\pagestyle{plain}\clearpage\centerline{\Huge Examples Class 2}
\section*{Calcium Chloride}

The orthorhombic unit cell of CaCl$_2$ has $a=0.624\,$nm, $b=0.643\,$nm and $c=0.420\,$nm with ion positions at

Ca: ~~~~~~0,0,0~~~~~~$\frac{1}{2},\frac{1}{2},\frac{1}{2}$.

Cl: ~~~~~~$x,y,0$~~~~~~$\overline x,\overline y,0$~~~~~~$\frac{1}{2}+x,\frac{1}{2}-y,\frac{1}{2}$~~~~~~$\frac{1}{2}-x,\frac{1}{2}+y,\frac{1}{2}$

with $x=0.325$ and y$=0.275$. Fig.~\ref{fig:CaCl} is an accurate projection of this structure on (001).


\begin{figure}[!ht]
  \begin{center}
    \begin{minipage}[t]{0.52\linewidth}
\raisebox{-4cm}{\includegraphics[width=0.95\linewidth]{QA/CaCl.eps}} 
    \end{minipage}\hfill
    \begin{minipage}[t]{0.45\linewidth}
      \caption{The structure projection for calcium chloride}  \label{fig:CaCl}
    \end{minipage}
  \end{center}  
\end{figure}

\begin{enumerate}
\item What is the Bravais lattice of this structure?

\item Locate all symmetry elements present in the structure and hence determine those along [100], [010] and [001]. What is the point group of the crystal structure?

\item Determine the point symmetries of the Ca and Cl ions and express them on sketch stereograms.

\end{enumerate}

\section*{Martensite}

Show diagramatically that it is impossible to obtain a fully coherent boundary between austenite and martensite.


\section*{Orientation relationships}

A rotation matrix can be used to describe the orientation relationship between two grains with identical crystal structure. It can also be described by an axis of rotation and a right--handed angle of rotation (an axis--angle pair).

The general rotation matrix relating the two cubic lattices for a right--handed rotation  $\theta$ about a unit axis $[u_1~u_2~u_3]$ is given by:
\begin{equation} ({\rm Y}~{\rm J}~{\rm X}) =
\left(\begin{array}{ccc} u_1 u_1(1-m)+m & u_1 u_2(1-m)+u_3n & u_1 u_3(1-m)-u_2n \cr
u_1 u_2(1-m)-u_3n & ~u_2u_2(1-m)+m & u_2u_3(1-m)+u_1 n \cr
u_1 u_3(1-m)+u_2n & u_2u_3(1-m)-u_1 n & u_3u_3(1-m)+m \end{array}\right) \end{equation}
where $m = \cos\theta$ and $n = \sin\theta$

Show how you might deduce the axis--angle pair from this matrix. Hence derive the axis--angle pair for the rotation matrix
\begin{equation} ({\rm Y}~{\rm J}~{\rm X}) =
\left(\begin{array}{ccc} 0 & 1 &0 \cr
\overline 1 & 0 & 0 \cr
0 & 0 & 1 \end{array}\right) \label{eqn:rotation2}\end{equation}
What is the $\Sigma$ value relating these two grains?
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