1 | \section{Photoelectric effect}\label{secphoto} |
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2 | |
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3 | \subsection{Total cross-section} |
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4 | |
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5 | The total photoelectric cross-section at a given energy, E, is calculated as |
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6 | described in section~\ref{subsubsigmatot}. Note that for this process the |
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7 | {\it MeanFreePathTable} is not built, since the cross-section is not a |
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8 | smooth function of the energy, therefore in all calculations the |
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9 | cross-section is used directly. |
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10 | |
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11 | \subsection{Sampling of the final state} |
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12 | |
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13 | The incident photon is absorbed and an electron is emitted. |
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14 | |
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15 | The electron kinetic energy is the difference between the incident photon |
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16 | energy and the binding energy of the electron before the interaction. |
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17 | The sub-shell, from which the electron is emitted, is randomly selected |
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18 | according to the relative cross-sections of all subshells, |
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19 | determined at the given energy, $T$, by interpolating the evaluated |
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20 | cross-section data from the EPDL97 data bank~\cite{pe-EPDL97}. |
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21 | |
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22 | The interaction leaves the atom in an excited state. |
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23 | The deexcitation of the atom is simulated as described in section~\ref{relax}. |
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24 | |
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25 | \subsection{Angular distribution of the emmited photoelectron} |
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26 | |
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27 | Three models are available to describe the direction of the emmited photoelectron: |
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28 | G4\-Photo\-Electric\-Angular\-Generator\-Simple, G4\-Photo\-Electric\-Angular\-Generator\-Sauter\-Gavrila and |
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29 | G4PhotoElectricAngularGeneratorPolarized. |
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30 | |
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31 | \subsubsection{G4PhotoElectricAngularGeneratorSimple} |
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32 | |
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33 | The default model assumes that the photoelectron direction is emmited in the same |
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34 | direction as the incident photon. |
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35 | |
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36 | \subsubsection{G4PhotoElectricAngularGeneratorSauterGavrila} |
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37 | |
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38 | This model implements the Sauter--Gavrilla distribution has |
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39 | presented in the Standard Photoelectric effect. |
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40 | |
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41 | \subsubsection{G4PhotoElectricAngularGeneratorPolarized} |
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42 | |
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43 | This model models the double differential cross section (for angles $\theta$ and $\phi$) and |
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44 | thus it is capable of account for polarization of the incident photon. The developed |
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45 | generator was based in the research of Sauter in 1931\cite{Sauter:1931}. The Sauter's formula was recalculated by Gavrila in |
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46 | 1959 for the K-shell~\cite{Gavrila:1959} and in 1961 for the L-shells~\cite{Gavrila:1961}. These new double differential formulas have some limitations, $\alpha$Z$<<$1 and have a range between 0.1$<\beta<$0.99 c. |
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47 | |
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48 | \subsubsection*{K--shell} |
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49 | |
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50 | The double differential photoeffect for K--shell can be written as~\cite{Gavrila:1959}: |
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51 | \begin{equation} |
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52 | \frac{d\sigma}{d \omega}(\theta,\phi) = \frac{4}{m^2}{\alpha^6}{Z^5}\frac{\beta^3(1-\beta^2)^3}{\left[1-(1-\beta^2)^{1/2}\right]} |
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53 | \left(F\left(1-\frac{\pi\alpha Z}{\beta}\right)+ \pi\alpha Z G\right) |
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54 | \end{equation} |
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55 | where |
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56 | \begin{eqnarray*} |
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57 | F &=& \frac{\sin^2 \theta \cos^2 \phi}{(1-\beta \cos \theta)^4} - \frac{1-(1-\beta^2)^{1/2}}{2(1-\beta^2)}\frac{\sin^2\theta\cos^2\phi}{(1-\beta\cos\theta)^3} \nonumber \\ &+&\frac{\left[1-(1-\beta^2)^{1/2}\right]^2}{4(1-\beta^2)^{3/2}}\frac{\sin^2\theta}{(1-\beta\cos\theta)^3} |
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58 | \end{eqnarray*} |
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59 | \begin{eqnarray*} |
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60 | G &=& \frac{[1-(1-\beta^2)^{1/2}]^{1/2}}{2^{7/2} \beta^2 (1-\beta \cos \theta)^{5/2}}\left[\frac{4\beta^2}{(1-\beta^2)^{1/2}} \frac{\sin^2 \theta \cos^2 \phi}{1-\beta\cos\theta} + \frac{4\beta}{1-\beta^2}\cos \theta \cos^2 \phi - \right.{} \nonumber \\ |
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61 | &-&4 \left.\frac{1-(1-\beta^2)^{1/2}}{1-\beta^2}(1-\cos^2\phi)-\beta^2\ \frac{1-(1-\beta^2)^{1/2}}{1-\beta^2} \frac{\sin^2 \theta}{1-\beta \cos \theta} - \right.{} \nonumber \\ |
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62 | &+& \left.4\beta^2\frac{1-(1-\beta^2)^{1/2}}{(1-\beta^2)^{3/2}} - 4\beta \frac{\left[ 1-(1-\beta^2)^{1/2}\right]^2}{(1-\beta^2)^{3/2}}\right] \nonumber \\ |
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63 | &+&\frac{1-(1-\beta^2)^{1/2}}{4\beta^2(1-\beta\cos\theta)^2}\left[\frac{\beta}{1-\beta^2}-\frac{2}{1-\beta^2}\cos\theta\cos^2\phi + \frac{1-(1-\beta^2)^{1/2}}{(1-\beta^2)^{3/2}}\cos\theta \right.{} \nonumber \\ |
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64 | &-& \left.\beta \frac{1-(1-\beta^2)^{1/2}}{(1-\beta^2)^{3/2}}\right] |
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65 | \end{eqnarray*} |
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66 | where $\beta$ is the electron velocity, $\alpha$ is the fine--structure constant, $Z$ is the atomic number of the material and $\theta$, $\phi$ are the emission angles with respect to the electron initial direction. |
