\chapter{Introduction}

%000000000000000000000000000000000000000000000000000000000000000
\section{Motivation}%concealed Uranium and Plutonium.
%\section{The Cargo Container Problem}%concealed Uranium and Plutonium.
\indent


\section{Emittance}

\subsection{What is Emittance}
\indent
In accelarator physis, Cartisian coordinate system was used to discribe motion of the accelarated particles. Usually the z-axis of Cartisian coordinate system is set to be along the electron beam line as longitudual beam direction. X-axis is set to be horizantal and perpendicular to the longitudual direction, as one of the transverse beam direction. Y-axis is set to be vertical and perpendicular to the longitudual direction, as another transverse beam direction.

For the conviniance of our representation, let's use "z" to repsent our transverse coordinates, while discussing emittance. And we would like to express longitual beam direction with "s". Our transverse beam profile changes along the beam line, it makes z is function of s, z(s).

The angle of a accelarated charge regarding the designed orbit can be defined as:

\begin{equation}
z' = \frac {dz}{ds}  \label{eq:beam-angle}
\end{equation}

If we plot $z$ vs. $z'$, we will get an ellipse. The area of the ellipse is an invariant, which is called Courant-Snyder invariant. The transverse emittance $\epsilon$ of the beam is defined to be the area of the ellipse, which contains $90\%$ of the particles.






\subsection{Emittance Experiment}
\indent
In July 2010y, Emittance measurement of HRRL was conducted at Beam Lab, at Physics Department of ISU. We installed a YAG crystal on the HRRL beam line to see electron beam. A quadrupole magnet was installed between HRRL gun and the YAG screen. We changed current on the quadrupole to control magnetic field strength of the quadrupole magnet, thus we changed electron beam shape on the YAG screen. 

\subsubsection{Experimental Setup}
\indent
We did quadrupole scan to measure emittance of the electron beam in HRRL. In quadrupole method, the strength of the quadrupole magnet was changed by changing the current go through quadrupole coils. The electron beam were coming out of the gun went through quadrupole first, then beam would enter a 3-way cross. Two end of the 3-way cross was installed on the beam line. The third end of the 3-way cross was placed upward and there was a actuator installed to it.  The YAG crystal was mounted in the actuator, which can put the YAG in the beam line or take it out of the beam line. A camera was placed inside the actuator to look through cacuum a window and to capture the image on the YAG crystal cread by electron beam. A faraday cup was mounted at the end of the beam line to measure the transmission of the  charge.




\begin{figure}[htbp] 
\centering
\includegraphics[scale=0.45]{./HRRL_July_Emit_Lay_Out.png}
\caption{Experiment set up of HRRL 2010 July emittance test.}
\label{fig:HRRL_July_Emit_Lay_Out} 
\end{figure}


 and the beam will change under the magnetic field of the 


\section{Positron Target}
\indent

\section{Positron Yield}
\indent

\section{Conclusion}
\indent









%~\cite{Sommerfeld1931}
%${\bf P_i}$
%${\bf {\theta}_f}$


%\begin{figure}[htbp] 
%\centering
%\includegraphics[scale=0.6]{1_brem_emission}
%\caption{Bremsstrahlung process and plane of emission in bremsstrahlung }
%\label{fig:brem_emission} 
%\end{figure}

%\begin{equation}
%\theta_c = \frac {m_ec^2}{E_0}  \label{eq:char-ang}
%\end{equation}

%\noindent  





%\begin{table}[ht]
%\caption{Angular distribution coefficients.}  
%\centering
%\begin{tabular}{|c|c|c|c|c|c|} % ccc means 3 columns, all centered; alternatives are l, r

%\hline % draws a line under the column headers
%         J & K &    a   		  &     b   &      c        &      d     \\ 
%\hline 1 & 0 &  $ -$ 	  &   3/4  &     $ - $       &      3/4  \\  
%\hline 1 & 1 & 3/4 		 &  $-$3/8  &     $ -  $      &     $-$3/8  \\ 
%\hline 2 & 0 &  $ -  $	 &          &  +15/16  &     15/4  \\ 
%\hline 2 & 1 & 5/4 		& $ -$5/8  &   $ -$5/8     &   $ -$15/8  \\ 
%\hline 2 & 2 &     		 &  +5/8  &  +5/32   &              \\ 
%\hline

%\end{tabular}      
%\label{tb:ang-dis-cof-abc}
%\end{table}


% \begin{center}                
 %                   $f_{2}(L,L)$=$\frac{1}{2} $ \\
  %                  $f_{2}(L,L)$=$\frac{1}{2} $  \\
  %                  $f_{4}(L,L)$=-$\frac{1}{2} $   \\
           
 %\end{center}


%\begin{sidewaysfigure}[htbp] 
%\centering
%  \includegraphics[scale=0.7]{1_Setup_abc}
%\caption{Experimental setup for photodisintegration experiment with HRRL whole view}
%\label{fig:1_Setup_abc} 
%\end{sidewaysfigure}


%Fig.~\ref{fig:1_Setup_abc} is the view of whole setup. 

