\chapter{Stereographic Projections}
\label{C2}


\section{Introduction}
We have seen already that the projection of a three-dimensional crystal structure  into two dimensions, along the $z$-coordinate, can simplify the perception and representation of atomic arrangements. There is no loss of information given that the fractional $z$-coordinate is clearly identified on the projection. This can be seen in Fig.~\ref{fig:diamond} where the three dimensional representation of the structure of diamond lacks clarity whereas the projected cell is readily visualised. The ease of interpretation using the projected structure is illustrated for the more complex structure of $\varepsilon$-carbide, which has a chemical formula between Fe$_2$C and Fe$_3$C, in Fig.~\ref{fig:epsilon}. There are six iron atoms in the cell (3 at $z=\frac{1}{2}$ and the six others that are shared with other cells at $z=0,1$ and therefore contributing a further 3 to the cell). The illustrated cell contains a full complement of carbon atoms but to achieve the composition Fe$_{2.4}$C, the sites coloured blue would only be occupied partially. 


\begin{figure}[!ht]
\centering
\subfigure(a){
\includegraphics[width=0.4\textwidth]{Stereograms/epsilon_3D.eps}}\hfill
\subfigure(b){
\includegraphics[width=0.4\textwidth]{Stereograms/epsilon_2D.eps}}
\caption{The structure of hexagonal $\varepsilon$-carbide with lattice parameters $a=0.4767\,$nm and $c=0.4353\,$nm. The large atoms are iron. The small atoms are carbon but not all the sites designated blue are occupied.  (a) The three dimensional cell with the $z$-axis vertical. (b) Projection of the cell along the $z$-axis, with the fractional $z$-coordinates listed.}
\label{fig:epsilon}
\end{figure}

The projection described in Fig.~\ref{fig:epsilon} is a straightforward linear operation and there are no distortions of that linearity. In contrast, Fig.~\ref{fig:circles} show non-linear projections of circular arcs, in one case a simple extrapolation on to a horizontal line, and in the other case via a pole located below the horizontal line where the projections are recorded. The purpose here is to record angles rather than spatial coordinates.

\begin{figure}[!ht]
\centering
\subfigure(a){
\includegraphics[width=0.2\textwidth]{Stereograms/circle_projection_1.eps}} \hskip 2cm
\subfigure(b){
\includegraphics[width=0.18\textwidth]{Stereograms/circle_projection_2.eps}}
\caption{Projections of angles made by lines intersecting a circular quadrant onto a horizontal line. }
\label{fig:circles}
\end{figure}
Imagine now that instead of quadrants of a circle, the diagrams in Fig.~\ref{fig:circles} represent the surfaces of spheres. A circle drawn on the surface of the sphere would in general project as an ellipse on the horizontal plane of Fig.~\ref{fig:circles}a, whereas it would project as a true circle in the case of the projection method used in Fig.~\ref{fig:circles}b.

\section{Angular truth}
Consider a sphere that has north and south poles.  If a circle is constructed on the northern hemisphere with each point on its locus projected on to the equatorial plane by lines originating from the south pole, then the projection itself will remain a circle. 
Fig~\ref{fig:small_circle}a illustrates the circle $ab$ on the surface of the sphere, together with its projection $a'b'$ on the equatorial plane. A cross-section of the same diagram through the centre of the circle $ab$ is shown in Fig~\ref{fig:small_circle}. The conjugate circular section $cd$ is symmetrically inclined to the axis $ST$.


The line $eb$ is constructed to be parallel to the equatorial plane. The arcs $eS$ and $Sb$ are equal in length so it follows that $\phi_1=\phi_2$. Since $cd$ is symmetrically inclined to $ST$, it follows that $\phi_3=\phi_2$ and therefore $\phi_3=\phi_1$. In other words, the circle $cd$ is parallel to the equatorial plane. If the circle $cd$ is now translated towards the equatorial plane, it remains a circle and ends up as a circle $a'b'$.


\begin{figure}[!ht]
\centering
\subfigure(a){
\includegraphics[width=0.41\textwidth]{./Stereograms/projection_smallcircle2.eps}}
\subfigure(b){
\includegraphics[width=0.49\textwidth]{./Stereograms/projection_smallcircle.eps}}
\caption{(a) Circle on the surface of a sphere projected on to the equatorial plane via the south pole. (b) Construction showing that the projection is also a circle.}
\label{fig:small_circle}
\end{figure}

This example shows that the {\em stereographic} projection method illustrated in Fig.~\ref{fig:small_circle} preserves angular truth; if two circles (e.g. a longitude and latitude)  inscribed on the sphere cross at 90$^\circ$ then the projections of those two circles will also cross at that angle. We shall now consider these stereographic projections in more detail, following a brief description of why they are of importance.

