[1211] | 1 | |
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| 2 | \section[Multiple Scattering]{Multiple Scattering} |
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| 3 | |
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| 4 | Elastic scattering of electrons and other charged particles is an important |
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| 5 | component of any transport code. Elastic cross section is huge when particle |
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| 6 | energy decreases, so multiple scattering (MSC) approach should be introduced in order |
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| 7 | to have acceptable CPU performance of the simulation. |
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| 8 | Geant4 uses a universal interface {\it G4VMultipleScattering} |
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| 9 | is used by all MSC processes: |
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| 10 | \begin{itemize} |
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| 11 | \item |
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| 12 | G4eMultipleScattering; |
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| 13 | \item |
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| 14 | G4hMultipleScattering; |
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| 15 | \item |
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| 16 | G4MuMultipleScattering. |
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| 17 | \end{itemize} |
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| 18 | For concrete simulation the {\it G4VMscModel} interface is used, which is an extension |
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| 19 | of the base {\it G4VEmModel} interface. The following models are available: |
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| 20 | \begin{itemize} |
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| 21 | \item |
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[1222] | 22 | G4UrbanMscModel90 - Geant4 v9.0 applied to muons, hadrons and ions; |
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[1211] | 23 | \item |
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[1222] | 24 | G4UrbanMscModel92 - Geant4 v9.2 (current default) applied for electron and positron; |
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[1211] | 25 | \item |
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[1222] | 26 | G4UrbanMscModel93 - Geant4 v9.3 applied for electron and positron for Option2, Option3 and other EM Physics Lists; |
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[1211] | 27 | \item |
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[1222] | 28 | G4GoudsmitSaundersonModel - for electrons and positrons (beta-version); |
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| 29 | \item |
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| 30 | G4WentzelVIModel - for muons and hadrons, should be included in Physics Lists together with G4CoulombScattering process; |
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[1211] | 31 | \end{itemize} |
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[1222] | 32 | The last models are not yet in the production mode, so below we will concentrate |
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| 33 | on models developed by L.~Urban \cite{msc.urban}. |
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[1211] | 34 | |
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| 35 | \subsection{Introduction} |
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| 36 | |
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| 37 | MSC simulation algorithms can be classified as either {\em detailed} or |
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| 38 | {\em condensed}. In the detailed algorithms, all the collisions/interactions |
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| 39 | experienced by the particle are simulated. This simulation can be considered |
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| 40 | as exact; it gives the same results as the solution of the transport equation. |
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| 41 | However, it can be used only if the number of collisions is not too large, a |
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| 42 | condition fulfilled only for special geometries (such as thin foils), or low |
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| 43 | enough kinetic energies. For larger kinetic energies the average number of |
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| 44 | collisions is very large and the detailed simulation becomes very inefficient. |
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| 45 | High energy simulation codes use condensed simulation algorithms, in which |
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| 46 | the global effects of the collisions are simulated at the end of a track |
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| 47 | segment. The global effects generally computed in these codes are the net |
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| 48 | displacement, energy loss, and change of direction of the charged particle. |
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| 49 | These quantities are computed from the multiple scattering theories used in |
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| 50 | the codes. The accuracy of the condensed simulations is limited by the |
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| 51 | approximations of the multiple scattering theories. \\ |
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| 52 | |
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| 53 | \noindent |
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| 54 | Most particle physics simulation codes use the multiple scattering theories |
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| 55 | of Moli\`ere \cite{msc.moliere}, Goudsmit and Saunderson \cite{msc.goudsmit} |
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| 56 | and Lewis \cite{msc.lewis}. The theories of Moli\`ere and Goudsmit-Saunderson |
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| 57 | give only the angular distribution after a step, while the Lewis theory |
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| 58 | computes the moments of the spatial distribution as well. None of these |
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| 59 | MSC theories gives the probability distribution of the spatial displacement. |
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| 60 | Therefore each of the MSC simulation codes incorporates its own algorithm to |
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| 61 | determine the spatial displacement of the charged particle after a given step. |
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| 62 | These algorithms are not exact, of course, and are responsible for most of the |
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| 63 | uncertainties in the MSC codes. Therefore the simulation results can depend |
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| 64 | on the value of the step length and generally one has to select the value of |
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| 65 | the step length carefully. \\ |
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| 66 | |
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| 67 | \noindent |
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| 68 | A new class of MSC simulation, the {\em mixed} simulation algorithms (see |
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| 69 | e.g.\cite{msc.fernandez}), appeared in the literature recently. The mixed |
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| 70 | algorithm simulates the {\em hard} collisions one by one and uses a MSC theory to |
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| 71 | treat the effects of the {\em soft} collisions at the end of a given step. Such |
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| 72 | algorithms can prevent the number of steps from becoming too large and also |
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| 73 | reduce the dependence on the step length. \\ |
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| 74 | |
