\chapter{Stereograms for Low Symmetry Systems}

\section{Introduction}

In the cubic system, a plane normal (pole) with indices $hkl$ is parallel to a direction $uvw$ when $h=u,~k=v,~l=w$; thus the normal to (123) is parallel to [123]. This is not generally the case for less symmetrical systems. Fig.~\ref{fig:directions_planes} illustrates this for a two--dimensional unit cell in which $b>a$.

\begin{figure}[!ht]
  \begin{center}
    \begin{minipage}[t]{0.32\linewidth}
\raisebox{-2.5cm}{\includegraphics[width=0.98\linewidth]{Stereograms_2/directions_planes.eps}} 
    \end{minipage}\hfill
    \begin{minipage}[t]{0.65\linewidth}
      \caption{The direction [11] is not parallel to the normal to the plane (11). On the other hand, the directions parallel to the cell edges, {\em i.e.} [01] and [10] are parallel to the normals of (01) and (10) respectively.}  \label{fig:directions_planes}
    \end{minipage}
  \end{center} \end{figure} 

The principles for the plotting of poles on stereographic projections are nevertheless identical for any lattice. However, the cubic system exhibits extraordinary symmetry so angular relationships on the stereographic projection are not dependent on the lattice parameter. This is not the case for other  lattices with lesser symmetry, the stereograms of which will not only appear different from a distance, but for the same lattice type, the angular positions become dependent on the lattice parameters. Fig.~\ref{fig:asymmetric}a,b shows how the projections differ in the case of  the cubic and orthorhombic lattices where the absence of tetrads in the latter is obvious. Fig.~\ref{fig:asymmetric}c shows how the projection changes when one of the lattice parameters of the orthorhombic lattice is changed relative to that illustrated in Fig.~\ref{fig:asymmetric}b.

The fact that directions are not in general parallel to plane normals with the same indices is apparent from a comparison of Fig.~\ref{fig:asymmetric}c,d.

\begin{figure}[!ht]
\centering
\subfigure(a){
\includegraphics[width=0.4\textwidth]{Stereograms_2/cubic_a.eps}} \hfill
\subfigure(b){
\includegraphics[width=0.4\textwidth]{Stereograms_2/orthorhombic_c.eps}}\\[10pt]
\subfigure(c){
\includegraphics[width=0.4\textwidth]{Stereograms_2/orthorhombic_a.eps}}\hfill
\subfigure(d){
\includegraphics[width=0.4\textwidth]{Stereograms_2/orthorhombic_b.eps}}
\caption{Stereographic projections centred on the 001 pole. (a) Cubic, lattice parameter 5 units. (b) Orthorhombic, lattice parameters 2, 3, 8 units. (c) Orthorhombic, lattice parameters 2, 3, 5 units. (d) The orthorhombic cell as in (c) but with directions plotted instead of poles.}
\label{fig:asymmetric}
\end{figure}

\newpage
\section{Hexagonal System}

The unit cell has $a=b\neq c$ and $\alpha=\beta=90^\circ$ with $\gamma=120^\circ$. The normals to the (100) and (010) planes therefore make an angle of 60$^\circ$ (Fig.~\ref{fig:asymmetric2}a). This is shown on the projection in Fig.~\ref{fig:asymmetric2}, which highlights a special difficulty with the hexagonal system.

\begin{figure}[!ht]
\centering
\subfigure(a){
\includegraphics[width=0.35\textwidth]{Stereograms_2/hexagonal_a.eps}} \hfill
\subfigure(b){
\includegraphics[width=0.45\textwidth]{Stereograms_2/hexagonal_b.eps}}\\[10pt]
\subfigure(c){
\includegraphics[width=0.35\textwidth]{Stereograms_2/hexagonal_c.eps}}\hfill
\subfigure(d){
\includegraphics[width=0.45\textwidth]{Stereograms_2/hexagonal_d.eps}}
\caption{(a) Showing the hexagonal cell with the normals to the (100) and (010) planes identified with coloured arrows. (b) Stereographic projection of hexagonal structure (lattice parameters $a=2$, $b=2$ and $c=3$ units with $\gamma=120^\circ$) centred on the 001 pole. (c) Showing the hexagonal cell, now in four index notation. (d) Corresponding stereogram in four index notation.}
\label{fig:asymmetric2}
\end{figure}

It is evident from Fig.~\ref{fig:asymmetric2}a that the (100), (010) and $(\overline 1 1 0)$ are planes belonging to the same form $\{100\}$, {\em i.e., } they are crystallographically equivalent. And yet, they have different indices. To eliminate this discrepancy, a four index system is frequently used for hexagonal lattices by introducing a fourth axis (labelled $d$ in Fig.~\ref{fig:asymmetric2}c), which gives plane indices as $hkil$, where from geometry, $i=-(h+k).$ This has the effect of giving equivalent planes the same combination of digits in their indices. Thus,  (100), (010) and $(\overline 1 1 0)$  become  $(10\overline 1 0)$,  
$(01\overline 1 0)$ and $(\overline 110 0)$ respectively.
 