\section{Utility of stereographic projections}

There is an excellent review of the history of the stereographic projection by Howarth, tracing the origin to the 2${\rm nd}$ century BCE in the context of astronomy. Howarth describes how projections were then used in mineralogy and structural geology, where `the spatial orientations of a crystal face, betting plane, fault surface etc. can be considered by imagining a plane passing through the centre of a sphere'. 

Stereographic projections  are two-dimensional representations which now have found numerous applications in materials science, with the method often embedded in the software that controls experimental crystallography, for example,
\begin{itemize}
\item the deformation of single crystals whence crystal planes rotate in order to comply with the external stress; such rotations in polycrystalline materials lead to non-random  aggregates of crystals, the properties of which are in between those of single and random polycrystals. In both of these cases, it becomes necessary to define the orientations of individual crystals relative to the deformation axes, which is where stereographic projections become useful. 
\item When phase transformations occur in the solid state, the probability is that the product phase will form such that the atomic arrangements match as much as possible, along the parent/product interface. This often leads to a reproducible orientation relationship between the two crystals, one which can be represented on a stereographic projection in order to  understand the mechanism of transformation or the deformation behaviour of the two-phase mixture. Stereographic projections can be used to decide whether the orientation relationship is reproducible or occurs by chance.
\item Diffraction data can be presented on stereographic projections. It is now routine to to examine both structure and crystallographic information from polycrystalline samples on a single image, with the crystallographic data presented in the form of corresponding colours on the microstructural image and stereographic projection. 
\end{itemize}

\section{Stereographic projection: construction and characteristics}

Imagine a sphere. Any plane that passes through its centre will intersect the surface of the sphere at a circle whose diameter is that of the sphere; this is known as a {\em great circle}, Fig.~\ref{fig:great}a. The primitive (great) circle is where the equatorial plane intersects the sphere Fig.~\ref{fig:great}b. The diameter of a great circle is the same as that of the sphere. Lines of longitude are all great circles in the spherical earth approximation.
\begin{figure}[!ht]
\centering
\subfigure(a){
\includegraphics[width=0.25\textwidth]{Stereograms/great_circle_2.eps}} 
\subfigure(b){
\includegraphics[width=0.25\textwidth]{Stereograms/small_circle_2.eps}}
\subfigure(c){
\includegraphics[width=0.32\textwidth]{Stereograms/globe.eps}}
\caption{Intersections of planes with sphere. (a) Great circle inclined with respect to the north and south poles of the sphere, (b) small circle; (c) latitude and longitude lines (image courtesy of vectortemplates.com). }
\label{fig:great}
\end{figure}


When an intersecting plane does not pass through the centre of the sphere it results in a  {\em small circle} at the intersection, as illustrated in Fig.~\ref{fig:great}c. Lines of latitude are in general small circles, the exception being the equator.

The shortest distance between two points on a sphere is along the great circle which passes through both of the points. 

Fig.~\ref{fig:poles}a shows a crystal with a cubic lattice, placed at the centre of a sphere with the $[100]\parallel x$-axis, $[010]\parallel y$-axis and $[001]\parallel z$-axis of the sphere; this is the conventional orientation of the crystal in the sphere. Similarly, the orientation of an arbitrary plane can be defined by its normal, which in general is known as the {\em pole} of the plane. Suppose now that the normal to the $(011)$ plane is projected so that it intersects the sphere at the point $a$ (Fig.~\ref{fig:poles}b), which then is projected to the south pole. The intersection on the equatorial plane being the point $b$ which represents the pole of the plane on the stereographic projection.

The plane (011) when extended so that it intersects the sphere will result in a great circle on the surface of the sphere, the trace of which can be projected on to the equatorial plane as illustrated in Fig.~\ref{fig:poles}d. The part of the plane that is in the northern hemisphere projects as a line, and that from the southern hemisphere as a dashed-line. All points on the trace are at 90$^\circ$ to the pole of (011) since the pole represents the normal to the plane, another indication of how the stereographic projection maintains angular truth.

\begin{figure}[!ht]
\centering
\subfigure(a){
\includegraphics[width=0.45\textwidth]{Stereograms/plot_1.eps}}
\subfigure(b){
\includegraphics[width=0.45\textwidth]{Stereograms/plot_2.eps}}
\\
\subfigure(c){
\includegraphics[width=0.45\textwidth]{Stereograms/plot_3.eps}}
\subfigure(d){
\includegraphics[width=0.45\textwidth]{Stereograms/plot_4.eps}}
\caption{Plotting plane normals on a stereographic projection. The insets, top-left in each case, show a plan view of the equatorial plane. (a) Crystal places at the centre of the sphere. (b) Projection of the pole of (011).  (c) Projection of $(0\overline 1\overline 1)$.  (d)  Projection of the trace of the great circle which is (011) on to the equatorial plane. The trace of the plane in the southern hemisphere is marked as a dashed curve. The crosses identify the intersections of the projection lines with the equatorial plane. }
\label{fig:poles}
\end{figure}

