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2 | \section[Photoelectric Effect]{PhotoElectric effect} |
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3 | |
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4 | The photoelectric effect is the ejection of an electron from a material after |
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5 | a photon has been absorbed by that material. It is simulated by using a |
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6 | parameterized photon absorption cross section to determine the mean free path, |
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7 | atomic shell data to determine the energy of the ejected electron, and the |
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8 | K-shell angular distribution to sample the direction of the electron. |
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9 | |
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10 | \subsection{Cross Section and Mean Free Path} |
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11 | |
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12 | The parameterization of the photoabsorption cross section proposed by |
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13 | Biggs et al. \cite{ph.sandia} was used : |
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14 | \begin{equation} \label{eqsandia} |
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15 | \sigma(Z,E_{\gamma}) = \frac{a(Z,E_{\gamma})}{E_{\gamma}} + |
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16 | \frac{b(Z,E_{\gamma})}{E_{\gamma}^2} + |
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17 | \frac{c(Z,E_{\gamma})}{E_{\gamma}^3} + |
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18 | \frac{d(Z,E_{\gamma})}{E_{\gamma}^4} |
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19 | \end{equation} |
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20 | |
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21 | \noindent |
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22 | Using the least-squares method, a separate fit of each of the coefficients |
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23 | $a,b,c,d$ to the experimental data was performed in several energy intervals |
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24 | \cite{ph.sandia.grich}. As a rule, the boundaries of these intervals were |
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25 | equal to the corresponding photoabsorption edges. |
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26 | |
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27 | \noindent |
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28 | In a given material the mean free path, $\lambda$, for a photon to interact |
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29 | via the photoelectric effect is given by : |
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30 | \begin{equation} \label{lambda} |
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31 | \lambda(E_{\gamma}) = |
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32 | \left( \sum_i n_{ati} \cdot \sigma (Z_i,E_{\gamma}) \right)^{-1} |
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33 | \end{equation} |
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34 | where $n_{ati}$ is the number of atoms per volume of the $i^{th}$ element |
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35 | of the material. The cross section and mean free path are |
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36 | discontinuous and must be computed 'on the fly' from the formulas |
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37 | \ref{eqsandia} and \ref{lambda}. |
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38 | |
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39 | \subsection{Final State} |
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40 | \subsubsection{Choosing an Element} |
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41 | The binding energies of the shells depend on the atomic number $Z$ of the |
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42 | material. In compound materials the $i^{th}$ element is chosen randomly |
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43 | according to the probability: |
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44 | \[ |
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45 | Prob(Z_i,E_{\gamma}) = |
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46 | \frac{n_{ati} \sigma(Z_i,E_{\gamma})} |
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47 | {\sum_i [ n_{ati} \cdot \sigma_i (E_{\gamma})]} . |
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48 | \] |
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49 | \subsubsection{Shell} |
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50 | A quantum can be absorbed if $E_{\gamma} > B_{shell}$ where the shell |
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51 | energies are taken from {\tt G4AtomicShells} data: the closest available |
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52 | atomic shell is chosen. The photoelectron is emitted with kinetic energy : |
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53 | \begin{equation} |
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54 | T_{photoelectron} = E_{\gamma}-B_{shell}(Z_i) |
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55 | \end{equation} |
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56 | |
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57 | \subsubsection{Theta Distribution of the Photoelectron} |
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58 | The polar angle of the photoelectron is sampled from the Sauter-Gavrila |
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59 | distribution (for K-shell) \cite{ph.cost}, which is correct only to zero order |
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60 | in $\alpha Z$ : |
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61 | \begin{equation} |
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62 | \frac{d\sigma}{d(\cos\theta)} \sim \frac{\sin^2\theta}{(1-\beta\cos\theta)^4} |
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63 | \left\lbrace 1 + \frac{1}{2} \gamma (\gamma-1)(\gamma-2)(1-\beta\cos\theta) |
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64 | \right\rbrace |
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65 | \end{equation} |
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66 | where $\beta$ and $\gamma$ are the Lorentz factors of the photoelectron. |
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67 | |
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68 | \noindent |
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69 | $\cos\theta$ is sampled from the probability density function : |
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70 | \begin{equation} |
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71 | f(\cos\theta) = \frac{1-\beta^2}{2\beta} \frac{1}{(1-\beta\cos\theta)^2} |
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72 | \hspace{5mm} \Longrightarrow \hspace{5mm} |
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73 | \cos\theta = \frac{(1-2r)+\beta}{(1-2r)\beta+1} |
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74 | \end{equation} |
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75 | The rejection function is : |
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76 | \begin{equation} |
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77 | g(\cos\theta) = \frac{1-\cos^2\theta}{(1-\beta\cos\theta)^2} |
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78 | \left\lbrack 1+b(1-\beta\cos\theta) \right\rbrack |
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79 | \end{equation} |
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80 | with $b=\gamma(\gamma-1)(\gamma-2)/2$ \\ |
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81 | It can be shown that $g(\cos\theta)$ is positive $\forall \cos\theta \in |
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82 | [-1,\ +1]$, and can be majored by : |
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83 | \begin{eqnarray} |
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84 | gsup&=&\gamma^2 \ \lbrack 1+b(1-\beta) \rbrack \mbox{ if } \gamma \in \ ]1,2] \\ |
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85 | &=&\gamma^2 \ \lbrack 1+b(1+\beta) \rbrack \mbox{ if } \gamma > 2 \nonumber |
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86 | \end{eqnarray} |
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87 | The efficiency of this method is $\sim 50\%$ if $\gamma < 2$, $\sim 25\%$ if |
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88 | $\gamma \in [2,\ 3]$. |
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89 | |
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90 | \subsubsection{Relaxation} |
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91 | In the current implementation the relaxation of the atom is not simulated, |
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92 | but instead is counted as a local energy deposit. |
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93 | |
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94 | \subsection{Status of this document} |
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95 | 09.10.98 created by M.Maire. \\ |
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96 | 08.01.02 updated by mma \\ |
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97 | 22.04.02 re-worded by D.H. Wright \\ |
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98 | 02.05.02 modifs in total cross section and final state (mma) \\ |
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99 | 15.11.02 introduction added by D.H. Wright \\ |
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100 | |
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101 | \begin{latexonly} |
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102 | |
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103 | \begin{thebibliography}{99} |
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104 | \bibitem{ph.sandia} Biggs F., and Lighthill R., |
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105 | {Preprint Sandia Laboratory, SAND 87-0070} (1990) |
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106 | \bibitem{ph.sandia.grich} Grichine V.M., Kostin A.P., Kotelnikov S.K. et al., |
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107 | {Bulletin of the Lebedev Institute no. 2-3, 34} (1994). |
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108 | \bibitem{ph.cost} Gavrila M. |
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109 | {Phys.Rev. 113, 514} (1959). |
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110 | \end{thebibliography} |
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111 | |
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112 | \end{latexonly} |
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113 | |
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114 | \begin{htmlonly} |
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115 | |
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116 | \subsection{Bibliography} |
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117 | |
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118 | \begin{enumerate} |
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119 | \item Biggs F., and Lighthill R., |
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120 | {Preprint Sandia Laboratory, SAND 87-0070} (1990) |
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121 | \item Grichine V.M., Kostin A.P., Kotelnikov S.K. et al., |
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122 | {Bulletin of the Lebedev Institute no. 2-3, 34} (1994). |
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123 | \item Gavrila M. |
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124 | {Phys.Rev. 113, 514} (1959). |
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125 | \end{enumerate} |
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126 | |
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127 | \end{htmlonly} |
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