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+\documentclass[./manual.tex]{subfiles}
+\begin{document}
+
+\chapter{Electron Diffraction}
+
+\textbf{Experiment objectives}:  observe diffraction of the beam of electrons on a graphitized carbon target and  calculate the intra-atomic spacings in the graphite.
+
+\section*{History}
+
+    A primary tenet of quantum mechanics is the wavelike
+    properties of matter. In 1924, graduate student Louis de
+    Broglie suggested in his dissertation that since light has
+    both particle-like {\bf and} wave-like properties, perhaps all
+    matter might also have wave-like properties. He postulated
+    that the wavelength of objects was given by $\lambda = h/p$, where $h$ is Planck's constant and $p =
+    mv$ is the momentum.  {\it This was quite a revolutionary idea},
+    since there was no evidence at the time that matter behaved
+    like waves. In 1927, however, Clinton Davisson and Lester
+    Germer discovered experimental proof of the wave-like
+    properties of matter --- particularly electrons. This discovery
+    was quite by mistake: while studying electron reflection
+    from a nickel target, they inadvertently crystallized their
+    target, while heating it, and discovered that the scattered
+    electron intensity  as a function of scattering angle showed
+    maxima and minima. That is, electrons were ``diffracting'' from
+    the crystal planes much like light diffracts from a grating,
+    leading to constructive and destructive interference. Not only
+    was this discovery important for the foundation of quantum
+    mechanics (Davisson and Germer won the Nobel Prize for their
+    discovery), but electron diffraction is an extremely important
+    tool used to study new materials.  In this lab you will study
+    electron diffraction from a graphite target, measuring the
+    spacing between the carbon atoms.
+
+
+\begin{figure}[h]
+\centering
+\includegraphics[width=\textwidth]{./pdf_figs/ed1_new} \caption{\label{ed1}Electron
+Diffraction from atomic layers in a crystal.}
+\end{figure}
+\section*{Theory}
+    Consider planes of atoms in a {\bf crystal} as shown in Fig.~\ref{ed1}
+separated by distance $d$. Electron ``waves'' reflect from each of these planes.
+Since the electron is wave-like, the combination of the reflections from each
+interface produces to an interference pattern. This is completely analogous to
+light interference, arising, for example, from different path lengths in the
+Fabry-Perot or Michelson interferometers. The de Broglie wavelength for the
+electron is given by $\lambda=h/p$, where $p$ can be calculated by knowing the
+energy of the electrons when they leave the ``electron gun'':
+\begin{equation}\label{Va}
+\frac{1}{2}mv^2=\frac{p^2}{2m}=eV_a,
+\end{equation}
+ where $V_a$ is the accelerating potential.  The condition for constructive
+interference is that the path length difference for the two waves
+shown in Fig. \ref{ed1} be a multiple of a wavelength. This leads to Bragg's
+Law:
+\begin{equation}\label{bragg}
+n\lambda=2d\sin\theta
+\end{equation}
+where $n = 1,2,\dots$ is integer. In this experiment, only the first order
+diffraction $n=1$ is observed. Therefore, the intra-atomic distance in a
+crystal can be calculated by measuring the angle of electron diffraction and
+their wavelength (\emph{i.e.} their momentum):
+\begin{equation}\label{bragg1}
+d=\frac{\lambda}{2\sin\theta} = \frac{1}{2\sin\theta}\frac{h}{\sqrt{2em_eV_a}}
+\end{equation}
+\noindent
+where $h$ is Planck's constant, $e$ is the electronic charge, $m_e$ is the
+electron's mass, and $V_a$ is the accelerating voltage.
+%
+%
+%
+%
+%     Knowing $\lambda$ and the angles $\theta$ for which
+%constructive interference occurs, the atomic spacing $d$ can be
+%extracted.
+
+\section*{Experimental Procedure}
+
+\textbf{Equipment needed}: Electron diffraction apparatus and power supply, ruler and thin receipt paper or masking tape.
+
+\subsection*{Safety}
+You will be working with high voltage. The power supply will be connected for you,
+but inspect the apparatus when you arrive \textbf{before} turning the power on.
+If any wires are unplugged, ask an instructor to reconnect them. Also,
+\textbf{before} turning the power on, identify the high voltage contacts on the
+electron diffraction tube, make sure these connections are well-protected and
+cannot be touched by accident while taking measurements.
