partmatter.tex 10.6 KB
 snuverink_j committed Sep 06, 2017 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 \input{header} \chapter{Physics Models Used in the Particle Matter Interaction Model} \label{chp:partmatt} \index{Particle Matter Interaction} The command to define the particle matter interacton is PARTICLEMATTERINTERACTION. \begin{description} \item[MATERIAL] \index{MATERIAL} The material of the surface. \item[ENABLERUTHERFORD] \index{ENABLERUTHERFORD} Switch to disable Rutherford scattering, default true. \end{description} The so defined instance has then to be added to an element using the attribute \section{The Energy Loss} The energy loss is simulated using the Bethe-Bloch equation. \label{eq:dEdx}  snuverink_j committed Sep 11, 2017 24 -\frac{\mathrm{d} E}{\mathrm{d} x}=\frac{K z^2 Z}{A \beta^2}\left[\frac{1}{2} \ln{\frac{2 m_e c^2\beta^2 \gamma^2 Tmax}{I^2}}-\beta^2 \right],  snuverink_j committed Sep 06, 2017 25 26 27 28 29 30 31  where $Z$ is the atomic number of absorber, $A$ is the atomic mass of absorber, $m_e$ is the electron mass, $z$ is the charge number of the incident particle, $K=4\pi N_Ar_e^2m_ec^2$, $r_e$ is the classical electron radius, $N_A$ is the Avogadro's number, $I$ is the mean excitation energy. $\beta$ and $\gamma$ are kinematic variables. $T_{max}$ is the maximum kinetic energy which can be imparted to a free electron in a single collision. T_{max}=\frac{2m_ec^2\beta^2\gamma^2}{1+2\gamma m_e/M+(m_e/M)^2}, where $M$ is the incident particle mass.  snuverink_j committed Sep 06, 2017 32 The stopping power is compared with PSTAR program of NIST in Figure~\ref{dEdx}.  snuverink_j committed Sep 06, 2017 33 34 35 36 37 38 39 40 41 \begin{figure}[h!] \begin{center} \includegraphics[width=0.5\textwidth]{figures/partmatter/dEdx} \end{center} \caption{The comparison of stopping power with PSTAR. } \label{fig:dEdx} \end{figure} Energy straggling: For relatively thick absorbers such that the number of collisions is large, the energy loss distribution is shown to be Gaussian in form.  snuverink_j committed Sep 08, 2017 42 For non-relativistic heavy particles the spread $\sigma_0$ of the Gaussian distribution is calculated by:  snuverink_j committed Sep 06, 2017 43   snuverink_j committed Sep 08, 2017 44 gma_0^2=4\pi N_Ar_e^2(m_ec^2)^2\rho\frac{Z}{A}\Delta s,  snuverink_j committed Sep 06, 2017 45 46 47 48 49 50 51 52  where $\rho$ is the density, $\Delta s$ is the thickness. \section{The Coulomb Scattering} The Coulomb scattering is treated as two independent events: the multiple Coulomb scattering and the large angle Rutherford scattering.\\ Using the distribution given in Classical Electrodynamics, by J. D. Jackson, the multiple- and single-scattering distributions can be written: \label{eq:PM}  snuverink_j committed Sep 11, 2017 53 P_M(\alpha) \;\mathrm{d} \alpha=\frac{1}{\sqrt{\pi}}e^{-\alpha^2}\;\mathrm{d}\alpha,  snuverink_j committed Sep 06, 2017 54 55 56  \label{eq:Ps}  snuverink_j committed Sep 08, 2017 57 P_S(\alpha) \;\mathrm{d} \alpha=\frac{1}{8 \ln(204 Z^{-1/3})} \frac{1}{\alpha^3}\;\mathrm{d}\alpha,  snuverink_j committed Sep 06, 2017 58 59 60 61 62 63  where $\alpha=\frac{\theta}{<\Theta^2>^{1/2}}=\frac{\theta}{\sqrt 2 \theta_0}$. \noindent The transition point is $\theta=2.5 \sqrt 2 \theta_0\approx3.5 \theta_0$, \label{eq:Multiple}  snuverink_j committed Sep 08, 2017 64 \theta_0=\frac{{13.6}{MeV}}{\beta c p} z \sqrt{\Delta s/X_0} [1+0.038 \ln(\Delta s/X_0)],  snuverink_j committed Sep 06, 2017 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91  where $p$ is the momentum, $\Delta s$ is the step size, and $X_0$ is the radiation length. \subsection{Multiple Coulomb Scattering} Generate two independent Gaussian random variables with mean zero and variance one: $z_1$ and $z_2$. If $z_2 \theta_0>3.5 \theta_0$, start over. Otherwise, \label{eq:Multiplex} x=x+\Delta s p_x+z_1 \Delta s \theta_0/\sqrt{12}+z_2 \Delta s \theta_0/2, \label{eq:Multiplepx} p_x=p_x+z_2 \theta_0. Generate two independent Gaussian random variables with mean zero and variance one: $z_3$ and $z_4$. If $z_4 \theta_0>3.5 \theta_0$, start over. Otherwise, \label{eq:Multipley} y=y+\Delta s p_y+z_3 \Delta s \theta_0/\sqrt{12}+z_4 \Delta s \theta_0/2, \label{eq:Multiplepy} p_y=p_y+z_4 \theta_0. \subsection{Large Angle Rutherford Scattering}  snuverink_j committed Sep 11, 2017 92 Generate a random number $\xi_1$, \textit{if} $\xi_1<\frac{\int_{2.5}^\infty P_S(\alpha)d\alpha}{\int_0^{2.5} P_M(\alpha)\;\mathrm{d}\alpha+\int_{2.5}^\infty P_S(\alpha)\;\mathrm{d}\alpha}=0.0047$, sampling the large angle  snuverink_j committed Sep 06, 2017 93 94 95 96 97 Rutherford scattering.