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| 75 | \noindent |
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| 76 | The MSC model used in Geant4 belongs to the class of condensed |
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| 77 | simulations. It uses model functions |
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| 78 | to determine the angular and spatial distributions after a step. The |
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| 79 | functions have been chosen in such a way as to give the same moments of the |
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| 80 | (angular and spatial) distributions as the Lewis theory \cite{msc.lewis}. |
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| 81 | |
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| 82 | \subsection{Definition of Terms} |
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| 83 | |
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| 84 | In simulation, a particle is transported by steps through the detector |
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| 85 | geometry. The shortest distance between the endpoints of a step is called |
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| 86 | the {\em geometrical path length}, $z$. |
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| 87 | In the absence of a magnetic field, this is a straight line. |
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| 88 | For non-zero fields, $z$ is the shortest distance along |
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| 89 | a curved trajectory. Constraints on $z$ are imposed when particle tracks |
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| 90 | cross volume boundaries. The path length of an actual particle, however, is |
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| 91 | usually longer than the geometrical path length, due to physical interactions |
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| 92 | like multiple scattering. |
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| 93 | This distance is called the {\em true path length}, $t$. |
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| 94 | Constraints on $t$ are imposed by the physical processes acting on the |
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| 95 | particle. \\ |
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| 96 | |
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| 97 | \noindent |
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| 98 | The properties of the multiple scattering process are completely determined |
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| 99 | by the {\em transport mean free paths}, $\lambda_k$, which are functions of the |
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| 100 | energy in a given material. The $k$-th transport mean free path is defined as |
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| 101 | |
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| 102 | \begin{equation} |
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| 103 | \frac {1}{\lambda_k} = 2 \pi n_a \int_{-1}^{1} \left[1 - P_k(cos\chi) \right] |
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| 104 | \frac{d\sigma(\chi)}{d\Omega} d(cos\chi) |
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| 105 | \label{msc.a1} |
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| 106 | \end{equation} |
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| 107 | |
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| 108 | \noindent |
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| 109 | where $d\sigma(\chi)/d\Omega$ is the differential cross section of the |
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| 110 | scattering, $P_k(cos\chi)$ is the $k$-th Legendre polynomial, and $n_a$ is the |
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| 111 | number of atoms per volume. \\ |
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| 112 | |
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| 113 | \noindent |
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| 114 | Most of the mean properties of MSC computed in the simulation codes depend |
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| 115 | only on the first and second transport mean free paths. The mean value of |
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| 116 | the geometrical path length (first moment) corresponding to a given true path |
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| 117 | length $t$ is given by |
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| 118 | \begin{equation} |
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| 119 | \langle z \rangle = |
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| 120 | \lambda_1 \left[ 1-\exp \left(-\frac{t}{\lambda_1} \right)\right] |
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| 121 | \label{msc.a} |
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| 122 | \end{equation} |
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| 123 | |
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| 124 | \noindent |
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| 125 | Eq.~\ref{msc.a} is an exact result for the mean value of $z$ if the |
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| 126 | differential cross section has axial symmetry and the energy loss can be |
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| 127 | neglected. The transformation between true and geometrical path lengths is |
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| 128 | called the {\em path length correction}. This formula and other expressions for |
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| 129 | the first moments of the spatial distribution were taken from either |
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| 130 | \cite{msc.fernandez} or \cite{msc.kawrakow}, but were originally |
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| 131 | calculated by Goudsmit and Saunderson \cite{msc.goudsmit} and Lewis |
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| 132 | \cite{msc.lewis}. \\ |
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| 133 | |
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| 134 | \noindent |
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| 135 | At the end of the true step length, $t$, the scattering angle is $\theta$. |
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| 136 | The mean value of $cos\theta$ is |
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| 137 | \begin{equation} |
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| 138 | \langle cos\theta \rangle = \exp \left[-\frac{t}{\lambda_1} \right] |
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| 139 | \label{msc.c} |
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| 140 | \end{equation} |
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| 141 | The variance of $cos\theta$ can be written as |
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| 142 | \begin{equation} |
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| 143 | \sigma^2 = \langle cos^2\theta \rangle - \langle cos\theta \rangle ^2 = |
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| 144 | \frac{1 + 2 e^{- 2 \kappa \tau}} {3} - e^{-2 \tau} |
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| 145 | \label{msc.c1} |
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| 146 | \end{equation} |
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| 147 | |
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| 148 | \noindent |
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| 149 | where $\tau = t/\lambda_1$ and $\kappa = \lambda_1/\lambda_2$. \\ |
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| 150 | |
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| 151 | \noindent |
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| 152 | The mean lateral displacement is given by a more complicated formula |
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| 153 | \cite{msc.fernandez}, but this quantity can also be calculated relatively |
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| 154 | easily and accurately. The square of the {\em mean lateral displacement} is |
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| 155 | \begin{equation} |