+\begin{figure}[h]
+\centering
+\includegraphics[width=6in]{./pdf_figs/ed2} \caption{\label{ed2}Electron Diffraction Apparatus.}
+\end{figure}
+\subsection*{Setup}
+
+The diagram of the apparatus is given in Fig.\ref{ed2}. An electron gun
+(consisting of a heated filament to boil electrons off a cathode and an anode
+to accelerate them, similar to the e/m experiment) ``shoots'' electrons at a
+carbon (graphite) target.
+
+The electrons diffract from the carbon target and the resulting interference
+pattern is viewed on a phosphorus screen.
+
+ The graphitized carbon is not completely crystalline but consists of crystal
+sheets in random orientations. Therefore, the constructive interference
+patterns will be seen as bright circular rings. For the carbon target, two
+rings (an outer and inner, corresponding to different crystal planes) will be
+seen, corresponding to two spacings between atoms in the graphite arrangement
+(see Fig.~\ref{ed3}).
+\begin{figure}
+\centering
+\includegraphics[width=4in]{./pdf_figs/ed3} \caption{\label{ed3}Spacing of
+carbon atoms. Here subscripts \textit{10} and \textit{11} correspond to the
+crystallographic directions in the graphite crystal.}
+\end{figure}
+
+\subsection*{Data acquisition}
+Acceptable power supply settings:
+\\\begin{tabular}{lll}
+Filament Voltage& $V_F$&6.3 V ac/dc (8.0 V max.)\\
+Anode Voltage & $V_A$& 1500 - 5000 V dc\\
+Anode Current & $I_A$& 0.15 mA at 4000 V ( 0.20 mA max.)
+\end{tabular}
+
+\begin{enumerate}
+\item  Switch on the heater and wait one minute for the oxide cathode to achieve thermal stability.
+\item  Slowly increase $V_a$ until you observe two rings appear around the direct beam.
+Slowly change the voltage and determine the highest achievable accelerating
+voltage, and the lowest voltage when the rings are visible.
+\item  Measure the diffraction angle $\theta$ for both inner and outer rings for 5-10 voltages from that range,
+using the same thin receipt paper (see procedure below). Each lab partner should
+repeat these measurements (using an individual length of the thin paper).
+\item Calculate the average value of $\theta$ from the individual measurements for each
+voltage $V_a$. Calculate the uncertainties for each $\theta$.
+\end{enumerate}
+
+\textbf{Measurement procedure for the diffraction angle $\theta$}
+
+To determine the crystalline structure of the target, one needs to carefully
+measure the diffraction angle $\theta$. It is easy to see (for example, from
+Fig.~\ref{ed1}) that the diffraction angle $\theta$ is 1/2 of the angle
+between the beam incident on the target and the diffracted beam to a ring,
+hence the $2\theta$ appearing in Fig.~\ref{ed4}. You are going to determine
+the diffraction angle $\theta$ for a given accelerated voltage from the
+approximate geometrical ratio
+\begin{equation}
+L\sin{2\theta} = R\sin\phi,
+\end{equation}
+where the distance between the target and the screen $L = 0.130$~m is controlled during the production process to have an accuracy better than 2\%. {\it Note, this means that the electron tubes are not quite spherical.}
+
+The ratio between the arc length $s$ and the
+radius of the curvature for the screen $R = 0.066$~m gives the angle $\phi$ in
+radians:  $\phi = s/2R$. To measure $\phi$ carefully place a piece of
+thin receipt paper on the tube so that it crosses the ring along the diameter.
+Mark the position of the ring for each accelerating voltage, and then
+remove the paper and measure the arc length $s$ corresponding to
+each ring. You can also make these markings on masking tape placed gently on the tube.