\\ The cumulative distribution function of the large angle Rutherford scattering is \label{eq:Fa}  snuverink_j committed Sep 11, 2017 98 F(\alpha)=\frac{\int_{2.5}^\alpha P_S(\alpha) \;\mathrm{d} \alpha}{\int_{2.5}^\infty P_S(\alpha) \;\mathrm{d} \alpha}=\xi,  snuverink_j committed Sep 06, 2017 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114  where $\xi$ is a random variable. So \label{eq:alpha} \alpha=\pm 2.5 \sqrt{\frac{1}{1-\xi}}=\pm 2.5 \sqrt{\frac{1}{\xi}}. Generate a random variable $P_3$,\\ \textit{if} $P_3>0.5$ \theta_{Ru}=2.5 \sqrt{\frac{1}{\xi}} \sqrt{2}\theta_0, \textit{else} \theta_{Ru}=-2.5 \sqrt{\frac{1}{\xi}} \sqrt{2}\theta_0.  snuverink_j committed Sep 06, 2017 115 The angle distribution after Coulomb scattering is shown in Figure~\ref{Coulomb}.  snuverink_j committed Sep 06, 2017 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 The line is from Jackson's formula, and the points are simulations with Matlab. For a thickness of $\Delta s=1e-4$ $m$, $\theta=0.5349 \alpha$ (in degree). \begin{figure}[ht!] \begin{center} \includegraphics[width=.8\textwidth]{figures/partmatter/10steps} \end{center} \caption{The comparison of Coulomb scattering with Jackson's book. } \label{fig:Coulomb} \end{figure} \section{The Flow Diagram of {\em CollimatorPhysics} Class in OPAL} \begin{figure}[ht!] \begin{center} \includegraphics[width=0.8\textwidth]{figures/partmatter/diagram} \end{center}  snuverink_j committed Sep 07, 2017 132 \caption{The diagram of CollimatorPhysics in \textit{OPAL}. }  snuverink_j committed Sep 06, 2017 133 134 135 136 137 138 \label{fig:diagram} \end{figure} \begin{figure}[ht!] \begin{center} \includegraphics[width=0.6\textwidth]{figures/partmatter/Diagram2} \end{center}  snuverink_j committed Sep 07, 2017 139 \caption{The diagram of CollimatorPhysics in \textit{OPAL} (continued). }  snuverink_j committed Sep 06, 2017 140 141 142 143 144 145 146 147 \label{fig:diagram2} \end{figure} \clearpage \subsection{The Substeps} Small step is needed in the routine of CollimatorPhysics.  snuverink_j committed Sep 08, 2017 148 If a large step is given in the main input file, in the file \textit{CollimatorPhysics.cpp},  snuverink_j committed Sep 06, 2017 149 it is divided by a integer number $n$ to make the step size using for the calculation of collimator physics less than 1.01e-12 s. As shown  snuverink_j committed Sep 06, 2017 150 by Figure~\ref{diagram,diagram2} in the previous section, first we track one step for the particles already in the  snuverink_j committed Sep 06, 2017 151 152 153 154 155 156 collimator and the newcomers, then another (n-1) steps to make sure the particles in the collimator experience the same time as the ones in the main bunch. Now, if the particle leave the collimator during the (n-1) steps, we track it as in a drift and put it back to the main bunch when finishing (n-1) steps.  snuverink_j committed Sep 07, 2017 157 \section{Available Materials in \textit{OPAL}}  snuverink_j committed Sep 06, 2017 158 159 \begin{table}[H]\footnotesize \centering  snuverink_j committed Sep 07, 2017 160  \caption{List of materials with their parameters implemented in \textit{OPAL}.