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% diploma_1.tex
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% Example 1 of the Debian package diploma
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% Copyright (C) 1999 Andreas Franzen
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% See the file copyright for details.

\documentclass[12pt]{article}
\usepackage[dvips]{graphicx}
\begin{document}

\title{A look into the atomic structure of table salt}
\author{Andreas Franzen}
\maketitle

\section{Introduction}

From the chemistry lessons at school you may know that the salt which you
can find in a kitchen consists of crystals of sodium chloride or NaCl.
In these crystals the atoms of sodium and
chlorine are arranged in a regular pattern. You may think of the crystal as
a stack of cubes where there is a sodium atom at each
corner of a cube and also in the center of each surface. Thus, each cube
contains 1/8 sodium atom at each of its 8 corners and 1/2 sodium atom at
each of its 6 surface squares. That is 4 sodium atoms per unit cell. There
are as many chlorine atoms in the crystal. These are located in the middle
of each edge of the cube and in the center of the cube. Each cube contains
1/4 chlorine atom at each of its 12 edges and the chlorine atom in its
center. This is also a total of 4 chlorine atoms per unit cell. This
structure is called face-centered-cubic.

What your teacher didn't tell you is how one can verify this. This paper
describes a simple way to scan the atomic structure of table salt. The
method is called x-ray diffraction or XRD.

When an x-ray interacts with the atoms of a crystal it is reflected by
lattice planes. A lattice plane is a plane through the crystal which
contains many atoms. The possible diffraction angle $2\Theta$ depends
on the wavelength $\lambda$ of the radiation and the distance $d$ between
the parallel lattice planes. The equation is quite simple.
\begin{equation}
2d\sin\Theta=n\lambda
\end{equation}
$n$ is a positive integer. We assume first order diffraction. That means $n$
is 1.

For a structure with cubic symmetry it is easy to calculate the possible
distances $d$
\begin{equation}
d=\frac{a}{\sqrt{h^2+k^2+l^2}}
\end{equation}
from the lattice parameter $a$. $a$ is the height of the unit cell. $h$,
$k$, and $l$ are small integers.

For a face-centered-cubic structure there is an additional requirement. All
of the three integers must be odd or all of them must be even.

\begin{figure}
\begin{center}
\includegraphics[width=80mm]{src/XRD.eps}
\caption{\label{FigXRD}
The XRD apparatus.}
\end{center}
\end{figure}

\section{Experimental arrangement}

An XRD apparatus in its basic form is quite simple.
\textbf{Figure~\ref{FigXRD}} shows a schematic drawing. The x-ray tube on
the left consist of the water cooled anode and the electrically heated
tungsten cathode. The electric potential of the cathode is
$-40\,\mathrm{kV}$ with respect to the anode which is at ambient potential.
Between cathode and anode is high vacuum. Thus, electrons can travel from
the heated cathode to the anode. The heating of the cathode is adjusted so
that the anode current is 40\,mA. The target area of the anode is made of
pure cobalt. It is 1\,mm wide and 10\,mm long. From the sample, this area
looks like a rectangle of 0.1\,mm width and 10\,mm length.

Half way from the anode to the sample there is a 1\,mm wide and 10\,mm long
slit made of lead. The beam is diffracted in the sample and passes a similar
slit half way between sample and detector. At the entry of the detector
there is a 0.2\,mm wide slit.

