[1211] | 1 | \section[Ionization]{Ionization} \label{sec:em.eion} |
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| 2 | |
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| 3 | \subsection{Method} |
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| 4 | |
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| 5 | The $G4eIonisation$ class provides the continuous and discrete |
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| 6 | energy losses of electrons and positrons due to ionization in a material |
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| 7 | according to the approach described in Section \ref{en_loss}. |
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| 8 | The value of the maximum energy transferable to a free electron $T_{max}$ |
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| 9 | is given by the following relation: |
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| 10 | \begin{equation} |
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| 11 | \label{eion.c} |
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| 12 | T_{max} = \left\{ \begin{array}{ll} |
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| 13 | E-mc^2 & {for \hspace{.2cm} e^+} \\ |
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| 14 | (E-mc^2)/2 & {for \hspace{.2cm} e^- } \\ |
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| 15 | \end{array} \right . |
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| 16 | \end{equation} |
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| 17 | where $mc^2$ is the electron mass. |
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| 18 | Above a given threshold energy the energy loss is simulated by the |
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| 19 | explicit production of delta rays by M\"{o}ller scattering ($e^- e^-$), or |
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| 20 | Bhabha scattering ($e^+ e^-$). Below the threshold the soft electrons |
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| 21 | ejected are simulated as continuous energy loss by the incident |
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| 22 | ${e^{\pm}}$. |
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| 23 | |
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| 24 | \subsection{Continuous Energy Loss} \label{seceloss} |
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| 25 | |
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| 26 | The integration of \ref{comion.a} leads to the Berger-Seltzer |
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| 27 | formula \cite{eion.messel}: |
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| 28 | \begin{equation} |
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| 29 | \label{eion.d e} |
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| 30 | \left. \frac{dE}{dx} \right]_{T < T_{cut}} = |
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| 31 | 2 \pi r_e^2 mc^2 n_{el} \frac{1}{\beta^2} |
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| 32 | \left [\ln \frac{2(\gamma + 1)} {(I/mc^2)^2}+ F^{\pm} (\tau , \tau_{up}) |
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| 33 | - \delta \right ] |
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| 34 | \end{equation} |
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| 35 | with |
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| 36 | \[ |
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| 37 | \begin{array}{ll} |
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| 38 | r_e & \mbox{classical electron radius:} |
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| 39 | \quad e^2/(4 \pi \epsilon_0 mc^2 ) \\ |
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| 40 | mc^2 & \mbox{mass energy of the electron} \\ |
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| 41 | n_{el} & \mbox{electron density in the material} \\ |
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| 42 | I & \mbox{mean excitation energy in the material}\\ |
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| 43 | \gamma & \mbox{$E/mc^2$} \\ |
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| 44 | \beta^2 & 1-(1/\gamma^2) \\ |
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| 45 | \tau & \gamma-1 \\ |
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| 46 | T_{cut} & \mbox{minimum energy cut for $\delta$ -ray production} \\ |
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| 47 | \tau_c & \mbox{$T_{cut}/mc^2$} \\ |
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| 48 | \tau_{max} & \mbox{maximum energy transfer: |
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| 49 | $\tau$ for $e^+$, $\tau/2$ for $e^-$} \\ |
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| 50 | \tau_{up} & \min(\tau_c,\tau_{max}) \\ |
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| 51 | \delta & \mbox{density effect function} . |
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| 52 | \end{array} |
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| 53 | \] |
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| 54 | In an elemental material the electron density is |
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| 55 | $$ n_{el} = Z \: n_{at} = Z \: \frac{\mathcal{N}_{av} \rho}{A} . $$ |
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| 56 | $\mathcal{N}_{av}$ is Avogadro's number, $\rho$ is the material density, |
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| 57 | and $A$ is the mass of a mole. In a compound material |
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| 58 | $$ |
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| 59 | n_{el} = \sum_i Z_i \: n_{ati} |
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| 60 | = \sum_i Z_i \: \frac{\mathcal{N}_{av} w_i \rho}{A_i} , |
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| 61 | $$ |
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| 62 | where $w_i$ is the proportion by mass of the $i^{th}$ element, with molar |