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| 156 | \langle x^2 + y^2 \rangle = \frac{4 \lambda_1^2}{3} \ \left[\tau |
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| 157 | - \frac{\kappa+1}{\kappa} |
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| 158 | + \frac{\kappa}{\kappa-1} e^{-\tau} - |
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| 159 | \frac{1}{\kappa (\kappa -1)} e^{-\kappa \tau} \right] |
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| 160 | \label{msc.e1} |
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| 161 | \end{equation} |
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| 162 | |
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| 163 | \noindent |
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| 164 | Here it is assumed that the initial particle direction is parallel to the |
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| 165 | the $z$ axis.\\ |
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| 166 | |
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| 167 | \noindent |
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| 168 | The lateral correlation is determined by the equation |
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| 169 | \begin{equation} |
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| 170 | \langle x v_x+y v_y \rangle = \frac{2 \lambda_1}{3} \ \left[1 |
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| 171 | - \frac{\kappa}{\kappa-1} e^{-\tau} + |
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| 172 | \frac{1}{\kappa-1} e^{-\kappa \tau} \right] |
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| 173 | \label{msc.e2} |
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| 174 | \end{equation} |
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| 175 | |
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| 176 | \noindent |
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| 177 | where $v_x$ and $v_y$ are the x and y components of the direction unit |
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| 178 | vector. |
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| 179 | This equation gives the correlation strength between the final lateral |
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| 180 | position and final direction.\\ |
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| 181 | |
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| 182 | \noindent |
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| 183 | The transport mean free path values have been calculated by Liljequist et al. |
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| 184 | \cite{msc.liljequist1}, \cite{msc.liljequist2} for electrons and positrons in |
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| 185 | the kinetic energy range \mbox{100 eV - 20 MeV} in 15 materials. The MSC |
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| 186 | model in Geant4 uses these values for kinetic energies below 10 MeV. |
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| 187 | For high energy particles (above 10 MeV) the transport mean free path |
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| 188 | values have been taken from a paper of R.Mayol and F.Salvat (\cite{msc.mayol}). |
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| 189 | When necessary, the model linearly interpolates |
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| 190 | or extrapolates the transport cross section, $\sigma_1 = 1 / \lambda_1$, in |
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| 191 | atomic number $Z$ and in the square of the particle velocity, $\beta^2$. |
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| 192 | The ratio $\kappa$ is a very slowly varying function of the energy: |
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| 193 | $\kappa > 2$ for $T >$ a few keV, and $\kappa \rightarrow 3$ for very high |
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| 194 | energies (see \cite{msc.kawrakow}). Hence, a constant value of 2.5 is used |
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| 195 | in the model.\\ |
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| 196 | |
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| 197 | \noindent |
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| 198 | Nuclear size effects are negligible for low energy particles and |
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| 199 | they are accounted for in the Born approximation in \cite{msc.mayol}, |
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| 200 | so there is no need for extra corrections of this kind in the model. |
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| 201 | |
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| 202 | \subsection{Path Length Correction} |
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| 203 | |
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| 204 | As mentioned above, the path length correction refers to the transformation |
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| 205 | true path length $\longrightarrow$ geometrical path length and its inverse. |
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| 206 | The true path length $\longrightarrow$ geometrical path length transformation |
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| 207 | is given by Eq.~\ref{msc.a} if the step is small and the energy loss can be |
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| 208 | neglected. If the step is not small the energy dependence makes the |
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| 209 | transformation more complicated. For this case Eqs. \ref{msc.c},\ref{msc.a} |
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| 210 | should be modified as |
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| 211 | \begin{equation} |
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| 212 | \langle cos\theta \rangle = \exp \left[-\int_0^t \frac{du}{\lambda_1 (u)} |
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| 213 | \right] |
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| 214 | \label{msc.ax} |
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| 215 | \end{equation} |
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| 216 | |
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| 217 | \begin{equation} |
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| 218 | \langle z \rangle = \int_0^t \langle cos\theta \rangle_u \ du |
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| 219 | \label{msc.bx} |
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| 220 | \end{equation} |
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| 221 | |
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| 222 | \noindent |
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| 223 | where $\theta$ is the scattering angle, $t$ and $z$ are the true and |
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| 224 | geometrical path lengths, and $\lambda_1$ is the transport mean free path. \\ |
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| 225 | |
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| 226 | \noindent |
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| 227 | In order to compute Eqs. \ref{msc.ax},\ref{msc.bx} the $t$ dependence of the |
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| 228 | transport mean free path must be known. $\lambda_1$ depends on the kinetic |
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| 229 | energy of the particle which decreases along the step. All computations in |
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| 230 | the model use a linear approximation for this $t$ dependence: |
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| 231 | \begin{equation} |
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| 232 | \lambda_1(t) = \lambda_{10} ( 1- \alpha t) \label{msc.cx} |
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| 233 | \end{equation} |
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| 234 | |
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| 235 | \noindent |
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| 236 | Here $\lambda_{10}$ denotes the value of $\lambda_1$ at the start of the step, |
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| 237 | and $\alpha $ is a constant. It is worth noting that Eq.~\ref{msc.cx} is |
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| 238 | \emph{not} |
