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87 | \left\{ |
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88 | \frac{\tilde{\sigma}_{\gamma}(\omega)} |
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89 | {\omega\left|\varepsilon(\omega)\right|^2} |
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90 | \left[ |
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91 | \ln\frac{2mv^2}{\omega\left|1-\beta^2\varepsilon\right|} - |
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92 | \right. \right. \nonumber \\ |
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93 | & & \left. \left. |
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94 | - \frac{\varepsilon_1-\beta^2\left|\varepsilon\right|^2}{\varepsilon_2} |
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95 | \arg(1-\beta^2\varepsilon^*) |
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96 | \right] |
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97 | + \frac{1}{\omega^2}\int_{I_1}^{\omega}\frac{\tilde{\sigma}_{\gamma}(\omega')} |
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98 | {\left|\varepsilon(\omega')\right|^2}d\omega' |
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99 | \right\} , |
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100 | \end{eqnarray} |
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101 | |
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102 | \[ |
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103 | \varepsilon_2(\omega) = \frac{N\hbar c}{\omega} |
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104 | \tilde{\sigma}_{\gamma}(\omega) , |
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105 | \] |
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106 | |
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107 | \[ |
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108 | \varepsilon_1(\omega) - 1 = \frac{2N\hbar c}{\pi}V.p.\int_{I_1}^{\omega_{max}} |
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109 | \frac{\tilde{\sigma}_{\gamma}(\omega')}{\omega'^2 - \omega^2}d\omega' . |
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110 | \] |
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111 | \\ |
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112 | |
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113 | \noindent |
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114 | For practical calculations using Eq.~\ref{PAI1} it is convenient to |
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115 | represent the photoabsorption cross section as a polynomial in $\omega^{-1}$ |
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116 | as was proposed in \cite{sandia}: |
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117 | |
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118 | \[ |
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119 | \sigma_{\gamma}(\omega) = \sum_{k=1}^{4}a_{k}^{(i)}\omega^{-k} , |
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120 | \] |
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121 | where the coefficients, $a_{k}^{(i)}$ result from a separate least-squares |
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122 | fit to experimental data in each energy interval $i$. As a rule the |
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123 | interval borders are equal to the corresponding photoabsorption edges. The |
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124 | dielectric constant can now be calculated analytically with elementary |
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125 | functions for all $\omega$, except near the photoabsorption edges where |
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126 | there are breaks in the photoabsorption cross section and the integral for |
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127 | the real part is not defined in the sense of the principal value. \\ |
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128 | |
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129 | \noindent |
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130 | The third term in Eq. (\ref{PAI1}), which can only be integrated |
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131 | numerically, results in a complex calculation of $d\sigma_i/d\omega$. |
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132 | However, this term is dominant for energy transfers $\omega > 10\ keV$, |
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133 | where the function $\left|\varepsilon(\omega)\right|^2 \sim 1$. This is |
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134 | clear from physical reasons, because the third term represents the |
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135 | Rutherford cross section on atomic electrons which can be considered as |
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136 | quasifree for a given energy transfer \cite{allis}. In addition, for high |
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137 | energy transfers, |
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138 | $\varepsilon(\omega) = 1 - \omega_{p}^{2}/\omega^2 \sim 1$, |
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139 | where $\omega_{p}$ is the plasma energy of the material. Therefore the |
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140 | factor $\left|\varepsilon(\omega)\right|^{-2}$ can be removed from under the |
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141 | integral and the differential cross section of ionizing collisions can be |
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142 | expressed as: |
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143 | |
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144 | \begin{eqnarray} |
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145 | \frac{d\sigma_i}{d\omega} |
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146 | & = &\frac{\alpha} |
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147 | {\pi\beta^2\left|\varepsilon(\omega)\right|^2} |
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148 | \left\{ |
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149 | \frac{\tilde{\sigma}_{\gamma}(\omega)}{\omega} |
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150 | \left[ |
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151 | \ln\frac{2mv^2}{\omega\left|1-\beta^2\varepsilon\right|} - |
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152 | \right. \right. \nonumber \\ |
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153 | & & \left. \left. |
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154 | - \frac{\varepsilon_1-\beta^2\left|\varepsilon\right|^2}{\varepsilon_2} |
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155 | \arg(1-\beta^2\varepsilon^*) |
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156 | \right] |
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157 | + \frac{1}{\omega^2}\int_{I_1}^{\omega}\tilde{\sigma}_{\gamma}(\omega')d\omega' |
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158 | \right\} . |
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159 | \end{eqnarray} |
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160 | This is especially simple in gases when |
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161 | $\left|\varepsilon(\omega)\right|^{-2} \sim 1$ for all $\omega > I_1$ |
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162 | \cite{allis}. |
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163 | |
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164 | \subsection{Energy Loss Simulation} |
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165 | |
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166 | For a given track length the number of ionizing collisions is simulated by |
