\chapter{Simulation}
The ratios of positrons to the electrons impinging the tungsten production target, T1,  was substantially smaller after they traversed the beam line than what was produced at the target (10$^{-15}$ instead of 10$^{-3}$). Simulations were performed using G4beamline to better understand the losses from the transportation of positrons to the experimental cell. ``G4beamline is a particle tracking and simulation program based on the GEANT4~\cite{geant4} toolkit that is specifically designed to easily simulate beamlines and other systems using single-particle tracking~\cite{muonsinc}.'' A sample G4beamline script for positron generation using the new HRRL beamline is given in the appendix F.

The simulation predicts that at least one positron per 1000 incident 10~MeV electrons is produced using a 2~mm thick tungsten target.
The simulation revealed that the number of electrons decreased by orders of magnitude as they were transported through some of the beamline magnets.
As a result of this beam loss, the simulation was divided into three steps to increase the beam line simulation efficiency. Each step generates particles at different locations along the beamline where the beam loss was found to be substantial. While the first simulation step used the measured electron energy profile, subsequent steps would generate particles based on the particle phase space observed at the end of previous step. The method decreased the simulation time so a sample of more than one million events could be produced within a single day.

The first step in the simulation generated an electron beam with the energy distribution observed in the experiment, see  Figure~\ref{fig:En-Scan}. The electrons were focused by three quadrupoles onto the positron production target T1 (see Figure~\ref{fig:app-hrrl-line}). Electrons traversing T1 produced bremsstrahlung photons of sufficient energy to produce $e^+e^-$ pairs that would escape the downstream side of the target and be collected by a second quadrupole triplet. The second step simulated the collection and transportation of positrons exiting T1 to the entrance of the first dipole D1. 
The last step transported positrons from the entrance of D1 all the way to the annihilation target T2, the interactions of positrons with T2, and the detection of the resulting 511~keV photon pairs. 

The two targets (T1 and T2) were positioned at different angles with respect to the incident beam momentum vector. The beamline coordinate system aligns the z-axis to point along the incident electrons momentum vector and the x-axis along the horizontal plane. The positron production target, T1, was placed such that the upstream side of T1 was facing vertically down at a 45$^{\circ}$ angle with respect to the vertical(rotated 45$^{\circ}$ counter clockwise about x-axis). The positron conversion target, T2, was rotated about two different axes. The first rotation positioned the upstream side of the target so it was facing down by 45$^{\circ}$ (rotated 45$^{\circ}$ clockwise about x-axis like T1). The second rotation was 45$^{\circ}$ about y-axis. As a result, target T2's upstream face was directed towards the beam right NaI detector.

\section{Step 1 - The Electron Beam Generation and Transportation to T1}
In the first simulation step, an electron beam was generated with an energy distribution that was observed in the experiment. The emittance, the Twiss parameters, and the energy distribution of the electron beam were measured experimentally. The energy distribution of the electron beam is shown in Figure~\ref{fig:En-Scan}. The distribution was fit using two skewed Gaussian distributions. The fit parameters given in Table~\ref{tab:En-Scan_resluts} were used by the simulation to generated electrons.

A series of virtual detectors were placed along the beamline to sample the beam. As an example, three virtual circular detectors and T1 are shown in Figure~\ref{fig:T1_UpD_DwD2}. The electron beam was detected by a virtual detector DUPT1 (Detector UPstream of T1) placed 25.52~mm upstream of T1. Positrons, electrons, and photons generated during the interaction of the electron beam with T1 were observed by virtual detectors DT1 (Detector of T1) and DDNT1 (Detector DowNstream of T1) placed 25.52~mm downstream of T1. 

\begin{figure}[htbp] 
\centering
\includegraphics[scale=0.55]{3-Simulation/Figures/sim_setup_T1_UpD_DwD2.png}
\caption{The T1 positron target and virtual detectors used in the simulation. DUPT1 is a virtual detector located upstream of T1 to detect the incoming electron beam. DDNT1 is a virtual detector downstream of T1. DT1 is a virtual detector that is placed right after T1 and parallel to it.}
\label{fig:T1_UpD_DwD2}
\end{figure}


\subsection{The Positron Beam on DDNT1}

In the first step, $1.38 \times 10^{10}$ electrons were generated with the energy distribution shown by the dotted-dashed line in Figure~\ref{fig:SimS1_T1UPDN}. These electrons were transported to T1 where they produced photons that produced the positron distribution shown by the solid the line in Figure~\ref{fig:SimS1_T1UPDN} by pair-production. The dashed line is the electron energy distribution observed by DDNT1. The simulation result using $1.38 \times 10^{7}$ electrons incident on T1 is drawn in Figure~\ref{fig:SimS1_T1UPDN}. The incident electrons were detected by the virtual detector DUPT1 and downstream positrons and electrons were detected by DDNT1. As shown in Figure~\ref{fig:SimS1_T1UPDN}, the electrons pass through T1 loosing approximately 4~MeV while the positrons escape the downstream side of T1 with a mean energy of about 3~MeV. The beam line was set to transport positrons with this mean energy as a result of this prediction.

\begin{figure}[htbp] 
\centering
\includegraphics[scale=0.75]{3-Simulation/Figures/s/s1/overlay9.eps}
\caption{The incident electron energy distribution (dotted dashed line), the distribution of electrons after T1 (dashed line), and the distribution of positrons produced (solid line). The incident electron distribution counts were weighted by 0.001.}
\label{fig:SimS1_T1UPDN}
\end{figure}

\begin{figure}[htbp] 
\begin{tabular}{cc}
{\scalebox{0.364} [0.364]{\includegraphics{3-Simulation/Figures/X_e+_DDNT1.eps}}}  & {\scalebox{0.364} [0.364]{\includegraphics{3-Simulation/Figures/Y_e+_DDNT1.eps}}}  \\
(a) $x$ projection.  &  (b) $y$  projection. \\
{\scalebox{0.364} [0.364]{\includegraphics{3-Simulation/Figures/XP_e+_DDNT1.eps}}}  & {\scalebox{0.364} [0.364]{\includegraphics{3-Simulation/Figures/YP_e+_DDNT1.eps}}}  \\
(c) $x$ projection of the divergence. &  (d) $y$ projection of the divergence.\\
{\scalebox{0.364} [0.364]{\includegraphics{3-Simulation/Figures/XY_e+_DDNT1.png}}}  & {\scalebox{0.364} [0.364]{\includegraphics{3-Simulation/Figures/XY_e+_DDNT1_zoom.png}}}  \\
(e) The transverse beam profile. &  (f) Zoom in of (e). \\
\end{tabular}
\caption{The transverse beam projections and angular distributions of positrons detected. The virtual detector diameter is 30~mm  while the beam pipe is only 24~mm.}
\label{fig:DDNT1_results}
\end{figure}
 