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| 239 | a crude approximation. It is rather good at low ($ < $ 1 MeV) energy. At |
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| 240 | higher energies the step is generally much smaller than the range of the |
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| 241 | particle, so the change in energy is small and so is the change in $\lambda_1$. |
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| 242 | Using Eqs.~\ref{msc.ax} - \ref{msc.cx} the explicit formula for |
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| 243 | $\langle cos\theta \rangle$ and $\langle z \rangle$ are : |
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| 244 | \begin{equation} |
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| 245 | \langle cos\theta \rangle = |
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| 246 | \left(1 - \alpha t \right)^{\frac{1}{ \alpha \lambda_{10}}} |
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| 247 | \label{msc.ff} |
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| 248 | \end{equation} |
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| 249 | |
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| 250 | \begin{equation} |
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| 251 | \langle z \rangle = \frac{1} { \alpha (1 + \frac{1}{\alpha \lambda_{10}})} |
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| 252 | \left[ 1 - (1 - \alpha t)^{1+ \frac{1}{ \alpha \lambda_{10}}} \right] |
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| 253 | \label{msc.dx} |
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| 254 | \end{equation} |
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| 255 | |
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| 256 | \noindent |
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| 257 | The value of the constant $\alpha $ can be expressed using $\lambda_{10}$ and |
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| 258 | $\lambda_{11}$ where $\lambda_{11}$ is the value of the transport mean free |
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| 259 | path at the end of the step |
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| 260 | \begin{equation} |
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| 261 | \alpha = \frac{\lambda_{10} - \lambda_{11}} {t \lambda_{10}} |
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| 262 | \label{msc.ex} |
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| 263 | \end{equation} |
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| 264 | |
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| 265 | \noindent |
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| 266 | At low energies ( $T_{kin} < M$ , M - particle mass) $\alpha $ has a simpler |
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| 267 | form: |
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| 268 | \begin{equation} |
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| 269 | \alpha = \frac{1} { r_0} \label{msc.fx} |
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| 270 | \end{equation} |
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| 271 | where $r_0$ denotes the range of the particle at the start of the step. |
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| 272 | |
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| 273 | \noindent |
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| 274 | It can easily be seen that for a small step (i.e. for a step with small |
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| 275 | relative energy loss) the formula of $\langle z \rangle$ is |
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| 276 | \begin{equation} |
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| 277 | \langle z \rangle = |
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| 278 | \lambda_{10} \left[ 1-\exp{\left( -\frac{t}{\lambda_{10}}\right)}\right] |
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| 279 | \label{msc.gx} |
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| 280 | \end{equation} |
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| 281 | |
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| 282 | \noindent |
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| 283 | Eq. \ref{msc.dx} or \ref{msc.gx} gives the mean value of the geometrical step |
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| 284 | length for a given true step length. \\ |
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| 285 | |
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| 286 | \noindent |
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| 287 | The actual geometrical path length is sampled in the |
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| 288 | model according to the simple probability density function defined for |
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| 289 | $v = z/t \in [0 , 1]$ : |
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| 290 | \begin{equation} |
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| 291 | f(v) = (k+1)(k+2) v^k (1-v) |
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| 292 | \label{msc.d2} |
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| 293 | \end{equation} |
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| 294 | |
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| 295 | \noindent |
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| 296 | The value of the exponent $k$ is computed from the requirement that $f(v)$ |
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| 297 | must give the same mean value for $z = v t$ as Eq.~\ref{msc.dx} or |
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| 298 | \ref{msc.gx}. Hence |
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| 299 | \begin{equation} |
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| 300 | k = \frac{3 \langle z \rangle - t}{t - \langle z \rangle} |
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| 301 | \label{msc.d3} |
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| 302 | \end{equation} |
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| 303 | |
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| 304 | \noindent |
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| 305 | The value of $z = v t$ is sampled using $f(v)$ if $k > 0$, otherwise |
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| 306 | $z = \langle z \rangle$ is used. \\ |
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| 307 | |
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| 308 | \noindent |
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| 309 | The geometrical path length $\longrightarrow$ true path length transformation |
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| 310 | is performed using the mean values. The transformation can be written as |
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| 311 | \begin{equation} |
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| 312 | t(z) = \langle t \rangle = -\lambda_1 \log\left(1-\frac{z}{\lambda_1}\right) |
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| 313 | \label{msc.d4} |
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| 314 | \end{equation} |
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| 315 | if the geometrical step is small and |
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| 316 | \begin{equation} |
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| 317 | t(z) = \frac{1}{\alpha} \left[ 1 - (1 - \alpha w z)^{\frac{1}{w}} \right] |
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| 318 | \label{msc.hx} |
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| 319 | \end{equation} |
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| 320 | where $$w = 1 + \frac{1}{\alpha \lambda_{10}}$$ if the step is not small, i.e. |
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| 321 | the energy loss should be taken into account. \\ |
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| 322 | |
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| 323 | \noindent |
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| 324 | This transformation is needed when the particle arrives at a volume boundary, |
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| 325 | causing the step to be geometry-limited. In this case the true path length |
