Sadiq Thesis Latex

Chapter 3: Data Analysis File:Sadiq thesis chapt 3.txt

\chapter{Data Analysis} In the first section of this chapter, the measurement of the HRRL emittance and Twiss parameters using the quadrupole scanning method is described. The current in the first quadrupole, Q1, was changed incrementally and the electron beam shapes were observed using the OTR screen located at the end of the 0 degree beamline. The emittance and Twiss parameters are obtained by fitting a parabolic function to the plot $\sigma_{x,y}^2$ vs K$_1$L (K$_1$ is quadrupole strength and L is the quadrupole effective length).

The second section describes the energy distribution measurement of the HRRL when it was tuned to generate an electron beam with its peak at 12~MeV. A Faraday cup, FC2, was placed at the end of the 45 degree beamline to measure the electron beam current when D1 was on and D2 was off. The energy scan data and the MATLAB script used for data analysis are given in the appendix A.

Last section is about the positron produced using the HRRL located in the Beam Lab of Physics Department at Idaho State University. The electron beam from the HRRL with 12~MeV peak incident on the tungsten production target (T1) produced positrons downstream and positrons were transported to annihilation target (T2), where they annihilate and crated back to back scattered 511~keV photon pairs. Two NaI detectors, their shielding, placement, and operation modes are described. The different methods to count positron rates are discussed. The electron beam current of the HRRL during the positron production was monitored by a scintillator. The calibration of the scintillator using a Faraday cup and oscilloscope is described. At last, the ratios of positrons detected using NaI detectors in coincidence mode to the electrons impinging on T1 are given for 1$\sim$5~MeV positron beam.

\section{Emittance Measurement} The HRRL beam emittance was measured by using an optical transition radiation (OTR) screen. This transition radiation was theoretically predicted by Ginzburg and Frank~\cite{Ginzburg-Frank} in 1946 to occur when a charged particle passes the boundary of two media. An OTR based viewer (31.75~mm in diameter and 10~$\mu$m thick aluminum) was installed to observe the electron beam size at 37.2~mA (higher currents might saturate the camera) electron peak currents, available using the HRRL at $15\pm1.5$~MeV with 200~ns macro pulse width. The visible light is produced when a relativistic electron beam crosses the boundary of two media (aluminum and air) with different dielectric constants.

Light is emitted in a conical shape at backward angles with the peak intensity at an apex angle of $\theta = 1/\gamma$ ($\gamma$ is relativistic factor) with respect to the incident electron's angle of reflection. A 15 MeV electron accelerated by the HRRL would emit light at $\theta = 2^\circ$. Orienting the OTR target at 45${^\circ}$ with respect to the incident electron beam will result in the high intensity photons being observed at an angle of 90${^\circ}$ with respect to the incident beam, see Figure~\ref{q-scan-layout}. These backward-emitted photons are observed using a digital camera and can be used to measure the shape and the intensity of the electron beam. Although an emittance measurement can be performed in several ways~\cite{emit-ways, sole-scan-Kim}, the quadrupole scanning method~\cite{quad-scan} was used to measure the emittance and Twiss parameters in this work.

\subsection{Emittance Measurement Using Quadrupole Scanning Method} Fig.~\ref{q-scan-layout} illustrates the beamline components used to measure the emittance for the quadrupole scanning method. A quadrupole is positioned at the exit of the linac to focus or de-focus the beam as observed on the OTR view screen. The 3.1~m distance between the quadrupole and the screen was chosen in order to minimize chromatic effects and to satisfy the thin lens approximation. %The quadrupole and the screen are located far away to minimize chromatic effects and to increase the veracity of the thin lens approximation used to calculate beam optics. \begin{figure}[htbp] \centering \includegraphics[scale=0.65]{1-Introduction/Figures/quad_scan_setup2.eps} \caption{Apparatus used to measure the beam emittance.} \label{q-scan-layout} \end{figure} Assuming the thin lens approximation, $\sqrt{k_1}L << 1$, is satisfied, the horizontal transfer matrix of a quadrupole magnet may be expressed as % thin lens approximation (sqrt{k1}*L << 1). In our case sqrt{k1}*L =0.07 \begin{equation} \label{quad-trans-matrix} \mathrm{\mathbf{Q}}=\Bigl(\begin{array}{cc} 1 & 0\\ -k_{1}L & 1 \end{array}\Bigr)=\Bigl(\begin{array}{cc} 1 & 0\\ -\frac{1}{f} & 1 \end{array}\Bigr), \end{equation} where $L$ is the length of quadrupole and $f$ is the focal length. $k_{1}$ is the quadrupole strength given by \begin{equation} \label{eq:k1} k_1 = 0.2998 \frac{g\text{(T/m)}}{p\text{(GeV/c)}} = 0.2998 \frac{ \frac{ B(I(\text{A})) }{ R_{\text{q}} } }{p\text{(GeV/c)}}, \end{equation} where $g$ is the gradient of the quadrupole with the Bore aperture radius $R_q$ at a given coil current $I$ and $p$ is the momentum of the electron beam. A matrix representing the drift space between the quadrupole and screen is given by \begin{equation} \label{drift-trans-matrix} \mathbf{\mathbf{S}}=\Bigl(\begin{array}{cc} 1 & l\\ 0 & 1 \end{array}\Bigr), \end{equation} where $l$ is the distance between the scanning quadrupole and the screen. The transfer matrix $\mathbf{M}$ of the scanning region is given by the matrix product $\mathbf{SQ}$. In the horizontal plane, the beam matrix at the screen ($\mathbf{\sigma_{s}}$) is related to the beam matrix of the quadrupole ($\mathbf{\sigma_{q}}$) using the similarity transformation \begin{equation} \mathbf{\mathbf{\sigma_{s}=M\mathrm{\mathbf{\mathbf{\sigma_{q}}}}}M}^{\mathrm{T}}. \end{equation} where the $\mathbf{\sigma_{s}}$ and $\mathbf{\sigma_{q}}$ are defined as~\cite{SYLee} \begin{equation} \mathbf{\mathbf{\sigma_{s,\mathnormal{x}}=}}\Bigl(\begin{array}{cc} \sigma_{\textnormal{s},x}^{2} & \sigma_{\textnormal{s},xx'} \\ \sigma_{\textnormal{s},xx'} & \sigma_{\textnormal{s},x'}^{2} \end{array}\Bigr) ,\; \mathbf{\mathbf{\sigma_{q,\mathnormal{x}}}}=\Bigl(\begin{array}{cc} \sigma_{\textnormal{q},x}^{2} & \sigma_{\textnormal{q},xx'}\\ \sigma_{\textnormal{q},xx'} & \sigma_{\textnormal{q},x'}^{2} \end{array}\Bigr). \end{equation} \noindent %By defining the new parameters~\cite{quad-scan}, $A \equiv \sigma_{11},~B \equiv \frac{\sigma_{12}}{\sigma_{11}},~C \equiv\frac{\epsilon_{x}^{2}}{\sigma_{11}}$ By defining the new parameters~\cite{quad-scan} \begin{equation} A \equiv l^2\sigma_{\textnormal{q},x}^{2},~B \equiv \frac{1}{l} + \frac{\sigma_{\textnormal{q},xx'}}{\sigma_{\textnormal{q},x}^{2}},~\text{and}~C \equiv l^2\frac{\epsilon_{x}^{2}}{\sigma_{\textnormal{q},x}^{2}}, \end{equation} the matrix element $\sigma_{\textnormal{s},x}^{2}$, the square of the beam size's rms at the screen, may be expressed as a parabolic function of the product of $k_1$ and $L$ \begin{equation} \sigma_{\textnormal{s},x}^{2}=A(k_{1}L)^{2}-2AB(k_{1}L)+(C+AB^{2}). \label{par_fit} \end{equation}

