Sadiq Thesis Latex

Chapter 1: Introduction File:Sadiq thesis chapt 1.txt

\chapter{Introduction}

The positron production efficiency using a proposed linac based quadrupole triplet collection system is measured in partnership with the Continuous Electron Beam Accelerator Facility (CEBAF) of Thomas Jefferson National Accelerator Facility (commonly referred to as Jefferson Lab or JLab) and the Idaho Accelerator Center (IAC) of Idaho State University (ISU). There has been substantial interest in developing a polarized positron source for the nuclear physics community~\cite{PolPos}. The main challenge has been increasing the intensity and polarization of the beam. This work investigates a proposed method for collecting positrons using a quadrupole triplet system.

\section{Different Approaches to Produce Positrons} Positrons are used in several disciplines of science, such as chemistry, physics, material science, surface science, biology and nanoscience~\cite{Chemerisov:2009zz}. Positrons may be obtained either from radioactive sources or from pair production of high energy photons. Radioactive sources like Co-58, Na-22, and Cu-64 emit positive beta particles $\beta^{+}$ when they decay. As given in Table~\ref{tab:e+_source}, the positron rate of radioactive sources is about $10^{7} \sim 10^{8}$ e$^{+}$/s because of the source activity limits of 100~mCi~\cite{PosSource1} that are imposed for radiation safety.

Positrons are available for use in experiments at the nuclear reactors~\cite{PosSource2} shown in Table~\ref{tab:e+_source}. Neutrons are used to produce positrons from the photons emitted when neutrons interact with cadmium. Photons pair produce positrons within the tungsten foils positioned in the photon flux. The positron beam created in the nuclear reactor has the intensity shown in Table~\ref{tab:e+_source}.

Another common method used to generate positrons is by using an electron linear accelerator (linac). In this method, electrons, from the linac, incident on a high Z material like tungsten produce positrons by bremsstrahlung and pair production. One of the advantages of the linac based positron beam is its variable energy, intensity, and the ability to pulse the source on and off at specific time intervals as given in Table~\ref{tab:e+_source}.

\begin{table} \centering \caption{Different Positron Sources~\cite{PosSource1,PosSource2}.} \begin{tabular}{llll} \toprule {} & {} & {19:09, 24 December 2013 (MST)$\beta^{+}$ sources} & {} \\ \midrule {Source} & {Activity} & {Rate (e$^{+}$/s)} & {Facility/Source Type} \\ \midrule %{Source} & {Energy Range} & {Intensity} & {Production Method} \\ {Co-58} & {100~mCi} & {$3\times10^{5}$ } & {W-moderator} \\ {Na-22} & {100~mCi} & {$2\times10^{6}$ } & {W-moderator} \\ {Cu-64} & {80~Ci/cm$^2$} &{$10^{7}\sim10^{8}$} & {Brookhaven, reactor beam} \\

\toprule {} & {} & {~~Pair-production} & {} \\ \midrule {Energy } & {} & {Rate (e$^{+}$/s)} & {Facility/Source Type} \\ \midrule {36~MeV} & {} & {$10^{8}$} & {Giessen, LINAC} \\ {100~MeV} & {} & {$10^{10}$} & {Livermore, LINAC} \\ {150~MeV} & {} & {$10^{8}$} & {Oak Ridge, LINAC} \\ {5~keV} & {} & {$5\times10^{6}$} & {NC State University, reactor} \\ \bottomrule \end{tabular} \label{tab:e+_source} \end{table}

\section{Positron Beam Generation from \\ Bremsstrahlung} When a moving charged particle interacts with the electric field of another charged particle, it can be deflected and lose energy in the form of photons, as shown in Figure~\ref{fig:Theo-Brem}. This interaction is known as the bremsstrahlung process. The probability of this interaction increases with the square of the atomic number of the material traversed by the incident charged particle. Figure~\ref{fig:Brems_photon_Ene} shows the photon energy distribution produced when the 12~MeV electron energy distribution from Figure~\ref{fig:Theo-Brems_ele_Ene} interacts with a 1~mm thick tungsten target. As shown in Figure~\ref{fig:Brems_photon_Ene}, the distribution peaks at 0.3~MeV.

\begin{figure}[htbp] \centering \includegraphics[scale=0.40]{1-Introduction/Figures/bremsstrahlung/brems2.eps} \caption{Photon emission from the bremsstrahlung processes.} \label{fig:Theo-Brem} \end{figure}

