\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. $$\label{eq:dEdx} -\frac{\mathrm{d} E}{\mathrm{d} x}=\frac{K z^2 Z}{A \beta^2}\left[\frac{1}{2} \ln{\frac{2 m_e c^2\beta^2 \gamma^2 Tmax}{I^2}}-\beta^2 \right],$$ 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. The stopping power is compared with PSTAR program of NIST in Figure~\ref{dEdx}. \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. For non-relativistic heavy particles the spread $\sigma_0$ of the Gaussian distribution is calculated by: $$gma_0^2=4\pi N_Ar_e^2(m_ec^2)^2\rho\frac{Z}{A}\Delta s,$$ 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: $$\label{eq:PM} P_M(\alpha) \;\mathrm{d} \alpha=\frac{1}{\sqrt{\pi}}e^{-\alpha^2}\;\mathrm{d}\alpha,$$ $$\label{eq:Ps} P_S(\alpha) \;\mathrm{d} \alpha=\frac{1}{8 \ln(204 Z^{-1/3})} \frac{1}{\alpha^3}\;\mathrm{d}\alpha,$$ where $\alpha=\frac{\theta}{<\Theta^2>^{1/2}}=\frac{\theta}{\sqrt 2 \theta_0}$. \noindent The transition point is $\theta=2.5 \sqrt 2 \theta_0\approx3.5 \theta_0$, $$\label{eq:Multiple} \theta_0=\frac{{13.6}{MeV}}{\beta c p} z \sqrt{\Delta s/X_0} [1+0.038 \ln(\Delta s/X_0)],$$ 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} 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 Rutherford scattering.\\ The cumulative distribution function of the large angle Rutherford scattering is $$\label{eq:Fa} F(\alpha)=\frac{\int_{2.5}^\alpha P_S(\alpha) \;\mathrm{d} \alpha}{\int_{2.5}^\infty P_S(\alpha) \;\mathrm{d} \alpha}=\xi,$$ 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} $$\theta_{Ru}=-2.5 \sqrt{\frac{1}{\xi}} \sqrt{2}\theta_0.$$ The angle distribution after Coulomb scattering is shown in Figure~\ref{Coulomb}. 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} \caption{The diagram of CollimatorPhysics in \textit{OPAL}. } \label{fig:diagram} \end{figure} \begin{figure}[ht!] \begin{center} \includegraphics[width=0.6\textwidth]{figures/partmatter/Diagram2} \end{center} \caption{The diagram of CollimatorPhysics in \textit{OPAL} (continued). } \label{fig:diagram2} \end{figure} \clearpage \subsection{The Substeps} Small step is needed in the routine of CollimatorPhysics. If a large step is given in the main input file, in the file \textit{CollimatorPhysics.cpp}, 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 by Figure~\ref{diagram,diagram2} in the previous section, first we track one step for the particles already in the 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. \section{Available Materials in \textit{OPAL}} \begin{table}[H]\footnotesize \centering \caption{List of materials with their parameters implemented in \textit{OPAL}.} \label{table:Materials} \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|} \hline \tabhead{Material & Z & A & $\rho$ [$g/cm^3$] & X0 [$g/cm^2$] & A2 & A3 & A4 & A5 & \textit{OPAL} Name} \hline Aluminum & 13 & 26.98 & 2.7 & 24.01 & 4.739 & 2766 & 164.5 & 2.023E-02 & \texttt{Aluminum }\\ %\hline AluminaAl2O3 & 50 & 101.96 & 3.97 & 27.94 & 7.227 & 11210 & 386.4 & 4.474e-3 & \texttt{AluminaAl2O3 }\\ %\hline Copper & 29 & 63.54 & 8.96 & 12.86 & 4.194 & 4649 & 81.13 & 2.242E-02 & \texttt{Copper}\\ %\hline Graphite & 6 & 12.0172 & 2.210 & 42.7 & 2.601 & 1701 & 1279 & 1.638E-02 & \texttt{Graphite }\\ %\hline GraphiteR6710 & 6 & 12.0172 & 1.88 & 42.7 & 2.601 & 1701 & 1279 & 1.638E-02 & \texttt{GraphiteR6710}\\ %\hline Titan & 22 & 47.8 & 4.54 & 16.16 & 5.489 & 5260 & 651.1 & 8.930E-03 & \texttt{Titan }\\ %\hline Air & 7 & 14 & 0.0012 & 37.99 & 3.350 & 1683 & 1900 & 2.513E-02 & \texttt{Air }\\ %\hline Kapton & 6 & 12 & 1.4 & 39.95 & 2.601 & 1701 & 1279 & 1.638E-02 & \texttt{Kapton }\\ %\hline Gold & 79 & 197 & 19.3 & 6.46 & 5.458 & 7852 & 975.8 & 2.077E-02 & \texttt{Gold }\\ %\hline Water & 10 & 18 & 1 & 36.08 & 2.199 & 2393 & 2699 & 1.568E-02 & \texttt{Water }\\ %\hline Mylar & 6.702 & 12.88 & 1.4 & 39.95 & 3.35 & 1683 & 1900 & 2.513E-02 & \texttt{Mylar }\\ %\hline Berilium & 4 & 9.012 & 1.848 & 65.19 & 2.590 & 966.0 & 153.8 & 3.475E-02 & \texttt{Berilium }\\ %\hline Molybdenum & 42 & 95.94 & 10.22 & 9.8 & 7.248 & 9545 & 480.2 & 5.376E-03 & \texttt{Molybdenum}\\ \hline \end{tabular} \end{table} \section{Example of an Input File} \examplefromfile{examples/particlematterinteraction.in} 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. \section{A Simple Test} A cold Gaussian beam with $\sigma_x=gma_y=5$ mm. 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} 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. \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}