The emittance measurement was performed by changing the quadrupole current, which changes $k_{1}L$, and measuring the corresponding beam image size on the view screen. The measured two-dimensional beam image was projected along the image's abscissa and ordinate axes. A super Gaussian fitting function is used on each projection to determine the rms value, $\sigma_{\textnormal{s}}$ in Eq.~(\ref{par_fit}). Measurements of $\sigma_{\textnormal{s}}$ for several quadrupole currents ($k_{1}L$) are then fit using the parabolic function in Eq.~(\ref{par_fit}) to determine the constants $A$, $B$, and $C$. The unnormalized projected rms emittance ($\epsilon$) and the Twiss parameters ($\alpha$ and $\beta$) can be found using Eq.~(\ref{emit-relation}) \begin{equation} \epsilon=\frac{\sqrt{AC}}{l^2},~\beta=\sqrt{\frac{A}{C}},~\alpha=\sqrt{\frac{A}{C}}(B+\frac{1}{l}). \label{emit-relation} \end{equation}

\subsection{HRRL Emittance Measurement Experiment} A quadrupole scanning method was used to measure the accelerator's emittance. The quadrupole current is changed to alter the strength and direction of the quadrupole magnetic field such that a measurable change in the beam shape is seen by the OTR system. Initially, the beam was steered by the quadrupole indicating that the beam was not entering along the quadrupole's central axis. Several magnetic elements upstream of this quadrupole were adjusted to align the incident electron beam with the quadrupole's central axis. First, the beam current observed by a Faraday cup located at the end of beam line was maximized using upstream steering coils within the linac nearest the gun. Second, the first solenoid nearest the linac gun was used to focus the electron beam on the OTR screen. Steering coils were adjusted to maximize the beam current to the Faraday cup and minimize the deflection of the beam by the quadrupole. A second solenoid and the last steering magnet shown in Figure~\ref{fig:app-hrrl-cavity}, both near the exit of the linac, were used in the final step to optimize the beam spot size on the OTR target and maximize the Faraday cup current. A configuration was found that minimized the electron beam deflection when the quadrupole current was altered during the emittance measurements.

The emittance measurement was performed using an electron beam energy of 15~MeV and a 200~ns long macro pulse of 40~mA electron current. The current in the first quadrupole after the exit of the linac was changed from $-$~5~A to $+$~5~A with an increment of 0.2~A. Seven measurements were taken at each current step in order to determine the average beam width and the variance. Background measurements were taken by turning the linac's electron gun off while keep the RF on. OTR images of the beam is shown in Figure~\ref{fig:eBeam}. The digitized OTR images before and after background subtraction are shown in Figure~\ref{bg}. A small dark current is visible in Figure~\ref{bg} (b) that is known to be generated when electrons are pulled off the cavity wall and accelerated.

\begin{figure}[htbp] \centering \includegraphics[scale=0.6]{4-Experiment/Figures/ElectronBeam26} \caption{The 15~MeV electron beam observed using the OTR screen when dipole coil current was at 0. The macro pulse was 200~ns and the electron peak current was 40~mA .} \label{fig:eBeam} \end{figure}

\begin{figure}[htbp] \begin{tabular}{ccc} \centerline{\scalebox{0.42} [0.42]{\includegraphics{4-Experiment/Figures/MOPPR087f4.eps}}} \\ (a)\\ \centerline{\scalebox{0.42} [0.42]{\includegraphics{4-Experiment/Figures/MOPPR087f5.eps}}}\\ (b)\\ \centerline{\scalebox{0.42} [0.42]{\includegraphics{4-Experiment/Figures/MOPPR087f6.eps}}}\\ (c) \end{tabular} \caption{Digital image from the OTR screen; (a) an image taken with the beam on, (b) a background image taken with the RF on but the electron gun off, (c) The background subtracted beam image ((a)-(b)).} \label{bg} \end{figure}

The electron beam energy was measured using a dipole magnet downstream of the quadrupole used for the emittance measurements. Prior to energizing the dipole, the electron micro-pulse bunch charge passing through the dipole was measured using a Faraday cup located approximately 50~cm downstream of the OTR screen. The dipole current was adjusted until a maximum beam current was observed on another Faraday cup located just after the 45 degree exit port of the dipole. A magnetic field map of the dipole indicates that the electron beam energy was 15~$\pm$~1.6~MeV.

%\subsection{Data Analysis and Results} Images from the JAI camera were calibrated using the OTR target frame. An LED was used to illuminate the OTR aluminum frame that has a known inner diameter of 31.75~mm. Image processing software was used to inscribe a circle on the image to measure the circular OTR inner frame in units of pixels. The scaling factor can be obtained by dividing this length with the number of pixels observed. The result is a horizontal scaling factor of 0.04327~$\pm$~0.00016 mm/pixel and vertical scaling factor of 0.04204~$\pm$~0.00018 mm/pixel. Digital images from the JAI camera were extracted in a matrix format in order to take projections on both axes and perform a Gaussian fit. The observed image profiles were not well described by a single Gaussian distribution.