The bremsstrahlung cross section for the energy range of this experiment given is by~\cite{brms-cors} %\begin{equation} %d \sigma = 4 Z^2r_e^2 \alpha \frac{d \nu}{\nu} \left \{ \left (1 + \left( \frac{E}{E_0} \right )^2 \right ) \left [ \frac{\phi_1(\gamma)}{4} - \frac{1}{3} \ln Z -f(Z)\right ] - \frac{2E}{3E_0} \left [ \frac{\phi_2(\gamma)}{4} - \frac{1}{3} \ln Z -f(Z)\right ] \right \}, %\label{eq:Brem-cross} %\end{equation} \begin{equation} \begin{array}{cl} d\sigma= & 4Z^{2}r_{e}^{2}\alpha\frac{d\nu}{\nu}\{\left(1+\left(\frac{E}{E_{0}}\right)^{2}\right)\left[\frac{\phi_{1}(\gamma)}{4}-\frac{1}{3}\ln Z-f(Z)\right]\\

& -\frac{2E}{3E_{0}}\left[\frac{\phi_{2}(\gamma)}{4}-\frac{1}{3}\ln Z-f(Z)\right]\},

\end{array}

\label{eq:Brem}

\end{equation} \noindent where $E_0$ is initial total energy of the electron, $E$ is the final total energy of the electron, $\nu = \frac{E_0-E}{h}$ is frequency of the emitted photon, and $Z$ is atomic number of the target. $\gamma = \frac{100 m_ec^2 h \nu}{E_0 E Z^{1/3}}$ is the charge screening parameter, and $f(Z)$ is given by \begin{equation} f(Z) = (Z \alpha)^2 \sum_1^{\infty} \frac{1}{ n [ n^2 + (Z \alpha)^2]}, \end{equation} \noindent where $\alpha = \frac{1}{137}$ is the fine-structure constant, $\phi_1$ and $\phi_2$ are screening functions that depend on Z. \begin{figure}[htbp] \centering \includegraphics[scale=0.75]{1-Introduction/Figures/En_photon_dnT1_logY_3.eps} \caption{Bremsstrahlung photon energy distribution produced when the 12~MeV electron energy distribution from Figure~\ref{fig:Theo-Brems_ele_Ene} interacts with the simulation's 1~mm thick tungsten target.} \label{fig:Brems_photon_Ene} \end{figure} \begin{figure}[htbp] \centering \includegraphics[scale=0.75]{1-Introduction/Figures/En_e_upT1.eps} \caption{The electron energy distribution incident on a tungsten foil in simulation.} \label{fig:Theo-Brems_ele_Ene} \end{figure}

There are three competing processes that a photon can undergo when interacting with matter. Figure~\ref{fig:Theo-3pro-in-W} illustrates the cross-sections for the different interactions that can occur when a photon traverses tungsten as a function of photon energy. At electron volt (eV) energies, which are comparable to the electron atomic binding energy, the dominant photon interaction is the photoelectric effect. As the photon energy increases up to the kilo electron volt (keV) range, the Compton scattering process starts to be more dominant. Although the photon is totally absorbed during the photoelectric effect, photons merely lose energy when undergoing Compton scattering. As the photon energy reaches twice the rest mass energy of the electron, $i.e.$ 2 \begin{math} \times \end{math} 511~keV, pair production begins to occur. Pair production becomes the dominant interaction process when photon energies are beyond 5~MeV~\cite{Krane}. In this process, a photon interacts with the electric field of the nucleus or the bound electrons and is converted into an electron and positron pair.

\begin{figure}[htbp] \centering \includegraphics[scale=0.5]{1-Introduction/Figures/xcom/10.eps} \caption{The cross-sections for different types of photon interactions with tungsten as a function of photon energy~\cite{nistxcom}.} \label{fig:Theo-3pro-in-W} \end{figure}

Using natural units, where \begin{math}c \equiv 1\end{math}, the differential cross-section for pair production can be expressed as \begin{equation} \begin{array}{l} \frac{d \sigma}{d \epsilon_1 d \theta_1 d \theta_2} = 8 \left ( \frac{\pi a}{\sinh (\pi a)} \right )^2 \frac{a^2}{2 \pi} \frac{e^2}{\hbar c} \left ( \frac{\hbar}{m_e c }\right )^2 \frac{\epsilon_1 \epsilon_2}{k^3} \theta_1 \theta_2 \\ \\ \times \{ \frac{V^2(x)}{q^4} \left [ k^2 (u^2 + v^2) \xi \eta - 2 \epsilon_1 \epsilon_2 (u^2 \xi^2 + v^2 \eta^2 ) + 2 (\epsilon_1^2 + \epsilon_2^2)uv \xi \eta cos(\phi) \right ] \\ \\ \left . + a^2W^2(x) \xi^2 \eta^2 \left [ k^2(1 - (u^2+v^2)\xi \eta - 2 \epsilon_1 \epsilon_2 (u^2 \xi^2 + v^2 \eta^2) \right . \right .\\ \\ \left . \left .- 2 (\epsilon_1^2 + \epsilon_2^2) u v \xi \eta \cos(\phi)\right ]\right \}, \\ \end{array} \end{equation}