A projection of  the longitudes and latitudes illustrated in Fig.~\ref{fig:great}c on to the equatorial plane would lead to a net, known as the `Wulff net' in which the resulting great circles are spaced at equal angles as illustrated in Fig.~\ref{fig:Wulff} where the angular intervals are 2$^\circ$. The net can be used to measure angles between two poles on a stereographic projection by rotating the net such that both poles lie on the same great circle, followed by counting the degrees separating them. Note that the angle between two great circles will be the same as the angle between their poles.

\begin{figure}[!ht]
\centering
\subfigure(a){
\includegraphics[width=0.45\textwidth]{Stereograms/Wulff_net_pole.eps}}
\subfigure(b){
\includegraphics[width=0.45\textwidth]{Stereograms/Wulff_net_rotated.eps}}\\
\subfigure(a){
\includegraphics[width=0.45\textwidth]{Stereograms/Wulff_net_small_circle.eps}}
\subfigure(b){
\includegraphics[width=0.45\textwidth]{Stereograms/Wulff_net_intersecting.eps}}
\caption{The Wulff net. (a) Relation between pole and trace of a plane. (b) The net is rotated until the poles $a$ and $b$ lie on the same great circle to measure the angle in between. (c) The geometrical centre $c$ of the small circle is different from its angular centre $b$. The angles $ab$ and $bd$ as measured on the great circle are identical. The distances $ac$ and $cd$ are identical when measured on a ruler. (d) The poles identified with arrows are both located at angles $\phi_1$ to $p_1$ and $\phi_2$ to $p_2$.} 
\label{fig:Wulff}

\end{figure}
\clearpage
The curve representing a great circle on a Wulff net is itself an arc of a circle, a result which follows from the fact that the projection method preserves angular truth. This arc is from a circle that in general has  a radius $r$ greater than that of the net ($r_{\rm o}$). We now derive the relationship between $r$, $r_{\rm o}$ and the offset $x$ of the trace of the great circle from the origin.

Referring to  the figure below, the distance $ab$ is $2r_{\rm o}$; the Pythagoras theorem then gives
\begin{eqnarray} 
& & r^2-r_{\rm o}^2 = (r-x)^2 \nonumber \\
\hbox{so it follows that} & & r=(x^2+r_{\rm o}^2)/2x \quad\hbox{with}\quad r_{\rm o}\leq r \leq \infty. \nonumber
\end{eqnarray}

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.41\textwidth]{./Stereograms/great_circle.eps}
\end{center}
\end{figure} 

Fig.~\ref{fig:cubic}a,b show the stereographic projection for the cubic system, with the rotational symmetry elements related to the crystallographic directions about which the rotations occur.  Fig.~\ref{fig:cubic}c shows the stereographic projection being used to show how the elastic modulus of a single crystal of ferritic iron varies between 131 and 284\,GPa as a function of the crystallographic orientation, in a manner consistent with its crystalline symmetry. Even aggregates of crystals often do not show collective macroscopic-properties that are isotropic because the crystals are not  distributed at random. Polycrystalline ferritic steel sheet in which the crystals are aligned with the $\langle001\rangle$ direction and with the $\{110\}$ planes aligned to the rolling plane, exhibit modulus variation in the range 140 to 210\,GPa. The level of anisotropy in aggregates can be controlled, as we shall see in later chapters, to suit specific purposes such as the formability of metal sheets or the magnetic properties of transformer steels. 

\begin{figure}[htp]
\centering
\subfigure(a){
\includegraphics[width=0.49\textwidth]{Stereograms/cubic_2.eps}}
\subfigure(b){
\includegraphics[width=0.41\textwidth]{Stereograms/cubic_1.eps}}\\
\subfigure(c){
\includegraphics[width=0.49\textwidth]{Stereograms/modulus_ferrite.eps}}
\caption{Stereographic projections for the cubic system. (a) Showing the standard way of drawing the projection with the locations of $[100]$ and $[010]$ on the perimeter and $[001]$ at the centre. (b) The symmetry elements. The filled squares are tetrads, the ellipses are diads and triangles are triads. The  tetrads are at $\langle001\rangle$, triads at $\langle111\rangle$ and diads at $\langle011\rangle$. Anything with cubic symmetry must have four triads. (c) Variation in the elastic modulus of a single crystal of bcc iron as a function of orientation. Diagram courtesy of Shaumik Lenka, constructed using data from Dieter (1988).}
\label{fig:cubic}
\end{figure}

\section{Stereographic representation of point groups}

We shall now show using sketch stereograms, that the operation of a diad followed by an inversion (i.e. $ \overline 2$) is equivalent to a mirror plane. Do the operations $\overline 3$ and $\overline 6$ lead to centres of symmetry?