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| 326 | should be computed in order to have the correct energy loss of the particle |
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| 327 | after the step. |
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| 328 | |
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| 329 | \subsection{Angular Distribution} |
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| 330 | |
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| 331 | The quantity $u = cos\theta$ is sampled according to a model function |
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| 332 | $g(u)$. The shape of this function has been chosen such that |
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| 333 | Eqs. \ref{msc.c} and \ref{msc.c1} are satisfied. The functional form of $g$ is |
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| 334 | \begin{equation} |
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| 335 | g(u) = q [p g_1(u) + (1-p) g_2(u)] + (1-q) g_3(u) |
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| 336 | \label{msc.d} |
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| 337 | \end{equation} |
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| 338 | |
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| 339 | \noindent |
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| 340 | where $ 0 \leq p,q \leq 1 $, and the $g_i$ are simple functions of |
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| 341 | $u = cos\theta$, normalized over the range $ u \in [-1,\ 1] $. The functions |
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| 342 | $g_i$ have been chosen as |
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| 343 | \begin{equation} |
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| 344 | g_1(u) = C_{1}\hspace{3mm} e^{-a (1-u)} \hspace{2cm} -1 \leq u_0 \leq u \leq 1 |
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| 345 | \label{msc.d5} |
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| 346 | \end{equation} |
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| 347 | \begin{equation} |
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| 348 | g_2(u) = C_{2}\hspace{3mm} \frac{1} { (b-u)^d} |
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| 349 | \hspace{2cm} -1 \leq u \leq u_0 \leq 1 |
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| 350 | \label{msc.d6} |
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| 351 | \end{equation} |
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| 352 | \begin{equation} |
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| 353 | g_3(u) = C_{3} \hspace{4.8cm} -1 \leq u \leq 1 |
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| 354 | \label{msc.d7} |
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| 355 | \end{equation} |
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| 356 | where $a > 0$, $b > 0$, $d > 0$ and $u_0$ are model parameters, and the $C_{i}$ |
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| 357 | are normalization constants. It is worth noting that for small scattering |
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| 358 | angles, $\theta$, $g_1(u)$ is nearly Gaussian ($exp(-\theta^2/2 \theta_0^2)$) |
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| 359 | if $\theta_0^2 \approx 1 / a$, while $g_2(u)$ has a Rutherford-like tail for |
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| 360 | large $\theta$, if $b \approx 1$ and $d$ is not far from 2 . |
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| 361 | |
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| 362 | |
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| 363 | \subsection{Determination of the Model Parameters} |
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| 364 | |
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| 365 | The parameters $a$, $b$, $d$, $u_0$ and $p$, $q$ are not independent. The |
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| 366 | requirement that the angular distribution function $g(u)$ and its first |
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| 367 | derivative be continuous at $u = u_0$ imposes two constraints on the |
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| 368 | parameters: |
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| 369 | \begin{equation} |
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| 370 | p\hspace{1mm} g_1(u_0) = (1-p)\hspace{1mm} g_2(u_0) |
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| 371 | \label{msc.p1} |
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| 372 | \end{equation} |
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| 373 | \begin{equation} |
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| 374 | p\hspace{1mm} a\hspace{1mm} g_1(u_0) = (1-p)\hspace{1mm} |
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| 375 | \frac{d}{b-u_0}\hspace{1mm} g_2(u_0) |
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| 376 | \label{msc.p2} |
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| 377 | \end{equation} |
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| 378 | |
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| 379 | \noindent |
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| 380 | A third constraint comes from Eq. \ref{msc.ax} : $g(u)$ must give the same |
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| 381 | mean value for $u$ as the theory. \\ |
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| 382 | |
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| 383 | \noindent |
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| 384 | It follows from Eqs. \ref{msc.ff} and \ref{msc.d} that |
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| 385 | \begin{equation} |
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| 386 | q \{ p \langle u \rangle_1 + (1-p) \langle u \rangle_2 \} = |
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| 387 | [ 1 - \alpha \ t ]\ ^{\frac{1}{\alpha \lambda_{10}}} |
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| 388 | \label{msc.par1} |
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| 389 | \end{equation} |
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| 390 | where $\langle u \rangle_i$ denotes the mean value of $u$ computed from the |
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| 391 | distribution $g_i(u)$. \\ |
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| 392 | |
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| 393 | |
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| 394 | |
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| 395 | \noindent |
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| 396 | The parameter $a$ was chosen according to a modified Highland-Lynch-Dahl formula |
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| 397 | for the width of the angular distribution \cite{msc.highland}, |
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| 398 | \cite{msc.lynch}. |
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| 399 | \begin{equation} |
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| 400 | a = \frac {0.5} {1 -cos(\theta_0)} |
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| 401 | \end{equation} |
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| 402 | where $\theta_0$ is |
---|
| 403 | \begin{equation} |
---|
| 404 | \theta_0 = \frac {13.6 MeV}{ \beta c p} z_{ch} \sqrt{\frac{t}{X_0}} |
---|
| 405 | \ \left[ 1 + h_c \ln \left(\frac{t}{X_0} \right)\ \right] |
---|
| 406 | \end{equation} |
---|
| 407 | |
---|
| 408 | \noindent |
---|
| 409 | when the original Highland-Lynch-Dahl formula is used. |
---|
| 410 | Here $\theta_0 = \theta^{rms}_{plane}$ is the width of the approximate |
---|
| 411 | Gaussian projected angle distribution, $p$, $\beta c$ and $z_{ch}$ are the |
---|
| 412 | momentum, |
---|
| 413 | velocity and charge number of the incident particle, and $t/X_0$ is the |
---|
| 414 | true path length in radiation length unit. The correction term $h_c$ = 0.038 |
---|
| 415 | in the formula. This value of |
---|
| 416 | $\theta_0$ is from a fit to the Moli\`ere distribution for singly charged |
---|
| 417 | particles with $\beta = 1$ for all Z, and is accurate to 11 $\%$ or better |
---|
| 418 | for $ 10^{-3} \leq t/X_0 \leq 100$ (see e.g. Rev. of Particle Properties, |