\noindent where $k$ is photon energy, $\theta_{1}$ and $\theta_2$ are the scattering angle of $e^+$ and $e^-$ respectively, $ \phi = \phi_1 - \phi_2$ is the angle between the $e^+$ and $e^-$ pair, $\epsilon_1$ and $\epsilon_2$ are the energy of the positron and electron respectively. Other constants are $u = \epsilon_1 \theta_1$, $v=\epsilon_2 \theta_2$, $\xi = \frac{1}{1+u^2}$, $\eta= \frac{1}{1+v^2}$, $q^2 = u^2 + v^2 + 2 u v \cos(\phi)$, $x= 1-q^2 \xi \eta$, $a = \frac{Ze^2}{\hbar c}$, $V(x) = 1 + \frac{a^2}{(1!)^2} + \frac{a^2 (1+a^2) x^2}{(2!)^2} + \frac{a^2 (1+a^2)(2^2+a^2)x^4 x^2}{(3!)^2} + \cdots$, and $W(x) = \frac{1}{a^2} \frac{d V(x)}{d x}$.

In pair production, positron and electron pairs are created back to back in the center of mass frame as given by \begin{equation} \gamma \rightarrow e^{+}+e^{-}. \label{eq:pair} \end{equation} In the lab frame, electrons and positrons tend to move in the direction of the photon, as shown in Figure~\ref{fig:Theo-pair-pro}. The positron and electron carry away the energy from the photon that is in excess of 1.022~MeV. In the center of mass frame, the kinetic energy is equally shared. Photons with an energy above 1.022~MeV in the bremsstrahlung spectrum of Figure~\ref{fig:Brems_photon_Ene} have the potential to create electron and positron pairs. Figure~\ref{fig:Theo-brem} is the simulation of 10 million 12~MeV mono energetic electrons impinging on a 1.016~mm thick tungsten target. Turning on the annihilation process resulted in a 511~keV peak on top of the bremsstrahlung spectrum. This 511~keV peak represents photons produced when the created positrons from the pair production annihilate with atomic electrons inside the tungsten target.

\begin{figure}[htbp] \centering \includegraphics[scale=0.40]{1-Introduction/Figures/Pair_Production/Pair_Production.eps} \caption{Pair production.} \label{fig:Theo-pair-pro} \end{figure} \begin{figure}[htbp] \centering \includegraphics[scale=0.7]{1-Introduction/Figures/pair/on/Eng3.eps} \\ \caption{Photon spectrum created by one million 12~MeV mono energy electrons impinging on a 1.016~mm thick tungsten target.} \label{fig:Theo-brem} \end{figure}

\section{Generation of Electron Beam Using a \\ Linac} Electrons need to be accelerated, using a linac (linear particle accelerator), up to at lest few mega electron volts (MeV) in order to generate bremsstrahlung photons with enough energy to produce electron and positron pairs. A linac is a device that accelerates charged particles to nearly the speed of light using an electromagnetic wave. The HRRL (High Repetition Rate Linac), located in the Beam Lab of ISU's Physics Department, can accelerate electrons up to 16~MeV peak energy, which is sufficient energy for positron production.

To optimize positron production, one needs to know the beam size and divergence of the electrons impinging the production target. The emittance and the Twiss parameters ($\alpha$, $\beta$, and $\gamma$) quantify these beam properties and are used as the input parameters for accelerator simulation tools. These tools can study beam transport and predict beam properties (beam size and divergence) along the beam line.

