[1211] | 1 | \section{Penelope physics} |
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| 2 | |
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| 3 | \subsection{Introduction} |
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| 4 | A new set of physics processes for photons, electrons and positrons is |
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| 5 | implemented in Geant4: it includes Compton scattering, photoelectric |
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| 6 | effect, Rayleigh scattering, gamma conversion, bremsstrahlung, ionization |
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| 7 | (to be released) and positron annihilation (to be released). These processes |
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| 8 | are the Geant4 implementation of the physics models developed |
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| 9 | for the PENELOPE code (PENetration and Energy LOss of Positrons and |
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| 10 | Electrons), version 2001, that are described in detail in Ref. \cite{uno}. |
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| 11 | The Penelope models have been specifically developed for Monte Carlo |
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| 12 | simulation and great care was given to the low energy description |
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| 13 | (i.e. atomic effects, etc.). Hence, these implementations provide reliable |
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| 14 | results for energies down to a few hundred eV and can be used up to |
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| 15 | $\sim$1 GeV \cite{uno,due}. For this reason, they may be used in Geant4 as |
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| 16 | an alternative to the Low Energy processes. For the same physics processes, |
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| 17 | the user now has more alternative descriptions from which to choose, including |
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| 18 | the cross section calculation and the final state sampling. |
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| 19 | |
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| 20 | \subsection{Compton scattering} |
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| 21 | \subsubsection{Total cross section} |
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| 22 | The total cross section of the Compton scattering process is determined from |
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| 23 | an analytical parameterization. For $\gamma$ energy $E$ greater than 5 MeV, |
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| 24 | the usual Klein-Nishina formula is used for $\sigma(E)$. For \mbox{$E<5$ MeV} |
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| 25 | a more accurate parameterization is used, which takes into account atomic |
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| 26 | binding effects and Doppler broadening \cite{tre}: |
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| 27 | \begin{eqnarray} |
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| 28 | \sigma(E) \ = \ 2 \pi \int_{-1}^{1} \frac{r_{e}^{2}}{2} \frac{E_{C}^{2}} |
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| 29 | {E^{2}} (\frac{E_{C}}{E} + \frac{E}{E_{C}} - \sin^{2} \theta) \cdot |
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| 30 | \nonumber \\ |
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| 31 | \sum_{shells} |
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| 32 | f_{i} \Theta(E-U_{i})n_{i}(p_{z}^{max}) \ d(\cos \theta) \label{equno} |
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| 33 | \end{eqnarray} |
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| 34 | where: \\ |
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| 35 | $r_{e}$ = classical radius of the electron; \\ |
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| 36 | $m_{e}$ = mass of the electron; \\ |
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| 37 | $\theta$ = scattering angle; \\ |
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| 38 | $E_{C}$ = Compton energy \\ |
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| 39 | \begin{displaymath} |
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| 40 | = \ \frac{E}{1+\frac{E}{m_{e}c^{2}}(1-\cos\theta)} |
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| 41 | \end{displaymath} \\ |
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| 42 | $f_{i}$ = number of electrons in the i-th atomic shell; \\ |
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| 43 | $U_{i}$ = ionisation energy of the i-th atomic shell; \\ |
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| 44 | $\Theta$ = Heaviside step function; \\ |
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| 45 | $p_{z}^{max}$ = highest possible value of $p_{z}$ (projection of the initial |
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| 46 | momentum of the electron in the direction of the scattering angle) \\ |
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| 47 | \begin{displaymath} |
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| 48 | = \ \frac{E(E-U_{i})(1-\cos\theta)-m_{e}c^{2}U_{i}}{c \sqrt{2E(E-U_{i})(1- |
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| 49 | \cos\theta)+U_{i}^{2}}}. |
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| 50 | \end{displaymath} |
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| 51 | Finally, |
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| 52 | \begin{equation} |
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| 53 | \begin{array}{rlll} |
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| 54 | n_{i}(x) = & & & \\ |
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| 55 | & \frac{1}{2} e^{[ \frac{1}{2}-( \frac{1}{2} - \sqrt{2} J_{i0}x )^{2}]} & |
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| 56 | \mbox{if} & x < 0 \\ |
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| 57 | & 1-\frac{1}{2} e^{[\frac{1}{2}-(\frac{1}{2}+\sqrt{2}J_{i0}x)^{2}]} & |
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| 58 | \mbox{if} & x > 0 \\ |
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| 59 | % \begin{cases} |
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| 60 | % \frac{1}{2} e^{[ \frac{1}{2}-( \frac{1}{2} - \sqrt{2} J_{i0}x )^{2}]} & |
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| 61 | % \textrm{if} \quad x<0\\ |
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| 62 | % 1-\frac{1}{2} e^{[\frac{1}{2}-(\frac{1}{2}+\sqrt{2}J_{i0}x)^{2}]} & |
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| 63 | % \textrm{if} \quad x>0\\ |
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| 64 | % \end{cases} |
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| 65 | \end{array} |
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| 66 | \end{equation} |
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| 67 | where $J_{i0}$ is the value of the $p_{z}$-distribution profile |
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| 68 | $J_{i}(p_{z})$ for the i-th atomic shell calculated in $p_{z}=0$. The values |
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| 69 | of $J_{i0}$ for the different shells of the different elements are |
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| 70 | tabulated from the Hartree-Fock atomic orbitals |
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| 71 | of Ref. \cite{quattro}.\\ |
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| 72 | The integration of Eq.(\ref{equno}) is performed numerically using the |
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| 73 | 20-point Gaussian method. For this reason, the initialization of the |
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| 74 | Penelope Compton process is somewhat slower than the Low Energy process. |
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| 75 | |
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| 76 | \subsubsection{Sampling of the final state} |
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| 77 | The polar deflection $\cos\theta$ is sampled from the probability density |
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| 78 | function |
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| 79 | \begin{equation} |
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| 80 | P(\cos\theta) \ = \frac{r_{e}^{2}}{2} \frac{E_{C}^{2}} |
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| 81 | {E^{2}} \Big( \frac{E_{C}}{E} + \frac{E}{E_{C}} - \sin^{2} \theta |
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| 82 | \Big) \sum_{shells} |
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| 83 | f_{i} \Theta(E-U_{i})n_{i}(p_{z}^{max}) \label{eqdue} |
