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\title{Lab 5: X-ray Diffractometer}
\author{Otho Ulrich, Mike Pirkola, Jacob Burke, Andrew Messecar}
\maketitle

\begin{abstract}
Western Michigan University's new X-ray Diffractometer is used to probe four materials. The lattice contant is computed for an NaCl sample. For three amorphous samples -- wood, grease, and SOMETHING -- the average distance between atoms is computed.
\end{abstract}


%─────────────
\section{Introduction: Bragg Diffraction and Materials Science}
    \label{sec:intro}
    packing fraction for salt, 60\% is about good

    densities of plastic are all around 2

    pick carbon, with occassional oxygens, so using just carbon you get ~$2grams/cm^3$

    Bragg diffraction of X-rays is a useful method for characterising the atomic and molecular structure of materials. Many mechanical and electric properties are functions of the structures that constitute materials. Bragg diffraction uses the wave theory of electromagnetic radiation to predict how X-rays will interact with the atomic lattice of a crystal. The spacing between atoms can be measured by inference, and the average spacing between atoms is often called the ``lattice constant''.\\

    We attempt to compute the lattice constant from an X-ray diffractometer reading of a sample of NaCl, or common salt. NaCl forms a cubic crystal structure, so it has a single lattice constant, and this is computed from the diffraction pattern and compared to known values. Three amorphous samples are also analyzed: plastic of an unknown type; grease; and plywood. These materials are not expected to have rigid crystal structures, but the average spacing between atoms can still be ascertained from the diffraction pattern.\\

    \subsection{X-ray Diffractometer}
    	\label{subsec:diffrator}
    	An Empyrean X-ray diffractometer by PANalytical \cite{empyrean} was used to collect a diffraction pattern from each sample. In this machine, an X-ray source emits onto a material sample, and a detector records X-rays diffracted at the angle of incidence; see Figure~\ref{fig:diffractometer}. X-rays are created by accelerating electrons toward a copper anode (Figure~\ref{fig:xraysource}). The X'Celerator detector is an X-ray sensor with a window size of $9 \textrm{mm} \times 15 \textrm{mm}$ and Copper K-$\alpha$ efficiency $>94\%$ \cite{xcelerator}.\\ 

    	\begin{figure}
    		\center
    		\includegraphics[width=3in]{empyrean_sample.jpg}
    		\includegraphics[width=3in]{empyrean_scanners.pdf}
    		\caption{The PANalytical Empyrean X-ray diffractometer. Sample are placed in a bin at the center. X-rays are generated in the arm on the left, diffracted by the sample at the center, and detected at an angle $\theta$ by the X'Celerator in the arm on the right. Each scan ran through $\theta = \{5\degree .. 45\degree\}$.}
    		\label{fig:diffractometer}
    	\end{figure}

    	\begin{figure}
    		\center
    		[A]\includegraphics[width=3in]{xraytube.png}
    		[B]\includegraphics[width=3in]{Copper_K_Rontgen.png}
    		\caption{X-rays are generated when collisions with accelerated electrons knock electrons in the copper atoms of the anode out of the K shell. When the electrons fall back to the K shell, X-rays are emitted with energies indicated in [B]. These energy values are important for predicting the X-ray diffraction pattern from Bragg diffraction. \cite{advlabxrd} \cite{xraytubephoto}}
    		\label{fig:xraysource}
    	\end{figure}

    \subsection{Bragg Diffraction}
        \label{subsec:bragg}

        The diffraction angle of X-rays by atoms in a crystal lattice or other molecule depends on the distance between atoms. The lattice constants of a crystal describe the distances and angles between atoms, but in the case of a cubic lattice such as NaCl, there is only one relevant lattice constant. Bragg diffraction predicts strong X-ray signals at diffraction angles that produce constructive interference. Figure~\ref{fig:bragg} illustrates the geometry of Bragg diffraction; the distance between planes d is the lattice constant, and the Bragg condition

            \begin{equation}
                \centering 2d\sin \theta = n\lambda,
            \end{equation}

        where $\lambda$ is the photon wavelength of the X-ray and $n$ a positive integer describes the angles at which strong signals should be detected. \cite{advlabxrd}

        \begin{figure}
            \center
            \includegraphics[width=3in]{braggdiffraction.png}
            \caption{Bragg diffraction from a cubic crystal lattice. Plane waves incident on a crystal lattice at angle $\theta$ are partially reflected by successive parallel crystal planes of spacing d. The superposed reflected waves interfere constructively if the Bragg condition $2d\sin \theta =n\lambda$ is satisfied. \cite{braggdiff_wiki}} 
            \label{fig:bragg}
        \end{figure}


        \begin{table}
        	\centering
        	\begin{tabular}{cccc}
        		Copper Line & Energy (eV) & Frequency (Hz) & Wavelength\\
        		\hline
        		K-$\alpha_1$ & 8046 & 1.946$\times 10^{18}$ & 1.541$\times 10^{-10}$\\
        		K-$\alpha_2$ & 8027 & 1.941$\times 10^{18}$ & 1.393$\times 10^{-10}$\\
        		K-$\beta$    & 8903 & 2.153$\times 10^{18}$ & 1.545$\times 10^{-10}$\\

        	\end{tabular}
        \end{table}

        The modern approach to analyzing materials by Bragg diffraction is to interpret the output as the reciprocal space representation of the lattice positions. An inverse Fourier transform then gives the positions that make the lattice. The HighScore Plus software, associated with the Empyrean XRD, is used to perform these operations. \cite{highscore} It computes a spacing constant in angstroms, which can be interpretted as the cubic lattice constant, and other quantities.





\section{Computational Details}
    \label{sec:compdets}
    The phase of the X-rays are not known, so the program determines the phase by fitting predicted profiles. A background is determined using the minimum 2nd derivative method with ``bending factor'' = 5, ``granularity'' = 20, and using smoothed input data. Peaks are located with ``minimum significant'' = 10.00, ``minimum tip width'' = 0.01, ``maximum tip width'' = 1.00, and ``peak base width'' = 2.00. The program was able to identify the copper Bragg diffraction pattern, which was very prominent, seen in Figure~\ref{fig:cudiff}.


    \begin{figure}
        \centering
        \includegraphics[width=4in]{}
    \end{figure}



%─────────────    
\section{Results}
    \label{sec:results}


%─────────────
\section{Conclusion}
    \label{sec:conclusion}



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