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| 63 | mass $A_i$ . |
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| 64 | \par |
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| 65 | \noindent |
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| 66 | The mean excitation energies $I$ for all elements are taken from |
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| 67 | \cite{ioni.icru1}. |
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| 68 | \par |
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| 69 | \noindent |
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| 70 | The functions $ F^{\pm}$ are given by : |
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| 71 | \begin{eqnarray} |
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| 72 | F^+ (\tau,\tau_{up}) & = &\ln(\tau\tau_{up} ) \\ |
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| 73 | & & -\frac{\tau_{up}^2}{\tau}\left[\tau + 2 \tau_{up} - |
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| 74 | \frac{3\tau_{up}^2 y } {2} -\left(\tau_{up} - \frac{\tau_{up}^3 }{3} \right) y^2 |
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| 75 | - \left (\frac{\tau_{up}^2}{2} - \tau |
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| 76 | \frac{\tau_{up}^3}{3} + \frac{\tau_{up}^4 } {4} \right) |
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| 77 | y^3 \right] \nonumber |
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| 78 | \end{eqnarray} |
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| 79 | |
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| 80 | \begin{eqnarray} |
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| 81 | F^- (\tau,\tau_{up} ) & = & -1 -\beta^2 \\ |
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| 82 | & & +\ln \left [(\tau - \tau_{up}) |
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| 83 | \tau_{up} \right ] + \frac{\tau}{\tau -\tau_{up}} |
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| 84 | + \frac{1}{\gamma^2} \left [ |
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| 85 | \frac{\tau_{up}^2}{2} + ( 2\tau +1) \ln |
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| 86 | \left (1- \frac{\tau_{up}}{\tau} \right ) \right ] \nonumber |
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| 87 | \end{eqnarray} |
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| 88 | where $y = 1/(\gamma+1)$. |
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| 89 | |
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| 90 | |
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| 91 | The density effect correction is calculated according to the formalism of |
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| 92 | Sternheimer \cite{eion.sternheimer}: |
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| 93 | \input{electromagnetic/utils/densityeffect} |
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| 94 | |
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| 95 | \subsection{Total Cross Section per Atom and Mean Free Path } \label{sectot} |
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| 96 | |
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| 97 | The total cross section per atom for M\"{o}ller scattering ($e^- e^-$) and |
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| 98 | Bhabha scattering ($e^+ e^-$) is obtained by integrating Eq.~\ref{comion.b}. |
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| 99 | In {\sc Geant4} $T_{cut}$ is always 1 keV or larger. For delta ray energies |
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| 100 | much larger than the excitation energy of the material ($T \gg I$), the |
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| 101 | total cross section becomes \cite{eion.messel} for M\"{o}ller scattering, |
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| 102 | \begin{eqnarray} |
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| 103 | \sigma ( Z,E,T_{cut} ) & = & \frac {2 \pi r_e^2 Z}{\beta^2(\gamma -1)} \times \\ |
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| 104 | & & \left[\frac{(\gamma-1)^2} {\gamma^2}\left(\frac{1}{2}-x\right) |
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| 105 | +\frac{1}{x}-\frac{1}{1-x}-\frac{2\gamma-1}{\gamma^2}\ln |
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| 106 | \frac{1-x}{x}\right] , \nonumber |
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| 107 | \end{eqnarray} |
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| 108 | and for Bhabha scattering ($e^+ e^-$), |
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| 109 | \begin{eqnarray} |
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| 110 | \sigma (Z,E,T_{cut}) & = & \frac{ 2 \pi r_e^2 Z }{(\gamma -1)} \times \\ |
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| 111 | & & \left [\frac {1 }{\beta^2} \left(\frac{1}{x}-1\right) |
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| 112 | + B_1 \ln x + B_2 (1-x) - |
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| 113 | \frac {B_3 } {2} ( 1-x^2 ) +\frac{B_4}{3}(1-x^3)\right] . \nonumber |
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| 114 | \end{eqnarray} |
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| 115 | Here |
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| 116 | \[ |
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| 117 | \begin{array}{lcllcl} |
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| 118 | \gamma & = & E/mc^2 & |
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| 119 | B_1 & = & 2-y^2 \\ |
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| 120 | \beta^2 & = & 1-(1/\gamma^2) & |
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| 121 | B_2 & = & (1-2y)(3+y^2 ) \\ |
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| 122 | x & = & T_{cut}/(E-mc^2) & |
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| 123 | B_3 & = & (1-2y)^2+(1-2y)^3 \\ |
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| 124 | y & = & 1/(\gamma + 1) & |