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| 84 | \end{equation} |
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| 85 | (see Ref. \cite{uno} for details on the sampling algorithm). Once the |
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| 86 | direction of the emerging photon has been set, the active electron shell $i$ |
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| 87 | is selected with relative probability equal to $Z_{i} |
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| 88 | \Theta(E-U_{i})n_{i}[p_{z}^{max}(E,\theta)]$. A random value of |
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| 89 | $p_{z}$ is generated from the analytical Compton profile \cite{quattro}. The |
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| 90 | energy of the emerging photon is |
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| 91 | \begin{equation} |
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| 92 | E' \ = \ \frac{E \tau}{1-\tau t} \ \Big[ (1-\tau t \cos\theta) + |
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| 93 | \frac{p_{z}}{|p_{z}|} \sqrt{(1-\tau t \cos\theta)^{2}-(1-t \tau^{2})(1-t)} |
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| 94 | \Big], |
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| 95 | \end{equation} |
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| 96 | where |
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| 97 | \begin{equation} |
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| 98 | t \ = \ \Big( \frac{p_{z}}{m_{e}c} \Big)^{2} \quad \textrm{and} \quad |
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| 99 | \tau \ = \ \frac{E_{C}}{E}. |
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| 100 | \end{equation} |
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| 101 | The azimuthal scattering angle $\phi$ of the photon is sampled uniformly in |
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| 102 | the interval (0,2$\pi$). It is assumed that the Compton electron is emitted |
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| 103 | with energy $E_{e} = E-E'-U_{i}$, |
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| 104 | with polar angle $\theta_{e}$ and azimuthal angle $\phi_{e} = |
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| 105 | \phi + \pi $, relative to the direction of the incident photon. |
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| 106 | In this case $\cos\theta_{e}$ is given by |
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| 107 | \begin{equation} |
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| 108 | \cos\theta_{e} \ = \ \frac{E-E'\cos\theta}{\sqrt{E^{2}+E^{'2}- |
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| 109 | 2EE' \cos\theta}}. |
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| 110 | \end{equation} |
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| 111 | Since the active electron shell is known, characteristic x-rays and |
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| 112 | electrons emitted in the de-excitation of the ionized atom can also be |
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| 113 | followed. The de-excitation is simulated as described in |
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| 114 | section~\ref{relax}. For further details see \cite{uno}.\\ |
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| 115 | |
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| 116 | \subsection{Rayleigh scattering} |
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| 117 | \subsubsection{Total cross section} |
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| 118 | The total cross section of the Rayleigh scattering process is determined from |
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| 119 | an analytical parameterization. The atomic cross section for coherent |
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| 120 | scattering is given approximately by \cite{cinque} |
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| 121 | \begin{equation} |
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| 122 | \sigma(E) \ = \ \pi r_{e}^{2} \int_{-1}^{1} \frac{1+\cos^{2}\theta}{2} |
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| 123 | [F(q,Z)]^{2} \ d \cos\theta, \label{eqtre} |
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| 124 | \end{equation} |
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| 125 | where $F(q,Z)$ is the atomic form factor, $Z$ is the atomic number and $q$ |
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| 126 | is the magnitude of the momentum transfer, i.e. |
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| 127 | \begin{equation} |
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| 128 | q \ = \ 2 \ \frac{E}{c} \ \sin \Big( \frac{\theta}{2} \Big). |
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| 129 | \end{equation} |
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| 130 | In the numerical calculation the following analytical approximations are |
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| 131 | used for the form factor: |
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| 132 | \begin{equation} |
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| 133 | \begin{array}{rlll} |
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| 134 | F(q,Z) = f(x,Z) = & & & \\ |
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| 135 | & Z \ \frac{1+a_{1}x^{2}+a_{2}x^{3}+a_{3}x^{4}}{(1+a_{4}x^{2}+a_{5}x^{4})^{2}} |
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| 136 | & \mbox{or} & \\ |
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| 137 | & \max[f(x,Z),F_{K}(x,Z)] & \mbox{if} \ Z>10 \ \mbox{and} \ f(x,Z) < 2 & \\ |
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| 138 | % \begin{cases} |
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| 139 | % f(x,Z) = Z \ \frac{1+a_{1}x^{2}+a_{2}x^{3}+a_{3}x^{4}}{(1+a_{4}x^{2}+a_{5} |
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| 140 | % x^{4})^{2}} & \\ |
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| 141 | % \max[f(x,Z),F_{K}(x,Z)] & \textrm{if} \ Z>10 \ \textrm{and} \ |
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| 142 | % f(x,Z)<2\\ |
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| 143 | % \end{cases} |
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| 144 | \end{array} |
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| 145 | \end{equation} |
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| 146 | where |
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| 147 | \begin{equation} |
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| 148 | F_{K}(x,Z) \ = \ \frac{\sin(2b \arctan Q)}{bQ(1+Q^{2})^{b}}, |
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| 149 | \end{equation} |
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| 150 | with |
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| 151 | \begin{equation} |
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| 152 | x = 20.6074 \frac{q}{m_{e}c}, \quad Q = \frac{q}{2m_{e}ca}, \quad |
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| 153 | b = \sqrt{1-a^{2}}, \quad a = \alpha \Big( Z-\frac{5}{16} \Big ), |
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| 154 | \end{equation} |
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| 155 | where $\alpha$ is the fine-structure constant. The function $F_{K}(x,Z)$ is |
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| 156 | the contribution to the atomic form factor due to the two K-shell electrons |
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| 157 | (see \cite{sei}). The parameters of expression $f(x,Z)$ have |
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| 158 | been determined in Ref. \cite{sei} for Z=1 to 92 by numerically fitting |
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| 159 | the atomic form factors tabulated in Ref. \cite{sette}. |
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| 160 | The integration of Eq.(\ref{eqtre}) is performed numerically using the |
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| 161 | 20-point Gaussian method. For this reason the initialization of the |
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| 162 | Penelope Rayleigh process is somewhat slower than the Low Energy process. |
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| 163 | |
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| 164 | \subsubsection{Sampling of the final state} |
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| 165 | The angular deflection $\cos\theta$ of the scattered photon is sampled from |
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| 166 | the probability distribution function |
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| 167 | \begin{equation} |