---|
| 419 | section 23.3). \\ |
---|
| 420 | \noindent |
---|
| 421 | The model uses a slightly modified Highland-Lynch-Dahl formula to |
---|
| 422 | compute $\theta_0$. |
---|
| 423 | |
---|
| 424 | For electrons/positrons the modified $\theta_0$ formula is (see G4UrbanMscModel2) |
---|
| 425 | \begin{equation} |
---|
| 426 | \theta_0 = \frac {13.6 MeV}{\beta c p} z_{ch} \sqrt{y } c |
---|
| 427 | \end{equation} |
---|
| 428 | where |
---|
| 429 | |
---|
| 430 | \begin{equation} |
---|
| 431 | y = \ln \left(\frac{t}{X_0}\right) |
---|
| 432 | \end{equation} |
---|
| 433 | |
---|
| 434 | The correction term $c$ is |
---|
| 435 | |
---|
| 436 | \begin{equation} |
---|
| 437 | c = c_1 + c_2 y |
---|
| 438 | \end{equation} |
---|
| 439 | for $y > -6.5$ and |
---|
| 440 | \begin{equation} |
---|
| 441 | c = c_1 + c_2 y - 0.011 (6.5 + y) |
---|
| 442 | \end{equation} |
---|
| 443 | for $y <= 6.5$. The coefficients $c_i$ are |
---|
| 444 | |
---|
| 445 | \begin{equation} |
---|
| 446 | c_1 = 0.885 + ln(Z) \left[ 0.104-0.0177 ln(Z) \right] |
---|
| 447 | \end{equation} |
---|
| 448 | |
---|
| 449 | \begin{equation} |
---|
| 450 | c_2 = 0.028 + ln(Z) \left[ 0.012-0.00125 ln(Z) \right] |
---|
| 451 | \end{equation} |
---|
| 452 | |
---|
| 453 | In the case of heavy charged particles (muons,hadrons) |
---|
| 454 | the modified formula for $\theta_0$ is (see G4UrbanMscModel90) |
---|
| 455 | |
---|
| 456 | \begin{equation} |
---|
| 457 | \theta_0 = \frac {13.6 MeV}{\beta c p} z_{ch} \sqrt{\frac{t}{X_0} } |
---|
| 458 | \left[ 1 + 0.105 \ln \left(\frac{t}{X_0}\right) |
---|
| 459 | + 0.0035 \left(\ln \left(\frac{t}{X_0}\right)\right)^2 |
---|
| 460 | \right] ^{\frac{1}{2}} f \left(Z \right) |
---|
| 461 | \end{equation} |
---|
| 462 | |
---|
| 463 | \noindent |
---|
| 464 | where |
---|
| 465 | \begin{equation} |
---|
| 466 | f \left(Z \right) = 1 - \frac{0.24}{ Z \left(Z + 1 \right) } |
---|
| 467 | \end{equation} |
---|
| 468 | \noindent |
---|
| 469 | is an empirical correction factor based on high energy proton scattering |
---|
| 470 | data \cite{msc.shen}. |
---|
| 471 | This formula gives a much smaller step dependence in the angular |
---|
| 472 | distribution than the Highland form. \\ |
---|
| 473 | |
---|
| 474 | \noindent |
---|
| 475 | The value of the parameter $u_0$ has been chosen as |
---|
| 476 | \begin{equation} |
---|
| 477 | u_0 \hspace{4mm} = \hspace{4mm} 1 - \frac{\xi}{a} |
---|
| 478 | \end{equation} |
---|
| 479 | where $\xi$ is a constant ($\xi = 3$). \\ |
---|
| 480 | |
---|
| 481 | For electrons/positrons the parameter $d$ is set to |
---|
| 482 | \begin{equation} |
---|
| 483 | d \hspace{4mm} = \hspace{4mm} d_1 |
---|
| 484 | \end{equation} |
---|
| 485 | for $y >= -13.5$ and |
---|
| 486 | \begin{equation} |
---|
| 487 | d \hspace{4mm} = \hspace{4mm} d_1 + d_2 (y+13.5)^3 |
---|
| 488 | \end{equation} |
---|
| 489 | for $y < 13.5$ |
---|
| 490 | where |
---|
| 491 | \begin{equation} |
---|
| 492 | d_1 \hspace{4mm} = \hspace{2mm} 2.134-\ln(Z) (0.1045-0.00602 ln(Z)) |
---|
| 493 | \end{equation} |
---|
| 494 | \begin{equation} |
---|
| 495 | d_2 \hspace{4mm} = \hspace{2mm} 0.001126-\ln(Z) (0.0001089+0.0000247 \ln(Z)) |
---|
| 496 | \end{equation} |
---|
| 497 | |
---|
| 498 | For heavy particle the parameter $d$ is set to |
---|
| 499 | \begin{equation} |
---|
| 500 | d \hspace{4mm} = \hspace{4mm} 2.40 - 0.027 \ Z^ \frac{2}{3} |
---|
| 501 | \end{equation} |
---|
| 502 | This (empirical) expression is obtained comparing the simulation |
---|
| 503 | results to the data of the MuScat experiment \cite{msc.attwood}. \\ |
---|
| 504 | |
---|
| 505 | \noindent |
---|
| 506 | The remaining three parameters can be computed from |
---|
| 507 | Eqs. \ref{msc.p1} - \ref{msc.par1}. |
---|
| 508 | The numerical value of the parameters can be found in the |
---|
| 509 | code. \\ |
---|
| 510 | |
---|
| 511 | |
---|
| 512 | \noindent |
---|
| 513 | It should be noted that in this model there is no step limitation |
---|
| 514 | originating from the multiple scattering process. Another important feature |
---|
| 515 | of this model is that the sum of the 'true' step lengths of the particle, that |
---|
| 516 | is, the total true path length, does not depend on the length of the steps. |
---|
| 517 | Most algorithms used in simulations do not have these properties. \\ |
---|
| 518 | |
---|
| 519 | \noindent |
---|
| 520 | In the case of heavy charged particles ($\mu$, $\pi$, $p$, etc.) the mean |
---|
| 521 | transport free path is calculated from the electron or positron $\lambda_1$ |
---|
| 522 | values with a 'scaling' applied. This is possible because the transport |
---|
| 523 | mean free path $\lambda_1$ depends only on the variable $P \beta c$, where |
---|
| 524 | $P$ is the momentum, and $\beta c$ is the velocity of the particle. \\ |
---|
| 525 | |
---|
| 526 | \noindent |
---|
| 527 | In its present form the model samples the path length correction and angular |
---|
| 528 | distribution from model functions, while for the lateral displacement and |
---|
| 529 | the lateral correlation only |
---|
| 530 | the mean values are used and all the other correlations are neglected. |
---|
| 531 | However, the model |
---|
| 532 | is general enough to incorporate other random quantities and correlations in |
---|
| 533 | the future. |
---|
| 534 | |
---|
| 535 | \subsection{The MSC Process in Geant4} |
---|
| 536 | |
---|
| 537 | The step length of the particles is determined by the physics processes or |
---|
| 538 | the geometry of the detectors. The tracking/stepping algorithm checks all the |
---|
| 539 | step lengths demanded by the (continuous or discrete) physics processes and |
---|
| 540 | determines the minimum of these step lengths. \\ |
---|
| 541 | |
---|
| 542 | \noindent |
---|
| 543 | Then, this minimum step length |
---|
| 544 | must be compared with the length determined by the geometry of the detectors |
---|
| 545 | and one has to select the minimum of the 'physics step length' and the |
---|
| 546 | 'geometrical step length' as the actual step length. \\ |
---|
| 547 | |
---|
| 548 | \noindent |
---|
| 549 | This is the point where the MSC model comes into the game. All the |
---|
| 550 | physics processes use the true path length $t$ to sample the interaction point, |
---|
| 551 | while the step limitation originated from the geometry is a |
---|
| 552 | geometrical path length $z$. The MSC algorithm transforms the 'physics step |
---|
| 553 | length' into a 'geometrical step length' before the comparison of the two |
---|
| 554 | lengths. This 't'\(\rightarrow\)'z' transformation can be called the |
---|
| 555 | inverse of the path length correction. \\ |
---|
| 556 | |
---|
| 557 | \noindent |
---|
| 558 | After the actual step length has been determined and the particle relocation |
---|
| 559 | has been performed the MSC performs the transformation 'z'\(\rightarrow\)'t', |
---|
| 560 | because the energy loss and scattering computation need the true step length |
---|
| 561 | 't'. \\ |
---|
| 562 | |
---|
| 563 | \noindent |
---|
| 564 | The scattering angle $\theta$ of the particle after the step of length 't' is |
---|
| 565 | sampled according to the model function given in Eq.~\ref{msc.d} . |
---|
| 566 | The azimuthal angle $\phi$ is generated uniformly in the range $[0, 2 \pi]$. \\ |
---|
| 567 | |
---|
| 568 | \noindent |
---|
| 569 | After the simulation of the scattering angle, the lateral displacement is |
---|
| 570 | computed using Eq.~\ref{msc.e1}. Then the correlation given by Eq.~\ref{msc.e2} |
---|
| 571 | is used to determine the direction of the lateral displacement. |
---|
| 572 | Before 'moving' the particle according to the displacement a check is performed |
---|
| 573 | to ensure that the relocation of the particle with the lateral displacement |
---|
| 574 | does not take the particle beyond the volume boundary. |
---|
| 575 | |
---|
| 576 | \subsection{Step Limitation Algorithm} |
---|
| 577 | |
---|
| 578 | In Geant4 the boundary crossing is treated by the transportation process. |
---|
| 579 | The transportation ensures that the |
---|
| 580 | particle does not penetrate in a new volume without stopping at the boundary, |
---|
| 581 | it restricts the step size when the particle leaves a volume. However, |
---|
| 582 | this step restriction can be rather weak in big volumes and this fact |
---|
| 583 | can result a not very good angular distribution after the volume. |
