Difference between revisions of "Sadiq Thesis Latex"

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=PDF=
 
  
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[[File:Sadiq_hesis_Latex.pdf]]
 
  
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= Text =
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Sadiq Thesis in pdf: [[File:Sadiq_hesis_Latex.pdf]]
  
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==Introduction==
 
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\chapter{Introduction}
 
  
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\section{Positron Beam}
 
  
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Positrons have many potentials in many discipline of science, like chemistry, physics, material science, surface science, biology and nanoscience~\cite{Chemerisov:2009zz}. There are many different ways to generate positrons, and the main challenge is increasing the intensity (or current) of the positron beam.
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=Chapter 1: Introduction [[File:sadiq_thesis_chapt_1.txt]]=
  
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\section{Motivation}
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=Chapter 2: Apparatus [[File:sadiq_thesis_chapt_2.txt]]=
  
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The nucleon electromagnetic form factors are fundamental  quantities that related to the charge and magnetization distribution in the nucleon. Conventionally, the nucleon form factors are measured using Rosenbluth Technique (RT)~\cite{Rosenbluth1950}. 
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=Chapter 3: Data Analysis [[File:sadiq_thesis_chapt_3.txt]]=
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The form factor scaling ratio, \begin{math}R=\mu _p G_{Ep} / G_{Mp}\end{math}, measured using this technique is around unity as shown in the figure below~\cite{PhysRevD.49.5671}.
 
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Since nighties, a technique using elastic electron-proton polarization transfer to measurement this ratio have been developed~\cite{PhysRevD.49.5671, PhysRevC.68.034325, WalkerThesis1989}. In this technique, form factor scaling ratio linearly decreases as the \begin{math}Q^2\end{math} increases, as shown in the Fig.~\ref{rosen-com-RPT}.
 
  
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\begin{figure}[htb]
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=Chapter 4: Simulation [[File:sadiq_thesis_chapt_4.txt]]=
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\centering
 
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\includegraphics[scale=0.70]{1-Introduction/Figures/Sadiq_thesis_mot_RT_RPT_1.png}
 
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\caption{Form factor ratio, obtained by Rosenbluth Technique (hollow square) and results from Recoil Polarization Technique~\cite{PhysRevC.68.034325}.}
 
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\label{rosen-com-RPT}
 
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\end{figure}
 
  
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=Chapter 5: Conclusion [[File:sadiq_thesis_chapt_5.txt]]=
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The disagreement could arise from the fact the Rosenbluth Techqniue assumes that One Photon Exchange (OPE) during the scattering while the two–photon exchange (TPE), which depends weakly on \begin{math}Q^2\end{math}, could also become considerable with increasing \begin{math}Q^2\end{math}~\cite{PhysRevC.68.034325}. The contribution of TPE can be obtained by comparing the ratio of \begin{math}e^+~p\end{math} to \begin{math}e^-~p\end{math} ratio. The interference of OPE and TPE can also be studied in the process \begin{math}e^+e^- \rightarrow p\bar p\end{math}
 
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==  Theory ==
 
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\chapter{Theory}
 
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\section{Positron Beam Generation from Bremsstrahlung}
 
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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 Fig.~\ref{fig:Theo-Brem}. This interaction is known as the Bremsstrahlung process. The probability of this interaction increases with the atomic number of the material traversed by the incident charged particle. Fig.~\ref{fig:Brems_photon_Ene} shows the photon energy distribution when a 12 MeV electron distribution from Fig.~\ref{fig:Theo-Brems_ele_Ene} interacts with a 1 mm thick Tungsten target. The number of photons in this example produced decreases as the energy of the produced photon increases. The Bremsstrahlung photons are also likely to interact with the material.
 
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\begin{figure}[htb]
 
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\centering
 
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\includegraphics[scale=0.40]{2-Theory/Figures/bremsstrahlung/brems.eps}
 
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\caption{Photon generation from Bremsstrahlung processes.}
 
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\label{fig:Theo-Brem}
 
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\end{figure}
 
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The cross section of Bremsstrahlung process is give by Eq.~\ref{eq:Brem-cross}~\cite{brms-cors}
 
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\begin{equation}
 
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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 \}
 
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\label{eq:Brem-cross}
 
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\end{equation}
 
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where, $E_0$ is initial total energy of the electron, $E$ is final total energy of the electron, $\nu = \frac{E_0-E}{h}$ is energy of the emitted photon, and $Z$ is atomic number. $\gamma = \frac{100 m_ec^2 h \nu}{E_0 E Z^{1/3}}$ is charge screening parameter and $f(Z)$ is given by~\cite{tf-wiki}
 
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\begin{equation}
 
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f(Z) = (Z \alpha)^2 \sum_1^{\infty} \frac{1}{ n [ n^2 + (Z \alpha)^2]}
 
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\end{equation}
 
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where $\alpha = \frac{1}{137}$ is fine-structure constant, $\phi_1$ and $\phi_2$ are screening functions that depend on Z
 
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\begin{figure}[htb]
 
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\centering
 
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\includegraphics[scale=0.50]{2-Theory/Figures/Brems_photon_Ene.png}
 
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\caption{Simulated Bremsstrahlung photon energy right after a tungsten foil.}
 
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\label{fig:Brems_photon_Ene}
 
