\pagestyle{plain}\clearpage\centerline{\Huge Examples Class 1}
\section*{Point groups and consequences}

\begin{description}
\item{a. } Draw structure projections of the unit cells of austenite and ferrite, and mark on these the locations of the octahedral interstices.

\item{b. }  Identify the point group symmetry of an octahedral interstice in austenitic iron. Repeat this for the octahedral interstice in ferritic iron. Identify using the table below, the symmetry class to which each interstice belongs. 

\item{c. }  Why does carbon strengthen ferrite much more than austenite? 

\end{description}

\begin{table}[!ht]\begin{center}
\begin{tabular}{lll}\midrule
Class & Non--centrosymmetric point groups & Centrosymmetric point groups \\[3pt]\midrule
Cubic & 23,  432, $\overline 4 3 m$ & $m3$, $m3m$ \\[3pt]
Hexagonal & 6, $\overline 6$,  622, $6mm$, $\overline 6 m2$& $6/m$, $6/mmm$ \\[3pt]
Trigonal & 3,  32, $3m$ &$\overline 3$, $\overline 3m$\\[3pt]
Tetragonal & $4,~\overline 4,~422,~ 4mm,~\overline 42m$ & $4/m,~~4/mmm$\\[3pt]
Orthorhombic & $222,~2mm$ & $mmm$\\[3pt]
Monoclinic & $2,~m$ & $2/m$ \\[3pt]
Triclinic &$1$ & $\overline 1$\\[3pt]\midrule
\end{tabular}
\end{center}
\label{tab:pg2}
\end{table}%


	
\newpage\section*{Sketch Sterograms}
	
The corners of a wooden cube are being cut off to make small equal faces of the form $\{111\}$. The first four new faces are cut in the order (a) (111); (b) $(1 1\overline 1)$; (c) $(\overline 111)$ and (d) $(\overline 1 1 \overline 1)$.

\begin{enumerate}

\item Sketch stereographic projections, displaying the faces and the symmetry elements of the model before cutting, and after each new face is made.

\item Using the table of crystal systems in which defining symmetries are given for each crystal system, name the 
crystal system to which the model will belong after each cut
\end{enumerate}


\begin{table}[!ht]
\caption{The crystal systems}
\begin{center}
\begin{tabular}{lccr}\midrule
System   &  \multicolumn{2}{c}{Conventional unit cell}  &   Defining symmetry \\[3pt]\midrule
Triclinic  &  ${ a_1} \neq { a_2} \neq { a_3}$ & $\alpha \neq \beta \neq \gamma$   &  monad \\[3pt]
Monoclinic  &  ${ a_1} \neq { a_2} \neq { a_3}$ &  $ \alpha = \gamma,~~ \beta \geq 90^\circ $ &  1 diad \\[3pt]
Orthorhombic  &  ${ a_1} \neq { a_2} \neq { a_3}$ &   $\alpha = \beta = \gamma = 90^\circ $ &  3 diads \\[3pt]
Tetragonal  &  ${ a_1} = { a_2} \neq { a_3}$ &   $\alpha = \beta = \gamma = 90^\circ $ &  1 tetrad \\[3pt]
Trigonal  &  ${ a_1} = { a_2} = { a_3}$ &   $\alpha = \beta = \gamma \neq 90^\circ $ &  1 triad \\[3pt]
Hexagonal  &  ${ a_1} = { a_2} \neq { a_3}$ &  $ \alpha = \beta = 90^\circ, \gamma = 120^\circ $ &  1 hexad \\[3pt]
Cubic  &  ${ a_1} = { a_2} = { a_3}$ &   $\alpha = \beta = \gamma = 90^\circ $ &  4 triads\\[3pt]\midrule
\end{tabular}
\end{center}
\label{tab:system}
\end{table}