The positron spatial and angular distribution detected by DDNT1 is shown in Figure~\ref{fig:DDNT1_results}. The $y$ vs. $x$ spatial distribution of the beam is shown in Figure~\ref{fig:DDNT1_results}~(e) and Figure~\ref{fig:DDNT1_results} (f). As can be seen from Figure~\ref{fig:DDNT1_results} (b) and (d), the $y$ spatial distribution and divergence, defined in equation~\ref{eq:divergence}, of the positron beam have a sharp drop in counts in the region between $-25.8$~mm and $-27.2$~mm from the beam center. 
Figure~\ref{fig:sim-DDNT1-T1-geo} shows the geometry and location of T1 and DDNT1. 
If the size of T1 were to be increased, it would eventually intersect with DDNT1 at a distance between $25.8$~mm and $27.2$~mm from the beam center, $i.e.$ the edge of the T1 is facing this $1.4$~mm wide low count area. 

\begin{figure}[htbp] 
\centering
\includegraphics[scale=1]{3-Simulation/Figures/sharp_drop2.eps}
\caption{The geometry of the target T1 and the virtual detector DDNT1.}
\label{fig:sim-DDNT1-T1-geo}
\end{figure}  

As shown in Figure~\ref{fig:DDNT1_YpY}, the $y$ distribution count decreases at $\theta$ = 45$^{\circ}$. Positrons were emitted from both the downstream and upstream side of T1. Positrons from the downstream side of T1 intersected the detector at angles below 45$^{\circ}$ while positrons from the upstream side of T1 begin to hit the detector at angles beyond 45$^{\circ}$. Neither positrons upstream nor downstream of T1 traveled to the 1.4~mm wide low count area. Only positrons created on the edge of T1 reached the low count area between 25.8~mm~$<x<$~27.2~mm. As a result, the counts in this area are comparatively lower.
  
\begin{figure}[htbp] 
\begin{tabular}{cc}
{\scalebox{0.39} [0.39]{\includegraphics{3-Simulation/Figures/STSimS1_YYP_DDNT1.png}}}  & {\scalebox{0.39} [0.39]{\includegraphics{3-Simulation/Figures/STSimS1_YYP_DDNT1_zoom.png}}}  \\
(a) y' $vs.$ y.  &  (b) y' $vs.$ y zoom. \\
\end{tabular}
\caption{The positron beam $y'$ $vs$. $y$ detected by DDNT1.}
\label{fig:DDNT1_YpY}
\end{figure}

\subsection{The Positron Beam on Virtual Detectors DQ4 and DD1}   
  
The positron beam energy distributions at the entrance of Q4 observed by virtual detector DQ4 is shown in Figure~\ref{STSimS1_En_DQ4_DD1}. The distribution of positron energies observed leaving the downstream side of the production target T1 as observed by virtual detector DD1UP are also shown. The ratio of positrons leaving the target T1 and entering the first quadrupole indicate that nearly 90\% of the positrons are lost. The large positron divergence is responsible for this loss. The median positron energy does not change suggesting that the beamline simulation is tuned to transport the optimal number of positrons emitted by the positron production target, T1.

\begin{figure}[htbp] 
\begin{tabular}{ccc}
\centerline{\scalebox{0.8} [0.8]{\includegraphics{3-Simulation/Figures/En_e+_DQ4.eps}}} \\
(a) The positron energy distribution on DQ4.  \\
\centerline{\scalebox{0.8} [0.8]{\includegraphics{3-Simulation/Figures/En_e+_DD1.eps}}}\\
(b) The positron energy distribution on DD1.  \\
\end{tabular}
\caption{The positron beam energy distribution detected downstream of T1 just before entering the first quadrupole (a) and after leaving the production target upstream of this quadrupole (b).}
\label{STSimS1_En_DQ4_DD1}
\end{figure}

\section{Step 2 - The Transportation of The Positron Beam from DDNT1 to The Entrance of The First Dipole} 

The simulation's second step was designed to predict the amount of beam loss that would result when the positron beam is transported from the entrance of Q4 to the entrance of the first energy selecting dipole D1.  The beam observed on detector DDNT1 in step 1 was used to generate positrons directed towards the entrance of the quadrupole Q4. At detector DDNT1, the higher energy positrons tend to have smaller polar angles and are closer to the beam center. Positrons were generated in 1~keV/c momentum bins with different weights, spatial, and angular distributions to reproduce the distributions observed in step 1. Figure~\ref{fig:STSimSetupS2} illustrates the quadrupole triplet system and the dipole magnet D1 used in the beamline.  Positrons generated at DDNT1 are  transported to the entrance of D1 through this quadrupole triplet system. The virtual detectors were placed at the entrance of Q4 (DQ4) and D1 (DD1UP) to track the positrons. The simulation predicts that the number of positrons decreases by a factor of ten as they travel through the quad triplet system to the entrance of the first Dipole D1, see Table 4.2.
  
\begin{figure}[htbp] 
\centering
\includegraphics[scale=0.75]{3-Simulation/Figures/STSimSetupS2_3.png}
\caption{The generation and transportation of the positron beam in step 2. The virtual detectors were used to track the positrons.}
\label{fig:STSimSetupS2}
\end{figure}

\section{Step 3 - The Transportation of Positrons from the Entrance of The First Dipole to T2 and The Detection of 511~keV Photons}

In this step, the positron beam was generated at the entrance of the first dipole D1 and observed using virtual detector DD1UP. The beam was deflected $45^{\circ}$ by D1 and passed through the energy slit. The positrons diverged along the horizon according to their energy when traversing D1's magnetic field. The energy slit constrained the spatial distribution of the beam by blocking the beam using a 34.8~mm wide gap oriented vertically. Quadrupole Q7, located after the energy slit, was set to 3.5~A (similar to the experiment) in order to focus the beam before it enters a second dipole, D2. The beam was deflected another $45^{\circ}$ by D2 and sent through three quadrupoles towards the annihilation target T2 located at the end of the beamline as shown in Figure~\ref{fig:T2}.

\begin{figure}[htbp] 
\centering
\includegraphics[scale=0.4]{3-Simulation/Figures/HRRL_T2_2.png}
\caption{T2 and virtual detectors located upstream (DT2UP) and downstream (DT2DN) of T2 are shown at the center of the figure. NaI detectors and Pb shielding are located horizontally on both sides.}
\label{fig:T2}
\end{figure}

As shown in Figure~\ref{fig:T2}, two virtual, circular detectors DT2UP and DT2DN with a 48~mm diameter (48~mm is the inner diameter of the beam pipe) were placed upstream and downstream of T2 to detect positrons. Two other virtual detectors, DT2L and DT2R (not shown) with the same diameters as the annihilation target T2 were placed on the left and right side of the beam and parallel to T2 to detect positrons. Two additional virtual detectors were placed horizontally at the locations of NaI detectors, which were 170~mm away from the beamline center, to detect photons. 
Two-inch-thick Pb bricks with 2-inch diameter circular openings were positioned between T2 and the NaI virtual detectors. Each detector was surrounded by 2$''$ of Pb. When a positron annihilates inside T2, two back-to-back 511~keV photons are produced. 
An event is registered in the simulation when both NaI detectors observe a 511~keV photon.