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\end{figure}
 
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\begin{figure}[htb]
 
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\centering
 
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\includegraphics[scale=0.50]{2-Theory/Figures/Brems_ele_Ene.png}
 
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\caption{Simulated electron energy distribution right before a tungsten foil.}
 
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\label{fig:Theo-Brems_ele_Ene}
 
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\end{figure}
 
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There are three competing processes that a photon can undergo when interacting with matter. At electron volt (eV) energies comparable to the electron atomic binding energy, the dominant photon interaction is via photoelectric effect. As the photon energy increases up to 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 electron, 2 \begin{math} \times \end{math} 511~keV, pair production begins to happen. Pair production becomes dominant interaction process only for energies above 5 MeV~\cite{Krane}. In this process, a photon interacts with the electric field of the nucleus or the bound electrons and decays into an electron and positron pair.
 
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\begin{figure}[htb]
 
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\centering
 
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\includegraphics[scale=0.50]{2-Theory/Figures/3pro-in-W.png}
 
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\caption{Cross section of processes that photons interacts with tungsten~\cite{nistxcom}}
 
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\label{fig:Theo-3pro-in-W}
 
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\end{figure}
 
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Using natural unit \begin{math}c \equiv 1\end{math}, the differential cross-section for pair production can be expressed as~\cite{pair-cors, tf-wiki},
 
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\begin{equation}
 
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\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
 
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\end{equation}
 
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\begin{description}
 
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\item[]
 
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\begin{math}
 
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\times \left \{ \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 ]  \right .
 
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\end{math}
 
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\end{description}
 
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\begin{equation}
 
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\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) -2 (\epsilon_1^2 + \epsilon_2^2) u v \xi \eta \cos(\phi)\right ]\right \}
 
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\end{equation}
 
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where $k$ is photon energy, $\theta_{1}1$ 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 $\epsilon_2$
 
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are the energy of the  positron and electron respectively. Other constants are
 
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\begin{equation}
 
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u = \epsilon_1 \theta_1
 
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\end{equation}
 
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\begin{equation}
 
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v=\epsilon_2 \theta_2
 
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\end{equation}
 
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\begin{equation}
 
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\xi = \frac{1}{1+u^2}
 
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\end{equation}
 
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\begin{equation}
 
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\eta= \frac{1}{1+v^2}
 
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\end{equation}
 
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\begin{equation}
 
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q^2 = u^2 + v^2 + 2 u v \cos(\phi)
 
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\end{equation}
 
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\begin{equation}
 
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x= 1-q^2 \xi \eta
 
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\end{equation}
 
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\begin{equation}
 
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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
 
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\end{equation}
 
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\begin{equation}
 
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W(x) = \frac{1}{a^2} \frac{d V(x)}{d x}
 
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\end{equation}
 
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\begin{equation}
 
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a = \frac{Ze^2}{\hbar c}
 
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\end{equation}
 
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The positron and electron pairs are created back to back in the center of mass frame. In the lab frame, electrons and positrons are boosted forward, as demonstrated in the Fig.~\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 Fig.~\ref{fig:Brem} have the potential to create electron and positron pairs. When the process of annihilation is included in the simulation, Fig.~\ref{fig:brem} becomes Fig.~\ref{fig:annih} showing a clear 511~keV peak on top of the bremsstrahlung spectrum. This 511 keV peak represents photon produced when the created positrons, from pair production, annihilates with an atomic electrons inside the tungsten target.
 
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\begin{figure}[htb]
 
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\centering
 
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\includegraphics[scale=0.650]{2-Theory/Figures/Pair_Production/Pair_Production.png}
 
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\caption{Pair production.}
 
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\label{fig:Theo-pair-pro}
 
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\end{figure}
 
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\begin{figure}[htb]
 
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\centering
 
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\includegraphics[scale=0.50]{2-Theory/Figures/Brems_photon_Ene_log.png}
 
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\caption{Simulated Bremsstrahlung photon energy right after a tungsten foil.}
 
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\label{fig:Theo-gamma-pair-pro}
 
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\end{figure}
 
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\begin{figure}[htb]
 
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\centering
 
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\includegraphics[scale=0.50]{2-Theory/Figures/Brems_photon_Ene_log.png}
 
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\caption{Simulated Bremsstrahlung photon energy right after a tungsten foil.}
 
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\label{fig:Theo-gamma-pair-pro}
 
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\label{fig:Brem}
 
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\end{figure}
 
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\begin{figure}[htb]
 
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\centering
 
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\includegraphics[scale=0.50]{2-Theory/Figures/Brems_photon_Ene_log.png}
 
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\caption{Simulated Bremsstrahlung photon energy right after a tungsten foil.}
 
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\label{fig:brem}
 
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\end{figure}
 
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\begin{figure}[htb]
 
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\centering
 
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\includegraphics[scale=0.50]{2-Theory/Figures/Brems_photon_Ene_log.png}
 
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\caption{Simulated Bremsstrahlung photon energy right after a tungsten foil.}
 
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\label{fig:annih}
 
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\end{figure}
 

Latest revision as of 17:42, 14 March 2014


Sadiq Thesis in pdf: File:Sadiq hesis Latex.pdf


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

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

Retrieved from "https://wiki.iac.isu.edu/index.php?title=Sadiq_Thesis_Latex&oldid=91648"