1 | |
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2 | \section[Single Scattering]{Single Scattering} |
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3 | |
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4 | Single elastic scattering process is an alternative to the |
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5 | multiple scattering process. The advantage of the single scattering process is |
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6 | in possibility of usage of theory based cross sections, in contrary to |
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7 | the Geant4 multiple scattering model \cite{singscat.urban}, which uses a number of |
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8 | phenomenological approximations on top of Lewis theory. |
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9 | The process $G4CoulombScattering$ was created |
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10 | for simulation of single scattering of muons, it also applicable with some physical |
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11 | limitations to electrons, muons and ions. |
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12 | Because each of elastic collisions are simulated the number of steps of charged |
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13 | particles significantly increasing in comparison with the multiple scattering |
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14 | approach, correspondingly its CPU performance is pure. However, in low-density media |
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15 | (vacuum, low-density gas) multiple scattering may provide wrong results and |
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16 | single scattering processes is more adequate. |
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17 | |
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18 | \subsection{Coulomb Scattering} |
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19 | |
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20 | The single scattering model of Wentzel \cite{singscat.wentzel} |
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21 | is used in many of multiple scattering models including Penelope code |
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22 | \cite{singscat.penelope}. The Wentzel for describing elastic |
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23 | scattering of particles with charge $ze$ ($z=-1$ for electron) |
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24 | by atomic nucleus with atomic number $Z$ |
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25 | based on simplified scattering potential |
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26 | \begin{equation} |
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27 | V(r) = \frac {zZe^2}{r}exp(-r/R), |
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28 | \label{singscat.a} |
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29 | \end{equation} |
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30 | where the exponential factor tries to reproduce the effect of screening. |
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31 | The parameter $R$ is a screening radius, which may be estimated from Thomas-Fermi model of |
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32 | the atom |
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33 | \begin{equation} |
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34 | R = 0.885 Z^{-1/3} r_B, |
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35 | \label{singscat.a1} |
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36 | \end{equation} |
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37 | where $r_B$ is the Bohr radius. In the first Born approximation the |
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38 | elastic scattering cross section $\sigma^(W)$ can be obtained as |
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39 | \begin{equation} |
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40 | \frac{d\sigma^{(W)}(\theta)}{d\Omega}= \frac{(zZe^2)^2}{(p\beta c)^2}\frac{1}{(2A + 1 - cos\theta)^2}, |
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41 | \label{singscat.a2} |
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42 | \end{equation} |
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43 | where $p$ is the momentum and $\beta$ is the velocity of the projectile particle. |
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44 | The screening parameter $A$ according to Moliere and Bethe \cite{singscat.bethe} |
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45 | \begin{equation} |
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46 | A = \left(\frac{\hslash}{2pR}\right)^2(1.13 + 3.76(\alpha Z/\beta)^2), |
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47 | \label{singscat.a3} |
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48 | \end{equation} |
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49 | where $\alpha$ is a fine structure constant and the factor in brackets |
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50 | is used to take into account second order corrections to |
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51 | the first Born approximation.\\ |
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52 | |
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53 | The total elastic cross section $\sigma$ can be expressed via Wentzel cross section |
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54 | (\ref{singscat.a2}) |
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55 | \begin{equation} |
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56 | \frac{d\sigma(\theta)}{d\Omega}= \frac{d\sigma^{(W)}(\theta)}{d\Omega}\left(\frac{1}{(1 + \frac{(qR_N)^2}{12})^2} + \frac{1}{Z}\right), |
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57 | \label{singscat.a4} |
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58 | \end{equation} |
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59 | where $q$ is momentum transfer to the nucleus, $R_N$ is nuclear radius. This term takes into |
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60 | account nuclear size effect \cite{singscat.kokoulin}, the second term takes into account scattering off |
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61 | electrons. The results of simulation with the single scattering model (Fig.\ref{plot:Alumin}) |
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62 | are competitive with the results of the multiple scattering. |
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63 | \begin{figure}[htbp] |
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64 | \center |
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65 | \includegraphics[scale=0.4]{electromagnetic/standard/al.eps} |
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66 | \caption{Scattering of muons off 1.5 mm aluminum foil: data \cite{singscat.attwood} - black squares; |
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67 | simulation - colored markers corresponding |
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68 | different options of multiple scattering and single scattering model; in the bottom |
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69 | plot - relative difference between the simulation and the data in percents; |
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70 | hashed area demonstrates one standard |
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71 | deviation of the data.} |
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72 | \label{plot:Alumin} |
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73 | \end{figure} |
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74 | \noindent |
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75 | |
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76 | \subsection{Implementation Details} |
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77 | |
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78 | The total cross section of the process is obtained as a result of integration |
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79 | of the differential cross section (\ref{singscat.a4}). The first term of this cross section |
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80 | is integrated in the interval $(0,\pi)$. The second term in the smaller interval $(0,\theta_m)$, |
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81 | where $\theta_m$ is the maximum scattering angle off electrons, which is determined using the cut value |
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82 | for the delta electron production. Before sampling of angular distribution the random |
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83 | choice is performed between scattering off the nucleus and off electrons. |
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84 | |
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85 | \subsection{Status of this document} |
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86 | 06.12.07 created by V. Ivanchenko \\ |
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87 | |
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88 | \begin{latexonly} |
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89 | |
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90 | \begin{thebibliography}{99} |
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91 | |
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92 | \bibitem{singscat.urban} L.~Urban, A multiple scattering model, |
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93 | {\em CERN-OPEN-2006-077, Dec 2006. 18 pp.} |
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94 | \bibitem{singscat.wentzel}G.~Wentzel, |
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95 | {\em Z. Phys. 40 (1927) 590.} |
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96 | \bibitem{singscat.bethe}H.A.~Bethe, |
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97 | {\em Phys. Rev. 89 (1953) 1256.} |
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98 | \bibitem{singscat.penelope}J.M.~Fernandez-Varea et al. |
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99 | {\em NIM B 73 (1993) 447.} |
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100 | \bibitem{singscat.kokoulin} A.V.~Butkevich et al., |
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101 | {\em NIM A 488 (2002) 282. } |
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102 | \bibitem{singscat.attwood} D.~Attwood et al. |
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103 | {\em NIM B 251 (2006) 41.} |
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104 | \end{thebibliography} |
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105 | |
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106 | \end{latexonly} |
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107 | |
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108 | \begin{htmlonly} |
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109 | |
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110 | \subsection{Bibliography} |
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111 | |
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112 | \begin{enumerate} |
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113 | |
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114 | \item L.~Urban, A multiple scattering model, |
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115 | {\em CERN-OPEN-2006-077, Dec 2006. 18 pp.} |
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116 | \item G.~Wentzel, |
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117 | {\em Z. Phys. 40 (1927) 590.} |
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118 | \item H.A.~Bethe, |
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119 | {\em Phys. Rev. 89 (1953) 1256.} |
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120 | \item J.M.~Fernandez-Varea et al. |
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121 | {\em NIM B 73 (1993) 447.} |
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122 | \item A.V.~Butkevich et al., |
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123 | {\em NIM A 488 (2002) 282. } |
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124 | \item D.~Attwood et al. |
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125 | {\em NIM B 251 (2006) 41.} |
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126 | \end{enumerate} |
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127 | |
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128 | \end{htmlonly} |
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129 | |
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130 | |
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131 | |
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