\subsection{Positrons Detected by The Detection System}
The photons observed by each NaI detector are subject to the detector efficiency.
The detector efficiency chart (shown in Figure~\ref{fig:NaI_Ef}), obtained from Saint-Gobain Crystals~\cite{NaI-Eff}, indicates that the NaI crystal used in this experiment has an efficiency of 68\% for 511~keV photons. If two detectors are operated in coincidence mode, the detection efficiency of the system is $68\% \times 68\%~=~46.24\%$. 

Figure~\ref{fig:e+_Generated_and_Detected} shows the number of 511~keV photon pairs detected in coincidence mode (multiplied by 46.24\%) and overlaid with the positrons detected on DDNT1. In step 3 of the simulation, the quadrupole current for Q7 was simulated for 0, 3.5, and 10~A to study the effect of Q7 on the positron energy distribution. As shown in Figure~\ref{fig:e+_Generated_and_Detected}, the shape of the predicted positron energy distribution more closely resembling the one observed in the experiment when Q7 was set to the value used in the actual experiment, 3.5~A. Fewer positrons are predicted to reach T2 when the Q7 currents are 0 and 10~A.
The simulation's prediction for the experiment based on the beamline parameters used during the experiment are given in Table~\ref{tab:Sim_e+/e-} as the ratio of the number of e$^+$ detected in coincidence mode by NaI detectors to the number of e$^-$ incident on T1 for the measured positron energies.  The values in Table~\ref{tab:Sim_e+/e-} are compared with the experiment in the next chapter.

\begin{figure}[htbp] 
\centering
\includegraphics[scale=0.53]{3-Simulation/Figures/NaI_Ef_3.png}
\caption{NaI detector efficiency obtained from Saint-Gobain Crystals~\cite{NaI-Eff}. The lines corresponding to the different crystal sizes (in inches) are shown on the right side of the figure.}
\label{fig:NaI_Ef}
\end{figure}

\begin{figure}[htbp] 
\centering
\includegraphics[scale=0.77]{3-Simulation/Figures/overlay/e+_Generated_and_Detected.eps}
\caption{Positrons detected on virtual detector DDNT1 and 511~keV photon pairs detected by the NaI detectors in coincidence mode when Q7 set to 0~A, 3.5~A and 10~A.}
\label{fig:e+_Generated_and_Detected}
\end{figure}

\begin{table}
\centering
\caption{Positron to Electron Ratio Estimated by the Simulation.}
\begin{tabular}{ll}
\toprule
{Energy (MeV)}        & {Positron to Electron Ratio}   \\
\midrule
$1.02 \pm 0.03$ &    $(8.90 \pm 5.1) \times10^{-17}$    \\
$2.15 \pm 0.06$ &    $(10.10 \pm 0.06) \times10^{-15}$  \\
$3.00 \pm 0.07$ &    $(9.96 \pm 0.06) \times10^{-15}$  \\
$4.02 \pm 0.07$ &    $(9.67 \pm 0.03) \times10^{-15}$  \\
$5.00 \pm 0.06$ &    $(7.80 \pm 0.05) \times10^{-15}$   \\      
\bottomrule
\end{tabular}
\label{tab:Sim_e+/e-}
\end{table}

The remaining sections of this chapter use the above simulation to investigate beam loss, an observed asymmetry in the photon detectors, and systematic errors. Understanding the source of the beam loss through the beam line was critical to evaluating the veracity of the data. The large discrepancy of nine orders of magnitude between the observed positron production rate and the expected rate was difficult to believe. As quantified in the sections below, the poor beam quality from a medical linac combined with a poor positron collection system has a large negative impact on the efficiency of producing positrons from electrons. There is also an indication that the beam line alignment plays an even more important role than one might naively expect.  The following sections attempt to explains these effects as well as investigate their impact on the systematic error.

\subsection{Beam Loss Study}
The three step simulation method described in the previous section was used to predict the amount of beam loss at several locations along the beam line. Table~\ref{tab:sim-S2E} quantifies these predictions. In particular, columns 2 and 3 in Table~\ref{tab:sim-S2E} represent the number of positrons that would need to be generated in order to observe the number of positrons in the remaining columns. The number of 511~keV photon pairs detected by NaI detectors in coincidence mode is shown in the last column of Table~\ref{tab:sim-S2E}. The number of positrons observed by virtual detectors which were placed along the beamline and 511~keV photon pairs detected are shown in Figure~\ref{fig:TransEff}.