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| 168 | P(\cos\theta) \ = \ \frac{1+\cos^{2}\theta}{2} [F(q,Z)]^{2}. |
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| 169 | \end{equation} |
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| 170 | For details on the sampling algorithm (which is quite heavy from the |
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| 171 | computational point of view) see Ref. \cite{uno}. The azimuthal scattering |
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| 172 | angle $\phi$ of the photon is sampled uniformly in the interval (0,2$\pi$). |
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| 173 | % |
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| 174 | \subsection{Gamma conversion} |
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| 175 | \subsubsection{Total cross section} |
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| 176 | The total cross section of the $\gamma$ conversion process is determined from |
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| 177 | the data \cite{otto}, as described in section~\ref{subsubsigmatot}. |
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| 178 | |
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| 179 | \subsubsection{Sampling of the final state} |
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| 180 | The energies $E_{-}$ and $E_{+}$ of the secondary electron and positron are |
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| 181 | sampled using the Bethe-Heitler cross section with the Coulomb correction, |
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| 182 | using the semiempirical model of Ref. \cite{sei}. If |
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| 183 | \begin{equation} |
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| 184 | \epsilon \ = \ \frac{E_{-}+m_{e}c^{2}}{E} |
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| 185 | \end{equation} |
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| 186 | is the fraction of the $\gamma$ energy $E$ which is taken away from the |
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| 187 | electron, |
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| 188 | \begin{equation} |
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| 189 | \kappa \ = \ \frac{E}{m_{e}c^{2}} \quad \textrm{and} \quad a = \alpha Z, |
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| 190 | \end{equation} |
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| 191 | the differential cross section, which includes a low-energy correction and a |
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| 192 | high-energy radiative correction, is |
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| 193 | \begin{equation} |
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| 194 | \frac{d\sigma}{d\epsilon} \ = \ r_{e}^{2} a (Z+\eta) C_{r} \frac{2}{3} |
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| 195 | \Big[ 2 \Big( \frac{1}{2} - \epsilon \Big)^{2}\phi_{1}(\epsilon)+ |
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| 196 | \phi_{2}(\epsilon) \Big], |
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| 197 | \label{eqquattro} |
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| 198 | \end{equation} |
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| 199 | where: |
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| 200 | \begin{eqnarray} |
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| 201 | \phi_{1}(\epsilon) \ = \ \frac{7}{3} - 2 \ln (1+b^{2}) |
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| 202 | -6b\arctan (b^{-1}) |
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| 203 | \nonumber \\ |
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| 204 | -b^{2}[4-4b \arctan(b^{-1})-3 \ln(1+b^{-2})] \nonumber \\ |
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| 205 | + 4\ln (R m_{e} c/\hbar) - 4f_{C}(Z) + F_{0}(\kappa,Z) |
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| 206 | \end{eqnarray} |
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| 207 | and |
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| 208 | \begin{eqnarray} |
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| 209 | \phi_{2}(\epsilon) \ = \ \frac{11}{6} - 2 \ln (1+b^{2}) |
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| 210 | -3b\arctan (b^{-1}) |
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| 211 | \nonumber \\ |
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| 212 | +\frac{1}{2}b^{2}[4-4b \arctan(b^{-1})-3 \ln(1+b^{-2})] \nonumber \\ |
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| 213 | + 4\ln (R m_{e} c/\hbar) - 4f_{C}(Z) + F_{0}(\kappa,Z), |
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| 214 | \end{eqnarray} |
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| 215 | with |
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| 216 | \begin{equation} |
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| 217 | b \ = \ \frac{Rm_{e}c}{\hbar} \frac{1}{2\kappa} \frac{1}{\epsilon(1-\epsilon)}. |
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| 218 | \end{equation} |
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| 219 | In this case $R$ is the screening radius for the atom $Z$ (tabulated in |
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| 220 | \cite{dieci} for Z=1 to 92) and $\eta$ is the contribution of pair |
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| 221 | production in the electron field (rather than in the nuclear field). The |
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| 222 | parameter $\eta$ is approximated as |
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| 223 | \begin{equation} |
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| 224 | \eta \ = \ \eta_{\infty}(1-e^{-v}), |
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| 225 | \end{equation} |
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| 226 | where |
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| 227 | \begin{eqnarray} |
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| 228 | v \ = \ (0.2840-0.1909a)\ln(4/\kappa)+(0.1095+0.2206a)\ln^{2}(4/\kappa) |
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| 229 | \nonumber \\ |
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| 230 | + (0.02888 - 0.04269a)\ln^{3}(4/\kappa) \nonumber \\ |
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| 231 | +(0.002527+0.002623)\ln^{4}(4/\kappa) |
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| 232 | \end{eqnarray} |
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| 233 | and $\eta_{\infty}$ is the contribution for the atom $Z$ in the high-energy |
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| 234 | limit and is tabulated for Z=1 to 92 in Ref. \cite{dieci}. |
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| 235 | In the Eq.(\ref{eqquattro}), the function $f_{C}(Z)$ is the high-energy |
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| 236 | Coulomb correction of Ref. \cite{nove}, given by |
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| 237 | \begin{eqnarray} |
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| 238 | f_{C}(Z) \ = \ a^{2}[(1+a^{2})^{-1}+0.202059-0.03693a^{2}+0.00835a^{4} |
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| 239 | \nonumber \\ |
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| 240 | -0.00201a^{6}+0.00049a^{8}-0.00012a^{10}+0.00003a^{12}]; |
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| 241 | \end{eqnarray} |
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| 242 | $C_{r} = 1.0093$ is the high-energy limit of Mork and Olsen's radiative |
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| 243 | correction (see Ref. \cite{dieci}); $F_{0}(\kappa,Z)$ is a Coulomb-like |
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| 244 | correction function, which has been analytically approximated as \cite{uno} |
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| 245 | \begin{eqnarray} |
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| 246 | F_{0}(\kappa,Z) \ = \ (-0.1774 - 12.10a + 11.18a^{2})(2/\kappa)^{1/2} |
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| 247 | \nonumber \\ |
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| 248 | + (8.523 + 73.26a - 44.41a^{2})(2/\kappa) \nonumber \\ |
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| 249 | - (13.52 + 121.1a - 96.41a^{2})(2/\kappa)^{3/2} \nonumber \\ |
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| 250 | + (8.946 + 62.05a - 63.41a^{2})(2/\kappa)^{2}. |
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| 251 | \end{eqnarray} |
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| 252 | The kinetic energy $E_{+}$ of the secondary positron is obtained as |
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| 253 | \begin{equation} |
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| 254 | E_{+} \ = \ E - E_{-} - 2m_{e}c^{2}. |