---|
| 584 | At the same time, there is no similar |
---|
| 585 | step limitation when a particle enters a volume and this fact does not allow |
---|
| 586 | a good backscattering simulation for low energy particles. Low energy particles |
---|
| 587 | penetrate too deeply into the volume in the first step and then - because |
---|
| 588 | of energy loss - they are not able to reach again the boundary in backward |
---|
| 589 | direction.\\ |
---|
| 590 | |
---|
| 591 | \noindent |
---|
| 592 | A very simple step limitation algorithm has been implemented in the |
---|
| 593 | MSC code to cure this situation. At the start of a track or after |
---|
| 594 | entering in a new volume, the algorithm restricts |
---|
| 595 | the step size to a value |
---|
| 596 | \begin{equation} |
---|
| 597 | f_r \cdot max\{r,\lambda_1\} |
---|
| 598 | \end{equation} |
---|
| 599 | where $r$ is the range of the particle, $f_r$ is a parameter $\in [0, 1]$, |
---|
| 600 | taking the max of $r$ and $\lambda_1$ is an empirical choice.The value of $f_r$ |
---|
| 601 | is constant for low energy particles while for particles with $\lambda_1 > \lambda_{lim}$ |
---|
| 602 | an effective value is used given by the scaling equation |
---|
| 603 | \begin{equation} |
---|
| 604 | f_{reff} = f_r \cdot \left[ 1 - sc + sc * \frac{\lambda_1}{\lambda_{lim}} \right] |
---|
| 605 | \end{equation} |
---|
| 606 | ( The numerical values $sc = 0.25$ and $\lambda_{lim} = 1 \hspace{2mm} mm$ are used in the equation.) |
---|
| 607 | In order not to |
---|
| 608 | use very small - unphysical - step sizes a lower limit is given for the step |
---|
| 609 | size as |
---|
| 610 | \begin{equation} |
---|
| 611 | tlimitmin = max\left[ \frac{\lambda_1}{nstepmax}, \lambda_{elastic} \right] |
---|
| 612 | \end{equation} |
---|
| 613 | with $nstepmax = 25$ and $\lambda_{elastic}$ is the elastic mean free path |
---|
| 614 | of the particle (see later).\\ |
---|
| 615 | |
---|
| 616 | \noindent |
---|
| 617 | It can be easily seen that this kind of step limitation poses a real constraint |
---|
| 618 | only for low energy particles. \\ |
---|
| 619 | |
---|
| 620 | \noindent |
---|
| 621 | In order to prevent a particle from crossing |
---|
| 622 | a volume in just one step, an additional limitation is imposed: |
---|
| 623 | after entering a volume |
---|
| 624 | the step size cannot be bigger than |
---|
| 625 | \begin{equation} |
---|
| 626 | \frac {d_{geom}}{f_g} |
---|
| 627 | \end{equation} |
---|
| 628 | where $d_{geom}$ is the distance to the next boundary (in the direction |
---|
| 629 | of the particle) and $f_g$ is a constant parameter. A similar restriction |
---|
| 630 | at the start of a track is |
---|
| 631 | \begin{equation} |
---|
| 632 | \frac {2 d_{geom}}{f_g} |
---|
| 633 | \end{equation} |
---|
| 634 | |
---|
| 635 | \noindent |
---|
| 636 | The choice of the parameters $f_r$ and $f_g$ is also |
---|
| 637 | related to performance. By default $f_r = 0.02$ and $f_g = 2.5$ |
---|
| 638 | are used, but these may |
---|
| 639 | be set to any other value in a simple way. One can get an |
---|
| 640 | approximate simulation of the backscattering with the default value, while |
---|
| 641 | if a better backscattering simulation is needed it is possible to get it |
---|
| 642 | using a smaller value for $f_r$. However, this model is very simple and |
---|
| 643 | it can only approximately reproduce the backscattering data. |
---|
| 644 | |
---|
| 645 | \subsection{Boundary Crossing Algorithm} |
---|
| 646 | |
---|
| 647 | A special stepping algorithm has been implemented recently (Autumn 2006) |
---|
| 648 | in order to improve the simulation around interfaces. |
---|
| 649 | This algorithm does not allow 'big' last steps in a volume and 'big' first steps |
---|
| 650 | in the next volume. The step length of these steps around a boundary crossing |
---|
| 651 | can not be bigger than the mean free path of the elastic scattering |
---|
| 652 | of the particle in the given volume (material). After these small steps the particle |
---|
| 653 | scattered according to a single scattering law (i.e. there is no multiple scattering |
---|
| 654 | very close to the boundary or at the boundary). \\ |
---|
| 655 | |
---|
| 656 | \noindent |
---|
| 657 | The key parameter of the algorithm is the variable called $skin$. The algorithm is |
---|
| 658 | not active for $skin \leq 0$, while for $skin > 0$ it is active in layers |
---|
| 659 | of thickness $skin \cdot \lambda_{elastic}$ before boundary crossing |
---|
| 660 | and of thickness $(skin-1) \cdot \lambda_{elastic}$ after |
---|
| 661 | boundary crossing (for $skin=1$ there is only one small step just before |
---|
| 662 | the boundary). In this active area the particle |
---|
| 663 | performs steps of length $\lambda_{elastic}$ (or smaller if the particle reaches the |
---|
| 664 | boundary traversing a smaller distance than this value). \\ |
---|
| 665 | |
---|
| 666 | \noindent |
---|
| 667 | The scattering at the end of a small step is single or |
---|
| 668 | plural and for these small steps there are no path length correction |
---|
| 669 | and lateral displacement computation. In other words the program works in this |
---|
| 670 | thin layer in 'microscopic mode'. \\ |
---|
| 671 | |
---|
| 672 | \noindent |
---|
| 673 | The elastic mean free path can be estimated as |
---|
| 674 | \begin{equation} |
---|
| 675 | \lambda_{elastic} = \lambda_1 \cdot rat \left( T_{kin} \right) |
---|
| 676 | \end{equation} |
---|
| 677 | where $rat(T_{kin})$ a simple empirical function computed from the elastic and |
---|
| 678 | first transport cross section values of Mayol and Salvat \cite{msc.mayol} |
---|
| 679 | |
---|
| 680 | \begin{equation} |
---|
| 681 | rat \left( T_{kin} \right) = \frac{0.001 (MeV)^2} |
---|
| 682 | { T_{kin} \left(T_{kin} + 10 MeV \right)} |
---|
| 683 | \end{equation} |
---|
| 684 | $T_{kin}$ is the kinetic energy of the particle. \\ |
---|
| 685 | |
---|
| 686 | \noindent |
---|
| 687 | At the end of a small step the number of scatterings is sampled according to |
---|
| 688 | the Poisson's distribution with a mean value $t/\lambda_{elastic}$ and in the |
---|
| 689 | case of plural scattering the final scattering angle is computed by summing |
---|
| 690 | the contributions of the individual scatterings. \\ |
---|
| 691 | \noindent |
---|
| 692 | The single scattering is determined by the distribution |
---|
| 693 | \begin{equation} |
---|
| 694 | g(u) = C \frac{1} {(2 a + 1 -u)^2} |
---|
| 695 | \end{equation} |
---|
| 696 | where $u = \cos(\theta)$ , $a$ is the screening parameter, |
---|
| 697 | $C$ is a normalization constant. The form of the screening parameter is |
---|
| 698 | the same as in the single scattering (see there). |
---|
| 699 | |
---|
| 700 | |
---|
| 701 | \subsection{Implementation Details} |
---|
| 702 | |
---|
| 703 | The concrete implementation of described algorithms of simulation is |
---|
| 704 | provided in {\it G4UrbanMscModel2, G4UrbanMscModel90}. |
---|
| 705 | Because multiple scattering is very similar for different particles the base |
---|
| 706 | class {\it G4VMultipleScattering} was created to collect and provide significant |
---|
| 707 | features of the calculations which are common to different particle types. \\ |
---|
| 708 | |
---|
| 709 | \noindent |
---|
| 710 | In the {\tt AlongStepGetPhysicalInteractionLength} method the minimum step |
---|
| 711 | size due to the physics processes is compared with the step size constraints |
---|
| 712 | imposed by the transportation process and the geometry. In order to do this, |
---|
| 713 | the \mbox{'t' step $\rightarrow$ 'z' step} transformation must be performed. |
---|
| 714 | Therefore, the method should be invoked after the |
---|
| 715 | {\tt GetPhysicalInteractionLength} methods of other physics processes, |
---|
| 716 | but before the same method of the transportation process. |
---|
| 717 | The reason for this ordering is that the physics processes 'feel' |
---|
| 718 | the true path length $t$ traveled by the particle, while the transportation |
---|
| 719 | process (geometry) uses the $z$ step length.\\ |
---|
| 720 | |
---|
| 721 | \noindent |
---|
| 722 | At this point the program also checks whether the particle has entered a |
---|
| 723 | new volume. If it has, the particle steps cannot be bigger than |
---|
