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\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.

\begin{equation}
\label{eq:dEdx}
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-\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],
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\end{equation}
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.
\begin{equation}
T_{max}=\frac{2m_ec^2\beta^2\gamma^2}{1+2\gamma m_e/M+(m_e/M)^2},
\end{equation}
where $M$ is the incident particle mass.

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The stopping power is compared with PSTAR program of NIST in Figure~\ref{dEdx}.
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\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.
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For non-relativistic heavy particles the spread $\sigma_0$ of the Gaussian distribution is calculated by:
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\begin{equation}
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gma_0^2=4\pi N_Ar_e^2(m_ec^2)^2\rho\frac{Z}{A}\Delta s,
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\end{equation}
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:
\begin{equation}
\label{eq:PM}
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P_M(\alpha) \;\mathrm{d} \alpha=\frac{1}{\sqrt{\pi}}e^{-\alpha^2}\;\mathrm{d}\alpha,
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\end{equation}
\begin{equation}
\label{eq:Ps}
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P_S(\alpha) \;\mathrm{d} \alpha=\frac{1}{8 \ln(204 Z^{-1/3})} \frac{1}{\alpha^3}\;\mathrm{d}\alpha,
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\end{equation}
where $\alpha=\frac{\theta}{<\Theta^2>^{1/2}}=\frac{\theta}{\sqrt 2 \theta_0}$.

\noindent The transition point is $\theta=2.5 \sqrt 2 \theta_0\approx3.5 \theta_0$,
\begin{equation}
\label{eq:Multiple}
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\theta_0=\frac{{13.6}{MeV}}{\beta c p} z \sqrt{\Delta s/X_0} [1+0.038 \ln(\Delta s/X_0)],
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\end{equation}
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,
\begin{equation}
\label{eq:Multiplex}
x=x+\Delta s p_x+z_1 \Delta s \theta_0/\sqrt{12}+z_2 \Delta s \theta_0/2,
\end{equation}
\begin{equation}
\label{eq:Multiplepx}
p_x=p_x+z_2 \theta_0.
\end{equation}
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,
\begin{equation}
\label{eq:Multipley}
y=y+\Delta s p_y+z_3 \Delta s \theta_0/\sqrt{12}+z_4 \Delta s \theta_0/2,
\end{equation}
\begin{equation}
\label{eq:Multiplepy}
p_y=p_y+z_4 \theta_0.
\end{equation}

\subsection{Large Angle Rutherford Scattering}

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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
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Rutherford scattering.\\
The cumulative distribution function of the large angle
Rutherford scattering is
\begin{equation}
\label{eq:Fa}
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F(\alpha)=\frac{\int_{2.5}^\alpha P_S(\alpha) \;\mathrm{d} \alpha}{\int_{2.5}^\infty P_S(\alpha) \;\mathrm{d} \alpha}=\xi,
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\end{equation}
where $\xi$ is a random variable. So
\begin{equation}
\label{eq:alpha}
\alpha=\pm 2.5 \sqrt{\frac{1}{1-\xi}}=\pm 2.5 \sqrt{\frac{1}{\xi}}.
\end{equation}
Generate a random variable $P_3$,\\
\textit{if} $P_3>0.5$
\begin{equation}
   \theta_{Ru}=2.5 \sqrt{\frac{1}{\xi}} \sqrt{2}\theta_0,
\end{equation}
\textit{else}
\begin{equation}
       \theta_{Ru}=-2.5 \sqrt{\frac{1}{\xi}} \sqrt{2}\theta_0.
\end{equation}

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The angle distribution after Coulomb scattering is shown in Figure~\ref{Coulomb}.
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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}
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\caption{The diagram of CollimatorPhysics in \textit{OPAL}. }
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\label{fig:diagram}
\end{figure}
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=0.6\textwidth]{figures/partmatter/Diagram2}
\end{center}
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\caption{The diagram of CollimatorPhysics in \textit{OPAL} (continued). }
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\label{fig:diagram2}
\end{figure}
\clearpage

\subsection{The Substeps}

Small step is needed in the routine of CollimatorPhysics.

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If a large step is given in the main input file, in the file \textit{CollimatorPhysics.cpp},
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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
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by  Figure~\ref{diagram,diagram2} in the previous section, first we track one step for the particles already in the
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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.

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\section{Available Materials in \textit{OPAL}}
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\begin{table}[H]\footnotesize
\centering
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  \caption{List of materials with their parameters implemented in \textit{OPAL}.}
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  \label{table:Materials}
  \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
  \hline
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  \tabhead{Material     & Z    &    A         &    $\rho$  [$g/cm^3$]     &    X0      [$g/cm^2$]     &    A2    &    A3    &    A4    &    A5 & \textit{OPAL} Name}
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    \hline
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    Aluminum        &    13        &          26.98        &              2.7        &    24.01        &    4.739        &    2766        &    164.5        &    2.023E-02 & \texttt{Aluminum }\\
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    %\hline
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    AluminaAl2O3        &    50        &          101.96        &              3.97        &    27.94        &    7.227        &    11210        &    386.4        &    4.474e-3 & \texttt{AluminaAl2O3 }\\
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    %\hline
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    Copper            &    29        &      63.54        &        8.96        &    12.86        &     4.194        &    4649        &    81.13        &    2.242E-02 & \texttt{Copper}\\
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    %\hline
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    Graphite            &    6        &       12.0172            &           2.210    &    42.7            &    2.601        &    1701        &    1279        &    1.638E-02 & \texttt{Graphite }\\
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    %\hline
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    GraphiteR6710        &    6        &       12.0172            &           1.88        &    42.7            &    2.601        &    1701        &    1279        &    1.638E-02 & \texttt{GraphiteR6710}\\
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    %\hline
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    Titan            &    22        &      47.8        &          4.54        &    16.16        &    5.489        &    5260        &    651.1        &    8.930E-03 & \texttt{Titan }\\
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    %\hline
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    Air                &    7        &    14            &        0.0012        &    37.99        &    3.350        &    1683        &    1900        &    2.513E-02 & \texttt{Air }\\
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    %\hline
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    Kapton            &    6        &    12            &    1.4            &    39.95        &    2.601        &    1701        &    1279        &    1.638E-02 & \texttt{Kapton }\\
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    %\hline
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    Gold                &    79        &    197            &    19.3            &    6.46            &    5.458        &    7852        &    975.8        &    2.077E-02 & \texttt{Gold }\\
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    %\hline
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    Water            &    10        &    18            &    1            &    36.08        &    2.199        &    2393        &    2699        &    1.568E-02 & \texttt{Water }\\
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    %\hline
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    Mylar            &    6.702    &    12.88            &    1.4            &    39.95        &    3.35        &    1683        &    1900        &     2.513E-02 & \texttt{Mylar }\\
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    %\hline
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    Berilium                 &    4        &    9.012        &    1.848        &    65.19        &    2.590        &    966.0        &    153.8        &     3.475E-02 & \texttt{Berilium }\\
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    %\hline
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    Molybdenum                &    42        &    95.94        &    10.22        &    9.8            &    7.248        &    9545        &    480.2        &     5.376E-03 &  \texttt{Molybdenum}\\
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    \hline
    \end{tabular}
\end{table}



\section{Example of an Input File}

\examplefromfile{examples/particlematterinteraction.in}

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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.
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\section{A Simple Test}
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A cold Gaussian beam with $\sigma_x=gma_y=5$ mm.
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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}
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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.
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\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}