The radiation of a cobalt anode exited with 40\,keV electrons consists
mainly of the Co-K$\alpha_1$ line with a wavelength of
$\input{lambdaKalpha1}$ and the Co-K$\alpha_2$ line with a
wavelength of $\input{lambdaKalpha2}$ and half of the intensity
of the Co-K$\alpha_1$ line. An iron foil between the two lead slits
on the detector side reduces the intensity of the weaker Co-K$\beta_{1,3}$
line at $\input{lambdaKbeta}$ relative to the Co-K$\alpha$ lines. There
is also continuous background radiation. \cite{Lid95}

\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{fig/salt.eps}
\caption{\label{FigSalt}Raw diffraction pattern of sodium chloride.}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{fig/salt2.eps}
\caption{\label{FigSalt2}Processed diffraction pattern of sodium chloride.}
\end{center}
\end{figure}

\section{Measurement}

The position of the sample and the detector are adjusted so that the
intensity of the diffracted radiation can be measured at different
diffraction angles. The measurement consisted of $\input{stepnumber}$ single
measurements with a step size of $\input{stepsize}$ and a measurement time
of 1\,s at each step. The diffraction angle was varied from
$\input{min2theta}$ to $\input{max2theta}$. The
measured value is the number of counts at each step.

The sample material was ordinary table salt obtained from a supermarket.
\textbf{Figure~\ref{FigSalt}} shows the result.

\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{|c|c|c|c|c|c|c|}
\cline{2-7}
\multicolumn{1}{c|}{}
& \multicolumn{3}{c|}{Co-K$\alpha_1$}
& \multicolumn{3}{c|}{Co-K$\beta_{1,3}$}\\
\hline
$2\Theta[^\circ]$&$d[\mathrm{nm}]$&$(hkl)$&$a[\mathrm{nm}]$&
$d[\mathrm{nm}]$&$(hkl)$&$a[\mathrm{nm}]$\\\hline
\input{res/table1}\hline
\end{tabular}
\end{center}
\caption{\label{tableoflines}Observed diffraction angle
$2\Theta$ of each line and the corresponding lattice spacing $d$. Each
line is characterized by three small integers $(hkl)$ which
are used to calculate the lattice parameter $a$.}
\end{table}

\section{Processing of the measurement data}

We are interested in the positions of the peaks in the diffraction pattern.
We can find these much better when we denoise the measurement data. Also, we
transform the step number $i$ into the diffraction angle $2\Theta$.
\begin{equation}
2\Theta=\frac{\input{max2theta}-\input{min2theta}}{\input{stepnumber}-1}
\cdot(i-1)+\input{min2theta}
\end{equation}
To denoise the pattern, the Poisson distributed noise of the raw counts per
step $x_i$ is approximately transformed into Gaussian distributed white noise
$y_i$.
\begin{equation}
y_i=2\sqrt{x_i+0.25109}
\end{equation}
The resulting diffraction pattern is shown in the first and second part of
\textbf{figure~\ref{FigSalt2}}. Then it is denoised with a wavelet denoising
method. This is done with the program \texttt{wzip} which is contained in
the package \texttt{wzip} in the section \texttt{math} of the Debian
GNU/Linux distribution. Hard denoising is used with a scale factor of 4.
The third part of figure~\ref{FigSalt2} shows the result.

There are 17 peaks visible in figure~\ref{FigSalt2}. Their
diffraction angles are shown in \textbf{table~\ref{tableoflines}}. The
corresponding lattice spacings $d$ have been calculated with the
wavelength of the Co-K$\alpha_1$ line or the Co-K$\beta_{1,3}$ line.
All peaks could be successfully indexed with $(hkl)$s. From $d$ and $(hkl)$
the lattice parameter $a$ can be calculated. All 17 measurements of $a$ are
in good agreement with each other. Both the geometric accuracy and the
resolution of the experimental arrangement rise with the diffraction angle.
Thus, the result of our measurement of the lattice parameter of sodium
chloride is
\begin{equation}
a_{\mathrm{NaCl}}=\input{res/result}\quad.
\end{equation}

\section{Acknowledgement}

Many thanks to the Department of Ferrous Metallurgy and Foundry Technology
of the Technical University of Clausthal, Clausthal-Zellerfeld, Germany for
the permission to use the XRD facility.

\begin{thebibliography}{Mmm00}

\bibitem[Lid95]{Lid95}
Lide, D.R.: CRC Handbook of Chemistry and Physics. Boca Raton 1995

\end{thebibliography}

\end{document}