| 724 | $t_{lim} = f_r\hspace{2mm} max( r, \lambda )$. This step limitation is |
---|
| 725 | governed by the physics, because $t_{lim}$ depends on the particle energy |
---|
| 726 | and the material. \\ |
---|
| 727 | |
---|
| 728 | \noindent |
---|
| 729 | The {\tt PostStepGetPhysicalInteractionLength} method of the multiple |
---|
| 730 | scattering process simply sets the force flag to 'Forced' in order to |
---|
| 731 | ensure that {\tt PostStepDoIt} is called at every step. It also returns a |
---|
| 732 | large value for the interaction length so that there is no step limitation |
---|
| 733 | at this level. \\ |
---|
| 734 | |
---|
| 735 | \noindent |
---|
| 736 | The {\tt AlongStepDoIt} function of the process performs the inverse,\linebreak |
---|
| 737 | \mbox{'z' $\rightarrow$ 't'} transformation. This function should be |
---|
| 738 | invoked after the \linebreak {\tt AlongStepDoIt} method of the transportation |
---|
| 739 | process, that is, after the particle relocation is determined by the |
---|
| 740 | geometrical step length, but before applying any other physics |
---|
| 741 | {\tt AlongStepDoIt}. \\ |
---|
| 742 | |
---|
| 743 | \noindent |
---|
| 744 | The {\tt PostStepDoIt} method of the process samples the scattering angle |
---|
| 745 | and performs the lateral displacement when the particle is not near a boundary. |
---|
| 746 | |
---|
| 747 | Default MSC parameter values optimized per particle type are shown in Table \ref{msc.t}. |
---|
| 748 | Note, that there is three types of step limitation by multiple scattering process: |
---|
| 749 | \begin{itemize} |
---|
| 750 | \item |
---|
| 751 | {\tt Minimal} - only $f_r$ parameter is used, was used for g4 7.1 release; |
---|
| 752 | \item |
---|
| 753 | {\tt UseSafety} or $skin=0$ - uses particle range and |
---|
| 754 | geometrical safety; |
---|
| 755 | \item |
---|
| 756 | {\tt UseDistanceToBoundary} - uses particle range, geometrical safety and linear distance to |
---|
| 757 | geometrical boundary. |
---|
| 758 | \end{itemize} |
---|
| 759 | \begin{table}[hbt] |
---|
| 760 | \begin{centering} |
---|
| 761 | \begin{tabular}{|c|c|c|c|} |
---|
| 762 | \hline |
---|
| 763 | particle & $e^+$, $e^-$ & {\it muons, hadrons} & {\it ions}\\ \hline |
---|
| 764 | {\it StepLimitType} & {\it fUseSafety} & {\it fMinimal} & {\it fMinimal}\\ \hline |
---|
| 765 | {\it skin} & 0 & 0 & 0 \\ \hline |
---|
| 766 | $f_r$ & 0.04 & 0.2 & 0.2\\ \hline |
---|
| 767 | $f_g$ & 2.5 & 0.1 & 0.1\\ \hline |
---|
| 768 | {\it LateralDisplacement} & {\it true} & {\it true}& {\it false}\\ \hline |
---|
| 769 | \end{tabular} |
---|
| 770 | \caption{The default values of parameters for different particle type.} |
---|
| 771 | \label{msc.t} |
---|
| 772 | \end{centering} |
---|
| 773 | \end{table} |
---|
| 774 | The parameters of the model can be changed via public functions of the base |
---|
| 775 | class {\it G4VMultipleSacttering}. They can be changed for all |
---|
| 776 | multiple scattering processes simultaneously via {\it G4EmProcessOptions} class or |
---|
| 777 | via Geant4 UI commands. The following commands are available:\\ |
---|
| 778 | \\ |
---|
| 779 | {\it /process/msc/StepLimit UseDistanceToBoundary\\ |
---|
| 780 | /process/msc/LateralDisplacement false\\ |
---|
| 781 | /process/msc/RangeFactor 0.02\\ |
---|
| 782 | /process/msc/GeomFactor 2.5\\ |
---|
| 783 | /process/msc/Skin 2} |
---|
| 784 | \\ |
---|
| 785 | |
---|
| 786 | |
---|
| 787 | \subsection{Status of this document} |
---|
| 788 | 09.10.98 created by L. Urb\'an. \\ |
---|
| 789 | 15.11.01 major revision by L. Urb\'an.\\ |
---|
| 790 | 18.04.02 updated by L. Urb\'an. \\ |
---|
| 791 | 25.04.02 re-worded by D.H. Wright \\ |
---|
| 792 | 07.06.02 major revision by L. Urb\'an. \\ |
---|
| 793 | 18.11.02 updated by L. Urb\'an, now it describes the new angle distribution. \\ |
---|
| 794 | 05.12.02 grammar check and parts re-written by D.H. Wright \\ |
---|
| 795 | 13.11.03 revision by L. Urb\'an. \\ |
---|
| 796 | 01.12.03 revision by V. Ivanchenko. \\ |
---|
| 797 | 17.05.04 revision by L.Urb\'an. \\ |
---|
| 798 | 01.12.04 updated by L.Urb\'an. \\ |
---|
| 799 | 18.03.05 sampling z + mistyping corrections (mma) \\ |
---|
| 800 | 22.06.05 grammar, spelling check by D.H. Wright \\ |
---|
| 801 | 12.12.05 revised by L. Urb\'an, according to Geant4 V8.0 \\ |
---|
| 802 | 14.12.05 updated implementation Details (mma) \\ |
---|
| 803 | 08.06.06 revised by L. Urb\'an, according to Geant4 V8.1 \\ |
---|
| 804 | 25.11.06 revised by L. Urb\'an, according to Geant4 V8.2 \\ |
---|
| 805 | 29.03.07 revised by L. Urb\'an, for Geant4 V8.3 \\ |
---|
| 806 | 13.06.07 modified introduction (mma) \\ |
---|
| 807 | 17.06.07 explain effective $F_R$ (L. Urb\'an) \\ |
---|
| 808 | 25.06.07 update description of options by V. Ivanchenko \\ |
---|
| 809 | 05.12.07 revised by L. Urb\'an, for Geant4 V9.1 \\ |
---|
| 810 | 08.12.08 revised by L. Urb\'an, for Geant4 V9.2 \\ |
---|
| 811 | 11.12.08 minor revision by V. Ivanchenko \\ |
---|
[1222] | 812 | 11.12.09 minor revision by V. Ivanchenko, for Geant4 v9.3 \\ |
---|
[1211] | 813 | |
---|
| 814 | \begin{latexonly} |
---|
| 815 | |
---|
| 816 | \begin{thebibliography}{99} |
---|
| 817 | |
---|
| 818 | \bibitem{msc.urban} L.~Urban, A multiple scattering model, |
---|
| 819 | {\em CERN-OPEN-2006-077, Dec 2006. 18 pp.} |
---|
| 820 | \bibitem{msc.moliere} G.Z.~Moli\`ere |
---|
| 821 | {\em Z. Naturforsch. 3a (1948) 78. } |
---|
| 822 | \bibitem{msc.lewis} H.W.~Lewis. |
---|
| 823 | {\em Phys. Rev. 78 (1950) 526. } |
---|
| 824 | \bibitem{msc.goudsmit}S.~Goudsmit and J.L.~Saunderson. |
---|
| 825 | {\em Phys. Rev. 57 (1940) 24. } |
---|
| 826 | \bibitem{msc.fernandez}J.M.~Fernandez-Varea et al. |
---|
| 827 | {\em NIM B73 (1993) 447.} |
---|
| 828 | \bibitem{msc.kawrakow} I.~Kawrakow and A.F.~Bielajew |
---|
| 829 | {\em NIM B 142 (1998) 253. } |
---|
| 830 | \bibitem{msc.liljequist1} D.~Liljequist and M.~Ismail. |
---|
| 831 | {\em J.Appl.Phys. 62 (1987) 342. } |
---|
| 832 | \bibitem{msc.liljequist2} D.~Liljequist et al. |
---|
| 833 | {\em J.Appl.Phys. 68 (1990) 3061. } |
---|
| 834 | \bibitem{msc.mayol} R.~Mayol and F.~Salvat |
---|
| 835 | {\em At.Data and Nucl.Data Tables 65 (1997) 55.}. |
---|
| 836 | \bibitem{msc.highland} V.L.~Highland |
---|
| 837 | {\em NIM 129 (1975) 497. } |
---|
| 838 | \bibitem{msc.lynch} G.R.~Lynch and O.I.~Dahl |
---|
| 839 | {\em NIM B58 (1991) 6. } |
---|
| 840 | \bibitem{msc.shen} G.~Shen et al. |
---|
| 841 | {\em Phys. Rev. D 20 (1979) 1584.} |
---|
| 842 | \bibitem{msc.attwood} D.~Attwood et al. |
---|
| 843 | {\em NIM B 251 (2006) 41.} |
---|
| 844 | \end{thebibliography} |
---|
| 845 | |
---|
| 846 | \end{latexonly} |
---|
| 847 | |
---|
| 848 | \begin{htmlonly} |
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| 849 | |
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| 850 | \subsection{Bibliography} |
---|
| 851 | |
---|
| 852 | \begin{enumerate} |
---|
| 853 | |
---|
| 854 | \item L.~Urban, A multiple scattering model, |
---|
| 855 | {\em CERN-OPEN-2006-077, Dec 2006. 18 pp.} |
---|
| 856 | \item G. Z. Moli\`ere |
---|
| 857 | {\em Z. Naturforsch. 3a (1948) 78. } |
---|
| 858 | \item H. W. Lewis |
---|
| 859 | {\em Phys. Rev. 78 (1950) 526. } |
---|
| 860 | \item S. Goudsmit and J. L. Saunderson. |
---|
| 861 | {\em Phys. Rev. 57 (1940) 24. } |
---|
| 862 | \item J. M. Fernandez-Varea et al. |
---|
| 863 | {\em NIM B73 (1993) 447.} |
---|
| 864 | \item I. Kawrakow and Alex F. Bielajew |
---|
| 865 | {\em NIM B 142 (1998) 253. } |
---|
| 866 | \item D. Liljequist and M. Ismail. |
---|
| 867 | {\em J.Appl.Phys. 62 (1987) 342. } |
---|
| 868 | \item D. Liljequist et al. |
---|
| 869 | {\em J.Appl.Phys. 68 (1990) 3061. } |
---|
| 870 | \item R. Mayol and F. Salvat |
---|
| 871 | {\em At.Data and Nucl.Data Tables 65 (1997) 55.}. |
---|
| 872 | \item V.L. Highland |
---|
| 873 | {\em NIM 129 (1975) 497. } |
---|
| 874 | \item G.R. Lynch and O.I. Dahl |
---|
| 875 | {\em NIM B58 (1991) 6. } |
---|
| 876 | \item{msc.shen} G. Shen et al. |
---|
| 877 | {\em Phys. Rev. D 20 (1979) 1584.} |
---|
| 878 | \item{msc.attwood} D. Attwood et al. |
---|
| 879 | {\em NIM B 251 (2006) 41.} |
---|
| 880 | \end{enumerate} |
---|
| 881 | |
---|
| 882 | \end{htmlonly} |
---|
| 883 | |
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| 884 | |
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| 885 | |
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