Sadiq Thesis Latex

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

\chapter{Introduction}

The positron production efficiency using a proposed 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 that uses 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 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 using electromagnetic wave to near speed of light. 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. This can be achieved if the emittance and the Twiss parameters ($\alpha$, $\beta$, and $\gamma$) are given at the exit of a linac. Emittance and Twiss parameters are the input parameters for accelerator simulation tools used to 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 effect, coherent synchrotron radiation, and wakefiled 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 electron beam gains energy during the acceleration, the divergence of the beam decreases because the momentum increases in forward direction. This emittance decreases 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 are 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 rates higher than reactors.

Chapter 2 discusses the hardware used in the experiment and the tools used to measure emittance, Twiss parameter, energy of the HRRL. The emittance, Twiss parameter and energy, as well as the positron production are discussed in the Chapter 3. The simulation of the positron production and transportation process are compared with measurement in Chapter 4. The conclusion from this work are presented in Chapter 5. %The HRRL beamline, beamline alignment, beamline elements, controllers and the imaging system used to measure emittance and Twiss parameters are described in details in Chapter 2. The positron detection system are also dIn the Chapter 3, the emittance/Twiss parameters measurement and energy scan of HRRL are discussed and results are shown. Th

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

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

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

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

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