\begin{sidewaystable}
\centering
\caption{Simulation of $7.253 \times 10^{16}$ Electrons Incident on the T1: The Number of Positrons Transported and the Number of 511~keV Photons Detected in Coincidence Mode.}
\begin{tabular}{lccccccccc}
%\begin{tabular}{lllllllllll}
\toprule
Energy 	  		  & e$^{+}$ on 					& e$^{+}$ enter 	 			& e$^{+}$ enter 				& e$^{+}$ exit 					& e$^{+}$ enter 				& e$^{+}$ enter     			& e$^{+}$ exit 					& e$^{+}$ reach   				& 511~keV $\gamma$ \\  
(MeV)  	  		  & DDNT1   			& Q4   	     			& D1    				& D1   					&  Q7       			& D2    				& D2   					& T2   	  				&  by NaI  \\
 &  &  &  &  &  &   &   &   &  detectors \\
\midrule
$1.02 \pm 0.25$   & $2.5 \times 10^{12} $ & $1.4 \times 10^{10}  $ & $1.4 \times 10^{9} $ & $1.2 \times 10^{8}$   & $2.6 \times 10^{7} $ & $2.7 \times 10^{6} $  & $1.2 \times 10^{6} $  &  $4.3 \times 10^{3} $ & $7$\\
$1.50 \pm 0.25$   & $4.9 \times 10^{12} $ & $2.7 \times 10^{10}  $ & $2.8 \times 10^{9} $ & $1.8 \times 10^{8}$   & $9.1 \times 10^{7} $ & $2.7 \times 10^{7} $  & $1.2 \times 10^{7} $  &  $5.1 \times 10^{4} $ & $97  $\\
$2.15 \pm 0.25$   & $6.5 \times 10^{12} $ & $3.8 \times 10^{10}  $ & $3.8 \times 10^{9} $ & $3.8 \times 10^{8}$   & $1.7 \times 10^{8} $ & $7.8 \times 10^{7} $  & $3.6 \times 10^{7} $  &  $4.4 \times 10^{5} $ & $734 $\\
$2.50 \pm 0.25$   & $6.8 \times 10^{12} $ & $4.1 \times 10^{10}  $ & $4.1 \times 10^{9} $ & $4.4 \times 10^{8}$   & $2.3 \times 10^{8} $ & $1.1 \times 10^{8} $  & $5.0 \times 10^{7} $  &  $4.6 \times 10^{5} $ & $794 $\\
$3.00 \pm 0.25$   & $6.6 \times 10^{12} $ & $4.1\times 10^{10}   $ & $4.1 \times 10^{9} $ & $4.6 \times 10^{8}$   & $2.7 \times 10^{8} $ & $1.3 \times 10^{8} $  & $6.1 \times 10^{7} $  &  $4.5 \times 10^{5} $ & $723 $\\
$3.50 \pm 0.25$   & $6.1 \times 10^{12} $ & $3.9 \times 10^{10}  $ & $3.9 \times 10^{9} $ & $4.4 \times 10^{8}$   & $2.8 \times 10^{8} $ & $1.4 \times 10^{8} $  & $6.7 \times 10^{7} $  &  $4.4 \times 10^{5} $ & $716 $\\
$4.02 \pm 0.25$   & $5.3 \times 10^{12} $ & $3.5 \times 10^{10}  $ & $3.5 \times 10^{9} $ & $4.1 \times 10^{8}$   & $2.7 \times 10^{8} $ & $1.4 \times 10^{8} $  & $6.9 \times 10^{7} $  &  $4.3 \times 10^{5} $ & $702 $\\
$4.50 \pm 0.25$   & $4.6 \times 10^{12} $ & $3.1 \times 10^{10}  $ & $3.1 \times 10^{9} $ & $3.6 \times 10^{8}$   & $2.5 \times 10^{8} $ & $1.3 \times 10^{8} $  & $6.7 \times 10^{7} $  &  $4.0 \times 10^{5} $ & $646 $\\
$5.00 \pm 0.25$   & $3.8 \times 10^{12} $ & $2.7 \times 10^{10}  $ & $2.7 \times 10^{9} $ & $3.1 \times 10^{8}$   & $2.2 \times 10^{8} $ & $1.2 \times 10^{8} $  & $6.2 \times 10^{7} $  &  $3.6 \times 10^{5} $ & $566 $\\
$5.50 \pm 0.25$   & $3.0 \times 10^{12} $ & $2.2 \times 10^{10}  $ & $2.2 \times 10^{9} $ & $2.6 \times 10^{8}$   & $1.9 \times 10^{8} $ & $1.0 \times 10^{8} $  & $5.6 \times 10^{7} $  &  $3.3 \times 10^{5} $ & $533 $\\
\bottomrule
\end{tabular}
\label{tab:sim-S2E}
\end{sidewaystable}
 
\begin{figure}[htbp] 
\centering
\includegraphics[scale=0.75]{3-Simulation/Figures/Transporation_Efficiency/Ef.eps}
\caption{Predicted number of positrons transported. Black squares: positrons incident on DDNT1. Red squares: positrons entered Q4. Blue squares: positrons entered D1. Magenta squares: positrons exited D1. Black circles: positrons entered Q7. Red circles: positrons entered D2. Blue circles: positrons exited D2. Magenta circles: positrons incident on DT2UP. Black triangles: 511~keV photons detected by NaI detectors in coincidence mode.}
\label{fig:TransEff}
\end{figure}

The simulation's beam loss prediction between the positron production target, T1, and the first collection quadrupole, Q4, may be compared to a simple solid angle argument to demonstrate the veracity of the prediction.   
As shown in Table~\ref{tab:sim-S2E}, the positron energy distribution is divided into 10 bins. Positrons were detected at both detector DDNT1 (28.5~mm downstream T1) and detector DQ4 (484.4~mm downstream T1). The ratio of positrons detected by DDNT1 to those detected by DQ4 is about 157. The solid angle that the entrance of Q4 makes with respect to T1, if one assumes that T1 is a point source of positrons is about 0.6$\pi$~steradians. The distance between T1 and the virtual detector DQ4 is $484.4$~mm and the radius of DQ4 is $24$~mm. The solid angle of the DQ4 is $\Omega_{\text{Q4}}=\frac{\pi r^2}{d^2}=\frac{\pi 24^2}{484.4^2}$~steradians. Positrons make up a cone with a 45$^\circ$ half angle, which is $\Omega_{\text{beam}}=0.6\pi$~steradians in solid angle. The ratio of the two solid angles, $\Omega_{\text{Q4}}/\Omega_{\text{beam}}$, is 1:244, $i.e.$ 1 out of 244 positrons makes it from T1 to DQ4, assuming that the positron beam is isotropic inside the cone. 
However, the positron beam peaks at a smaller angle as shown in Figure~\ref{fig:DDNT1_results} (c) and (d), and as a result more positrons are transported from DDNT1 to DQ4 which may result in a ratio closer to 1:157. 

%In a simulation where the dipoles were set to transport 3~MeV positron, the beam energy distribution at the exit of D1, entrance %of Q7, exit of Q7, entrance of D2, exit of D2, and T2 are shown in Figure~\ref{fig:dipole-trans} and Table~\ref{tab:pos-beam-loss}. 
%In Table~\ref{tab:pos-beam-loss}, the relative counts reported in the last column are obtained by dividing the second %column over 723, the number of 511~keV photons detected by NaI detectors in coincidence mode.