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| 255 | \end{equation} |
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| 256 | The polar angles $\theta_{-}$ and $\theta_{+}$ of the directions of |
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| 257 | movement of the electron and the positron, relative to the direction of the |
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| 258 | incident photon, are sampled from the leading term of the expression |
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| 259 | obtained from high-energy theory (see Ref. \cite{undici}) |
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| 260 | \begin{equation} |
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| 261 | p(\cos\theta_{\pm}) \ = \ a(1-\beta_{\pm}\cos\theta_{\pm})^{-2}, |
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| 262 | \end{equation} |
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| 263 | where $a$ is the a normalization constant and $\beta_{\pm}$ is the particle |
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| 264 | velocity in units of the speed of light. As the directions of the produced |
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| 265 | particles and of the incident photon are not necessarily coplanar, the |
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| 266 | azimuthal angles $\phi_{-}$ and $\phi_{+}$ of the electron and of the |
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| 267 | positron are sampled independently and uniformly in the interval (0,2$\pi$). |
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| 268 | % |
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| 269 | \subsection{Photoelectric effect} |
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| 270 | \subsubsection{Total cross section} |
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| 271 | The total photoelectric cross section at a given photon energy $E$ is |
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| 272 | calculated from the data \cite{dodici}, as described in |
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| 273 | section~\ref{subsubsigmatot}. |
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| 274 | |
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| 275 | \subsubsection{Sampling of the final state} |
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| 276 | The incident photon is absorbed and one electron is emitted. The direction |
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| 277 | of the electron is sampled according to the Sauter |
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| 278 | distribution \cite{dodicibis}. |
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| 279 | Introducing the variable $\nu = 1 - \cos\theta_{e}$, the angular distribution |
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| 280 | can be expressed as |
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| 281 | \begin{equation} |
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| 282 | p(\nu) \ = \ (2-\nu) \Big[ \frac{1}{A+\nu} + \frac{1}{2} \beta \gamma |
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| 283 | (\gamma - 1)(\gamma -2) \Big] \frac{\nu}{(A+\nu)^{3}}, |
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| 284 | \end{equation} |
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| 285 | where |
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| 286 | \begin{equation} |
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| 287 | \gamma = 1 + \frac{E_{e}}{m_{e}c^{2}}, \quad A = \frac{1}{\beta} - 1, |
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| 288 | \end{equation} |
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| 289 | $E_{e}$ is the electron energy, $m_{e}$ its rest mass and $\beta$ its velocity |
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| 290 | in units of the speed of light $c$. |
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| 291 | Though the Sauter distribution, strictly speaking, is adequate only for |
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| 292 | ionisation of the K-shell by high-energy photons, in many practical |
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| 293 | simulations it does not introduce appreciable errors in the description of any |
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| 294 | photoionisation event, irrespective of the atomic shell or of the photon |
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| 295 | energy.\\ |
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| 296 | %in the same |
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| 297 | %direction as the primary photon. |
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| 298 | The subshell from which the electron is emitted is randomly selected |
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| 299 | according to the relative cross sections of subshells, determined at the |
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| 300 | energy $E$ by interpolation of the data of Ref. \cite{undici}. The electron |
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| 301 | kinetic energy is the difference between the incident photon energy and |
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| 302 | the binding energy of the electron before the interaction in the sampled |
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| 303 | shell. The interaction leaves the atom in an excited state; the subsequent |
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| 304 | de-excitation is simulated as described in section~\ref{relax}.\\ |
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| 305 | |
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| 306 | \subsection{Bremsstrahlung} |
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| 307 | \subsubsection{Introduction} |
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| 308 | The class {\tt G4PenelopeBremsstrahlung} calculates the continuous energy loss |
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| 309 | due to soft $\gamma$ emission and simulates the photon production by |
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| 310 | electrons and positrons. As usual, the gamma production threshold $T_{c}$ for |
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| 311 | a given material is used to separate the continuous and the discrete parts of |
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| 312 | the process. |
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| 313 | |
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| 314 | \subsubsection{Electrons} |
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| 315 | The total cross sections are calculated from the data \cite{quattordici}, as |
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| 316 | described in sections~\ref{subsubsigmatot} and \ref{lowebrems}.\\ |
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| 317 | The energy distribution $\frac{d\sigma}{dW}(E)$, i.e. the probability of the |
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| 318 | emission of a photon with energy $W$ given an incident electron of |
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| 319 | kinetic energy $E$, is generated according to the formula |
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| 320 | \begin{equation} |
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| 321 | \frac{d\sigma}{dW}(E) \ = \ \frac{F(\kappa)}{\kappa}, \quad |
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| 322 | \kappa \ = \ \frac{W}{E}. |
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| 323 | \end{equation} |
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| 324 | The functions $F(\kappa)$ describing the energy spectra of the outgoing |
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| 325 | photons are taken from Ref. \cite{tredici}. For each element $Z$ from 1 to 92, |
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| 326 | 32 points in $\kappa$, ranging from $10^{-12}$ to 1, are used for the linear |
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| 327 | interpolation of this function. $F(\kappa)$ is normalized using the condition |
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| 328 | $F(10^{-12})=1$. The energy distribution of the emitted photons is available |
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| 329 | in the library \cite{tredici} for 57 energies of the incident electron |
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| 330 | between 1 keV and 100 GeV. For other primary energies, logarithmic |
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| 331 | interpolation is used to obtain the values of the function $F(\kappa)$.\\ |
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| 332 | The direction of the emitted bremsstrahlung photon is determined by the polar |
