[1211] | 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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