}  snuverink_j committed Sep 06, 2017 161 162 163  \label{table:Materials} \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|} \hline  snuverink_j committed Sep 07, 2017 164  \tabhead{Material & Z & A & $\rho$ [$g/cm^3$] & X0 [$g/cm^2$] & A2 & A3 & A4 & A5 & \textit{OPAL} Name}  snuverink_j committed Sep 06, 2017 165  \hline  snuverink_j committed Sep 06, 2017 166  Aluminum & 13 & 26.98 & 2.7 & 24.01 & 4.739 & 2766 & 164.5 & 2.023E-02 & \texttt{Aluminum }\\  snuverink_j committed Sep 06, 2017 167  %\hline  snuverink_j committed Sep 06, 2017 168  AluminaAl2O3 & 50 & 101.96 & 3.97 & 27.94 & 7.227 & 11210 & 386.4 & 4.474e-3 & \texttt{AluminaAl2O3 }\\  snuverink_j committed Sep 06, 2017 169  %\hline  snuverink_j committed Sep 06, 2017 170  Copper & 29 & 63.54 & 8.96 & 12.86 & 4.194 & 4649 & 81.13 & 2.242E-02 & \texttt{Copper}\\  snuverink_j committed Sep 06, 2017 171  %\hline  snuverink_j committed Sep 06, 2017 172  Graphite & 6 & 12.0172 & 2.210 & 42.7 & 2.601 & 1701 & 1279 & 1.638E-02 & \texttt{Graphite }\\  snuverink_j committed Sep 06, 2017 173  %\hline  snuverink_j committed Sep 06, 2017 174  GraphiteR6710 & 6 & 12.0172 & 1.88 & 42.7 & 2.601 & 1701 & 1279 & 1.638E-02 & \texttt{GraphiteR6710}\\  snuverink_j committed Sep 06, 2017 175  %\hline  snuverink_j committed Sep 06, 2017 176  Titan & 22 & 47.8 & 4.54 & 16.16 & 5.489 & 5260 & 651.1 & 8.930E-03 & \texttt{Titan }\\  snuverink_j committed Sep 06, 2017 177  %\hline  snuverink_j committed Sep 06, 2017 178  Air & 7 & 14 & 0.0012 & 37.99 & 3.350 & 1683 & 1900 & 2.513E-02 & \texttt{Air }\\  snuverink_j committed Sep 06, 2017 179  %\hline  snuverink_j committed Sep 06, 2017 180  Kapton & 6 & 12 & 1.4 & 39.95 & 2.601 & 1701 & 1279 & 1.638E-02 & \texttt{Kapton }\\  snuverink_j committed Sep 06, 2017 181  %\hline  snuverink_j committed Sep 06, 2017 182  Gold & 79 & 197 & 19.3 & 6.46 & 5.458 & 7852 & 975.8 & 2.077E-02 & \texttt{Gold }\\  snuverink_j committed Sep 06, 2017 183  %\hline  snuverink_j committed Sep 06, 2017 184  Water & 10 & 18 & 1 & 36.08 & 2.199 & 2393 & 2699 & 1.568E-02 & \texttt{Water }\\  snuverink_j committed Sep 06, 2017 185  %\hline  snuverink_j committed Sep 06, 2017 186  Mylar & 6.702 & 12.88 & 1.4 & 39.95 & 3.35 & 1683 & 1900 & 2.513E-02 & \texttt{Mylar }\\  snuverink_j committed Sep 06, 2017 187  %\hline  snuverink_j committed Sep 06, 2017 188  Berilium & 4 & 9.012 & 1.848 & 65.19 & 2.590 & 966.0 & 153.8 & 3.475E-02 & \texttt{Berilium }\\  snuverink_j committed Sep 06, 2017 189  %\hline  snuverink_j committed Sep 06, 2017 190  Molybdenum & 42 & 95.94 & 10.22 & 9.8 & 7.248 & 9545 & 480.2 & 5.376E-03 & \texttt{Molybdenum}\\  snuverink_j committed Sep 06, 2017 191 192 193 194 195 196 197 198 199 200  \hline \end{tabular} \end{table} \section{Example of an Input File} \examplefromfile{examples/particlematterinteraction.in}  snuverink_j committed Sep 06, 2017 201 202 FX5 is a slit in x direction, the \texttt{APERTURE} is \textbf{POSITIVE}, the first value in \texttt{APERTURE} is the left part, the second value is the right part. FX16 is a slit in y direction, the \texttt{APERTURE} is \textbf{NEGATIVE}, the first value in \texttt{APERTURE} is the down part, the second value is the up part.  snuverink_j committed Sep 06, 2017 203 204  \section{A Simple Test}  snuverink_j committed Sep 08, 2017 205 A cold Gaussian beam with $\sigma_x=gma_y=5$ mm.  snuverink_j committed Sep 06, 2017 206 The position of the collimator is from 0.01 m to 0.1 m, the half aperture in y direction is $3$ mm. Figure~\ref{longcoll}  snuverink_j committed Sep 07, 2017 207 shows the trajectory of particles which are either absorbed or deflected by a copper slit. As a benchmark of the collimator model in \textit{OPAL}, Figure~\ref{Espectrum} shows the energy spectrum and angle deviation at z=0.1 m after an elliptic collimator.  snuverink_j committed Sep 06, 2017 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 \begin{figure}[ht!] \begin{center} \includegraphics[width=0.8\textwidth]{figures/partmatter/longcoll6} \end{center} \caption{The passage of protons through the collimator. } \label{fig:longcoll} \end{figure} \begin{figure}[ht!] \begin{center} \includegraphics[width=0.8\textwidth]{figures/partmatter/spectandscatter} \end{center} \caption{The energy spectrum and scattering angle at z=0.1 m} \label{fig:Espectrum} \end{figure} \input{footer}