A similar conversion for directions from $uvw$ to $UVJW$  is a little more difficult because it is  designed in such a way that $J=-(U+V)$ and at the same time, the Weiss zone law in which a plane $hkl$ containing a direction $uvw$ is  satisfied
\begin{equation}
hu+kv+lw=0 \label{eqn:weiss}\end{equation} so that
\begin{equation}
hU+kV+iJ+lW \qquad\hbox{should equal zero.} \nonumber\end{equation}
On substituting for $i=-(h+k)$ we get, 
\begin{eqnarray}
 hU + kV + (-h-k)J + lW = 0\\\nonumber
 h(U-J) + k(V-J) + lW = 0\nonumber
 \end{eqnarray}
 On comparing the coefficients of $h$, $k$ and $l$, and noting that $J=-(U+V)$, 
\setlength{\extrarowheight}{0.3cm} \begin{equation}
\begin{array}{lll} U-J=u \\ V-J=v\\ W=w \end{array} \rightarrow \begin{array}{lll} u=2U+V\\v=U+2V\\w=W \end{array} \quad\hbox{and}\quad \begin{array}{lll}U=\frac{1}{3}(2u-v)\\ V=\frac{1}{3}(2v-u)\\ J=-\frac{1}{3}(u+v)\end{array}\end{equation}
\begin{table}[!ht]
\caption{Directions of the same form now have similar combinations of digits in the four figure notation.}
\begin{center}
\begin{tabular}{cc}\midrule
Three-index notation	& Four-index notation\\[3pt]
$[100]$ & $\frac{1}{3}[2\overline 1 \overline 1 0]$\\[3pt]
[010]	 & $\frac{1}{3}[\overline 1 2 \overline 1 0]$\\[3pt]
[110]	 & $\frac{1}{3}[1 1 \overline 2 0]$\\[3pt] 
[uvw]	  & $\frac{1}{3}[2u-v,~2v-u,~-(u+v),~3w]$\\[3pt]\midrule
\end{tabular}
\end{center}
\label{tab:four}
\end{table}%



Iron, by far the most used metallic material in the world, under ambient conditions is cubic-I  with one iron atom located at each lattice point. The common slip system is $\{0\overline 11\}\langle 1 1 1\rangle$, each containing two independent slip directions of the form $\langle 1 1 1\rangle$. There are therefore,  twelve independent slip systems, making the iron ductile in polycrystalline form. The same can be demonstrated for the cubic-I form of iron. This makes iron very ductile. 

The cubic-I form of iron is stabilised under ambient conditions by its ferromagnetic properties; ruthenium and osmium, which in terms of outer electronic structure are iron analogues in the periodic table, do not exhibit ferromagnetism and hence have hexagonal crystal structures. In the absence of its ferromagnetic properties, the stable form would be hexagonal with a motif  of a pair of identical Fe-atoms, one at 0,0,0 and another at $\frac{1}{3},\frac{2}{3},\frac{1}{2}$. The easy-slip system is then $\{0001\}\langle 11\overline 20\rangle$. Since there is only one (0001) plane in which there are three directions of the form $\langle 11\overline 20\rangle$, polycrystalline hexagonal iron is relatively brittle since the number of slip systems in each crystal is less than the required five. So in the absence of ferromagnetism, we would not have civilisation in the form that we know today.

There are caveats to this story. Deformation can occur by mechanisms other than slip, for example, mechanical twinning, so that the five independent slip system requirement is not sufficient to determine the ductility of a polycrystalline material. And iron below its Curie temperature is not strictly cubic, but tetragonal because the magnetic spins in any given crystal tend to be aligned along a $\langle 100\rangle$ axis. However, the tetragonality is quite small and can be neglected in most experiments, although it does manifest on a macroscopic scale via the magnetostrictive effect, which is the variation in the length of a sample subjected to a magnetic field. 

\section{Summary}

Complications arise once the defining symmetry of the cube, i.e., four triads, is lost. Directions and plane normals with the same indices are then not necessarily parallel and hence have to be distinguished on a stereographic projection. Furthermore, the extent of the discrepancy relative to the cubic system depends on the anisotropy of the lattice parameters and the angles between the basis vectors of the system concerned.

The hexagonal system poses a particular problem of communication because of the nature of the special symmetry of the hexad. For example, the crystallographically equivalent planes (100), (010) and $(1\overline 10)$ have indices that are not permutations of the same set of digits. The Miller-Bravais four-index system eliminates this ambiguity but requires additional work to ensure that the Weiss zone rule is satisfied. Whether the intellectual cost of the four-index system over the usual three-index notation is justified by its elegance is a matter of taste.