The profiles may be described using a Lorentzian distribution, however, the rms of the Lorentzian function is not defined. A super Gaussian distribution was used~\cite{sup-Gau}, because it has a sharper distribution than Gaussian and, unlike Lorentzian, the rms values may be directly extracted. The super Gaussian reduced the Chi-square per degree of freedom by a factor of ten compared to a typical Gaussian fit. The beam spot, beam projections, and fits are shown in Figure~\ref{Gau-SupGaus-fits}. In a typical Gaussian distribution, $\phi(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2}$, the independent variable $x$ is raised to the 2nd power while in the super Gaussian it is raised to a power smaller than 2. For example, the beam projections shown in Figure~\ref{Gau-SupGaus-fits} were fitted with the super Gaussian distributions. The variable $x$ on the exponent was raised to $0.9053$ ($x$-projection) and $1.0427$ ($y$-projection).

\begin{figure}[htbp] \begin{tabular}{lr} {\scalebox{0.38} [0.38]{\includegraphics{4-Experiment/Figures/Gau_SupGau/Gau_ChiSqaure.eps}}} {\scalebox{0.38} [0.38]{\includegraphics{4-Experiment/Figures/Gau_SupGau/SupGau_ChiSqaure.eps}}} \end{tabular} \caption{Gaussian and super Gaussian fits for beam projections. The beam images is background subtracted image and taken when quadrupole magnets are turned off. Left image is Gaussian fit and right image is super Gaussian fit.} \label{Gau-SupGaus-fits} \end{figure}

Figure~\ref{fig:par-fit} shows the square of the rms ($\sigma^2_{\textnormal{s}}$) $vs$ $k_1L$ for $x$ (horizontal) and $y$ (vertical) beam projections along with the parabolic fits using Eq.~\ref{fig:par-fit}. The emittance and Twiss parameters from these fits are summarized in Table~\ref{tab:results}. The MATLAB~\cite{MATLAB} scripts used to calculate emittance and Twiss parameters are given in appendix B. \begin{figure}[htbp] \begin{tabular}{lr} {\scalebox{0.27} [0.27]{\includegraphics{4-Experiment/Figures/par_fit_x.eps}}} {\scalebox{0.27} [0.27]{\includegraphics{4-Experiment/Figures/par_fit_y.eps}}} \end{tabular} \caption{Square of rms values and parabolic fittings. As the quadrupole current changes, so does quadrupole strength times quadrupole legnth, k$_1$L, and the square of the beam rms changes accordingly.} \label{fig:par-fit} \end{figure}

\begin{table} \centering \caption{Emittance Measurement Results} \begin{tabular}{lcc} \toprule {Parameter} & {Unit} & {Value} \\ \midrule unpolarized projected emittance $\epsilon_x$ & $\mu$m & $0.37 \pm 0.02$ \\ unpolarized projected emittance $\epsilon_y$ & $\mu$m & $0.30 \pm 0.04$ \\ % normalized \footnote{normalization procedure assumes appropriate beam chromaticity.} emittance $\epsilon_{n,x}$ & $\mu$m & $10.10 \pm 0.51$ \\ %normalized emittance $\epsilon_{n,y}$ & $\mu$m & $8.06 \pm 1.1$ \\ $\beta_x$-function & m & $1.40 \pm 0.06$ \\ $\beta_y$-function & m & $1.17 \pm 0.13$ \\ $\alpha_x$-function & rad & $0.97 \pm 0.06$ \\ $\alpha_y$-function & rad & $0.24 \pm 0.07$ \\ micro-pulse charge & pC & 11 \\ micro-pulse length & ps & 35 \\ energy of the beam $E$ & MeV & 15 $\pm$ 1.6 \\ relative energy spread $\Delta E/E$ & \% & 10.4 \\ \bottomrule \end{tabular} \label{tab:results} \end{table}

\section{Measurement of HRRL Electron Beam \\Energy Spread at 12~MeV} The HRRL energy profile was measured when it was tuned to accelerate electrons to 12~MeV peak energy. A Faraday cup, FC2, was placed at the end of the 45 degree beamline to measure the electron beam current when D1 was on and D2 was off. The dipole coil current for D1 was changed in 1~A increments. As the dipole coil current was changed, the energy of the electrons transported to the Faraday cup would change as described in Appendix A. Figure~\ref{fig:En-Scan} illustrates a measurement of the 12~MeV peak illustrating the observed low energy tail. The HRRL energy profile can be described by overlapping two skewed Gaussian fits~\cite{sup-Gau}. The fit function is given by \begin{equation} G(En)=A_{1}e^{\frac{-\left(En-\mu_{1}\right)^{2}}{2\left\{ \sigma_{1}\left[1+sgn\left(En-\mu_{1}\right)\right]\right\} ^{2}}}+A_{2}e^{\frac{-\left(En-\mu_{2}\right)^{2}}{2\left\{ \sigma_{2}\left[1+sgn\left(En-\mu_{2}\right)\right]\right\} ^{2}}}, \label{eq:skew-Gau} \end{equation} where $sgn$ is the sign function, that is defined as $$ sgn=\left\{ \begin{array}{ll}

  \mbox -1\ & {(x<0)} \\
  \mbox 0 \ & {(x=0)} \\
  \mbox 1 \ & {(x>0)}.\\
                  \end{array}\right.

$$ The other variables are defined as $\sigma_{1}=\frac{\sigma_{r,1}+\sigma_{l,1}}{2}$, $\sigma_{2}=\frac{\sigma_{r,2}+\sigma_{l,2}}{2}$, $E_{1}=\frac{\sigma_{r,1}-\sigma_{l,1}}{\sigma_{r,1}+\sigma_{l,1}}$, and $E_{2}=\frac{\sigma_{r,2}-\sigma_{l,2}}{\sigma_{r,2}+\sigma_{l,2}}$. The measurement results and fits are shown in Figure~\ref{fig:En-Scan} and in Table~\ref{tab:En-Scan_resluts}. \begin{figure}[htbp] \centering \includegraphics[scale=0.5]{4-Experiment/Figures/En_Fit_Assym_Gau.eps} \caption{HRRL energy scan (dots) and fit (line) with two skewed Gaussian distribution.} \label{fig:En-Scan} \end{figure} \begin{table} \centering \caption{Fit Parameters for Two Skewed Gaussian.} \begin{tabular}{lclcc} \toprule {Parameter} & Notation & Unit & {First Gaussian} & {Second Gaussian} \\ \midrule amplitude & A & mA & ~2.14 & 10.88 \\ mean & $\mu$ & MeV & 12.07 & 12.32 \\ sigma left & $\sigma_L$ & MeV & ~4.47 & ~0.70 \\ sigma right & $\sigma_R$ & MeV & ~1.20 & ~0.45 \\ \bottomrule \end{tabular} \label{tab:En-Scan_resluts} \end{table}

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % \section{Positron Production Experiment} Positron produced using the HRRL located in the Beam Lab of Physics Department at Idaho State University. The electron beam of the HRRL was incident on T1, positron production target, and the positrons produced downstream of T1 was transported to T2, annihilation target, where they annihilate and crated back to back scattered 511~keV photon pairs. Two NaI detectors, shielded with Pb bricks, were placed horizontally and operated in coincidence with electron electron gun pulse to detect 511~keV photon pairs came out of T2. The electron beam current of the HRRL was monitored by a scintillator was placed between Q9 and Q10. The ROOT script used to calculate number/charge of positrons/electrons and the ratio of the two is given in the appendix.