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| 333 | angle $\theta$ and the azimuthal angle $\phi$. For isotropic media, with |
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| 334 | randomly oriented atoms, the bremsstrahlung differential cross section is |
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| 335 | independent of $\phi$ and can be expressed as |
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| 336 | \begin{equation} |
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| 337 | \frac{d^{2} \sigma}{dW d\cos\theta} \ = \ \frac{d\sigma}{dW} p(Z,E,\kappa; |
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| 338 | \cos\theta). |
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| 339 | \end{equation} |
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| 340 | Numerical values of the ``shape function'' $p(Z,E,\kappa;\cos\theta)$, |
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| 341 | calculated by partial-wave methods, have been published in |
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| 342 | Ref. \cite{quindici} for the |
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| 343 | following benchmark cases: $Z$= 2, 8, 13, 47, 79 and 92; $E$= 1, 5, 10, 50, |
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| 344 | 100 and 500 keV; $\kappa$= 0, 0.6, 0.8 and 0.95. It was found in |
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| 345 | Ref. \cite{uno} that the benchmark partial-wave shape function of |
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| 346 | Ref. \cite{quindici} can be closely approximated by the analytical form |
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| 347 | (obtained in the Lorentz-dipole approximation) |
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| 348 | \begin{eqnarray} |
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| 349 | p(\cos\theta) = A \frac{3}{8} \Big[ 1+\Big( \frac{\cos\theta - \beta'} |
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| 350 | {1-\beta' \cos |
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| 351 | \theta} \Big)^{2} \Big] \frac{1-\beta^{'2}}{(1-\beta'\cos\theta)^{2}} |
---|
| 352 | \nonumber \\ |
---|
| 353 | + (1-A) \frac{3}{4} \Big[ 1- \Big( \frac{\cos\theta - \beta'}{1-\beta' \cos |
---|
| 354 | \theta}m \Big)^{2} \Big] \frac{1-\beta^{'2}}{(1-\beta'\cos\theta)^{2}}, |
---|
| 355 | \end{eqnarray} |
---|
| 356 | with $\beta' = \beta (1+B)$, if one considers $A$ and $B$ as adjustable |
---|
| 357 | parameters. The parameters $A$ and $B$ have been determined, by least squares |
---|
| 358 | fitting, for the 144 combinations of atomic numbers, electron energies and |
---|
| 359 | reduced photon energies corresponding to the benchmark shape functions |
---|
| 360 | tabulated in \cite{quindici}. The quantities $\ln(AZ\beta)$ and $B\beta$ vary |
---|
| 361 | smoothly with Z, $\beta$ and $\kappa$ and can be obtained by cubic spline |
---|
| 362 | interpolation of their values for the benchmark cases. This permits the fast |
---|
| 363 | evaluation of the shape function |
---|
| 364 | $p(Z,E,\kappa;\cos\theta)$ for any combination of $Z$, $\beta$ and $\kappa$. \\ |
---|
| 365 | The stopping power $\frac{dE}{dx}$ due to soft bremsstrahlung is |
---|
| 366 | calculated by interpolating in $E$ and $\kappa$ the numerical data of scaled |
---|
| 367 | cross sections of Ref. \cite{sedici}. The energy and the direction of the |
---|
| 368 | outgoing electron are determined by using energy-momentum balance. |
---|
| 369 | |
---|
| 370 | \subsubsection{Positrons} |
---|
| 371 | The radiative differential cross section $\frac{d\sigma^{+}}{dW} (E)$ |
---|
| 372 | for positrons reduces to that for electrons in the high-energy limit, but is |
---|
| 373 | smaller for intermediate and low energies. Owing to the lack of more accurate |
---|
| 374 | calculations, the differential cross section for positrons is obtained by |
---|
| 375 | multiplying the electron differential cross section |
---|
| 376 | $\frac{d\sigma^{-}}{dW} (E)$ |
---|
| 377 | by a $\kappa -$indendent factor, i.e. |
---|
| 378 | \begin{equation} |
---|
| 379 | \frac{d\sigma^{+}}{dW} \ = \ F_{p}(Z,E) \frac{d\sigma^{-}}{dW}. |
---|
| 380 | \end{equation} |
---|
| 381 | The factor $F_{p}(Z,E)$ is set equal to the ratio of the radiative stopping |
---|
| 382 | powers for positrons and electrons, which has been calculated in Ref. |
---|
| 383 | \cite{diciassette}. For the actual calculation, the following analytical |
---|
| 384 | approximation is used: |
---|
| 385 | \begin{eqnarray} |
---|
| 386 | F_{p}(Z,E) \ = \ 1-\exp(-1.2359 \cdot 10^{-1} t + 6.1274 \cdot 10^{-2} t^{2} |
---|
| 387 | - 3.1516 \cdot 10^{-2} t^{3} \nonumber \\ |
---|
| 388 | + 7.7446 \cdot 10^{-3} t^{4} - 1.0595 \cdot 10^{-3} t^{5} + 7.0568 |
---|
| 389 | \cdot 10^{-5} t^{6} \nonumber \\ |
---|
| 390 | -1.8080 \cdot 10^{-6} t^{7}), |
---|
| 391 | \end{eqnarray} |
---|
| 392 | where |
---|
| 393 | \begin{equation} |
---|
| 394 | t \ = \ \ln \Big( 1+ \frac{10^{6}}{Z^{2}} \frac{E}{m_{e}c^{2}} \Big). |
---|
| 395 | \end{equation} |
---|
| 396 | Because the factor $F_{p}(Z,E)$ is independent on $\kappa$, the energy |
---|
| 397 | distribution of the secondary $\gamma$'s has the same shape as electron |
---|
| 398 | bremsstrahlung. Similarly, owing to the lack of numerical data for positrons, |
---|
| 399 | it is assumed that the shape of the angular distribution |
---|
| 400 | $p(Z,E,\kappa;\cos\theta)$ of the bremsstrahlung photons for positrons is the |
---|
| 401 | same as for the electrons.\\ |
---|
| 402 | The energy and direction of the outgoing positron are determined from |
---|
| 403 | energy-momentum balance. |
---|
| 404 | % |
---|
| 405 | \subsection{Ionisation} |
---|
| 406 | |
---|
| 407 | The {\tt G4PenelopeIonisation} class calculates the continuous energy loss due |
---|
| 408 | to electron and positron ionisation and simulates the $\delta$-ray production |
---|
| 409 | by electrons and positrons. The electron production threshold $T_{c}$ for a |
---|
| 410 | given material is used to separate the continuous and the discrete parts of the |
---|
| 411 | process.\\ |
---|
| 412 | The simulation of inelastic collisions of electrons and positrons is |
---|
| 413 | performed on the basis of a Generalized Oscillation Strength (GOS) model |
---|
| 414 | (see Ref. \cite{uno} for a complete description). It is assumed that GOS |
---|
| 415 | splits into contributions from the different atomic electron shells. |
---|
| 416 | % |
---|
| 417 | \subsubsection{Electrons} \label{ionelect} |
---|
| 418 | The total cross section $\sigma^{-} (E)$ for the inelastic collision of |
---|
| 419 | electrons of energy $E$ is calculated analytically. It can be split into |
---|
| 420 | contributions from distant longitudinal, distant transverse and close |
---|
| 421 | interactions, |
---|
| 422 | \begin{equation} |
---|
| 423 | \sigma^{-} (E) \ = \ \sigma_{dis,l} + \sigma_{dis,t} + \sigma^{-}_{clo}. |
---|
| 424 | \end{equation} |
---|
| 425 | The contributions from distant longitudinal and transverse interactions are |
---|
| 426 | \begin{equation} |
---|
| 427 | \sigma_{dis,l} \ = \ |
---|
| 428 | \frac{2 \pi e^{4}}{m_{e}v^{2}} \sum_{shells} f_{k} \frac{1}{W_{k}} |
---|
| 429 | \ln \Big( \frac{W_{k}}{Q^{min}_{k}} \ |
---|
| 430 | \frac{Q^{min}_{k}+2m_{e}c^{2}}{W_{k}+2m_{e}c^{2}} \Big) \Theta (E-W_{k}) |
---|
| 431 | \label{dist1} |
---|
| 432 | \end{equation} |
---|
| 433 | and |
---|
| 434 | \begin{equation} |
---|
| 435 | \sigma_{dis,t} \ = \ |
---|
| 436 | \frac{2 \pi e^{4}}{m_{e}v^{2}} \sum_{shells} f_{k} \frac{1}{W_{k}} |
---|
| 437 | \Big[ \ln \Big( \frac{1}{1-\beta^{2}} \Big) - \beta^{2}-\delta_{F} \Big] |
---|
| 438 | \Theta (E-W_{k}) \label{dist2} |
---|
| 439 | \end{equation} |
---|
| 440 | respectively, where: \\ |
---|
| 441 | $m_{e}$ = mass of the electron; \\ |
---|
| 442 | $v$ = velocity of the electron; \\ |
---|
| 443 | $\beta$ = velocity of the electron in units of $c$; \\ |
---|
| 444 | $f_{k}$ = number of electrons in the $k$-th atomic shell; \\ |
---|
| 445 | $\Theta$ = Heaviside step function; \\ |
---|
| 446 | $W_{k}$ = resonance energy of the $k$-th atomic shell oscillator;\\ |
---|
| 447 | $Q^{min}_{k}$ = minimum kinematically allowed recoil energy for energy transfer $W_{k}$ |
---|
| 448 | \\ |
---|
| 449 | \begin{displaymath} |
---|
| 450 | = \ \sqrt{\Big[ \sqrt{E(E+2m_{e}c^{2})}-\sqrt{(E-W_{k})(E-W_{k}+ |
---|
| 451 | 2m_{e}c^{2})} \Big]^{2}+m_{e}^{2}c^{4}}-m_{e}c^{2}; |
---|
| 452 | \end{displaymath} \\ |
---|
| 453 | $\delta_{F}$ = Fermi density effect correction, computed as described in Ref. |
---|
| 454 | \cite{diciotto}. |
---|
| 455 | % |
---|
| 456 | |
---|
| 457 | The value of $W_{k}$ is calculated from the ionisation energy $U_{k}$ of |
---|
| 458 | the $k$-th shell as \mbox{$W_{k}=1.65 \ U_{k}$}. This relation is derived from |