+
+
+\begin{figure}
+\centering
+\includegraphics[width=4in]{./pdf_figs/edfig4} \caption{\label{ed4}Geometry of the experiment.}
+\end{figure}
+
+\section*{Data analysis}
+
+Use the graphical method to find the average values for the distances between
+the atomic planes in the graphite crystal $d_{11}$ (outer ring) and $d_{10}$
+(inner ring). To determine the combination of the experimental parameters that
+is proportional to $d$, one needs to substitute the expression for the
+electron's velocity  Eq.(\ref{Va}) into the diffraction condition given by
+Eq.(\ref{bragg}):
+\begin{equation}\label{bragg.analysis}
+\sin\theta=\frac{h}{2\sqrt{2m_ee}}\frac{1}{d}\frac{1}{\sqrt{V_a}}
+\end{equation}
+
+Make a plot of $\sin\theta$ (y-axis, with uncertainty) vs $1/\sqrt{V_a}$ (x-axis) for the inner and outer rings
+(both curves can be on the same graph). Fit the linear dependence and measure
+the slope for both lines. From the values of the slopes find the distances
+between atomic layers $d_{inner}$ and $d_{outer}$.
+
+Compare your measurements to the accepted values: $d_{inner}=d_{10} = .213$~nm
+and $d_{outer}=d_{11}=0.123$~nm.
+
+\section*{Seeing with Electrons}
+	
+The resolution of ordinary optical microscopes
+is limited (the diffraction limit) by the wavelength of light ($\approx$ 400
+nm). This means that we cannot resolve anything smaller than this by looking at
+it with light (even if we had no limitation on our optical instruments). Since
+the electron wavelength is only a couple of angstroms ($10^{-10}$~m), with
+electrons as your ``light source'' you can resolve features to the angstrom
+scale. This is why ``scanning electron microscopes'' (SEMs) are used to look at
+very small features. The SEM is very similar to an optical microscope, except
+that ``light'' in SEMs is electrons and the lenses used are made of magnetic
+fields, not glass.
+
+%\begin{tabular}{lll}
+%Filament Voltage& $V_F$&6.3 V ac/dc (8.0 V max.)\\
+%Anode Voltage & $V_A$& 2500 - 5000 V dc\\
+%Anode Current & $I_A$& 0.15 mA at 4000 V ( 0.20 mA max.)
+%\end{tabular}
+%
+%\newpage
+%\noindent
+%Section: $\underline{\hskip 1in}$\\
+%Name: $\underline{\hskip 1in}$\\
+%Partners: $\underline{\hskip 1in}$\\
+%$\hskip 1in\underline{\hskip 1in}$\\
+%\section*{Electron Diffraction}
+%Step through the electron accelerating potential from 5 kV to 2 kV in
+%steps of 0.5 kV. Record $V_a$ (from needle reading on power supply) and
+%measure the diameter of the two rings. In measuring the diameter,
+%either try to pick the middle of the ring, or measure the inside and
+%outside of that  ring
+%and average. Each lab partner should measure the diameter, and then average
+%the result. {\bf Make sure you give units in the following tables}
+%
+%\begin{center}
+%{\bf Final data -- fill out preliminary chart on next page first}
+%\end{center}
+%
+%\begin{tabular}{|p{20mm}|p{20mm}|p{20mm}|p{20mm}|p{20mm}|}\hline
+%$V_a$&(inner s)&$d_{inner}$&outer s &$d_{outer}$\\\hline
+%&&&&\\\hline
+%&&&&\\\hline
+%&&&&\\\hline
+%&&&&\\\hline
+%&&&&\\\hline
+%&&&&\\\hline
+%\multicolumn{2}{|l|}{Average $d_{inner}\pm\sigma=$}
+%&\multicolumn{2}{|l}{Average $d_{outer}\pm\sigma=$}
+%&\multicolumn{1}{l|}{}\\\hline
+%\end{tabular}
+%
+%
+%
+%
+%Make a plot of $1/sqrt{V_a}$ versus $s$ for the inner and outer rings
+%{both curves can be on the same graph).
+%
+%Compare your measurements to the known spacing below.$d_{outer}=$\\
+%True values are: $d_{inner}=.213 nm$ and $d_{outer}=0.123 nm $.
+%
+%\begin{boxedminipage}{\linewidth}
+%
+%
+%Turn in this whole stapled report, including your data tables. Attach your
+%plot(s) at the end.
+%\end{boxedminipage}
+%\newpage
+%\begin{boxedminipage}{\linewidth}
+%Note: start with the External Bias at 30 V. Decrease the bias if you need
+%to, to increase the intensity of the rings. DO NOT EXCEED 0.2 mA on ammeter!
+%\end{boxedminipage}
+
+%\newpage
+
+\end{document}
+