%Positrons transported to the scintillator located at the end of 90 degree the beamline. \subsection{The Electron Beam Current Measurement} A scintillator was placed between Q9 and Q10, as shown in Figure~\ref{fig:Scint_e-}, to monitor the electron beam current. The electron beam current was changed incrementally to measure the correlation between the scintillator and the accelerated electron beam. The beam current was measured using FC1 and the output was integrated using an oscilloscope. The scintillator output was integrated using an ADC (CAEN Mod. V792). As the electron beam current was decreased, the signal observed on the oscilloscope decreased and the ADC measured less charge from the scintillator as shown in Figure~\ref{fig:ADC-CH9}. A linear fit to data resulted a linear relation \begin{equation} Q_{\text{e}^-}(i) = (0.0186 \pm 0.0028)i + 2.79 \pm 0.08~\text{C}, \end{equation} where $i$ is ADC channel number and $Q$ is the accelerated electron beam charge. The fit is shown in Figure~\ref{fig:calb-fit}.

\begin{figure}[htbp] \centering \includegraphics[scale=0.4]{4-Experiment/Figures/HRRL_line.eps} \caption{The electron beam monitor.} \label{fig:Scint_e-} \end{figure}

\begin{figure}[htbp] \centering \includegraphics[scale=0.76]{4-Experiment/Figures/ScintilatorCalibration/ADC_CHAN4.eps} \caption{The photon flux detected using scintillator. The mean of the ADC channel decreased linearly as the electron beam current was decreased. The electron beam current was measured using the Faraday cup at the end of the 0 degree beamline, FC1, and integrated using oscilloscope.} \label{fig:ADC-CH9} \end{figure}

\begin{figure}[htbp] \centering \includegraphics[scale=0.70]{4-Experiment/Figures/ScintilatorCalibration/calb.eps} \caption{Fit for accelerator beam current $v.s.$ the ADC channel.} \label{fig:calb-fit} \end{figure}

\begin{table} \centering \caption{Scintillator Calibration Data.} \begin{tabular}{lll} \toprule {Run Number} & {Faraday Cup} & {Mean of ADC} \\ { } & {Charge Area (nVs)} & {Channel 9} \\ \midrule 3703 & $1201 \pm 10$ & $1126 \pm 0.8$ \\ 3705 & $777 \pm 110$ & $791.8 \pm 0.6$ \\ 3706 & $367.7 \pm 2.3$ & $242.1 \pm 0.3$ \\ \bottomrule \end{tabular} \label{tab:scint_calb} \end{table}

When running in coincidence mode, the electron beam current is sampled by the scintillator only when a coincidence event occurs causing a trigger that gates the ADC and measures the scintillator output for that positron coincidence event. The charge measured by the ADC is the total charge of the electron pulses that created the positron events. The number of beam pulses are counted using a scaler. The total charge on T1 for the entire run is estimated using \begin{equation} Q_{\text{C}} = \left ( \overset{N}{\underset{i}{\sum}} 0.0186i\times(\text{bin~content}[i]) + 2.79 \right ) \times \frac{\text{(\# of beam pulses)}}{\text{(\# of events)}}. \label{eq:q_calc1} \end{equation} \noindent

\subsection{NaI Detectors} The NaI detectors were calibrated using radioactive sources as mention in the Chapter 2. However, in the positron production experiment conducted using HRRL, the photon peak observed by the NaI detector was not at the channels corresponding the 511~keV energy region (according to the calibration with sources). To determine the peak observed is the 511~keV peak, the Na-22 source was placed between two NaI detectors to measure the spectrum when the RF was on, represented by dotted line (no electrons were fired from accelerator gun), and off, represented by dashed line in Fig~\ref{fig:NaI-peak-shift}. The solid line in Fig~\ref{fig:NaI-peak-shift} represents the photon spectrum created by 3~MeV positrons impinging on T2.

When the RF was on, no difference was observed for photon peaks created by the 3~MeV positrons and the 511~keV photon peak from Na-22 source. However, when the RF was off, the 511~keV photon peak from Na-22 source occupied at different channels. It seems the 511~keV peak would shift when the RF is turned on. The 511 keV peak in the right NaI detector was shifted to the right side of the spectrum by 17 channels when the RF was on as shown in Figure~\ref{fig:NaI-peak-shift} (a) while in the left NaI detector the peak shifted to the left by 28 channels as shown if Figure~\ref{fig:NaI-peak-shift} (b).

\begin{figure}[htbp] \begin{tabular}{ccc} \centerline{\scalebox{0.6} [0.6]{\includegraphics{4-Experiment/Figures/RunDec2012/PeakShiftNaIL9.eps}}}\\ (a) Left NaI detector. \\ \\ \\ \centerline{\scalebox{0.6} [0.6]{\includegraphics{4-Experiment/Figures/RunDec2012/PeakShiftNaIR9.eps}}} \\ (b) Right NaI detector.\\ \end{tabular} \caption{The 511~keV peak observed using NaI detectors shifted when accelerator RF was on. The spectrum were taken with RF on (dotted line) and with RF off (dashed line). The solid line represents the photon spectrum created by 3~MeV positrons impinging on T2.} \label{fig:NaI-peak-shift} \end{figure}

\subsection{Positron Rate Estimation in AND Mode}

The background subtracted and normalized photon energy spectra observed using two NaI detectors for $3.00 \pm 0.07$~MeV positrons are shown in Figure~\ref{fig:pos_NaILR}. Figure~\ref{fig:pos_NaILR} (a) and (b) are the background subtracted spectrum with no coincidence or energy cut (OR mode). In OR mode, no coincidence between two detectors are required. Figure~\ref{fig:pos_NaILR} (c) and (d) illustrate events observed in coincidence and within a energy window of $511\pm75$~keV for both detectors (AND or coincidence mode). The measured positron rate using NaI detectors in AND mode was 0.25~Hz for $3.00 \pm 0.07$~MeV positrons when HRRL was operated at 300~Hz repetition rate, 100~mA peak current, and 300~ns (FWHM) macro pulse length. %The uncertainty of the power supplies used for dipoles is 0.1~A which creates 0.06~MeV uncertainty in energy of the beam bent by dipoles. \begin{figure}[htbp] \centering \begin{tabular}{cc} {\scalebox{0.74} [0.74]{\includegraphics[scale=0.40]{4-Experiment/Figures/subtract/whole/NaI_L/r3735_sub_r3736.png}}} & {\scalebox{0.74} [0.74]{\includegraphics[scale=0.40]{4-Experiment/Figures/subtract/whole/NaI_R/r3735_sub_r3736.png}}} \\ (a) & (b) \\ & \\ & \\ {\scalebox{0.74} [0.74]{\includegraphics[scale=0.40]{4-Experiment/Figures/subtract/511_peak/NaI_L/r3735_sub_r3736.png}}} & {\scalebox{0.74} [0.74]{\includegraphics[scale=0.40]{4-Experiment/Figures/subtract/511_peak/NaI_R/r3735_sub_r3736.png}}} \\ (c) & (d) \\

& \\

\end{tabular} \caption{Photon spectrum of NaI detectors after background subtraction created by 3~MeV positrons incident on T2. (a) and (b) are spectrum after background subtractions. (c) and (d) are the spectrum of events coincident in both detectors in 511~keV peaks.} \label{fig:pos_NaILR} \end{figure}

%The electron beam was transported to a phosphorous screen at the end of the 90 degree beamline to find the errors on the positron beam energy. The positron beam current was too low to be observed on phosphorous screen. %The beam centered on the phosphorous screen and then steered to the edge by changing current of D2 by 0.1~A. %The phosphorous screen is twice as large as T2 in horizantal direction.

