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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 to 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 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!). They were 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} \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 will lead 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{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, tape, ruler.
+
+\subsection*{Safety}
+You will be working with high voltage. Power supply will be connected for you,
+but inspect the apparatus when you arrive \textbf{before} turning the power on.
+In 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 to, 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 to 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 masking tape (see procedure below). Each lab partner should
+repeat these measurements (using an individual length of the masking tape).
+\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 simple 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 and the distance between the target and the
+radius of the curvature of the screen $R = 0.066$~m gives the angle $\phi$ in
+radian:  $\phi = s/2R$. To measure $\phi$ carefully place a piece of masking tape 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 masking tape and measure the arc length $s$ corresponding to each ring. You can also make these markings by using the thin paper which cash register receipts are printed on.
+
+
+\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 need 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}
+2d\sin\theta=\lambda=\frac{h}{\sqrt{2m_ee}}\frac{1}{\sqrt{V_a}}
+\end{equation}
+
+Make a plot of $1/\sqrt{V_a}$ versus $\sin\theta$ 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 slope find the distance
+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.
+
+ \fbox{Looking with Electrons}
+
+\noindent
+\begin{boxedminipage}{\linewidth} 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.
+\end{boxedminipage}
+
+%\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}
+