The first dipole is another region where substantial beam loss is predicted by the simulation. The positron counts dropped one order of magnitude when positrons are transported from the entrance of D1 to the exit of D1. This could be explained by beam scraping on the vacuum chamber. The width of the dipole chamber is 18~mm and the beam pipe diameter is 48~mm. Positrons scraping on the top and bottom of the chamber would be lost. When the dipole was set to transport 3~MeV positrons, positrons having an energy range between 2.8~MeV and 3.3~MeV are transported to the exit D1 as shown in Figure~\ref{fig:dipole-trans}. This small energy range of 0.5 MeV is only one tenth of the full range subtended somewhat uniformly by the positrons and could easily explain the order of magnitude drop in the number of positrons that traverse the first dipole.  

\begin{table}
\centering
\caption{The Positron Beam Loss along the Beamline When Dipoles Set to Transport 3~MeV Positrons.}
\begin{tabular}{lcc}
\toprule
{Beam Sample} & {Absolute Counts}     &    {Relative Counts}    \\
{Locations}  & {}     				&    {}    \\
\midrule
e$^+$ on DDNT1	&	$6.6\times10^{12}$	&	$9.4\times10^{10}$	   \\
e$^+$ enter Q4	&	$4.1\times10^{10}$ 	&	$5.8\times10^{7}$	   \\
e$^+$ enter D1	&	$4.1\times10^{9}$	&	$5.8\times10^{6}$	   \\
e$^+$ exit D1	&	$4.6\times10^{8}$	&	$6.6\times10^{5}$	   \\
e$^+$ enter Q7	&	$2.7\times10^{8}$	&	$3.8\times10^{5}$	   \\
e$^+$ enter D2	&	$1.3\times10^{8}$	&	$1.9\times10^{5}$	   \\
e$^+$ exit D2	&	$6.1\times10^{7}$	&	$8.7\times10^{4}$	   \\
e$^+$ on T2		&	$4.4\times10^{5}$	&	$6.3\times10^{2}$	   \\
e$^+$ annihilated in T2 &	$2.5\times10^{5}$&	$3.6\times10^{2}$  \\
511~keV photons on NaI	&	723		&	1	   				   \\
	
\bottomrule
\end{tabular}
\label{tab:pos-beam-loss}
\end{table}

%The dipoles in the simulation were set to transport 3~MeV positrons to the annihilation target T2, where 58\% of the positrons %annihilated and created 511~keV photon pairs. 0.7\% of the photon pairs are lost when passing through the vacuum windows. The left %NaI detector observed 2897 511~keV photons and the right one detected 4100. The ratio of 511~keV photon pairs created in T2 to %those that reached the right and left NaI are 88:1 and 62:1 respectively. This is comparable to the ratio made by out-going %photons to the solid angle made by a NaI detector. 

%The distance between the NaI detectors and the beamline center is 170~cm. The Pb shielding has a 2-inch-diameter hole facing T2. %Assuming that positrons annihilated at the center of T2, the solid angle made by a NaI detector is $\Omega=\frac{\pi %r^2}{d^2}=\frac{\pi 25.4^2}{170^2}$~steradian. The 511~keV photons created during the annihilation are opposite in direction and %emitted from the two surfaces of T2 (ignoring the ones escaping from the edge). Photons emitted in each side make a solid angle of %a hemisphere, $2\pi$~steradian. The ratio of $2\pi$~steradian to solid angle of the NaI detector is about 90:1.

%1518 511~keV photon pairs are detected by NaI detectors in coincidence mode. The ratio of 511~keV photon pairs created to the ones %detected in coincidence mode is 168:1. If one counts photons in coincidence mode with the 46.24\% detection efficiency of the %system, it cuts the rate in half. 

\begin{figure}[htbp] 
\centering
\includegraphics[scale=0.74]{3-Simulation/Figures/Transporation_Efficiency/En2.eps}
\caption{Energy distribution of positrons transported when dipoles were set to bend 3~MeV positrons along the beamline.}
\label{fig:dipole-trans}
\end{figure}


\section{Analysis of the Photon Count Asymmetry in NaI Detectors}

The beam right NaI detector observed higher rates that the beam left detector in both the experiment and the simulation.  
Eight virtual NaI detectors were placed as shown in Figure~\ref{fig:NaI-detectors} to study this count asymmetry. 
The orientation of T2 with respect to the incident positron beam was simulated for three cases. 
In the first case, the photon count rate was predicted by the simulation when the T2 is perpendicular to the beam (its area vector is parallel to the beam).
In the second case, T2 was rotated clockwise (starting with the T2 location in the first case) about the x-axis (the axis pointing beam left) by 45$^\circ$ making the upstream side of T2 face downward as shown in Figure~\ref{fig:NaI-detectors} (a). 
In the third case, T2 was rotated clockwise about the y-axis (axis points beam up) by 45$^\circ$ so it is facing the beam right NaI detector as shown in Figure~\ref{fig:NaI-detectors} (b). 
The setup in the third case is similar to the setup in the experiment and the simulation described in the beginning of this chapter. 

\begin{figure}[htbp] 
\begin{tabular}{ccc}
\centerline{\scalebox{0.45} [0.45]{\includegraphics{3-Simulation/Figures/Photon_Counts_Asymmetry/T2FaceDown.png}}}\\
(a)  \\
\centerline{\scalebox{0.45} [0.45]{\includegraphics{3-Simulation/Figures/Photon_Counts_Asymmetry/T2Exp.png}}} \\
(b)\\
\end{tabular}
\caption{NaI detector locations around T2. The positron beam (blue line) is traveling along the z-axis (into the paper in the right figures). (a) T2 was rotated counter-clockwise about the x-axis by 45$^\circ$ positioning the upstream side of the T2 such that it faces the bottom NaI detector. (b) T2 was positioned as in the experiment. It was first rotated to the position as in (a), then it was rotated clockwise about the y-axis by 45$^\circ$, positioning the upstream side of the T2 such that it faces the beam right NaI detector.}
\label{fig:NaI-detectors}
\end{figure}

\begin{table}
\centering
\caption{Number of 511~keV photons observed by the NaI detectors.}
\begin{tabular}{lcccccc}
\toprule
{T2 Placement} 	  & Exp.  & Perp.  & Face    & Face    & Face    & Face    \\
			 	  &       & to Beam& Down    & Down    & Down    & Down    \\
\midrule
{Energy}      	  & 3~MeV & 3~MeV & 3~MeV    & 1~MeV   & 6~MeV   & 10~MeV  \\
\midrule
{NaI Right}  	  & 18085 & 7610  & 7160      &  10315   &  6209    &  4436    \\
{NaI Left}  	  & 12798 & 7651  & 7114      &  10254   &  6111    &  4487    \\
{NaI Top Right}   & 7050  & 7580  & 12964     &  12371   &  14698   &  10636    \\
{NaI Bottom Left} & 7084  & 7609  & 18563     &  20238   &  15989   &  10239    \\
{NaI Top}		  & 12687 & 7599  & 14810     &  14332   &  16131   &  11479    \\
{NaI Bottom}	  & 18008 & 7609  & 18874     &  20193   &  16950   &  11181    \\
{NaI Top Left}	  & 14632 & 7656  & 12818     &  12268   &  14812   &  10735    \\
{NaI Bottom Right}& 18764 & 7623  & 18415     &  20197   &  16004   &  10317    \\  
\bottomrule
\end{tabular}
\label{tab:photon-counts-asym}
\end{table}