\subsection{The Positron Production Runs} The annihilation target T2 is insertable into the beamline and placed inside a 6-way cross that has horizontal sides vacuum sealed with 1~mil stainless steel windows. Positrons intercepting T2 thermalise, annihilate, and produce 511~keV photon pairs back-to-back. These photons are detected by two NaI detectors facing T2 and shielded with Pb bricks as shown in Figure~\ref{fig:HRRL-En-Scan}. The background was measured by retracting T2 thereby allowing positrons to exit the beamline and be absorbed by the beam dump.

\begin{figure}[htbp] \centering \includegraphics[scale=0.58]{4-Experiment/Figures/PositronDetection/NaI_Setup4.png} \caption{Positron detection using T2 and NaI detectors.} \label{fig:HRRL-En-Scan} \end{figure}

As shown in Figure~\ref{fig:Dipole-in}, a 511~keV peak was observed (solid line) when T2 was in. A permanent dipole magnet was placed on the beamline after Q10 to deflect charged particles on the accelerator side preventing them from entering the shielded cell. The peak was not observed (dashed line) when T2 was in and the permanent magnet was used. The peak was also not observed (doted dashed line) when T2 and the permanent magnet were removed. Thus, one may argue that the observed peak is due to positrons annihilating in T2.

\begin{figure}[htbp] \centering \begin{tabular}{cc} {\scalebox{0.319} [0.319]{\includegraphics{4-Experiment/Figures/SweepingDipole/LNaI3.png}}} & {\scalebox{0.319} [0.319]{\includegraphics{4-Experiment/Figures/SweepingDipole/RNaI3.png}}} \\ (a) Original spectrum on left NaI. & (b) Original spectrum on right NaI.\\ \end{tabular} \caption{Photon spectrum when a permanent dipole magnet is inserted along with T2 (dashed line), dipole out and T2 in (solid line), and dipole removed and T2 out (dotted dashed line). The positron energy incident on the T2 was $2.15\pm0.06$~MeV.} \label{fig:Dipole-in} \end{figure}

The normalized photon energy spectra observed by two NaI detectors are shown in Figure~\ref{fig:in-out-runs} when T2 was both in (signal) and out (background) of the beamline. Figure~\ref{fig:in-out-runs} (c) and (d) illustrate the events observed when 511~keV photons were required in both detectors. The signal is indicated by the solid line and background by dashed line.

\begin{figure}[htbp] \centering \begin{tabular}{ll} {\scalebox{0.29} [0.29]{\includegraphics{4-Experiment/Figures/NaI_L1/r3735_sub_r3736_2.png}}} & {\scalebox{0.29} [0.29]{\includegraphics{4-Experiment/Figures/NaI_R1/r3735_sub_r3736_2.png}}} \\ (a) Original spectrum on left NaI. & (b) Original spectrum on right NaI.\\ & \\ & \\ {\scalebox{0.28} [0.28]{\includegraphics{4-Experiment/Figures/NaI_L2/r3735_sub_r3736_2.png}}} & {\scalebox{0.28} [0.28]{\includegraphics{4-Experiment/Figures/NaI_R2/r3735_sub_r3736_2.png}}} \\

(c) Spectrum with cut on left NaI. & (d) Spectrum with cut on right NaI.\\ & \\ \end{tabular} \caption{The time normalized spectra of photons created by 3~MeV positrons incident on T2. In the top row are original spectrum and in the bottom row are spectrum of incidents happened in the 511~keV peak coincidently in both detectors. The positron beam energy incident on the T2 was $3.00\pm0.06$~MeV.} \label{fig:in-out-runs} \end{figure}

%\begin{table} %\centering %\caption{Run Parameters of The Run No. 3735.} %\begin{tabular}{lll} %\toprule %{Parameter} & {Unit} & {Value} \\ %\midrule %run number & & 3735 \\ %repetition rate & Hz & 300 \\ %run time & s & 1002 \\ %pulses & & 301462 \\ %events & & 9045 \\ %e$^+$ Counts on NaI Detectors & & 256 $\pm$ 16\\ %\bottomrule %\end{tabular} %\label{tab:run3735} %\end{table}

\subsection{Positron to Electron Ratio in AND Mode} Gaussian distributions were fit to 511~keV photon peaks observed on NaI detectors as shown in Figure~\ref{fig:511Fit} and the fits result in $\sigma_{NaI, 511} = 37.5 \pm 3.0$~keV. Events in the experiment was observed within a 2$\sigma_{NaI, 511}$ energy window (436$-$586~keV) for both detectors. The ratios of positrons, detected using NaI detectors in coincidence mode, to the electrons impinging on T1 at different energies are given in Table~\ref{tab:e+2e-} and the errors on the ratio are statistical. The systematic errors are discussed in the nest section and results are plotted in Figure~\ref{fig:e+2e-}. %error on total counts = sqt(total counts). Rate = Counts/time. Error on rates = sqrt(d(rate)/d(counts)^2*(error on the counts)^2)=sqrt((error on the counts)^2/time^2)=sqrt(counts/time^2)=sqrt(rate/time). \begin{figure}[htbp] \centering \begin{tabular}{cc} {\scalebox{0.76} [0.76]{\includegraphics[scale=0.48]{4-Experiment/Figures/systimatics/NaI_L/3MeV_511Fit_r3735_sub_r3736L.eps}}} & {\scalebox{0.76} [0.76]{\includegraphics[scale=0.48]{4-Experiment/Figures/systimatics/NaI_R/3MeV_511Fit_r3737_sub_r3736R.eps}}} \\ (a) Left NaI detector.& (b) Right NaI detector.\\ & \\ \end{tabular} \caption{} \label{fig:511Fit} \end{figure}

\begin{table} \centering \caption{Positron to Electron Rate Ratio.} \begin{tabular}{cc} \toprule {Energy} & {Positron to Electron Ratio} \\ \midrule $1.02 \pm 0.03$ & $(1.7 \pm 0.6)\times10^{-16}$ \\ $2.15 \pm 0.06$ & $(8.6 \pm 1.5) \times10^{-16}$ \\ $3.00 \pm 0.07$ & $(8.52 \pm 0.54)\times10^{-15}$ \\ $4.02 \pm 0.07$ & $(3.11 \pm 0.31)\times10^{-15}$ \\ $5.00 \pm 0.06$ & $(3.32 \pm 0.89)\times10^{-15}$ \\ \bottomrule \end{tabular} \label{tab:e+2e-} \end{table}