\subsection{Emittance and Twiss Parameters} In accelerator physics, a Cartesian coordinate system is used to describe the motion of the accelerated particles. As shown in Figure~\ref{fig:coordinates}, the $s$-axis of the Cartesian coordinate system is defined as the natural coordinate that is oriented along the same direction as the beam momentum. The $x$-axis and $y$-axis are horizontal and vertical coordinates which constitute the transverse beam profile. The transverse beam profiles are described as a function of the longitudinal coordinates, $x(s)$ and $y(s)$.

\begin{figure}[htbp] \centering \includegraphics[scale=0.7]{1-Introduction/Figures/coordinates.eps} \caption{Coordinated system and reference orbit (dashed line)~\cite{Conte}.} \label{fig:coordinates} \end{figure}

The horizontal phase space $x'$ vs. $x$ (similar for $y$ projection), shown in Figure~\ref{fig:phase-space}, of the beam is an ellipse with invariant area along the beamline (under conditions that space charge effects, coherent synchrotron radiation, and wakefield are ignorable)~\cite{Conte}. Here $x'$ is defined as \begin{equation} x'=\frac{dx}{ds}. \label{eq:divergence} \end{equation} \noindent The area of the ellipse is \begin{equation} A = \pi w = \pi (\gamma z^2 + 2\alpha z z^{\prime} + \beta z^{\prime 2}). \label{eq:el-area} \end{equation} \noindent Here $w$ is called the Courant-Snyder invariant~\cite{Conte} and $\alpha$, $\beta$, and $\gamma$ are called Twiss parameters. The transverse emittance $\epsilon$ of the beam is defined to be the area of the ellipse that contains fraction of the particles. The units for the emittance are m$\cdot$rad and mm$\cdot$mrad. Conventionally used unit for emittance is $\mu$m, which is similar as mm$\cdot$mrad (1 $\mu$m = 1 mm$\cdot$mrad = $10^{-6}$ m$\cdot$rad). Twiss parameters are useful because they are related to the beam size and divergence by \begin{equation} \sigma_{x}(s)=\sqrt{\epsilon _x (s) \beta _x (s)},~ \sigma_{x'}(s)=\sqrt{\epsilon _x (s) \gamma _x (s)}, \label{eq:twiss-emit} \end{equation} \noindent where $\epsilon_{x}$ is beam's horizontal emittance, $\sigma_{x}$ is the horizontal rms beamsize, $\sigma_{x'}$ is the horizontal rms beam divergence, and $\beta _x $ and $\gamma _x$ are two of the three Twiss parameters (same for vertical projection). The Twiss parameters are related by \begin{equation} \gamma = \frac{1 + \alpha^2}{\beta}. %\text{ or } \beta _x (s) \gamma _x (s) - \alpha _x (s)^2 = 1. \label{eq:twiss} \end{equation} \begin{figure}[htbp] \centering \includegraphics[scale=0.5]{1-Introduction/Figures/Emttance_Ellipse.eps} \caption{Phase space ellipse and its relation to the emittance and the Twiss parameters $\alpha$,$\beta$, and $\gamma$~\cite{Conte}.} \label{fig:phase-space} \end{figure} The beam emittance is inversely proportional to the beam energy. As the electron beam gains energy during acceleration, the divergence of the beam decreases because the momentum increases in forward direction. This emittance decrease due to the energy gain is called adiabatic damping. Thus, it is more practical to use normalized emittance which is defined as \begin{equation} \epsilon_{\text{n}} = \beta\gamma\epsilon, \label{eq:nor_emi} \end{equation} where $\beta$ is $c/v$, $\gamma$ is relativistic (Lorentz) factor, and $\epsilon$ unnormalized emittance.

In this chapter, the possible ways to generate positrons were described. Positrons from radioactive sources have lower intensities due to safety limits placed on the source activity. Positrons from reactors have intensities comparable to radioactive sources but are low in energy. Generating positrons using a linac can produce higher energy positrons at higher fluxes than reactors.

Chapter 2 discusses the hardware used in this experiment and the tools used to measure emittance, Twiss parameter, and energy of the HRRL. The measured emittance, Twiss parameter and energy, as well as the positron production efficiency are discussed in the Chapter 3. The simulation of the positron production efficiency and transportation process are compared with measurement in Chapter 4. The conclusion from this work is presented in Chapter 5.

Chapter 2: Apparatus File:Sadiq thesis chapt 2.txt

\chapter{Apparatus} This chapter describes the apparatus and associated hardware used in the experiment to produce positrons as described in Chapter 1. The new HRRL beamline was constructed using two dipoles as well as ten quadrupoles to optimize the beam transportation. Beamline elements were aligned using a laser alignment system. Additional beamline elements such as an energy slit, beam viewers, and Faraday cups were added. Two NaI detectors were installed at the end of the beamline and were used to detect 511~keV photons emitted when positrons annihilate in a second tungsten target.

