\documentclass[12pt]{article}
\usepackage{fullpage}
\usepackage{setspace}
\usepackage{parskip}
\usepackage{titlesec}
\usepackage[section]{placeins}
\usepackage{xcolor}
\usepackage{breakcites}
\usepackage{lineno}
\usepackage{hyphenat}
\PassOptionsToPackage{hyphens}{url}
\usepackage[colorlinks = true,
linkcolor = blue,
urlcolor = blue,
citecolor = blue,
anchorcolor = blue]{hyperref}
\usepackage{etoolbox}
\makeatletter
\patchcmd\@combinedblfloats{\box\@outputbox}{\unvbox\@outputbox}{}{%
\errmessage{\noexpand\@combinedblfloats could not be patched}%
}%
\makeatother
\usepackage[round]{natbib}
\let\cite\citep
\renewenvironment{abstract}
{{\bfseries\noindent{\abstractname}\par\nobreak}\footnotesize}
{\bigskip}
\titlespacing{\section}{0pt}{*3}{*1}
\titlespacing{\subsection}{0pt}{*2}{*0.5}
\titlespacing{\subsubsection}{0pt}{*1.5}{0pt}
\usepackage{authblk}
\usepackage{graphicx}
\usepackage[space]{grffile}
\usepackage{latexsym}
\usepackage{textcomp}
\usepackage{longtable}
\usepackage{tabulary}
\usepackage{booktabs,array,multirow}
\usepackage{amsfonts,amsmath,amssymb}
\providecommand\citet{\cite}
\providecommand\citep{\cite}
\providecommand\citealt{\cite}
% You can conditionalize code for latexml or normal latex using this.
\newif\iflatexml\latexmlfalse
\providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}%
\AtBeginDocument{\DeclareGraphicsExtensions{.pdf,.PDF,.eps,.EPS,.png,.PNG,.tif,.TIF,.jpg,.JPG,.jpeg,.JPEG}}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage{float}
\begin{document}
\title{Compton Effect}
\author[1]{Forrest Bullard}%
\affil[1]{California State University, Chico}%
\vspace{-1em}
\date{\today}
\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup
\sloppy
\section*{Introduction}
{\label{650319}}
In this experiment we will explore the particle nature of light. In
particular we will see that scattered photons have less energy than
unscattered photons in accordance with the same equations that can be
used in classical elastic collisions. In general we will show that this
collision conserves both energy and momentum.
\section*{Math}
{\label{785945}}
To derive the equation necessary to model Compton scattering we will
need to use both conservation of energy and momentum.~ Conservation of
momentum shows us that the initial momentum of the incoming
photon~~\(\vec{p_1}\) is equal to the final momentum of the
photon~~\(\vec{p_2}\) plus the momentum of the
electron~~\(\vec{p_e}\) .
\begin{equation} \label{eq:1}
\vec{p_1} = \vec{p_2} + \vec{p_e}
\end{equation}
Now we will need to find the magnitude of the~ momentum of the electron
for later use. A simple dot product will do.
\begin{equation} \label{eq:2}
{p_e}^2 = \vec{p_e}\cdot \vec{p_e} = {p_1}^2 + {p_2}^2 - 2{p_1}{p_2}\cos(\theta)
\end{equation}
Conservation of energy gives us
\begin{equation} \label{eq:2}
E_1 + E_0 = E_2 + \sqrt{{E_0}^2 + {p_e}^2{c}^2}
\end{equation}
where~\(E_0\) is the rest energy of the electron and the
notations 1 and 2 are used for first the initial energy of the incoming
photon and then the scattered photon. Rearranging and squaring both
sides gives us
\begin{equation} \label{eq:4}
{E_0}^2 + 2E_0 (E_1 - E_2) + (E_1 + E_2)^2 = {E_0}^2 + {p_e}^2c^2 .
\end{equation}
Now substitute in equation 2 to obtain
\begin{equation} \label{eq:5}
{E_0}^2 + 2E_0 (E_1 - E_2) + (E_1 + E_2)^2 = {E_0}^2 + c^2[{p_1}^2 + {p_2}^2 - 2{p_1}{p_2}\cos(\theta)] ,
\end{equation}
with the substitution~\(E=pc\)~ and a bit more algebra we will
get Compton's equation in a more reasonable format for energy
measurements:
\begin{equation} \label{eq:6}
\frac{1}{E_2} = \frac{1}{E_0}(1 - \cos(\theta)) + \frac{1}{E_1}.