\subsection{Positron Rate Estimation in Software OR Mode}

The ratio of 511~keV photons counted in software OR mode to the electrons incident on T1 are given in Table~\ref{tab:OrMode_e+/e-}. In these runs listed in the table, NaI detectors were operated in triple coincidence between NaI detectors and gun pulse. In the software of OR mode, there is no software coincidence are required between two NaI detectors. In OR mode, left and right NaI detectors observe different 511~keV photon rates (see section 4.4 of Chapter 4 for the detailed analysis on the count asymmetry in NaI detectors). In the experiment, the upstream side of the T2 was facing right NaI detector and right NaI detected more 511~keV photons. The asymmetry is defined as \begin{equation} \text{Asymmetry} = \frac{N_{r}-N_{l}}{N_{r}+N_{l}}\times 100\%, \end{equation} where $N_{r}$ and $N_{l}$ are the number of 511~keV photons detected by right and left NaI detectors respectively. The 511~keV photons are counted in 1, 2, and 3 $\sigma_{NaI, 511}$ ($\sigma_{NaI, 511} = 37.5 \pm 3.0$~keV) energy windows. \begin{table} \centering \caption{The Ratio of 511~keV Photons Measured in OR Mode to the Electrons Incident on T1 Measured in Different $\sigma_{NaI, 511}$ Energy Windows.} \begin{tabular}{cccc} \toprule 1 $\sigma_{NaI, 511}$ Cut \\ \midrule {Energy (MeV)} & {e$^{+}$/e$^{-}$ on NaI Right} & {e$^{+}$/e$^{-}$ on NaI Left} & {Asymmetry} \\ \midrule $1.02 \pm 0.03$ & $( 2.5 \pm 0.3 ) \times10^{-15}$ & $( 0.0 \pm 0.0 ) \times10^{-15}$ & $ 96.0 \pm 2.8 \%$ \\ $2.15 \pm 0.06$ & $( 8.5 \pm 0.6 ) \times10^{-15}$ & $( 1.0 \pm 0.2 ) \times10^{-15}$ & $ 79.2 \pm 4.0 \%$ \\ $3.00 \pm 0.07$ & $( 75.1\pm 1.5 ) \times10^{-15}$ & $( 14.2\pm 0.6 ) \times10^{-15}$ & $ 68.2 \pm 1.3 \%$ \\ $4.02 \pm 0.07$ & $( 42.0\pm 1.2 ) \times10^{-15}$ & $( 6.0 \pm 0.4 ) \times10^{-15}$ & $ 75.1 \pm 1.7 \%$ \\ $5.00 \pm 0.06$ & $( 21.5\pm 2.3 ) \times10^{-15}$ & $( 4.0 \pm 1.0 ) \times10^{-15}$ & $ 68.5 \pm 7.0 \%$ \\ \midrule

2 $\sigma_{NaI, 511}$ Cut  \\

\midrule {Energy (MeV)} & {e$^{+}$/e$^{-}$ on NaI Right} & {e$^{+}$/e$^{-}$ on NaI Left} & {Asymmetry} \\ \midrule $1.02 \pm 0.03$ & $( 2.9 \pm 0.3 ) \times10^{-15}$ & $( 0.3 \pm 0.1 ) \times10^{-15}$ & $ 81.5 \pm 5.2 \%$ \\ $2.15 \pm 0.06$ & $( 11.8 \pm 0.7 ) \times10^{-15}$ & $( 1.4 \pm 0.2 ) \times10^{-15}$ & $ 79.0 \pm 3.4 \%$ \\ $3.00 \pm 0.07$ & $( 98.0 \pm 1.7 ) \times10^{-15}$ & $( 19.0 \pm 0.7 ) \times10^{-15}$ & $ 67.6 \pm 1.2 \%$ \\ $4.02 \pm 0.07$ & $( 53.9 \pm 1.3 ) \times10^{-15}$ & $( 8.1 \pm 0.5 ) \times10^{-15}$ & $ 73.9 \pm 1.5 \%$ \\ $5.00 \pm 0.06$ & $( 28.6 \pm 2.6 ) \times10^{-15}$ & $( 5.5 \pm 1.1 ) \times10^{-15}$ & $ 67.6 \pm 6.1 \%$ \\ \midrule

3 $\sigma_{NaI, 511}$ Cut  \\

\midrule {Energy (MeV)} & {e$^{+}$/e$^{-}$ on NaI Right} & {e$^{+}$/e$^{-}$ on NaI Left} & {Asymmetry} \\ \midrule $1.02 \pm 0.03$ & $( 3.4 \pm 0.3 ) \times10^{-15}$ & $( 0.4 \pm 0.1 ) \times10^{-15}$ & $ 78.4 \pm 5.1 \%$ \\ $2.15 \pm 0.06$ & $( 13.4\pm 0.7 ) \times10^{-15}$ & $( 1.5 \pm 0.2 ) \times10^{-15}$ & $ 79.3 \pm 3.1 \%$ \\ $3.00 \pm 0.07$ & $( 106.3\pm1.8 ) \times10^{-15}$ & $( 21.4\pm 0.8 ) \times10^{-15}$ & $ 66.5 \pm 1.1 \%$ \\ $4.02 \pm 0.07$ & $( 58.9\pm 1.4 ) \times10^{-15}$ & $( 9.4 \pm 0.5 ) \times10^{-15}$ & $ 72.5 \pm 1.5 \%$ \\ $5.00 \pm 0.06$ & $( 16.4\pm 1.4 ) \times10^{-15}$ & $( 2.9 \pm 0.6 ) \times10^{-15}$ & $ 70.0 \pm 5.6 \%$ \\ \bottomrule \end{tabular} \label{tab:OrMode_e+/e-} \end{table}

The errors given in Table~\ref{tab:OrMode_e+/e-} are statistical. Counting positrons in AND mode underestimates the positron rates by factor of 10 approximately compared to the OR mode. The asymmetry tends to decrease as the positron energy increases. The average ratios are given in Table~\ref{tab:Sys_OrMode_e+/e-}. In the table, the first error on ratios are systematic and second ones are statistical.