The 511~keV photon counts observed by eight virtual NaI detectors for one million positrons are given in Table~\ref{tab:photon-counts-asym} for each configuration. 
When T2 was placed perpendicular to the incoming positron beam, the 511~keV photons created inside T2 have the same probability to escape from T2 and reach any one of the detectors. As shown in the third column of Table~\ref{tab:photon-counts-asym}, all eight detectors observed a similar number of photons. 
In a separate simulation, T2 was positioned as in the first case and impinged by 1, 3, 6, 10~MeV positrons. The average distance traveled by positrons inside T2 before annihilation was $0.0847 \pm 0.0001$, $0.3006 \pm 0.0004$, $0.511 \pm 0.001$, $0.564 \pm 0.008$~mm for the four energies respectively. 
In another GEANT4 simulation, T2 was positioned according to the second case and two virtual NaI detectors were placed on both the top and the bottom of T2. 
The bottom detector observed more photons than the top one when 511~keV photons were generated 0.3006~mm inside T2 isotropically. Photons are more likely to reach the bottom detector because they would travel through a thinner layer of tungsten to reach it.

In the second case, shown in Figure~\ref{fig:NaI-detectors} (a), the left and the right NaI detectors had the lowest counts for 3~MeV positrons as shown in the fourth column of Table~\ref{tab:photon-counts-asym}. There is less detection probability for a photon traversing T2 in the radial direction towards the left/right detectors than the top/bottom surfaces of T2 due to the amount of material. 
The average distance traveled by 3~MeV positrons inside T2 before annihilation was $0.3006 \pm 0.0004$~mm. In this case, positrons annihilated near the upstream face of T2.
For this reason, the bottom, bottom right, and bottom left NaI detectors (facing the upstream side of T2) observed more photons than the top, top right, and top left as shown in the fourth column of Table~\ref{tab:photon-counts-asym}.

The lower the positron beam energy, the shorter the annihilation depth, and the bigger asymmetry in the counts. As shown in the fourth and fifth columns of Table~\ref{tab:photon-counts-asym}, more/less 511~keV photons were observed on the bottom/top detectors with the 1~MeV positron beam than with the 3~MeV. As the positron beam energy increases, as shown in the sixth and seventh columns of Table~\ref{tab:photon-counts-asym}, the top and bottom detectors observed a similar number of photons, because positrons annihilate more uniformly inside T2 and the asymmetry in the counts decreases. With the increasing positron beam energy, fewer positrons were annihilated inside T2 and more penetrated through which was shown in the top two rows of Table~\ref{tab:photon-counts-asym}. 

For the third case, shown in Figure~\ref{fig:NaI-detectors} (b), the top right and left bottom detectors observed the lowest counts, because a photon would need to travel in the radial direction to reach these two detectors as shown in the second column of Table~\ref{tab:photon-counts-asym}. The right, bottom and bottom right (facing upstream face of T2) observed more photons than the left, top, top left.

According to the simulation, the asymmetry in the photon counts was due to the average positron annihilation depth and the photon attenuation inside T2. Low energy positrons tend to annihilate and produce photon pairs near the incident surface. The created photons are more likely to be detected from the incident surface. 
In the experiment and the simulation, the right NaI detector was facing the upstream side of T2 and observed more 511~keV photons than the left. 

\section{Quadrupole Triplet Collection Efficiency 
\\Study}

A G4beamline simulation was carried out to study the collection efficiency of the second quadrupole triplet (Q4, Q5, and Q6). 
%There is no magnet scanning option in G4beamline as in the ELEGANT code. 
The second quadrapole triplet magnets were set to the similar setting as in the experiment. 
In this simulation, 5,475,869,400 positrons were generated at DDNT1 and transported to DD1 to study the quadrupole triplet positron collection and transportation efficiency. Six quadrupole current settings of the triplet system were simulated as shown in Table~\ref{tab:triplet-eff}. For different quadrupole current settings, no significant differences were observed in the number of positrons, transverse beam profiles, and momentum distributions. The ratio of positrons generated at DDNT1 to the ones that enter D1 is 1525:1.

\begin{sidewaystable}
\centering
\caption{Quadrupole Triplet System Collection and Transportation Efficiency Data.}
\begin{tabular}{cccccccccccccc}
\toprule
Q4   &  Q5  &   Q6 & Entries &  $x$ &  $\sigma_{x} $ &   y  &  $\sigma_{y}$  &  $P_{x}$ &  $\sigma_{P_{x}}$ &   $P_{y}$ &  $\sigma_{P_{y}}$  & $P_{z}$ &  $\sigma_{P_{z}}$ \\
\midrule
A  &  A &  A &         & mm &   mm         & mm &     mm      &  MeV &   MeV &    MeV &   MeV &  MeV &  MeV \\
\midrule
-1   &  2   &  -1   & 3587220 &  -0.005   & 12 & 0.030 &  12  & $~~3.0 \times 10^{-5}$   & 0.0461 & -0.00216 &  0.04553  &  3.848   & 1.875 \\
-2   &  4   &  -2   & 3591423 &  -0.012   & 12 & 0.049 &  12  & $~~2.2 \times 10^{-5}$   & 0.0461 & -0.00211 &  0.04554  &  3.848   & 1.875 \\
1    & -2   &   1   & 3591509 &  -0.009   & 12 & 0.040 &  12  & $-1.5 \times 10^{-5}$   & 0.0462 & -0.00216 &  0.04557  &  3.849   & 1.876 \\
1    &  2   &   1   & 3589854 &  -0.005   & 12 & 0.034 &  12  & $-1.8 \times 10^{-5}$   & 0.0462 & -0.00216 &  0.04556  &  3.849   & 1.876 \\
2    & -4   &   2   & 3592977 &  -0.007   & 12 & 0.032 &  12  & $-3.3 \times 10^{-6}$   & 0.0462 & -0.00217 &  0.04549  &  3.849   & 1.876 \\ 
2    &  4   &   2   & 3589495 &  -0.004   & 12 & 0.033 &  12  & $~~3.0 \times 10^{-6}$   & 0.0462 & -0.00218 &  0.04554  &  3.849   & 1.875 \\  
%    &     &      &   &       &   &   &     &      &   &   &     &      &  \\                   
\bottomrule
\end{tabular}
\label{tab:triplet-eff}
\end{sidewaystable}


\section{Systematic Errors Study using Simulation}
Different sources of systematic errors in the positron production experiment was studied using G4beamline simulation. The power supply of the magnets might fluctuate $\pm$0.1~A which will change the magnetic field strength of the magnets. The misalignment of the magnets would also contribute to the systematic error of the experiment.

