\pagestyle{plain}\clearpage\centerline{\Huge Question Sheet 2}

\section*{Question 1}

A thin film of silver is deposited on to a single--crystal ceramic substrate and their relative orientations measured. Fig.~\ref{fig:Ag} shows the 111 poles from the silver together with the 110 poles from the ceramic. Both silver and the ceramic have a cubic lattice, with parameters 0.4086\,nm and 0.5760\,nm respectively.

Label the poles and determine the orientation relationship. Why might this orientation relationship be adopted?

\begin{figure}[!ht]
  \begin{center}
    \begin{minipage}[t]{0.72\linewidth}
\raisebox{-4cm}{\includegraphics[width=0.98\linewidth]{QA/Ag.eps}} 
    \end{minipage}\hfill
    \begin{minipage}[t]{0.25\linewidth}
      \caption{Orientation relationship between Ag deposited on to a cubic ceramic substrate. }  \label{fig:Ag}
    \end{minipage}
  \end{center} \end{figure} 

\clearpage\section*{Question 2}

Describe the Ewald sphere construction and using a sketch, explain why in transmission electron microscopy, it is possible to observe diffracted intensity even when the crystal is not exactly at the Bragg orientation.

Figure~\ref{fig:ed} shows an electron diffraction pattern from a body-centred cubic crystal structure. Examine the symmetry of this pattern and deduce the direction of the electron beam. Hence label each of spots on the pattern. Why are some of the spots more intense than others?



\begin{figure}[!ht]
  \begin{center}
    \begin{minipage}[t]{0.52\linewidth}
\raisebox{-4cm}{\includegraphics[width=0.98\linewidth]{QA/ed.eps}} 
    \end{minipage}\hfill
    \begin{minipage}[t]{0.35\linewidth}
      \caption{An electron diffraction pattern from a body--centred cubic crystal.}  \label{fig:ed}
    \end{minipage}
  \end{center}  
\end{figure}

\section*{Question 3}

Using real and reciprocal lattice vectors, prove the Weiss zone rule that when a direction with real space indices $[u~v~w]$ lies in a plane with reciprocal space indices $(h~k~l)$,
\begin{equation}
uh+vk+wl = 0
\end{equation}
irrespective of the nature of the unit cell.

\section*{Question 4}


Silicon has a cubic--F lattice with a motif of atoms at (0,0,0) and ($\frac{1}{4},\frac{1}{4},\frac{1}{4})$. Draw a projection of the cell and identify on this the diamond glide plane, indicating the translations involved. Indicate a possible Burgers vector of a dislocation whose motion would leave the lattice unchanged.

\section*{Question 5}

Explain why in  Fig.~\ref{fig:laue} there is a central region of strong reflections surrounded by two annular rings.  Verify that the  electron beam is parallel to a $<210>$ direction by indexing the spots in the central region. 


Show that the radius of the first bright ring in reciprocal space is given by 
\begin{equation}
{\mathbf r^*}=\sqrt{\biggl(\frac{1}{\lambda}\biggr)^2 - \biggl(\frac{1}{\lambda} - \frac{1}{t}\biggr)^2}
 \end{equation}
where $t$ is the spacing of the planes normal to the $<210>$ vector.


\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.78\linewidth]{QA/laue.eps}
\caption{Electron diffraction pattern from a cubic--I crystal of lattice parameter 0.89\,nm}
\label{fig:laue}
\end{center}
\end{figure} 