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67 | |
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68 | \subsubsection*{L1--shell} |
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69 | |
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70 | The double differential photoeffect distribution for L1--shell is the same as |
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71 | for K--shell despising a constant~\cite{Gavrila:1961}: |
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72 | \begin{equation} |
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73 | B = \xi \frac{1}{8} |
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74 | \end{equation} |
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75 | where $\xi$ is equal to 1 when working with unscreened Coulomb wave functions as it is done in this development. |
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76 | |
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77 | \subsubsection*{The generation of the photoelectron distribution} |
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78 | |
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79 | Since the polarized Gavrila cross--section is a 2--dimensional non--factorized distribution an acceptance--rejection technique was the adopted~\cite{Peralta:2003}. For the Gravrila distribution, two functions |
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80 | were defined $g_1(\phi)$ and $g_2(\theta)$: |
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81 | \begin{eqnarray} |
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82 | g_1(\phi) &=& a \\ |
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83 | g_2(\theta) &=& \frac{\theta}{1+c\theta^2} |
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84 | \end{eqnarray} |
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85 | such that: |
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86 | \begin{equation} |
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87 | A g_1(\phi)g_2(\theta) \ge \frac{d^2 \sigma}{d\phi d\theta} |
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88 | \end{equation} |
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89 | where A is a global constant. The method used to calculate the distribution is |
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90 | the same as the one used in Low Energy 2BN Bremsstrahlung Generator, being the |
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91 | difference $g_1(\phi) = a$. |
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92 | |
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93 | \subsection{Status of the document} |
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94 | |
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95 | \noindent |
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96 | 30.09.1999 created by Alessandra Forti\\ |
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97 | 07.02.2000 modified by V\'eronique Lef\'ebure\\ |
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98 | 08.03.2000 reviewed by Petteri Nieminen and Maria Grazia Pia\\ |
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99 | 13.05.2002 modified by Vladimir Ivanchenko |
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100 | 01.05.2006 modified by Ana Farinha, Andreia Trindade, Lu\'{\i}s Peralta and Pedro Rodrigues\\ |
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101 | |
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102 | \begin{latexonly} |
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103 | |
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104 | \begin{thebibliography}{99} |
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105 | \bibitem{pe-EPDL97} |
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106 | %http://reddog1.llnl.gov/homepage.red/photon.htm |
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107 | ``EPDL97: the Evaluated Photon Data Library, '97 version", |
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108 | D.Cullen, J.H.Hubbell, L.Kissel, |
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109 | UCRL--50400, Vol.6, Rev.5 |
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110 | \bibitem{Sauter:1931} |
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111 | ``K--Shell Photoelectric Cross Sections from 200 keV to 2 MeV", R H Pratt, R D Levee, |
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112 | R L Pexton and W Aron, Phys. Rev. 134 (1964) 4A |
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113 | \bibitem{Gavrila:1959} |
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114 | ``Relativistic K--Shell Photoeffect", M. Gavrila, Phys. Rev. 113 (1959) 2 |
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115 | \bibitem{Gavrila:1961} |
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116 | ``Relativistic L--Shell Photoeffect", M. Gavrila, Phys. Rev. 124 (1961) 4 |
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117 | \bibitem{Peralta:2003} |
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118 | ``Monte Carlo Generation of 2BNBremsstrahlung Distribution", |
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119 | L. Peralta, P. Rodrigues, A. Trindade |
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120 | CERN EXT--2004--039 (July, 2003) |
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121 | \end{thebibliography} |
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122 | |
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123 | \end{latexonly} |
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124 | |
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125 | \begin{htmlonly} |
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126 | |
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127 | \subsection{Bibliography} |
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128 | |
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129 | \begin{enumerate} |
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130 | \item |
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131 | %http://reddog1.llnl.gov/homepage.red/photon.htm |
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132 | ``EPDL97: the Evaluated Photon Data Library, '97 version", |
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133 | D.Cullen, J.H.Hubbell, L.Kissel, |
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134 | UCRL--50400, Vol.6, Rev.5 |
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135 | \item |
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136 | ``K--Shell Photoelectric Cross Sections from 200 keV to 2 MeV", R H Pratt, R D Levee, |
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137 | R L Pexton and W Aron, Phys. Rev. 134 (1964) 4A |
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138 | \item |
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139 | ``Relativistic K--Shell Photoeffect", M. Gavrila, Phys. Rev. 113 (1959) 2 |
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140 | \item |
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141 | ``Relativistic L--Shell Photoeffect", M. Gavrila, Phys. Rev. 124 (1961) 4 |
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142 | \item |
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143 | ``Monte Carlo Generation of 2BNBremsstrahlung Distribution", |
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144 | L. Peralta, P. Rodrigues, A. Trindade |
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145 | CERN EXT--2004--039 (July, 2003) |
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146 | \end{enumerate} |
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147 | |
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148 | \end{htmlonly} |
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