\subsection{Systematic Error Created by The Uncertainty in The Magnetic Fields Strength of The Magnets}
The field strength of the magnets are dependent on the current provided by the power supply. The uncertainty of the magnet power supply is 0.1~A.
Systematic errors for the positron counts were estimated by carrying out simulations with different magnetic field settings as shown in Table~\ref{tab:sim-error}.  

The magnet settings are indicated in the top three rows of Table~\ref{tab:sim-error}. The transported positron energy is given by the first column. 
The 511~keV photon pairs counted in coincidence mode ($i.e.$ the original count multiplied by 46.42\%) for different magnet settings are given in the corresponding columns below.
The ``Max''/``Min'' in the table refers to the Maximum/Minimum magnetic field strength when the magnet coil current is at $I_{\text{max}}$/$I_{\text{min}}$, where $I_{\text{max}}$ = $I_{\text{def}} + 0.1$~A and $I_{\text{min}}$ = $I_{\text{def}} - 0.1$~A. ``def'' refers to the default magnetic field strength of the magnet.

The average counts and fractional errors are shown in the last two columns of Table~\ref{tab:sim-error}. 
The fractional error in the 511~keV photon pair counts is calculated by dividing the standard deviation of counts by the counts in default magnet setting, $i.e.$ $\frac{\text{standard deviation of counts}}{\text{counts in default setting}}$.
%The fractional error in the 511~keV photon pair counts is calculated by dividing the standard deviation of counts at different settings by the counts in default magnet setting, $i.e.$ $\frac{\text{standard deviation of counts}}{\text{counts in default setting}}$.

\begin{sidewaystable}
\centering
\caption{Systematic Error Study: Counts of 511~keV Photon Pairs for Different Magnet Settings.}
%\begin{tabular}{cccccccccccc}
\begin{tabular}{lllllllllllll}
\toprule
D1          &  Max   & Min   & Max   & Min   & Max   & Max   & Min   & Min   & Def   &      &    \\
Q7          &  Def   & Def   & Def   & Def   & Max   & Min   & Min   & Def   & Def   &      &    \\
D2	        &  Max   & Min   & Def   & Def   & Def   & Def   & Def   & Def   & Def   &      &    \\
\midrule
Energy (MeV)&   	 &  	 &  	 &    	 &     	 & 	     & 	     & 	     &       &  Average     & Fractional Error \\
\midrule
0.765~1.265	 &	12	 &	2	 &	3	 &	1	 &	1	 &	3	 &	3	 &	0	 &	6	 &	4	 &	57.3	\%	  \\
1.25~-~1.75	 &	95	 &	75	 &	70	 &	86	 &	85	 &	92	 &	129	 &	86	 &	97	 &	90	 &	17.6	\%	  \\
1.85~-~2.35	 &	674	 &	724	 &	737	 &	769	 &	686	 &	747	 &	803	 &	681	 &	734	 &	728	 &	5.8	\%	  \\
2.25~-~2.75	 &	781	 &	755	 &	753	 &	759	 &	772	 &	963	 &	794	 &	763	 &	794	 &	793	 &	8.3	\%	  \\
2.75~-~3.25	 &	739	 &	737	 &	765	 &	752	 &	738	 &	698	 &	738	 &	757	 &	723	 &	739	 &	2.7	\%	  \\
3.25~-~3.75	 &	713	 &	699	 &	747	 &	712	 &	707	 &	751	 &	718	 &	705	 &	716	 &	719	 &	2.5	\%	  \\
3.77~-~4.27	 &	690	 &	708	 &	715	 &	676	 &	704	 &	658	 &	706	 &	715	 &	701	 &	697	 &	2.7	\%	  \\
4.25~-~4.75	 &	627	 &	666	 &	634	 &	646	 &	635	 &	675	 &	637	 &	653	 &	646	 &	646	 &	2.5	\%	  \\
4.75~-~5.25	 &	558	 &	595	 &	594	 &	603	 &	606	 &	688	 &	582	 &	582	 &	566	 &	597	 &	6.7	\%	  \\
5.25~-~5.75	 &	513	 &	536	 &	535	 &	522	 &	525	 &	535	 &	518	 &	544	 &	533	 &	529	 &	1.9	\%	  \\

\bottomrule
\end{tabular}
\label{tab:sim-error}
\end{sidewaystable}
 
\subsection{Systematic Error Introduced by The Misalignment of The Magnets and The Uncertainty in The Electron Beam}
The misalignment of the magnets are one of the main sources of systematic error. The 0 and 90 degree beamlines were aligned with a laser beam, while the 45 degree beamline was placed without any reference laser. Therefore, a misalignment was most likely to occur at the 45 degree beamline. The misalignment was estimated to be 5~mm.
To study the effect of misalignments from different sources, beam line elements were shifted as shown in Table~\ref{tab:sys-sim-single}. The measured positron to electron ratios were given as well. Among the different sources, the systematic error produced when the first dipole, D1, is misaligned appears to have the largest impact. 

\begin{longtable}{|c|c|c|}

\caption{Positron to Electron Ratio When a Single Beamline Element Is Misaligned and the Percent Error Compared to the Ratio When No Misalignment.} \label{tab:sys-sim-single}\\
\hline

{Energy (MeV)} & {45 degree line moved } & {percent error}  \\
{}   & {to beam right by 3~mm} & { }  \\

\hline
\endfirsthead
\multicolumn{3}{c}%
{\tablename\ \thetable\ -- \textit{Continued from previous page}} \\
%\hline
%{Energy (MeV)} & { } & {percent error}  \\
%{}   & {} & { }  \\
%\hline
\endhead
\hline \multicolumn{3}{r}{\textit{Continued on next page}} \\
\endfoot
\hline
\endlastfoot
$1.02 \pm 0.03$ &    $(	8.3	 \pm 3.4) \times10^{-17}$  &	-6.7	\%	 \\
$2.15 \pm 0.06$ &    $(	7.4	 \pm 0.3) \times10^{-15}$  &	-26.7	\%	 \\
$3.00 \pm 0.07$ &    $(	8.1	 \pm 0.3) \times10^{-15}$  &	-18.7	\%	 \\
$4.02 \pm 0.07$ &    $(	8.2	 \pm 0.3) \times10^{-15}$  &	-15.2	\%	 \\
$5.00 \pm 0.06$ &    $(	7.2	 \pm 0.3) \times10^{-15}$  &	-7.7	\%	 \\ 

