[1211] | 1 | \section{Electron ionisation} \label{secioni2} |
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| 2 | |
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| 3 | |
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| 4 | The class G4LowEnergyIonisation 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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