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| 125 | B_4 & = & (1-2y)^3 . |
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| 126 | \end{array} |
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| 127 | \] |
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| 128 | The above formulas give the total cross section for scattering above the |
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| 129 | threshold energies |
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| 130 | |
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| 131 | \begin{equation} |
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| 132 | T_{\rm Moller}^{\rm thr} =2T_{cut} \mbox{\hspace{2cm}and\hspace{2cm}} |
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| 133 | T_{\rm Bhabha}^{\rm thr} = T_{cut} . |
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| 134 | \end{equation} |
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| 135 | |
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| 136 | \noindent |
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| 137 | In a given material the mean free path is then |
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| 138 | \begin{equation} |
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| 139 | \begin{array}{lll} |
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| 140 | \lambda = (n_{at} \cdot \sigma)^{-1} & or & |
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| 141 | \lambda = \left( \sum_i n_{ati} \cdot \sigma_i \right)^{-1} . |
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| 142 | \end{array} |
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| 143 | \end{equation} |
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| 144 | |
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| 145 | \subsection{Simulation of Delta-ray Production} |
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| 146 | \subsubsection{Differential Cross Section} |
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| 147 | |
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| 148 | For $T \gg I$ the differential cross section per atom becomes |
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| 149 | \cite{eion.messel} for M\"{o}ller scattering, |
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| 150 | \begin{eqnarray} |
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| 151 | \label{eion.i} |
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| 152 | \frac{d\sigma }{d \epsilon } &=& \frac{2 \pi r_e^2 Z}{\beta^2 (\gamma -1)} |
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| 153 | \times \\ |
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| 154 | & & \left[ \frac{(\gamma -1 )^2} {\gamma^2 }+\frac{1}{\epsilon} |
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| 155 | \left(\frac{1}{\epsilon}-\frac{2 \gamma -1 } {\gamma^2 } \right) + |
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| 156 | \frac{1}{1- \epsilon}\left(\frac{1} {1- \epsilon} - \frac{2 \gamma - 1} |
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| 157 | {\gamma^2 }\right) \right] \nonumber |
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| 158 | \end{eqnarray} |
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| 159 | and for Bhabha scattering, |
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| 160 | \begin{equation} |
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| 161 | \label{eion.j} |
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| 162 | \frac{d \sigma}{d \epsilon}=\frac{2 \pi r_e^2 Z}{(\gamma -1)}\left[ |
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| 163 | \frac{1} {\beta^2 \epsilon^2}-\frac{B_1}{\epsilon}+B_2 - B_3 \epsilon |
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| 164 | + B_4 \epsilon^2\right] . |
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| 165 | \end{equation} |
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| 166 | Here $\epsilon = T/(E-mc^2)$. The kinematical limits of $\epsilon$ are |
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| 167 | \[ |
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| 168 | \epsilon_0 = \frac{T_{cut}}{E-mc^2} \leq \epsilon \leq \frac{1}{2} |
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| 169 | \mbox{\hspace{.2cm} for $e^- e^-$} \hspace{2cm} |
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| 170 | \epsilon_0 = \frac{T_{cut}}{E-mc^2} \leq \epsilon \leq 1 |
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| 171 | \mbox{\hspace{.2cm} for $e^+ e^-$} . |
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| 172 | \] |
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| 173 | |
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| 174 | \subsubsection{Sampling} |
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| 175 | The delta ray energy is sampled according to methods discussed in |
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| 176 | Chapter \ref{secmessel}. Apart from normalization, the cross section can |
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| 177 | be factorized as |
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| 178 | \begin{equation} |
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| 179 | \frac{d\sigma}{d\epsilon}=f(\epsilon) g(\epsilon) . |
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| 180 | \end{equation} |
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| 181 | For $e^- e^-$ scattering |
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| 182 | \begin{eqnarray} |
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| 183 | f(\epsilon)&=&\frac{1}{\epsilon^2} \frac{\epsilon_0 }{1- 2\epsilon_0} \\ |
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| 184 | g(\epsilon)&=&\frac{4}{9\gamma^2 - 10 \gamma + 5}\left[(\gamma -1)^2 |
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| 185 | \epsilon^2 - (2 \gamma^2 +2\gamma -1) \frac{\epsilon} {1- \epsilon }+ |
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| 186 | \frac{\gamma^2}{(1- \epsilon )^2 }\right] |