\section{HRRL Beamline} A 16~MeV S-band (2856~MHz RF frequency) standing-wave High Repetition Rate Linac (HRRL) located in the Department of Physics Beam Lab at Idaho State University was used to impinge a 12~MeV electron beam onto a tungsten foil. The energy of the HRRL is tunable between 3 to 16~MeV and its repetition rate is variable from 1 to 300~Hz. The operating parameters of the HRRL is given in Table~\ref{tab:hrrl-par}. As shown in Figure~\ref{fig:app-hrrl-cavity}, the HRRL has a thermionic gun, vertical and horizontal steering magnet sets on two ends, and two solenoid magnets. The startup, shutdown, and beam optimization procedure of the HRRL is given in the Appendix C.

\begin{table} \centering \caption{The Basic Parameters of the HRRL.} \begin{tabular}{lcc} \toprule {Parameter} & {Unit} & {Value} \\ \midrule maximum energy & MeV & 16 \\ peak current & mA & 100 \\ repetition rate & Hz & 300 \\ absolute energy spread & MeV & 25\% \\ macro pulse length & ns & $>$50 \\ RF Frequency & MHz & 2856 \\ \bottomrule \end{tabular} \label{tab:hrrl-par} \end{table}

\begin{figure}[htbp] \centering \includegraphics[scale=0.74]{2-Apparatus/Figures/HRRL_Cavity4.png} \caption{The configuration of the HRRL cavity.} \label{fig:app-hrrl-cavity} \end{figure}

The accelerator's cavity was relocated to the position shown in Figure~\ref{fig:app-hrrl-line} to provide enough space for a beam line that can transport either positrons or electrons. The beam elements are described in Table~\ref{tab:app-hrrl-coordinates}. Quadrupole and dipole magnets, described in Appendix D, were added to the new beam line as well as an OTR and YAG view screen to measure and monitor beam profile and position. Faraday cups and toroids were installed to measure the electron beam current. Energy slits were installed to control the energy/momentum spread of the beam after the first dipole (D1).

A 1.016~mm thick retractable tungsten (99.95\%) foil target (T1) was placed between the 1st and 2nd quadrupole triplets and used to produce positrons when the electron beam interacts with it. The room where the HRRL is located is divided by a wall into two parts; the accelerator side and the experimental cell. A beam pipe at the end of the 90 degree beamline goes through a hole in the wall and delivers the beam from the accelerator side to the experimental cell. The positron detection system consisting of two NaI detectors was placed at the end of the beamline in the experimental cell side as shown in Figure~\ref{fig:app-hrrl-line}.

\begin{sidewaysfigure} \centering \includegraphics[scale=0.265]{2-Apparatus/Figures/HRRL_line2.eps} \caption{The HRRL beamline layout and parts.} \label{fig:app-hrrl-line} \end{sidewaysfigure}

\begin{table} \centering \caption{The HRRL Beamline Parts and Coordinates.} \begin{tabular}{llll} \toprule {Label} & {Beamline Element} & {Distance from} & {} \\ {} & {} & {Linac Exit (mm)} & \\ \midrule Q1 & quadrupole & 335 & \\ Q2 & quadrupole & 575 & \\ Q3 & quadrupole & 813 & \\ T1 & e$^+$ production target & 1204 & \\ Q4 & quadrupole & 1763 & \\ Q5 & quadrupole & 2013 & \\ Q6 & quadrupole & 2250 & \\ D1 & dipole & 2680 & \\ S1 & OTR screen & 3570 & \\ FC1 & Faraday cup & 3740 & \\ EnS & energy slit & 3050 & \\ S2 & YAG screen & 3410 & \\ Q7 & quadrupole & 3275 & \\ D2 & dipole & 3842 & \\ FC2 & Faraday cup & 4142 & \\ Q8 & quadrupole & 4044 & \\ Q9 & quadrupole & 4281 & \\ Q10 & quadrupole & 4571 & \\ T2 & annihilation target & 7381 & \\ \bottomrule \end{tabular} \label{tab:app-hrrl-coordinates} \end{table} The positron production target, T1 was shielded with 8 inches of Fe bricks and 4 inches of Pb bricks to lower photon and electron background. The two dipoles were also shielded with Pb bricks because the beam scrapes the dipole vaccuum chamber when it is transported. An eight inch thick Pb wall was installed both sides of the wall dividing the accelerator and the experimental cell to shield the cell from background generated on the accelerator side. The NaI detectors were shielded with Pb brick to lower background. Positioning the NaI detector in a separate room from the positron production target T1 was done to decrease the high background that is historically present in the accelerator room. While this choice reduces the background there is also substantial signal loss, as will be shown in subsequent chapters, transporting a chromatic positron sample to the experimental cell.