\end{equation}
Which we have used in the form
\begin{equation} \label{eq:1}
\frac{1}{E_2} = \frac{2}{E_0}\sin(\theta)^2 + \frac{1}{E_1}.
\end{equation}
\section*{Procedure}
{\label{733322}}
We will be using a NaI detector on a goniometer arm so that more precise
angles can be made with correlation to the direction of incoming
photon's incidence with the aluminum target. For this experiment we use
as a target an aluminum sphere around one inch in diameter. Aluminum is
a good target as it maintains a large number of valence electrons with
small ionization energies so that we can treat these electrons as being
free. The goniometer arm is able to be rotated from zero degrees
(directly in line with the direction of the incoming photons) to 90
degrees (perpendicular to the direction of the incoming photons with
target as our axis). Scattered photons from the target will then
incidence with our detector with about 30\% efficiency. This incidence
will be measured and amplified by a photomultiplier tube (PMT) which
will then be amplified further before reaching our multi-channel
analyzer (MCA). The size of the electrical pulse arriving at the~MCA
will be proportional to the energy of the photon incident with the
detector so after some calibration we will be able to measure the energy
of incoming scattered photons. To calibrate the MCA we used a source
with known gamma emission, namely , directly in line with our detector
then set the two channel calibration on the MCA program to be in line
with the two known gamma emissions in the sodium-22 spectrum. After
everything is aligned we specify a time interval to take data over then
collect data for multiple angles to check that the energy of the photons
arriving at the detector are in line with what is believed from
Compton's equation. Here we used once more a sodium-22 source and looked
for scattered 511 MeV photons. Angles close to zero degrees are left out
of the data collection as these angles receive a large number of photons
directly from the source.
\section*{Data}
{\label{780336}}
Data was taken at multiple angles all over intervals of two hours then
regions of interest (ROI) were issued to the area about where the peak
occurred. Our MCA program was then able to give an approximation of the
center of our peak along with the uncertainty in that peak which I have
taken as the full width half max (FWHM) of each given peak.\selectlanguage{english}
\begin{table}[H]
\centering
\normalsize\begin{tabulary}{1.0\textwidth}{CCCC}
Angles (degrees) & Net (Counts) & Centroid (MeV) & FWHM (MeV) \\
90 & 1102 & 0.25 & 0.01 \\
80 & 1687 & 0.27 & 0.01 \\
70 & 2171 & 0.3 & 0.01 \\
60 & 1581 & 0.33 & 0.01 \\
50 & 2346 & 0.37 & 0.01 \\
40 & 2924 & 0.41 & 0.02 \\
30 & 2838 & 0.44 & 0.02 \\
\end{tabulary}
\caption{{Angular Correlation data
{\label{232354}}%
}}
\end{table}This data was then used to create a weighted best fit with Compton's
equation in it's linear form with~\(E_0\)
and~\(E_1\) as our fit parameters as shown below.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/compton-scatter/compton-scatter}
\caption{{Best Fit and Residuals
{\label{753763}}%
}}
\end{center}
\end{figure}
It can be seen that this is a reasonable fit for the data however
estimation of the rest energy of the electron is largely outside of the
expected value for the rest energy of an electron given our small
uncertainty in this value. This I believe can be explained by the
increasing amount of noise that we received near the low energy side of
the window set for our MCA. An attempt was made to limit the effect of
this noise by making runs of 2 hour intervals without a target to check
radiation coming from unintentional scattering, then subtracting this
background noise from runs made with the target in. This however did not
drastically reduce the shift in energies believe to be caused by the
background noise. A second attempt could be made to reduce this noise by
increasing the amount of shielding between possible unnecessary
scattering alignments from the source to the detector.
\section*{Conclusion}
{\label{610025}}
Given the information obtained from our residuals we find a reduced
chi-squared value of 1.5 which allows us to be 20\% confident in our
fit. It is possible however that error here has been underestimated by
our inability to properly account for error in our independent variable,
namely the angle at which we were collecting data, which if accounted
for is likely to improve our confidence. We conclude that treating
photon-electron interactions as elastic collisions is a proper way to
model this interaction.
\par\null
\selectlanguage{english}
\FloatBarrier
\end{document}