---|
| 459 | the hydrogenic model, which is valid for the innermost shells. In this model, |
---|
| 460 | the shell ionisation cross sections are only roughly approximated; nevertheless |
---|
| 461 | the ionisation of inner shells is a low-probability process and the |
---|
| 462 | approximation has a weak effect on the global transport |
---|
| 463 | properties\footnote{In cases where inner-shell ionisation is directly observed, |
---|
| 464 | a more accurate description of the process should be used.}. \\ |
---|
| 465 | The integrated cross section for close collisions is the M\o ller cross |
---|
| 466 | section |
---|
| 467 | \begin{equation} |
---|
| 468 | \sigma^{-}_{clo} \ = \ |
---|
| 469 | \frac{2 \pi e^{4}}{m_{e}v^{2}} \sum_{shells} f_{k} \int_{W_{k}}^{\frac{E}{2}} |
---|
| 470 | \frac{1}{W^{2}} F^{-}(E,W) dW, \label{close} |
---|
| 471 | \end{equation} |
---|
| 472 | where |
---|
| 473 | \begin{equation} |
---|
| 474 | F^{-}(E,W) \ = \ 1+ \Big( \frac{W}{E-W} \Big)^{2} - \frac{W}{E-W} |
---|
| 475 | + \Big( \frac{E}{E+m_{e}c^{2}} \Big)^{2} \Big( \frac{W}{E-W} + |
---|
| 476 | \frac{W^{2}}{E^{2}} \Big). |
---|
| 477 | \end{equation} |
---|
| 478 | The integral of Eq.(\ref{close}) can be evaluated analytically. In the final |
---|
| 479 | state there are two indistinguishable free electrons and the fastest one |
---|
| 480 | is considered as the ``primary''; accordingly, the maximum allowed energy |
---|
| 481 | transfer in close collisions is $\frac{E}{2}$.\\ |
---|
| 482 | The GOS model also allows evaluation of the spectrum |
---|
| 483 | $\frac{d \sigma^{-}}{d W}$ of the energy $W$ lost by the primary electron |
---|
| 484 | as the sum of distant longitudinal, distant transverse and close interaction |
---|
| 485 | contributions, |
---|
| 486 | \begin{equation} |
---|
| 487 | \frac{d\sigma^{-}}{dW} \ = \ \frac{d\sigma^{-}_{clo}}{dW} + |
---|
| 488 | \frac{d\sigma_{dis,l}}{dW} + \frac{d\sigma_{dis,t}}{dW}. \label{aaa} |
---|
| 489 | \end{equation} |
---|
| 490 | In particular, |
---|
| 491 | \begin{equation} |
---|
| 492 | \frac{d\sigma_{dis,l}}{dW} \ = \ |
---|
| 493 | \frac{2 \pi e^{4}}{m_{e}v^{2}} \sum_{shells} f_{k} \frac{1}{W_{k}} |
---|
| 494 | \ln \Big( \frac{W_{k}}{Q_{-}} \ |
---|
| 495 | \frac{Q_{-}+2m_{e}c^{2}}{W_{k}+2m_{e}c^{2}} \Big) \delta(W-W_{k}) |
---|
| 496 | \Theta (E-W_{k}), \label{ddist1} |
---|
| 497 | \end{equation} |
---|
| 498 | where |
---|
| 499 | \begin{equation} |
---|
| 500 | Q_{-} \ = \ \sqrt{\Big[ \sqrt{E(E+2m_{e}c^{2})}-\sqrt{(E-W)(E-W+ |
---|
| 501 | 2m_{e}c^{2})} \Big]^{2}+m_{e}^{2}c^{4}}-m_{e}c^{2}, |
---|
| 502 | \end{equation} |
---|
| 503 | \begin{eqnarray} |
---|
| 504 | \frac{d\sigma_{dis,t}}{dW} \ = \ |
---|
| 505 | \frac{2 \pi e^{4}}{m_{e}v^{2}} \sum_{shells} f_{k} \frac{1}{W_{k}} |
---|
| 506 | \Big[ \ln \Big( \frac{1}{1-\beta^{2}} \Big) - \beta^{2}-\delta_{F} \Big] |
---|
| 507 | \nonumber \\ |
---|
| 508 | \Theta (E-W_{k}) \delta(W-W_{k}) \label{ddist2} |
---|
| 509 | \end{eqnarray} |
---|
| 510 | and |
---|
| 511 | \begin{equation} |
---|
| 512 | \frac{d \sigma^{-}_{clo}}{dW} \ = \ |
---|
| 513 | \frac{2 \pi e^{4}}{m_{e}v^{2}} \sum_{shells} |
---|
| 514 | f_{k} \frac{1}{W^{2}} F^{-}(E,W) \Theta (W-W_{k}). \label{dclose} |
---|
| 515 | \end{equation} |
---|
| 516 | Eqs. (\ref{dist1}), (\ref{dist2}) and (\ref{close}) derive respectively |
---|
| 517 | from the integration in $dW$ of Eqs. (\ref{ddist1}), (\ref{ddist2}) and |
---|
| 518 | (\ref{dclose}) in the interval [0,$W_{max}$], where $W_{max}=E$ for distant |
---|
| 519 | interactions and $W_{max}=\frac{E}{2}$ for close. The analytical GOS model |
---|
| 520 | provides an accurate \emph{average} description of inelastic collisions. |
---|
| 521 | However, the continuous energy loss spectrum associated with single distant |
---|
| 522 | excitations of a given atomic shell is approximated as a single resonance |
---|
| 523 | (a $\delta$ distribution). As a consequence, the simulated energy loss spectra |
---|
| 524 | show unphysical narrow peaks at energy losses that are multiples of the |
---|
| 525 | resonance energies. These spurious peaks are automatically smoothed out after |
---|
| 526 | multiple inelastic collisions. \\ |
---|
| 527 | The explicit expression of $\frac{d\sigma^{-}}{dW}$, Eq. (\ref{aaa}), |
---|
| 528 | allows the analytic calculation of the partial cross sections for soft and |
---|
| 529 | hard ionisation events, i.e. |
---|
| 530 | \begin{equation} |
---|
| 531 | \sigma^{-}_{soft} \ = \ \int_{0}^{T_{c}} \frac{d\sigma^{-}}{dW} dW |
---|
| 532 | \quad \textrm{and} \quad |
---|
| 533 | \sigma^{-}_{hard} \ = \ \int_{T_{c}}^{W_{max}} \frac{d\sigma^{-}}{dW} dW. |
---|
| 534 | \end{equation} |
---|
| 535 | |
---|
| 536 | The first stage of the simulation is the selection of the active oscillator |
---|
| 537 | $k$ and the oscillator branch (distant or close). \\ |
---|
| 538 | In distant interactions with the $k$-th oscillator, the energy loss $W$ of the |
---|
| 539 | primary electron corresponds to the excitation energy $W_{k}$, i.e. |
---|
| 540 | $W$=$W_{k}$. If the interaction is transverse, the angular deflection of the |
---|
| 541 | projectile is neglected, i.e. $\cos \theta$=1. For longitudinal collisions, |
---|
| 542 | the distribution of the recoil energy $Q$ is given by |
---|
| 543 | \begin{equation} |
---|
| 544 | \begin{array}{rlll} |
---|
| 545 | P_{k}(Q) = & & & \\ |
---|
| 546 | & \frac{1}{Q [1+Q/(2m_{e}c^{2})]} & \textrm{if} \ Q_{-} < Q < W_{max} & \\ |
---|
| 547 | & 0 & \textrm{otherwise} & \\ \label{ele1} |
---|
| 548 | \end{array} |
---|
| 549 | %P_{k}(Q) = |
---|
| 550 | %\begin{cases} |
---|
| 551 | %\frac{1}{Q [1+Q/(2m_{e}c^{2})]} & |
---|
| 552 | %\textrm{if} \quad Q_{-} < Q < W_{max} \\ |
---|
| 553 | %0 & \textrm{otherwise} \label{ele1} |
---|
| 554 | %\end{cases}. |
---|
| 555 | \end{equation} |
---|
| 556 | Once the energy loss $W$ and the recoil energy $Q$ have been sampled, the |
---|
| 557 | polar scattering angle is determined as |
---|
| 558 | \begin{equation} |
---|
| 559 | \cos \theta \ = \ \frac{E(E+2m_{e}c^{2})+(E-W)(E-W+2m_{e}c^{2})- |
---|
| 560 | Q(Q+2m_{e}c^{2})}{2\sqrt{E(E+2m_{e}c^{2})(E-W)(E-W+2m_{e}c^{2})}}. \label{ele2} |
---|
| 561 | \end{equation} |
---|
| 562 | The azimuthal scattering angle $\phi$ is sampled uniformly in the interval |
---|
| 563 | (0,2$\pi$). \\ |
---|
| 564 | For close interactions, the distributions for the reduced energy loss |
---|
| 565 | $\kappa \equiv W/E$ for electrons are |
---|
| 566 | \begin{eqnarray} |
---|
| 567 | P^{-}_{k}(\kappa) \ = \ \Big[ \frac{1}{\kappa^{2}}+\frac{1}{(1-\kappa)^2} - |
---|
| 568 | \frac{1}{\kappa(1-\kappa)} + \Big( \frac{E}{E+m_{e}c^{2}} \Big)^{2} |
---|
| 569 | \Big( 1+\frac{1}{\kappa(1-\kappa)} \Big) \Big] \nonumber \\ |
---|
| 570 | \Theta(\kappa-\kappa_{c}) |
---|
| 571 | \Theta(\frac{1}{2}-\kappa) \label{closed} |
---|
| 572 | \end{eqnarray} |
---|
| 573 | with $\kappa_{c} = \max(W_{k},T_{c})/E$. The maximum allowed value of $\kappa$ |
---|
| 574 | is 1/2, consistent with the indistinguishability of the electrons in the |
---|
| 575 | final state. After the sampling of the energy loss $W= \kappa E$, the polar |
---|
| 576 | scattering angle $\theta$ is obtained as |
---|
| 577 | \begin{equation} |
---|
| 578 | \cos^{2} \theta \ = \ \frac{E-W}{E} \ \frac{E+2m_{e}c^{2}}{E-W+2m_{e}c^{2}}. |
---|
| 579 | \end{equation} |
---|
| 580 | The azimuthal scattering angle $\phi$ is sampled uniformly in the interval |
---|
| 581 | (0,2$\pi$). \\ |
---|
| 582 | According to the GOS model, each oscillator $W_{k}$ corresponds to an atomic |
---|
| 583 | shell with $f_{k}$ electrons and ionisation energy $U_{k}$. In the case of |
---|
| 584 | ionisation of an inner shell $i$ (K or L), a secondary electron |
---|
| 585 | ($\delta$-ray) |
---|
| 586 | is emitted with energy $E_{s}=W-U_{i}$ and the residual ion is left with |
---|
| 587 | a vacancy in the shell (which is then filled with the emission of fluorescence |
---|
| 588 | x-rays and/or Auger electrons). In the case of ionisation of outer shells, |
---|
| 589 | the simulated $\delta$-ray is emitted with kinetic energy $E_{s}=W$ and the |
---|
| 590 | target atom is assumed to remain in its ground state. The polar angle of |
---|
| 591 | emission of the secondary electron is calculated as |
---|
| 592 | \begin{equation} |
---|
| 593 | \cos^{2} \theta_{s} \ = \ \frac{W^{2}/\beta^{2}}{Q(Q+2m_{e}c^{2})} |
---|
| 594 | \Big[ 1+ \frac{Q(Q+2m_{e}c^{2})-W^{2}}{2W(E+m_{e}c^{2})} \Big]^{2} |
---|
| 595 | \end{equation} |
---|
| 596 | (for close collisions $Q=W$), while the azimuthal angle is |
---|
| 597 | $\phi_{s} = \phi + \pi$. In this model, the Doppler effects on the angular |
---|
| 598 | distribution of the $\delta$ rays are neglected. \\ |
---|
| 599 | The stopping power due to soft interactions of electrons, which is used |
---|