\section{HRRL Beamline Alignment Using Laser} The HRRL beamline was aligned using a laser beam as shown in Figure~\ref{fig:alignment}. The gun of the HRRL was removed so that the laser beam from a laser placed on a table in the experimental side of the HRRL cell would be directed through the cavity. Mirrors were mounted onto holders with horizontal and vertical adjustments. The laser was first adjust with two mirrors on the laser table and focused using two focusing lenses. Two mirrors, one in the experimental side and one at on the linac side, reflected the laser beam through the center of the linac.

\begin{figure}[htbp] \centering \includegraphics[scale=0.24]{2-Apparatus/Figures/HRRL_Alignment.png} \caption{HRRL beamline alignment using laser.} \label{fig:alignment} \end{figure}

The laser beam was shot through the center of the HRRL cavity and the geometrical center of the 0 degree beamline magnets (quadrupoles Q1$\sim$Q6, first dipole D1) downstream were aligned according to the laser beam. The laser beam was reflected by a mirror mounted on a rotator that reflected the beam by 90$^{\circ}$ to the 90 degree beamline and the quadrupole magnets in the 90 degree beamline were aligned relative to the laser beam reference point.

Two irises were placed at the end of the 90 degree beamline and centered on the reflected 90 degree laser beam. A second laser was mounted on the wall in the experimental cell side. The laser on the wall was positioned to pass through the center of the two irises placed at the end of the 90 degree beamline. Thus, the laser on the wall was aligned to the center of the 90 degree beamline and can be used as a reference as shown in Figure~\ref{fig:laser}. \begin{figure}[htbp] \centering \includegraphics[scale=0.23]{2-Apparatus/Figures/BeamlineParts/laser.jpg} \caption{The laser mounted on the wall in experimental cell side. This laser was aligned to the center of the 90 degree beamline.} \label{fig:laser} \end{figure}

At last, the second dipole, D2, was connected to the 90 degree beampipe and beamline elements between D1 and D2 (energy slits, Q7, and 5-way cross holds YAG screen) were placed one after another.

\section{Energy Slit and Flag Controller} A control box was built to open and close the energy slit as well as control the three flags shown in Figure~\ref{fig:controller}. The slit control is on the right side of the box and the maximum width of the slit is 3.47~cm as shown in Figure~\ref{fig:controller}. On the left of the controller are the switches for three flags. Power supplies were installed inside the box and controlled by these switches to remove the targets from the beamline, turn on/off the cameras of the flags, and turn on/off the lights within each flag.

\subsection{Energy Slit Controller} The controller was built to open or close the energy slit (Danfysik water cooled slit model 563 system 5000) based on the design from Danfysik~\cite{Danfysik} as shown in Figure~\ref{fig:controller} (on right side). The wiring diagram of the controller is shown in Figure~\ref{fig:controllerConnection}. When the energy slit is fully open, the width between slits is 3.47~cm as indicated by the LED number display. When the slit is fully open/closed, one of the two LED lights will light up and the motor will stop.

The power source of the energy slit provides 12~VDC, 1.8~A max current, 20~W power, and takes 85$\sim$264~VAC input. The motor and the relay switch both takes 12~VDC. The LED indicating lights take 12~V/50~mA current and the 57~$\Omega$ resistors take 0.5~W power.

A potentiometer is placed inside the energy slit and the resistance of the potentiometer changes with the width of the slit. A 370~mV voltage is applied to the potentiometer. An LED number display is connected to the potentiometer so that the voltage change in the potentiometer is numerically indicated by the LED display. The slit width is indicated by the potentiometer resistance in the circuit.The potentiometer resistance in the circuit is indicated by the voltage across the resistor in the circuit which is displayed on the LED number display. For example, when the width of the energy slit is 3.24~cm, the voltage on the potentiometer read by the LED display is 324~mV, and the LED number display indicates 3.24. The cm unit is labeled on the right of LED display. \begin{figure}[htbp] \centering \includegraphics[scale=0.29]{2-Apparatus/Figures/BeamlineParts/controller.jpg} \caption{Front panel of the energy slit controller (on the right) and flag controllers (on the left). } \label{fig:controller} \end{figure} \begin{figure}[htbp] \centering \includegraphics[scale=0.43]{2-Apparatus/Figures/controllerConnection.png} \caption{Controller wiring diagram of energy slit controller (modified from the design given in the Danfysik water cooled slit model 563 system 5000 manual)~\cite{Danfysik}.} \label{fig:controllerConnection} \end{figure}

\subsection{Flag Controller} On the left of the controller box are the switches for three flags (0 degree/OTR flag, 45 degree/YAG flag, and 90 degree/annihilation target flag) as shown in Figure~\ref{fig:controller}. Four power supplies (one back up) were installed inside the box and controlled by these switches. The switches in the top row insert or remove the targets from the beamline and the ones in the middle turn on/off the cameras of the flags. The bottom row switches turn on/off the lights of the flag which lights up the target.