\midrule
{Energy (MeV)} & {D1 raised up by 5~mm } & {percent error}  \\
\midrule
$1.02 \pm 0.03$ &    $(	3.2	 \pm 5.7) \times10^{-17}$  &	-64.0	\%	 \\
$2.15 \pm 0.06$ &    $(	6.1	 \pm 0.9) \times10^{-15}$  &	-39.6	\%	 \\
$3.00 \pm 0.07$ &    $(	7.2	 \pm 1.0) \times10^{-15}$  &	-27.7	\%	 \\
$4.02 \pm 0.07$ &    $(	5.5	 \pm 0.9) \times10^{-15}$  &	-43.1	\%	 \\
$5.00 \pm 0.06$ &    $(	4.8	 \pm 0.8) \times10^{-15}$  &	-38.5	\%	 \\


\midrule
{Energy (MeV)} & {D2 lowered down by 5~mm } & {percent error}  \\
\midrule
$1.02 \pm 0.03$ &    $(	1.6	 \pm 2.3) \times10^{-17}$  &	-82.0	\%	 \\
$2.15 \pm 0.06$ &    $(	6.9	 \pm 1.0) \times10^{-15}$  &	-31.7	\%	 \\
$3.00 \pm 0.07$ &    $(	8.6	 \pm 1.1) \times10^{-15}$  &	-13.7	\%	 \\
$4.02 \pm 0.07$ &    $(	6	 \pm 0.9) \times10^{-15}$  &	-38.0	\%	 \\
$5.00 \pm 0.06$ &    $(	7.7	 \pm 1.0) \times10^{-15}$  &	-1.3	\%	 \\


\midrule
{Energy (MeV)} & {detectors moved up by 3~mm, downstream } & {percent error}  \\
{}   & {by 3~mm, and right by 3~mm} & { }  \\
\midrule
$1.02 \pm 0.03$ &    $(	9.6	 \pm 8.1) \times10^{-17}$  &	7.9	\%	 \\
$2.15 \pm 0.06$ &    $(	7.4	 \pm 1.0) \times10^{-15}$  &	-26.7	\%	 \\
$3.00 \pm 0.07$ &    $(	8.2	 \pm 1.1) \times10^{-15}$  &	-17.7	\%	 \\
$4.02 \pm 0.07$ &    $(	9.6	 \pm 1.2) \times10^{-15}$  &	-0.7	\%	 \\
$5.00 \pm 0.06$ &    $(	7.7	 \pm 1.0) \times10^{-15}$  &	-1.3	\%	 \\


\midrule
{Energy (MeV)} & {T2 moved up by 3~mm, downstream } & {percent error}  \\
{}   & {by 3~mm, and right by 3~mm} & { }  \\
\midrule
$1.02 \pm 0.03$ &    $(	6.4	 \pm 6.6) \times10^{-17}$  &	-28.1	\%	 \\
$2.15 \pm 0.06$ &    $(	7.7	 \pm 1.0) \times10^{-15}$  &	-23.8	\%	 \\
$3.00 \pm 0.07$ &    $(	7.9	 \pm 1.1) \times10^{-15}$  &	-20.7	\%	 \\
$4.02 \pm 0.07$ &    $(	7.9	 \pm 1.0) \times10^{-15}$  &	-18.3	\%	 \\
$5.00 \pm 0.06$ &    $(	7.7	 \pm 1.0) \times10^{-15}$  &	-1.3	\%	 \\


\midrule
{Energy (MeV)} & {T2 rotated c.w. about x-axis by 5$^\circ$ } & {percent error}  \\
{}   & {and c.c.w. about y-axis by 5$^\circ$} & { }  \\
\midrule
$1.02 \pm 0.03$ &    $(	6.4	 \pm 6.6) \times10^{-17}$  &	-28.1	\%	 \\
$2.15 \pm 0.06$ &    $(	7.2	 \pm 1.0) \times10^{-15}$  &	-28.7	\%	 \\
$3.00 \pm 0.07$ &    $(	6.4	 \pm 0.9) \times10^{-15}$  &	-35.7	\%	 \\
$4.02 \pm 0.07$ &    $(	7.2	 \pm 1.0) \times10^{-15}$  &	-25.5	\%	 \\
$5.00 \pm 0.06$ &    $(	6.1	 \pm 0.9) \times10^{-15}$  &	-21.8	\%	 \\
\end{longtable}


There are many combinations of magnet misalignments (Q1$-$Q10, D1, and D1), the ``worst case scenario" is considered here to estimate the largest source of systematic error. 
A ``worst case scenario" was created by misaligning the beamline as given in Table~\ref{tab:mis-sim}. The results is given in Table~\ref{tab:sys-sim} in terms of the ratio of the number of e$^+$ detected in coincidence mode by NaI detectors to the number of e$^-$ incident on the T1. Beam line misalignments far beyond what can be reasonably argued indicate drops in positron rate dropped by 60\% to 80\%. A more realistic misalignment of beamline elements dropped this rate around 20\%. Since the chance of the beamline misalignment is large, the systematic error of the simulation is argued to be at least around 20\%.

\begin{table}
\centering
\caption{Misalignment of The Beam in The Worst Case Scenario.}
\begin{tabular}{ll}
\toprule
{Beam Element}        & {Placement}          \\
\midrule
D1 &    raised up by 5~mm    \\
D2 &    lowered down by 5~mm  \\
45 degree line  & shifted to beam right by 3~mm     \\
NaI detectors  &   raised up by 3~mm, moved 3~mm beam downstream and 3~mm  \\
  &  to beam right     \\ 
T2&   raised up by 3~mm, moved 3~mm beam downstream and 3~mm \\
  &  to beam right, rotated clockwise 5$^\circ$ about x-axis and counter \\
  &  clockwise 5$^\circ$ about y-axis    \\ 
\bottomrule
\end{tabular}
\label{tab:mis-sim}
\end{table} 

\begin{table}
\centering
\caption{Positron to Electron Ratio Predicted When Beamline is Misaligned as Given in Worst Case Scenario and the Percent Error Compared to the Ratio When No Misalignment.}
\begin{tabular}{lll}
\toprule
{Energy (MeV)}  & 	 {Positron to Electron Ratio}      &   percent error \\
\midrule
$1.02 \pm 0.03$ &    $(	3.4	 \pm 3.5)  \times10^{-18}$ & 	-61.8	\%	 \\	
$2.15 \pm 0.06$ &    $(	1.79 \pm 0.50) \times10^{-15}$ & 	-83.2	\%	  \\
$3.00 \pm 0.07$ &    $(	3.39 \pm 0.68) \times10^{-15}$& 	-66.9	\%	  \\	
$4.02 \pm 0.07$ &    $(	3.14 \pm 0.66) \times10^{-15}$& 	-67.9	\%	  \\	
$5.00 \pm 0.06$ &    $(	2.82 \pm 0.62) \times10^{-15}$& 	-64.1	\%	  \\
\bottomrule
\end{tabular}
\label{tab:sys-sim}
\end{table}