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| 187 | \end{eqnarray} |
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| 188 | and for $e^+ e^-$ scattering |
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| 189 | \begin{eqnarray} |
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| 190 | f(\epsilon)&=&\frac{1}{\epsilon^2} \frac{\epsilon_0}{1- \epsilon_0 } \\ |
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| 191 | g(\epsilon)&=&\frac{B_0 -B_1 \epsilon +B_2 \epsilon^2 |
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| 192 | -B_3 \epsilon^3 +B_4 \epsilon ^4}{B_ 0-B_1\epsilon_0 |
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| 193 | +B_2\epsilon_0^2 |
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| 194 | -B_3 \epsilon_0^3 +B_4 \epsilon_0^4} . |
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| 195 | \end{eqnarray} |
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| 196 | Here $ B_0=\gamma^2/(\gamma^2-1)$ and all other quantities have been defined |
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| 197 | above. |
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| 198 | |
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| 199 | |
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| 200 | To choose $\epsilon$, and hence the delta ray energy, |
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| 201 | \begin{enumerate} |
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| 202 | \item $\epsilon$ is sampled from $f(\epsilon)$ |
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| 203 | \item the rejection function $g(\epsilon)$ is calculated using the sampled |
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| 204 | value of $\epsilon$ |
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| 205 | \item $\epsilon$ is accepted with probability $g(\epsilon)$. |
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| 206 | \end{enumerate} |
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| 207 | After the successful sampling of $\epsilon$, the direction of the ejected |
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| 208 | electron is generated with respect to the direction of the incident |
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| 209 | particle. The azimuthal angle $\phi$ is generated isotropically and the |
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| 210 | polar angle $\theta$ is calculated from energy-momentum conservation. |
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| 211 | This information is used to calculate the energy and momentum of both the |
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| 212 | scattered incident particle and the ejected electron, and to transform them |
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| 213 | to the global coordinate system. |
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| 214 | |
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| 215 | \subsection{Status of this document} |
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| 216 | \ 9.10.98 created by L. Urb\'an. \\ |
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| 217 | 29.07.01 revised by M.Maire. \\ |
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| 218 | 13.12.01 minor cosmetic by M.Maire. \\ |
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| 219 | 24.05.02 re-written by D.H. Wright. \\ |
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| 220 | 01.12.03 revised by V. Ivanchenko. \\ |
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| 221 | |
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| 222 | \begin{latexonly} |
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| 223 | |
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| 224 | \begin{thebibliography}{99} |
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| 225 | |
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| 226 | \bibitem{eion.messel} |
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| 227 | H.~Messel and D.F.~Crawford. {\em Pergamon Press, Oxford (1970).} |
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| 228 | \bibitem{ioni.icru1} |
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| 229 | ICRU (A.~Allisy et al), Stopping Powers for Electrons and Positrons, |
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| 230 | {\em ICRU Report No.37 (1984).} |
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| 231 | \bibitem{eion.sternheimer} |
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| 232 | R.M.~Sternheimer. {\em Phys.Rev. B3 (1971) 3681.} |
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| 233 | \end{thebibliography} |
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| 234 | |
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| 235 | \end{latexonly} |
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| 236 | |
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| 237 | \begin{htmlonly} |
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| 238 | |
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| 239 | \subsection{Bibliography} |
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| 240 | |
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| 241 | \begin{enumerate} |
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| 242 | \item H.~Messel and D.F.~Crawford. {\em Pergamon Press, Oxford (1970).} |
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| 243 | \item ICRU (A.~Allisy et al), Stopping Powers for Electrons and Positrons, |
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| 244 | {\em ICRU Report No.37 (1984).} |
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| 245 | \item R.M.~Sternheimer. {\em Phys.Rev. B3 (1971) 3681.} |
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| 246 | \end{enumerate} |
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| 247 | |
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| 248 | \end{htmlonly} |
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