\section{The OTR Imaging System} The OTR target is a 10~$\mu$m thick aluminum foil with a 1.25 inch diameter. A moving charged particle produces electromagnetic fields that are dependent on the dielectric constant of the media the particle traverses. When a moving charge particle cross the boundary between two media (vacuum and aluminum in this case), electromagnetic radiation is emitted to reorganize the fields in the new media. The radiation is emitted in both forward (beam direction) and backward (image charge direction) as shown in Figure~\ref{fig:otr}. \begin{figure}[htbp] \centering \includegraphics[scale=0.6]{2-Apparatus/Figures/OTR.eps} \caption{OTR pattern when the incident beam is at $45^{\circ}$ angle with respect to the foil~\cite{OTR-Gitter}.} \label{fig:otr} \end{figure}

Three two inch diameter lenses were used for the imaging system to avoid optical distortion at lower electron energies. The focal lengths and positions of the lenses, shown in Figure~\ref{image_sys}, were calculated with the thin lens equation to produce the desired image magnification. The lenses, assembly rods, and lens holder plates for the camera cage system were purchased from Thorlabs, Inc.~\cite{thorlabs}.

The camera used is a JAI CV-A10GE digital 1/2" progressive scan camera with a 767 (horizontal) by 576 (vertical) pixel area and 6.49 (horizontal) by 4.83 (vertical) mm sensing area. It has high speed shutter from 1/60 to 1/300,000 second, edge pre-select, pulse width trigger modes, auto shutter, and smear-less mode. The images were taken by triggering the camera (in edge pre-select mode) synchronously with the electron gun. \begin{figure}[htbp] \centering {\scalebox{0.46} [0.46]{\includegraphics{2-Apparatus/image_sys2.eps}}} {\scalebox{0.5} [0.5]{\includegraphics{2-Apparatus/MOPPR087f3}}} \caption{The OTR imaging system.} \label{image_sys} \end{figure}

\section{Positron Detection} When the electron beam is incident on T1, photons and secondary electrons are created along with positrons. These particles are the main source of noise in the experiment. The positrons were transported to the second tungsten target (T2) which was shielded from this background by the concrete wall and Pb bricks. The setup is shown in Figure~\ref{fig:HRRL-pos-det-setup}. A 6-way cross was placed at the end of the beamline to hold T2. The 6-way cross has three 1~mil (0.0254 millimeters) thick stainless steel windows. The two horizontal windows perpendicular to the beamline allowed the 511~keV photons created from the positron annihilation to escape the beamline with a limited attenuation. A third window at the end of the 90 degree beamline was used as the beam exit. Two NaI detectors were placed facing the two exit windows to detect the photons produced in T2. A 2 inch thick lead brick collimator with a 2 inch diameter hole was placed between the exit window and NaI detector. A scintillator (Scint) and a Faraday cup (FC3) were placed at the end of the beamline and were used to tune the electron and positron beam. When positrons reach T2, they can thermalize and annihilate inside T2. During thermalization, a positron looses its kinetic energy. When it annihilates with an electron, two 511~keV photons are emitted back to back. A triplet coincidence is required between the accelerator RF pulse and the detection of a photon in each NaI detector.

\begin{figure}[htbp] \centering \includegraphics[scale=0.50]{2-Apparatus/HRRL_Pos_detection2.eps} \caption{The positron detection system: T2 was placed with 45$^\circ$ angle to the horizontal plane first, then rotated 45$^\circ$ along the vertical axis.} \label{fig:HRRL-pos-det-setup} \end{figure}

\subsection{NaI Detectors}

NaI crystals, shown in Figure~\ref{fig:PMT}, were used to detect 511~keV photons from positron annihilation. Originally, the detectors had pulse signal lengths around 400~$\mu$s. New PMT base were built to use the HV divider shown in Figure~\ref{fig:PMT_base}. A picture of the constructed bases is shown in Figure~\ref{fig:new_base_made}. The pulse length of the new PMT base is about 1~$\mu$s. The NaI crystal is from Saint-Gobain Crystal \& Detectors (Mod. 3M3/3) with a dimension of $3 \times 3$. Operating high voltage of -1150~V would position the 511~keV photons within the range of the charge sensing ADC (CAEN Mod. V792).