| 600 | for the computation of the continuous part of the process, is analytically |
---|
| 601 | calculated as |
---|
| 602 | \begin{equation} |
---|
| 603 | S^{-}_{in} \ = \ N \int_{0}^{T_{c}} W \frac{d\sigma^{-}}{dW} dW |
---|
| 604 | \end{equation} |
---|
| 605 | from the expression (\ref{aaa}), where $N$ is the number of scattering centers |
---|
| 606 | (atoms or molecules) per unit volume. \\ |
---|
| 607 | % |
---|
| 608 | \subsubsection{Positrons} |
---|
| 609 | The total cross section $\sigma^{+} (E)$ for the inelastic collision of |
---|
| 610 | positrons of energy $E$ is calculated analytically. As in the case of |
---|
| 611 | electrons, it can be split into contributions from distant longitudinal, |
---|
| 612 | distant transverse and close interactions, |
---|
| 613 | \begin{equation} |
---|
| 614 | \sigma^{+} (E) \ = \ \sigma_{dis,l} + \sigma_{dis,t} + \sigma^{+}_{clo}. |
---|
| 615 | \end{equation} |
---|
| 616 | The contributions from distant longitudinal and transverse interactions are |
---|
| 617 | the same as for electrons, Eq. (\ref{dist1}) and (\ref{dist2}), while the |
---|
| 618 | integrated cross section for close collisions is the Bhabha cross |
---|
| 619 | section |
---|
| 620 | \begin{equation} |
---|
| 621 | \sigma^{+}_{clo} \ = \ |
---|
| 622 | \frac{2 \pi e^{4}}{m_{e}v^{2}} \sum_{shells} f_{k} \int_{W_{k}}^{E} |
---|
| 623 | \frac{1}{W^{2}} F^{+}(E,W) dW, \label{closepos} |
---|
| 624 | \end{equation} |
---|
| 625 | where |
---|
| 626 | \begin{equation} |
---|
| 627 | F^{+}(E,W) \ = 1- b_{1}\frac{W}{E} + b_{2} \frac{W^{2}}{E^{2}} - |
---|
| 628 | b_{3} \frac{W^{3}}{E^{3}} + b_{4} \frac{W^{4}}{E^{4}}; |
---|
| 629 | \end{equation} |
---|
| 630 | the Bhabha factors are |
---|
| 631 | \begin{eqnarray} |
---|
| 632 | b_{1} = \Big( \frac{\gamma-1}{\gamma} \Big)^{2} \ \frac{2(\gamma+1)^{2}-1} |
---|
| 633 | {\gamma^{2}-1} & & |
---|
| 634 | b_{2} = \Big( \frac{\gamma-1}{\gamma} \Big)^{2} \ \frac{3(\gamma+1)^{2}+1} |
---|
| 635 | {(\gamma+1)^{2}}, \nonumber \\ |
---|
| 636 | b_{3} = \Big( \frac{\gamma-1}{\gamma} \Big)^{2} \ \frac{2(\gamma-1)\gamma} |
---|
| 637 | {(\gamma+1)^{2}}, & & |
---|
| 638 | b_{4} = \Big( \frac{\gamma-1}{\gamma} \Big)^{2} \ \frac{(\gamma-1)^{2}} |
---|
| 639 | {(\gamma+1)^{2}}, \\ |
---|
| 640 | \end{eqnarray} |
---|
| 641 | and $\gamma$ is the Lorentz factor of the positron. The integral of |
---|
| 642 | Eq. (\ref{closepos}) can be evaluated analytically. The particles in the |
---|
| 643 | final state are not undistinguishable so the maximum energy transfer $W_{max}$ |
---|
| 644 | in close collisions is $E$.\\ |
---|
| 645 | As for electrons, the GOS model allows the evaluation of the spectrum |
---|
| 646 | $\frac{d \sigma^{+}}{d W}$ of the energy $W$ lost by the primary positron |
---|
| 647 | as the sum of distant longitudinal, distant transverse and close interaction |
---|
| 648 | contributions, |
---|
| 649 | \begin{equation} |
---|
| 650 | \frac{d\sigma^{+}}{dW} \ = \ \frac{d\sigma^{+}_{clo}}{dW} + |
---|
| 651 | \frac{d\sigma_{dis,l}}{dW} + \frac{d\sigma_{dis,t}}{dW}, \label{bbb} |
---|
| 652 | \end{equation} |
---|
| 653 | where the distant terms $\frac{d\sigma_{dis,l}}{dW}$ and |
---|
| 654 | $\frac{d\sigma_{dis,t}}{dW}$ are those from Eqs. (\ref{ddist1}) and |
---|
| 655 | (\ref{ddist2}), while the close contribution is |
---|
| 656 | \begin{equation} |
---|
| 657 | \frac{d \sigma^{+}_{clo}}{dW} \ = \ |
---|
| 658 | \frac{2 \pi e^{4}}{m_{e}v^{2}} \sum_{shells} |
---|
| 659 | f_{k} \frac{1}{W^{2}} F^{+}(E,W) \Theta (W-W_{k}). \label{dclosepos} |
---|
| 660 | \end{equation} |
---|
| 661 | Also in this case, the explicit expression of $\frac{d\sigma^{+}}{dW}$, |
---|
| 662 | Eq. (\ref{bbb}), allows an analytic calculation of the partial cross |
---|
| 663 | sections for soft and hard ionisation events, i.e. |
---|
| 664 | \begin{equation} |
---|
| 665 | \sigma^{+}_{soft} \ = \ \int_{0}^{T_{c}} \frac{d\sigma^{+}}{dW} dW |
---|
| 666 | \quad \textrm{and} \quad |
---|
| 667 | \sigma^{+}_{hard} \ = \ \int_{T_{c}}^{E} \frac{d\sigma^{+}}{dW} dW. |
---|
| 668 | \end{equation} |
---|
| 669 | The sampling of the final state in the case of distant interactions |
---|
| 670 | (transverse or longitudinal) is performed in the same way as for |
---|
| 671 | primary electrons, see section~\ref{ionelect}. For close positron |
---|
| 672 | interactions with the $k$-th oscillator, the distribution for the reduced |
---|
| 673 | energy loss $\kappa \equiv W/E$ is |
---|
| 674 | \begin{eqnarray} |
---|
| 675 | P^{+}_{k}(\kappa) \ = \ \Big[\frac{1}{\kappa^{2}} - \frac{b_{1}}{\kappa}+b_{2} |
---|
| 676 | -b_{3}\kappa + b_{4} \kappa^{2} \Big] \Theta(\kappa-\kappa_{c}) |
---|
| 677 | \Theta(1-\kappa) \label{closedpos} |
---|
| 678 | \end{eqnarray} |
---|
| 679 | with $\kappa_{c} = \max(W_{k},T_{c})/E$. In this case, the maximum allowed |
---|
| 680 | reduced energy loss $\kappa$ is 1. After sampling the energy loss |
---|
| 681 | $W= \kappa E$, the polar angle $\theta$ and the azimuthal |
---|
| 682 | angle $\phi$ are obtained using the equations introduced for electrons |
---|
| 683 | in section~\ref{ionelect}. Similarly, the generation of $\delta$ rays is |
---|
| 684 | performed in the same way as for electrons.\\ |
---|
| 685 | Finally, the stopping power due to soft interactions of positrons, |
---|
| 686 | which is used for the computation of the continuous part of the process, |
---|
| 687 | is analytically calculated as |
---|
| 688 | \begin{equation} |
---|
| 689 | S^{+}_{in} \ = \ N \int_{0}^{T_{c}} W \frac{d\sigma^{+}}{dW} dW |
---|
| 690 | \end{equation} |
---|
| 691 | from the expression (\ref{bbb}), where $N$ is the number of scattering centers |
---|
| 692 | per unit volume. \\ |
---|
| 693 | % |
---|
| 694 | \subsection{Positron Annihilation} |
---|
| 695 | |
---|
| 696 | \subsubsection{Total Cross Section} |
---|
| 697 | The total cross section (per target electron) for the annihilation of |
---|
| 698 | a positron of energy $E$ into two photons is evaluated from the |
---|
| 699 | analytical formula \cite{diciannove,venti} |
---|
| 700 | \begin{eqnarray} |
---|
| 701 | \lefteqn{\sigma(E) \ = \ |
---|
| 702 | \frac{\pi r_{e}^{2}}{(\gamma+1)(\gamma^{2}-1)} \quad \times} \nonumber \\ |
---|
| 703 | & & \Big{\{} (\gamma^{2}+4\gamma+1) \ln \Big[ \gamma + |
---|
| 704 | \sqrt{\gamma^{2}-1} \Big] |
---|
| 705 | -(3+\gamma)\sqrt{\gamma^{2}-1} \Big{\}}. |
---|
| 706 | \end{eqnarray} |
---|
| 707 | where \\ |
---|
| 708 | $r_{e}$ = classical radius of the electron, and \\ |
---|
| 709 | $\gamma$ = Lorentz factor of the positron. \\ |
---|
| 710 | % |
---|
| 711 | \subsubsection{Sampling of the Final State} |
---|
| 712 | The target electrons are assumed to be free and at rest: binding effects, |
---|
| 713 | that enable one-photon annihilation \cite{diciannove}, are neglected. |
---|
| 714 | When the annihilation occurs in flight, the two photons may have different |
---|
| 715 | energies, say $E_{-}$ and $E_{+}$ (the photon |
---|
| 716 | with lower energy is denoted by the superscript ``$-$''), |
---|
| 717 | whose sum is $E+2m_{e}c^{2}$. Each annihilation event is completely |
---|
| 718 | characterized by the quantity |
---|
| 719 | \begin{equation} |
---|
| 720 | \zeta \ = \ \frac{E_{-}}{E+2m_{e}c^{2}}, |
---|
| 721 | \end{equation} |
---|
| 722 | which is in the interval $\zeta_{min} \le \zeta \le \frac{1}{2}$, with |
---|
| 723 | \begin{equation} |
---|
| 724 | \zeta_{min} \ = \ \frac{1}{\gamma + 1 + \sqrt{\gamma^{2}-1}}. |
---|
| 725 | \end{equation} |
---|
| 726 | The parameter $\zeta$ is sampled from the differential distribution |
---|
| 727 | \begin{equation} |
---|
| 728 | P(\zeta) \ = \ \frac{\pi r_{e}^{2}}{(\gamma+1)(\gamma^{2}-1)} |
---|
| 729 | [S(\zeta)+S(1-\zeta)], |
---|
| 730 | \end{equation} |
---|
| 731 | where $\gamma$ is the Lorentz factor and |
---|
| 732 | \begin{equation} |
---|
| 733 | S(\zeta) \ = \ -(\gamma+1)^{2}+(\gamma^{2}+4\gamma+1) |
---|
| 734 | \frac{1}{\zeta}-\frac{1}{\zeta^{2}}. |
---|
| 735 | \end{equation} |
---|
| 736 | From conservation of energy and momentum, it follows that the two photons |
---|
| 737 | are emitted in directions with polar angles |
---|
| 738 | \begin{equation} |
---|
| 739 | \cos \theta_{-} \ = \ \frac{1}{\sqrt{\gamma^{2}-1}} |
---|
| 740 | \Big( \gamma+1-\frac{1}{\zeta} \Big) |
---|
| 741 | \end{equation} |
---|
| 742 | and |
---|
| 743 | \begin{equation} |
---|
| 744 | \cos \theta_{+} \ = \ \frac{1}{\sqrt{\gamma^{2}-1}} |
---|
| 745 | \Big( \gamma+1-\frac{1}{1-\zeta} \Big) |
---|
| 746 | \end{equation} |
---|
| 747 | that are completely determined by $\zeta$; in particuar, when |
---|
| 748 | $\zeta=\zeta_{min}$, $\cos\theta_{-}=-1$. |
---|
| 749 | The azimuthal angles are $\phi_{-}$ and |
---|
| 750 | $\phi_{+} = \phi_{-} + \pi$; owing to the axial symmetry of the process, |
---|
| 751 | the angle $\phi_{-}$ is uniformly distributed in $(0,2\pi)$. |
---|
| 752 | % |
---|
| 753 | |