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167 | a Poisson distribution whose mean is proportional to the total cross |
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168 | section of ionizing collisions: |
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169 | |
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170 | \[ |
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171 | \sigma_i = \int_{I_1}^{\omega_{max}}\frac{d\sigma(\omega')}{d\omega'}d\omega' . |
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172 | \] |
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173 | The energy transfer in each collision is simulated according to a |
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174 | distribution proportional to |
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175 | |
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176 | \[ |
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177 | \sigma_i(>\omega) = \int_{\omega}^{\omega_{max}} |
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178 | \frac{d\sigma(\omega')}{d\omega'}d\omega' . |
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179 | \] |
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180 | The sum of the energy transfers is equal to the energy loss. PAI ionisation is implemented |
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181 | according to the model approach (class G4PAIModel) allowing a user to select specific |
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182 | models in different regions. Here is an example physics list: |
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183 | \begin{verbatim} |
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184 | const G4RegionStore* theRegionStore = G4RegionStore::GetInstance(); |
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185 | G4Region* gas = theRegionStore->GetRegion("VertexDetector"); |
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186 | ... |
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187 | if (particleName == "e-") |
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188 | { |
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189 | G4eIonisation* eion = new G4eIonisation(); |
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190 | G4PAIModel* pai = new G4PAIModel(particle, |
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191 | "PAIModel"); |
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192 | // set energy limits where 'pai' is active |
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193 | pai->SetLowEnergyLimit(0.1*keV); |
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194 | pai->SetHighEnergyLimit(100.0*TeV); |
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195 | |
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196 | // here 0 is the highest priority in region 'gas' |
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197 | eion->AddEmModel(0,pai,pai,gas); |
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198 | |
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199 | pmanager->AddProcess(eion,-1, 2, 2); |
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200 | pmanager->AddProcess(new G4MultipleScattering, -1, 1,1); |
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201 | pmanager->AddProcess(new G4eBremsstrahlung,-1,1,3); |
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202 | } |
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203 | \end{verbatim} |
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204 | It shows how to select the G4PAIModel to be the preferred ionisation model for electrons |
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205 | in a G4Region named VertexDetector. The first argument in AddEmModel is 0 which means |
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206 | highest priority. |
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207 | |
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208 | The class G4PAIPhotonModel generates both $\delta$-electrons and photons as secondaries |
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209 | and can be used for more detailed descriptions of ionisation space distribution around |
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210 | the particle trajectory. |
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211 | |
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212 | \subsection{Status of this document} |
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213 | |
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214 | 01.12.05 expanded discussion by V. Grichine \\ |
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215 | 08.05.02 re-written by D.H. Wright \\ |
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216 | 16.11.98 created by V. Grichine \\ |
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217 | |
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218 | \begin{latexonly} |
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219 | |
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220 | \begin{thebibliography}{99} |
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221 | \bibitem{pai.asosk} Asoskov V.S., Chechin V.A., Grichine V.M. at el, |
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222 | {Lebedev Institute annual report, v. 140, p. 3} (1982) |
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223 | \bibitem{pai.fano} Fano U., and Cooper J.W. |
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224 | {Rev.Mod.Phys., v. 40, p. 441} (1968) |
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225 | \bibitem{sandia} Biggs F., and Lighthill R., |
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226 | {Preprint Sandia Laboratory, SAND 87-0070} (1990) |
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227 | \bibitem{allis} Allison W.W.M., and Cobb J. |
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228 | {Ann.Rev.Nucl.Part.Sci., v.30,p.253} (1980) |
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229 | \end{thebibliography} |
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230 | |
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231 | \end{latexonly} |
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232 | |
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233 | \begin{htmlonly} |
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234 | |
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235 | \subsection{Bibliography} |
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236 | |
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237 | \begin{enumerate} |
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238 | \item Asoskov V.S., Chechin V.A., Grichine V.M. at el, |
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239 | {Lebedev Institute annual report, v. 140, p. 3} (1982) |
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240 | \item Fano U., and Cooper J.W. |
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241 | {Rev.Mod.Phys., v. 40, p. 441} (1968) |
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242 | \item Biggs F., and Lighthill R., |
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243 | {Preprint Sandia Laboratory, SAND 87-0070} (1990) |
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244 | \item Allison W.W.M., and Cobb J. |
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245 | {Ann.Rev.Nucl.Part.Sci., v.30,p.253} (1980) |
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246 | \end{enumerate} |
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247 | |
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248 | \end{htmlonly} |
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249 | |
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250 | |
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