\begin{table} \centering \caption{The Radioactive Sources and Corresponding Photon Peaks.} \begin{tabular}{lccc} \toprule {Radioactive Sources} & Unit & First Peak & Second Peak \\ \midrule Co-60 & keV & 1173 & 1332 \\ Na-22 & keV & 511 & 1275 \\ \bottomrule \end{tabular} \label{tab:Na22_Co60} \end{table}

\begin{figure}[htbp] 
\centering
\includegraphics[scale=0.52]{2-Apparatus/SAINT-GOBAIN_3M33.png}
\caption{The NaI crystal dimension.}
\label{fig:PMT}
\end{figure}

\begin{figure}[htbp] \centering \includegraphics[scale=0.75]{2-Apparatus/Modified_PMT.png} \caption{The modified PMT base design.} \label{fig:PMT_base} \end{figure}

\begin{figure}[htbp] \centering \includegraphics[scale=0.13]{2-Apparatus/IAC_NaI.png} \caption{The NaI crystals and new bases.} \label{fig:new_base_made} \end{figure}

The NaI detectors were calibrated using a Na-22 and a Co-60 source with the photon peaks indicated in Table~\ref{tab:Na22_Co60}. Figure~\ref{fig:NaI_Co60_Scope} is the oscilloscope image of several Co-60 photon pulses observed by the detector with the new PMT. The calibrated NaI detector spectrum from the Na-22 and Co-60 sources is shown in Figure~\ref{fig:NaI-Calb}. The rms values of the fits on the four peaks shown in Figure~\ref{fig:NaI-Calb} are $\sigma_{Na, 511}=18.28\pm0.04$~keV, $\sigma_{Na, 1275}=44.51\pm0.27$~keV, $\sigma_{Co, 1173}=42.49\pm0.24$~keV, and $\sigma_{Co, 1332}=50.30\pm0.39$~keV.

\begin{figure}[htbp] \centering \includegraphics[scale=0.5]{2-Apparatus/NaI_Co60_Scope.png} \caption{Detector output pulses using the Co-60 source and new PMT. The amplitude of the pulse is about 60~mV. The rise time of the pulse is larger than 50~ns, and the fall time is larger than 700~ns.} \label{fig:NaI_Co60_Scope} \end{figure}

\begin{figure}[htbp] \centering \includegraphics[scale=0.77]{2-Apparatus/Figures/NaI_Calbration/NaI_Calb8.eps} \caption{The calibrated NaI spectrum of Na-22 and Co-60 sources.} \label{fig:NaI-Calb} \end{figure}

\subsection{The DAQ Setup}

The data acquisition (DAQ) setup and timing diagram is shown in Figure~\ref{fig:daq-setup}. The last dynode signals from left and right NaI detectors were inverted using a ORTEC 474 inverting amplifier and sent to a CAEN Mod. N842 constant fraction discriminator (CFD). The electron gun pulse from the linac was used to generate a VETO for the CFD to prevent the RF noise from triggering the CFD, otherwise the CFD would generate multiple digital pulses. The PMT output itself could produce multiple CFD output pulses due to the voltage fluctuations within a single pulse. To prevent multiple CFD outputs, a GG~8000-01 octal gate generator was used to create a single 1~$\mu$s wide pulse from the CFD first pulse in order to ignore the multiple CFD pulses produced by a single analog output pulse from the detector. A triple coincidence was formed between the gun pulse and the 1~$\mu$s wide pulse from each detector using a LeCroy model 622 logic module. %\begin{equation} %\text{NaI~Left~\&\&~NaI~Right~\&\&~Gun~Trigger}. %\end{equation}

The ADC requires 5.7~$\mu$s to convert the analog signal to a digital signal. The logic module output was delayed 6~$\mu$s by a dual timer (CAEN Mod. N93B) to accommodate the ADC's conversion time and trigger the DAQ. The ADC (CAEN Mod. V792) converted the NaI detector's analog signals to digital when a 1~$\mu$s gate, created by the gun pulse using a dual timer, was present as shown in the lower part of Figure~\ref{fig:daq-setup}. The ADC was fast cleared unless it received a veto signal from the inverted output of logic module created using a dual timer.

\begin{sidewaysfigure}[htbp] \centering \includegraphics[scale=0.8]{2-Apparatus/Figures/DAQ_Logic_all.eps} \caption{The DAQ setup and timing diagram.} \label{fig:daq-setup} \end{sidewaysfigure}

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 5: Conclusion File:Sadiq thesis chapt 5.txt

pdf file: File:Sadiq hesis Latex.pdf