\begin{sidewaystable} \centering \caption{The Ratio of 511~keV Photons Measured in OR Mode to the Electrons Incident on T1.} \begin{tabular}{cccc} \toprule {Energy (MeV)} & {e$^{+}$/e$^{-}$ on NaI Right} & {e$^{+}$/e$^{-}$ on NaI Left} & {Asymmetry} \\ \midrule

$1.02 \pm 0.03$ & $(2.9\pm 0.5 \pm0.3) \times10^{-15}$ & $(0.3 \pm 0.2 \pm 0.1) \times10^{-15}$ & $85.3\pm 9.4\pm5.2\%$ \\ $2.15 \pm 0.06$ & $(11.2\pm2.5 \pm0.7) \times10^{-15}$ & $(1.3 \pm 0.3 \pm 0.2) \times10^{-15}$ & $79.2\pm 0.2\pm3.4\%$ \\ $3.00 \pm 0.07$ & $(93.1\pm16\pm1.7) \times10^{-15}$ & $(18.2\pm 3.7 \pm 0.7) \times10^{-15}$ & $67.4\pm 0.9\pm1.2\%$ \\ $4.02 \pm 0.07$ & $(51.6\pm8.7 \pm1.3) \times10^{-15}$ & $(7.8 \pm 1.7 \pm 0.5) \times10^{-15}$ & $73.8\pm 1.3\pm1.5\%$ \\ $5.00 \pm 0.06$ & $(22.2\pm6.1 \pm2.6) \times10^{-15}$ & $(4.1 \pm 1.3 \pm 1.1) \times10^{-15}$ & $68.7\pm 1.2\pm6.1\%$ \\

\bottomrule \end{tabular} \label{tab:SysOrMode_e+/e-} \end{sidewaystable}

\subsection{Positron Rate Estimation in Hardware OR Mode} The positron rate was also measured by running NaI detectors in OR mode and the photon spectrum are shown in Figure~\ref{fig:Or-mode} for $2.15 \pm 0.06$~MeV positrons incident on T2. Two NaI detectors were operated in hardware OR mode in these runs. The rate was $0.21$~Hz for left and $0.35$~Hz right NaI detectors in OR mode. However, when the software coincidence are required between two detectors by only drawing the events happened in 511~keV peaks of both detectors, the rate dropped $0.028$~Hz.

\begin{figure}[htbp] \centering \begin{tabular}{cc} {\scalebox{0.76} [0.76]{\includegraphics[scale=0.40]{4-Experiment/Figures/NaIOrMode/r3679_sub_r3680_NaI_L.png}}} & {\scalebox{0.76} [0.76]{\includegraphics[scale=0.40]{4-Experiment/Figures/NaIOrMode/r3679_sub_r3680_NaI_R.png}}} \\ (a) Left NaI detector.& (b) Right NaI detector.\\ & \\ \end{tabular} \caption{The photon rates for $2.15 \pm 0.06$~MeV incident positrons measured by running NaI detectors in OR mode. The rate was $0.21$~Hz for left and $0.35$~Hz right NaI detectors in OR mode while the rate in coincidence (AND) mode was $0.028$~Hz.} \label{fig:Or-mode} \end{figure}

\section{Systematic Error Analysis for AND Mode Rate} The systematic errors in the positron production experiment could be introduced by the uncertainty in the magnetic fields and misalignment of the quadrupoles. These systematic errors are discussed in section 6 of Chapter 4.%The uncertainty of the emittances and Twiss parameters of the electron beam would also create systematic error as well. The positron and electron rates are normalized by time which can introduce systematic error as well. However, the uncertainty in run duration time is very small (0.1$-$0.2$\%$) and can be ignored (the duration of positron production runs are around 500$\sim$1000 seconds and the uncertainty is about 1 second). The uncertainty in the 511~keV peaks detected by the NaI detectors would also contribute to the systematic error.

%\subsection{Systematic Error Introduced by the Uncertainty of The 511~keV Peak} Gaussian distributions were fit to the 511~keV photon peaks observed on NaI detectors as shown in Figure~\ref{fig:511Fit} and the fits result in $\sigma_{NaI, 511} = 37.5 \pm 3.0$~keV. Events in the experiment was observed in coincidence and within a 2$\sigma_{NaI, 511}$ energy window (436$-$586~keV) for both detectors. The systematic error of the 511~keV photon counts introduced by the uncertainty in the 511~keV peak is studied by counting the photons with 1$\sigma_{NaI, 511}$ (473.5$-$548.5~keV) and 3$\sigma_{NaI, 511}$ (398.5$-$623.5~keV) energy windows. The positron to electron rate ratio for different energy windows are given in Table~\ref{tab:sys2}. The results are plotted in Figure~\ref{fig:e+2e-}.

\begin{sidewaystable} \centering \caption{Positron to Electron Rate Ratio: Systematic Error Introduced by The Uncertainty of The 511~keV Peak.} \begin{tabular}{llll} \toprule {Energy} & {Positron to Electron Ratio} & {Positron to Electron Ratio} & {Positron to Electron Ratio} \\ {(MeV)} & {Energy Window: 2$\sigma_{NaI, 511}$} & {Energy Window: 3$\sigma_{NaI, 511}$} & {Energy Window: 1$\sigma_{NaI, 511}$} \\ \midrule $1.02 \pm 0.03$ & $(1.7 \pm 0.6) \times10^{-16}$ & $(1.7 \pm 0.6) \times10^{-16}$ & $(0.5 \pm 0.4) \times10^{-16}$ \\ $2.15 \pm 0.06$ & $(8.6 \pm 1.5) \times10^{-16}$ & $(9.2 \pm 2.0) \times10^{-16}$ & $(3.0 \pm 0.9) \times10^{-16}$ \\ $3.00 \pm 0.07$ & $(8.52 \pm 0.54)\times10^{-15}$ & $(9.90 \pm 0.58)\times10^{-15}$ & $(4.20 \pm 0.35)\times10^{-15}$ \\ $4.02 \pm 0.07$ & $(3.11 \pm 0.31)\times10^{-15}$ & $(3.66 \pm 0.34)\times10^{-15}$ & $(2.07 \pm 0.26)\times10^{-15}$ \\ $5.00 \pm 0.06$ & $(3.32 \pm 0.89)\times10^{-15}$ & $(3.55 \pm 0.92)\times10^{-15}$ & $(1.66 \pm 0.63)\times10^{-15}$ \\

\bottomrule \end{tabular} \label{tab:sys2} \end{sidewaystable}

\begin{figure}[htbp] \centering \includegraphics[scale=0.84]{4-Experiment/Figures/Ratio/R.eps} \caption{The ratios of positrons detected by NaI detectors in coincidence mode to the electrons impinging T1. The solid error bars statistical and the dashed ones are systematic.} \label{fig:e+2e-} \end{figure}

