1 | \section{Electron ionisation} \label{secioni2} |
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2 | |
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3 | |
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4 | The class G4LivermoreIonisationModel calculates the continuous energy loss |
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5 | due to electron ionisation and |
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6 | simulates |
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7 | $\delta$-ray production by electrons. |
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8 | The $delta$-electron |
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9 | production |
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10 | threshold for a given material, $T_c$, is used to separate |
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11 | the continuous and the discrete parts of the process. The energy loss |
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12 | of an electron with the incident energy, $T$, is expressed |
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13 | via the sum over all atomic shells, $s$, and the integral over the energy, $t$, |
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14 | of $delta$-electrons: |
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15 | \begin{equation} |
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16 | {dE\over dx}=\sum_{s}\left(\sigma_s(T){{\int^{T_c}_{0.1eV}t{d\sigma\over dt}dt} \over{\int^{T_{max}}_{0.1eV}{d\sigma\over dt}dt}}\right), |
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17 | \end{equation} |
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18 | where $T_{max} = 0.5T$ is the maximum energy transfered to a $\delta$-electron, |
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19 | $\sigma_s(T)$ is |
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20 | the total cross-section for the shell, $s$, at a given |
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21 | incident kinetic energy, $T$, and $0.1eV$ is the low energy limit |
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22 | of the EEDL data. |
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23 | The $\delta$-electron production cross-section is a complimentary |
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24 | function: |
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25 | \begin{equation} |
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26 | \sigma(T)=\sum_{s}\left(\sigma_s(T){{\int^{T_{max}}_{T_c}{d\sigma\over dt}dt}\over {\int^{T_{max}}_{0.1eV}{d\sigma\over dt}dt}}\right). |
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27 | \end{equation} |
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28 | The partial sub-shell cross-sections, $\sigma_s$, are obtained |
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29 | from an interpolation of the evaluated cross-section data in the EEDL |
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30 | library~\cite{io-EEDL}, |
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31 | according to the formula (\ref{eqloglog}) in Section~\ref{subsubsigmatot}. |
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32 | |
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33 | The probability of emission of a $\delta$-electron with kinetic energy, |
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34 | $t$, from a sub-shell, $s$, of binding energy, $B_s$, as the |
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35 | result of the interaction of |
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36 | an incoming electron with kinetic energy, $T$, |
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37 | is described by: |
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38 | \begin{equation} |
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39 | {d\sigma \over dt} = {P(x) \over x^2}, \;\; \mbox{with} x={t + B_s \over T + B_s}, |
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40 | \end{equation} |
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41 | where the parameter $x$ is varied from $x_{min} = (0.1eV + B_s)/(T + B_s)$ to |
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42 | 0.5. The function, $P(x)$, is parametrised differently in 3 regions of $x$: from $x_{min}$ |
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43 | to $x_1$ the linear interpolation with linear scale of 4 points is used; from $x_1$ to $x_2$ |
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44 | the linear interpolation with logarithmic scale of 16 points is used; from $x_2$ to $0.5$ |
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45 | the following interpolation is applied: |
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46 | \begin{equation} |
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47 | \label{io-ff} |
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48 | P(x) = 1 - gx +(1 - g)x^2 + {x^2 \over 1-x}({1 \over 1-x} - g) + A*(0.5 - x)/x, |
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49 | \end{equation} |
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50 | where $A$ is a fit coefficient, $g$ is expressed via the gamma factor |
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51 | of the incoming electron: |
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52 | \begin{equation} |
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53 | \label{ff} |
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54 | g = (2\gamma - 1) / \gamma^2. |
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55 | \end{equation} |
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56 | For the high energy case ($x >> 1$) the formula (\ref{io-ff}) is transformed |
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57 | to the M\"{o}ller electron-electron scattering formula \cite{io-g3,io-messel}. |
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58 | |
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59 | The value of the coefficient, $A$, for each element is obtained as a result |
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60 | of the fit on the spectrum from the EEDL data for those |
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61 | energies which are available in the database. |
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62 | The values of $x_1$ and $x_2$ are chosen for each atomic shell |
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63 | according to the spectrum of $\delta$-electrons in this shell. Note that $x_1$ |
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64 | corresponds to the maximum of the spectrum, if the maximum does not coincide |
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65 | with $x_{min}$. |
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66 | The dependence of all 24 parameters on the incident energy, $T$, |
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67 | is evaluated from a logarithmic interpolation (\ref{eqloglog}). |
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68 | |
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69 | The sampling of the final state proceeds in three steps. |
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70 | First a shell is randomly selected, then the energy of the |
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71 | $delta$-electron is sampled, finally |
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72 | the angle of emission of the scattered electron and of the $\delta$-ray |
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73 | is determined by energy-momentum conservation taken into account |
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74 | electron motion on the atomic orbit. |
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75 | |
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76 | The interaction leaves the atom in an excited state. |
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77 | The deexcitation of the atom is simulated as |
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78 | described in section~\ref{relax}. Sampling of the excitations is carried out |
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79 | for both the continuous and the discrete parts |
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80 | of the process. |
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81 | |
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82 | \subsection{Status of the document} |
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83 | |
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84 | \noindent |
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85 | 30.09.1999 created by Alessandra Forti\\ |
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86 | 07.02.2000 modified by V\'eronique Lef\'ebure\\ |
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87 | 08.03.2000 reviewed by Petteri Nieminen and Maria Grazia Pia\\ |
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88 | 05.12.2001 modified by Vladimir Ivanchenko \\ |
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89 | 13.05.2002 modified by Vladimir Ivanchenko |
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90 | |
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91 | \begin{latexonly} |
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92 | |
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93 | \begin{thebibliography}{99} |
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94 | \bibitem{io-EEDL} |
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95 | %http://reddog1.llnl.gov/homepage.red/Electron.htm |
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96 | ``Tables and Graphs of Electron-Interaction Cross-Sections from 10~eV to 100~GeV Derived from |
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97 | the LLNL Evaluated Electron Data Library (EEDL), Z=1-100" |
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98 | S.T.Perkins, D.E.Cullen, S.M.Seltzer, |
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99 | UCRL-50400 Vol.31 |
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100 | \bibitem{io-g3} |
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101 | G\textsc{eant3} manual ,CERN Program Library Long Writeup W5013 (October 1994). |
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102 | \bibitem{io-messel} |
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103 | H.Messel and D.F.Crawford. Pergamon Press,Oxford,1970. |
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104 | \end{thebibliography} |
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105 | |
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106 | \end{latexonly} |
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107 | |
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108 | \begin{htmlonly} |
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109 | |
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110 | \subsection{Bibliography} |
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111 | |
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112 | \begin{enumerate} |
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113 | \item |
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114 | %http://reddog1.llnl.gov/homepage.red/Electron.htm |
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115 | ``Tables and Graphs of Electron-Interaction Cross-Sections from 10~eV to 100~GeV Derived from |
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116 | the LLNL Evaluated Electron Data Library (EEDL), Z=1-100" |
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117 | S.T.Perkins, D.E.Cullen, S.M.Seltzer, |
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118 | UCRL-50400 Vol.31 |
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119 | \item |
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120 | G\textsc{eant3} manual ,CERN Program Library Long Writeup W5013 (October 1994). |
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121 | \item |
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122 | H.Messel and D.F.Crawford. Pergamon Press,Oxford,1970. |
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123 | \end{enumerate} |
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124 | |
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125 | \end{htmlonly} |
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126 | |
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