---|
| 754 | \subsection{Status of the document} |
---|
| 755 | 09.06.2003 created by L.~Pandola \\ |
---|
| 756 | 20.06.2003 spelling and grammar check by D.H.~Wright\\ |
---|
| 757 | 07.11.2003 Ionisation and Annihilation section added by L.~Pandola\\ |
---|
| 758 | 01.06.2005 Added text in the PhotoElectric effect section, L.~Pandola \\ |
---|
| 759 | % |
---|
| 760 | |
---|
| 761 | \begin{latexonly} |
---|
| 762 | |
---|
| 763 | \begin{thebibliography}{99} |
---|
| 764 | \bibitem{uno} \emph{Penelope - A Code System for Monte Carlo Simulation of |
---|
| 765 | Electron and Photon Transport}, Workshop Proceedings Issy-les-Moulineaux, |
---|
| 766 | France, 5$-$7 November 2001, AEN-NEA; |
---|
| 767 | \bibitem{due} J.Sempau \emph{et al.}, \emph{Experimental benchmarks of the |
---|
| 768 | Monte Carlo code PENELOPE}, submitted to NIM B (2002); |
---|
| 769 | \bibitem{tre} D.Brusa \emph{et al.}, \emph{Fast sampling algorithm for the |
---|
| 770 | simulation of photon Compton scattering}, NIM A379,167 (1996); |
---|
| 771 | \bibitem{quattro} F.Biggs \emph{et al.}, \emph{Hartree-Fock Compton profiles |
---|
| 772 | for the elements}, At.Data Nucl.Data Tables 16,201 (1975); |
---|
| 773 | \bibitem{cinque} M.Born, \emph{Atomic physics}, Ed. Blackie and Sons (1969); |
---|
| 774 | \bibitem{sei} J.Bar\'o \emph{et al.}, \emph{Analytical cross sections |
---|
| 775 | for Monte Carlo simulation of photon transport}, Radiat.Phys.Chem. 44,531 |
---|
| 776 | (1994); |
---|
| 777 | \bibitem{sette} J.H.Hubbel \emph{et al.}, \emph{Atomic form factors, |
---|
| 778 | incoherent scattering functions and photon scattering cross sections}, J. |
---|
| 779 | Phys.Chem.Ref.Data 4,471 (1975). Erratum: \emph{ibid.} 6,615 (1977); |
---|
| 780 | \bibitem{otto} M.J.Berger and J.H.Hubbel, \emph{XCOM: photom cross sections |
---|
| 781 | on a personal computer}, Report NBSIR 87-3597 (National Bureau of Standards) |
---|
| 782 | (1987); |
---|
| 783 | \bibitem{nove} H.Davies \emph{et al.}, \emph{Theory of bremsstrahlung and |
---|
| 784 | pair production. II.Integral cross section for pair production}, Phys.Rev. |
---|
| 785 | 93,788 (1954); |
---|
| 786 | \bibitem{dieci} J.H.Hubbel \emph{et al.}, \emph{Pair, triplet and total |
---|
| 787 | atomic cross sections (and mass attenuation coefficients) for 1 MeV $-$ 100 |
---|
| 788 | GeV photons in element Z=1 to 100}, J.Phys.Chem.Ref.Data 9,1023 (1980); |
---|
| 789 | \bibitem{undici} J.W.Motz \emph{et al.}, \emph{Pair production by |
---|
| 790 | photons}, Rev.Mod.Phys 41,581 (1969); |
---|
| 791 | \bibitem{dodici} D.E.Cullen \emph{et al.}, \emph{Tables and graphs of |
---|
| 792 | photon-interaction cross sections from 10 eV to 100 GeV derived from the |
---|
| 793 | LLNL evaluated photon data library (EPDL)}, Report UCRL-50400 (Lawrence |
---|
| 794 | Livermore National Laboratory) (1989); |
---|
| 795 | \bibitem{dodicibis}, F. Sauter, Ann. Phys. 11 (1931) 454 |
---|
| 796 | \bibitem{tredici} S.M.Seltzer and M.J.Berger, \emph{Bremsstrahlung energy |
---|
| 797 | spectra from electrons with kinetic energy 1 keV - 100 GeV incident on |
---|
| 798 | screened nuclei and orbital electrons of neutral atoms with Z=1-100}, |
---|
| 799 | At.Data Nucl.Data Tables 35,345 (1986); |
---|
| 800 | \bibitem{quattordici} D.E.Cullen \emph{et al.}, \emph{Tables and graphs of |
---|
| 801 | electron-interaction cross sections from 10 eV to 100 GeV derived from the |
---|
| 802 | LLNL evaluated photon data library (EEDL)}, Report UCRL-50400 (Lawrence |
---|
| 803 | Livermore National Laboratory) (1989); |
---|
| 804 | \bibitem{quindici} L.Kissel \emph{et al.}, \emph{Shape functions for |
---|
| 805 | atomic-field bremsstrahlung from electron of kinetic energy 1$-$500 keV on |
---|
| 806 | selected neutral atoms $1 \le Z \le 92$}, At.Data Nucl.Data.Tab. 28,381 |
---|
| 807 | (1983); |
---|
| 808 | \bibitem{sedici} M.J.Berger and S.M.Seltzer, \emph{Stopping power of |
---|
| 809 | electrons and positrons}, Report NBSIR 82-2550 (National Bureau of |
---|
| 810 | Standards) (1982); |
---|
| 811 | \bibitem{diciassette} L.Kim \emph{et al.}, \emph{Ratio of positron to electron |
---|
| 812 | bremsstrahlung energy loss: an approximate scaling law}, Phys.Rev.A 33,3002 |
---|
| 813 | (1986); |
---|
| 814 | \bibitem{diciotto} U.Fano, \emph{Penetration of protons, alpha particles |
---|
| 815 | and mesons}, Ann.Rev.Nucl.Sci. 13,1 (1963); |
---|
| 816 | \bibitem{diciannove} W.Heitler, \emph{The quantum theory of radiation}, |
---|
| 817 | Oxford University Press, London (1954); |
---|
| 818 | \bibitem{venti} W.R.Nelson \emph{et al.}, \emph{The EGS4 code system}, |
---|
| 819 | Report SLAC-265 (1985). |
---|
| 820 | \end{thebibliography} |
---|
| 821 | |
---|
| 822 | \end{latexonly} |
---|
| 823 | |
---|
| 824 | \begin{htmlonly} |
---|
| 825 | |
---|
| 826 | \subsection{Bibliography} |
---|
| 827 | |
---|
| 828 | \begin{enumerate} |
---|
| 829 | \item \emph{Penelope - A Code System for Monte Carlo Simulation of |
---|
| 830 | Electron and Photon Transport}, Workshop Proceedings Issy-les-Moulineaux, |
---|
| 831 | France, 5$-$7 November 2001, AEN-NEA; |
---|
| 832 | \item J.Sempau \emph{et al.}, \emph{Experimental benchmarks of the |
---|
| 833 | Monte Carlo code PENELOPE}, submitted to NIM B (2002); |
---|
| 834 | \item D.Brusa \emph{et al.}, \emph{Fast sampling algorithm for the |
---|
| 835 | simulation of photon Compton scattering}, NIM A379,167 (1996); |
---|
| 836 | \item F.Biggs \emph{et al.}, \emph{Hartree-Fock Compton profiles |
---|
| 837 | for the elements}, At.Data Nucl.Data Tables 16,201 (1975); |
---|
| 838 | \item M.Born, \emph{Atomic physics}, Ed. Blackie and Sons (1969); |
---|
| 839 | \item J.Bar\'o \emph{et al.}, \emph{Analytical cross sections |
---|
| 840 | for Monte Carlo simulation of photon transport}, Radiat.Phys.Chem. 44,531 |
---|
| 841 | (1994); |
---|
| 842 | \item J.H.Hubbel \emph{et al.}, \emph{Atomic form factors, |
---|
| 843 | incoherent scattering functions and photon scattering cross sections}, J. |
---|
| 844 | Phys.Chem.Ref.Data 4,471 (1975). Erratum: \emph{ibid.} 6,615 (1977); |
---|
| 845 | \item M.J.Berger and J.H.Hubbel, \emph{XCOM: photom cross sections |
---|
| 846 | on a personal computer}, Report NBSIR 87-3597 (National Bureau of Standards) |
---|
| 847 | (1987); |
---|
| 848 | \item H.Davies \emph{et al.}, \emph{Theory of bremsstrahlung and |
---|
| 849 | pair production. II.Integral cross section for pair production}, Phys.Rev. |
---|
| 850 | 93,788 (1954); |
---|
| 851 | \item J.H.Hubbel \emph{et al.}, \emph{Pair, triplet and total |
---|
| 852 | atomic cross sections (and mass attenuation coefficients) for 1 MeV $-$ 100 |
---|
| 853 | GeV photons in element Z=1 to 100}, J.Phys.Chem.Ref.Data 9,1023 (1980); |
---|
| 854 | \item J.W.Motz \emph{et al.}, \emph{Pair production by |
---|
| 855 | photons}, Rev.Mod.Phys 41,581 (1969); |
---|
| 856 | \item D.E.Cullen \emph{et al.}, \emph{Tables and graphs of |
---|
| 857 | photon-interaction cross sections from 10 eV to 100 GeV derived from the |
---|
| 858 | LLNL evaluated photon data library (EPDL)}, Report UCRL-50400 (Lawrence |
---|
| 859 | Livermore National Laboratory) (1989); |
---|
| 860 | \item S.M.Seltzer and M.J.Berger, \emph{Bremsstrahlung energy |
---|
| 861 | spectra from electrons with kinetic energy 1 keV - 100 GeV incident on |
---|
| 862 | screened nuclei and orbital electrons of neutral atoms with Z=1-100}, |
---|
| 863 | At.Data Nucl.Data Tables 35,345 (1986); |
---|
| 864 | \item D.E.Cullen \emph{et al.}, \emph{Tables and graphs of |
---|
| 865 | electron-interaction cross sections from 10 eV to 100 GeV derived from the |
---|
| 866 | LLNL evaluated photon data library (EEDL)}, Report UCRL-50400 (Lawrence |
---|
| 867 | Livermore National Laboratory) (1989); |
---|
| 868 | \item L.Kissel \emph{et al.}, \emph{Shape functions for |
---|
| 869 | atomic-field bremsstrahlung from electron of kinetic energy 1$-$500 keV on |
---|
| 870 | selected neutral atoms $1 \le Z \le 92$}, At.Data Nucl.Data.Tab. 28,381 |
---|
| 871 | (1983); |
---|
| 872 | \item M.J.Berger and S.M.Seltzer, \emph{Stopping power of |
---|
| 873 | electrons and positrons}, Report NBSIR 82-2550 (National Bureau of |
---|
| 874 | Standards) (1982); |
---|
| 875 | \item L.Kim \emph{et al.}, \emph{Ratio of positron to electron |
---|
| 876 | bremsstrahlung energy loss: an approximate scaling law}, Phys.Rev.A 33,3002 |
---|
| 877 | (1986); |
---|
| 878 | \item U.Fano, \emph{Penetration of protons, alpha particles |
---|
| 879 | and mesons}, Ann.Rev.Nucl.Sci. 13,1 (1963); |
---|
| 880 | \item W.Heitler, \emph{The quantum theory of radiation}, |
---|
| 881 | Oxford University Press, London (1954); |
---|
| 882 | \item W.R.Nelson \emph{et al.}, \emph{The EGS4 code system}, |
---|
| 883 | Report SLAC-265 (1985). |
---|
| 884 | \end{enumerate} |
---|
| 885 | |
---|
| 886 | \end{htmlonly} |
---|
| 887 | |
---|