Chapter 4: Simulation File:Sadiq thesis chapt 4.txt

\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 2mm thick tungsten target. The simulation revealed that the number of electrons decreased by orders of magnitude as they were transported through 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 procuded 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$^{\circ}$ clockwise 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. %In Figure~\ref{fig:SimS1_pos_En_DDNT1}, $13.8 \times 10^{10}$ electrons shot at T1 and generated positrons positrons shown in blue.

\begin{figure}[htbp] \centering \includegraphics[scale=0.55]{3-Simulation/Figures/sim_setup_T1_UpD_DwD2.png} \caption{T1 is the positron production target. 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/overlay8.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. %This is the result of the target's thickness of 1.016 mm and the 45$^{\circ}$ angle of intersection ($1.016\sqrt{2}=1.44$). The edge of the target does not produce many positrons compared to the face of the target. \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 positrons 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 simlation's second step was designed to predict the amount of beam loss that would result when the postiron beam is transported from the entrace 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 number of positrons is predicted to decrease by a factor of ten as they travel through the quad triple system to the entrance of the 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 at two 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 crysta,l 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 as at 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 cube: positrons incident on DDNT1. Red cube: positrons entered Q4. Blue cube: positrons entered D1. Magenta cube: positrons exited D1. Black circle: positrons entered Q7. Red circle: positrons entered D2. Blue circle: positrons exited D2. Magenta circle: positrons incident on DT2UP. Black triangle: 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 $\Omega_{\text{Q4}}=\frac{\pi r^2}{d^2}=\frac{\pi 24^2}{484.4^2}$~steradian. Positrons make up a cone with a 45$^\circ$ half angle, which is $\Omega_{\text{beam}}=0.6\pi$~steradian 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 travers 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 Mis-alignment 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 appeard to have the largest impact.

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

\caption{Positron to Electron Ratio When a Single Beamline Element Is Mis-Aligned and the Percent Error Compared to the Ratio When No Mis-Alignment.} \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 mis-aligning 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 mis-alignment 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{Mis-alignment 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 Mis-aligned 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}

Chapter 5: Conclusion File:Sadiq thesis chapt 5.txt

\chapter{Conclusions and Suggestions} A new High Repetition Rate Linac (HRRL) beamline located in ISU's Physics Department beam lab has been successfully reconstructed to produce and transport positrons to the experimental cell. The electron beam energy profile and emittance of the HRRL were measured using a Faraday cup and an OTR based diagnostic system. The positron production rate was measured for positron energies between 1 and 5 MeV. The results are shown in Figure~\ref{fig:e+2e-exp-sim} along with the prediction made by a GEANT4 simulation of the beamline.

The production of positrons using an electron linac was done in several steps. First, positrons are emitted from the downstream side of a tungsten target (T1) when electrons impinge on the upstream side and produce photons of sufficient energy to pair produce within the tungsten target. Postrons escaping the downstream side of the target were collected by the quadrupole triplet. The positrons are then deflected by two dipoles in order to measure the positron rate as a function of the positron energy. Positrons that traversed the two dipoles would annihilate in a second tungsten target (T2) producing back-to-back 511 keV photons that were measured using two NaI detectors. The positron rate was measured by requiring a coincidence between both NaI detectors and the electron beam pulse.

The positron beam creation, beam loss in the transportation, and detection process were studied using the simulation package G4beamline and compared to this experiment. The simulated e$^+$/e$^-$ ratios are shown in Figure~\ref{fig:e+2e-exp-sim} for the energies measured in this experiment. The simulation includes electron beam generation with the measured electron energy profile, beam losses during transportation, positron annihilation in the tungsten target (T2), and the detection of 511~keV photons in coincidence by the two NaI detectors. While the simulation result agrees with the experiment in that the peak energy distribution is near 3~MeV, it predicts a higher positron to electron ratio as shown in Figure~\ref{fig:e+2e-exp-sim}.

The simulation was used to study the systematic errors in the experiment. The simulation predicts that a realistic misalignment of the beamline can reduce the e$^+$/e$^-$ ratios by 20\% to 30\%. In the worst case scenario, the ratios dropped by 60\% to 80\%. The systematic errors in the experiment bring it into agreement with the simulation.

The ratio of the positrons contained within the 90 degree beampipe to the 511~keV photons detected in coincidence mode when the dipoles were set to bend 3~MeV positrons is predicted to be 1655:1 by the simulation. The ratio of the positrons on T2 to the 511~keV photons is 620:1 under the same conditions as above. The 3~MeV positron rate measured in the experiment was $0.25\pm0.2$~Hz when the HRRL was operated at a 300~Hz repetition rate, 100~mA peak current, and 300~ns (FWHM) RF macro pulse length. Based on this simulation, a measured $0.25 \pm 0.02$~Hz coincidence rate by the NaI detectors would correspond to a $155 \pm 12$~Hz positron rate incident on T2.

In the simulation, the number of positrons collected was insensitive to the quadrupole triplet collection field setting (see section 4.5). The ratio of solid angles subtended by the quadrupole (Q4) and dipole (D1) entrance windows approximated the ratios of positron transported. Dipoles defocus in one plane and defocus in the other. Thus, one can only collect positrons in one plane while loses occur in the other. Solenoids, on other hand, focus in both planes. A solenoid may be a better option to improve the collection efficiency. Positioning the target T1 at the entrance of the solenoid may be the optimal choice for capturing positrons.

%7. Experimental results show quadrupole magnets are not efficient in collecting positrons, since positrons have large angular distribution. Solenoid might be able to improve the collection efficiency of positrons~\cite{kim-bindu-solenoid} and should be placed as close the production target as possible for better efficiency.

\begin{figure}[htbp] \includegraphics[scale=0.79]{5-Conclusion/Figures/Overlay_Exp-Sim-Ratio/R.eps} \caption{Ratio of positrons detected to electrons measured in the experiment (hollow diamond) and simulation (full circle) in coincidence mode. The black solid error bars are statistical and dashed ones are systematic. The experimental systematic errors (red dashed lines) are discussed in section 3.4 of Chapter 3 and the systematic errors in simulation (red dashed lines) estimation is described section 4.6 of Chapter 4.} \label{fig:e+2e-exp-sim} \end{figure} %An OTR based diagnostic tool was designed, constructed, and used to measure the beam emittance of the HRRL. The electron spatial profile measured using the OTR system was not described by a Gaussian distribution but by a super Gaussian or Lorentzian distribution. The unnormalized projected emittances of the HRRL were measured to be less than 0.4~$\mu$m by the OTR based tool using the quadrupole scanning method when accelerating electrons to an energy of 15~MeV. %OTR used, Not Guassian, Changed magnet, measured emttiance

pdf file: File:Sadiq hesis Latex.pdf