A general pole on a stereographic projection is one that is not located at particular symmetry elements such as the rotation axis or on a mirror plane. The black dot on Fig.~\ref{fig:bar2}a is a general pole which on the operation of a diad results in a second pole as shown in Fig.~\ref{fig:bar2}b. When inverted through the centre, this second pole intersects the stereographic sphere in the southern hemisphere and hence is shown as an open circle in Fig.~\ref{fig:bar2}c, which shows that the net operation of a diad followed by an inversion is a mirror plane parallel to the plane of the diagram. It is seen that $\overline 2\equiv m$.

The operation of a triad on a general pole leads to three such poles, which after inversion result in the situation illustrated in Fig.~\ref{fig:bar2}e, which shows that the point group $\overline 3$ belongs to a class that have centres of symmetry whereas $\overline 2$ and $\overline 6$ do not.
 

\begin{figure}[!ht]
\centering
(a) \includegraphics[width=0.2\textwidth]{./Stereograms/mirror_1.eps} \hskip 10mm
(b) \includegraphics[width=0.2\textwidth]{./Stereograms/mirror_2.eps} \hskip 10mm
(c) \includegraphics[width=0.2\textwidth]{./Stereograms/mirror_3.eps}\\
(d) \includegraphics[width=0.2\textwidth]{./Stereograms/mirror_4.eps}  \hskip 10mm
(e) \includegraphics[width=0.2\textwidth]{./Stereograms/mirror_5.eps} \hskip 10mm
(f) \includegraphics[width=0.2\textwidth]{./Stereograms/mirror_6.eps}
\caption{(a) An arbitrary pole is rotated by 180$^\circ$ about the diad normal to the plane of the diagram, to generate the second pole illustrated in (b). This second pole is inverted through the centre to lead to the result in (c) to represent the full operation of the inversion diad $\overline 2$. (d) Triad, (e) inversion triad and (f) inversion hexad. Mirror planes in (c) and (f) are identified using bold circles.}
\label{fig:bar2}\end{figure}


\section{The point groups $3m$ and $m3$}

Referring to Table~\ref{tab:pg}, we now explain using sketch stereograms, the difference between the point groups $3m$ and $m3$.

According to the conventions described on page~\pageref{page:pg}, the point group $3m$ implies that there is a single triad with the secondary axis being the normal of the mirror plane. The triad ensures that there would be three such mirror planes.   There is no mirror plane normal to the triad (Fig.~\ref{fig:3m}a), which means that the atomic arrangement along one end of the triad is different from the opposite end. Such a crystal may display spontaneous polarisation and the point group $3m$ is said to be a {\em polar point group}, which include 1 (triclinic); 2, $m$ (monoclinic); $mm2$ (orthorhombic); 4, $4mm$ (tetragonal); 3, $3m$ (trigonal); 6, $6mm$ (hexagonal). There are no polar groups in cubic crystals. Gallium nitride as a polar material $(6mm)$ is of huge interest in the manufacture of light emitting diodes -- Fig.~\ref{fig:3m}b shows that the top surface normal to the hexad is different from the bottom surface. During deposition of GaN films on substrates, the layers deposit predominantly with uniform polarity except some regions which have opposite polarity and are regarded as defects that lead to poor control of electrical properties.


\begin{figure}[!ht]
\centering
(a) \includegraphics[width=0.33\textwidth]{./Stereograms/3m_trigonal.eps} \hfill
(b) \includegraphics[width=0.53\textwidth]{./Stereograms/GaN_b.eps}\\[10pt]
(c) \includegraphics[width=0.42\textwidth]{./Stereograms/m3_cubic.eps} 
(c) \includegraphics[width=0.46\textwidth]{./Stereograms/Pyrite_Structure.eps} 
\caption{Sketch stereograms showing what happens to a general pole on the operation of the following point groups (bold lines represent mirror planes): (a) trigonal $3m$.  (b) The structure of GaN with point group symmetry $6mm$, viewed with the 6-fold axis  along $z$. The gallium atoms are labelled green. (c) Cubic $m3$. (d) Projection of the unit cell of pyrite (FeS$_2$) with the fractional $z$ coordinates indicated, glide planes marked with dashed lines and screw diads. Not all the symmetry elements are illustrated for simplicity.}
\label{fig:3m}
\end{figure}


The point group $m3$ belongs to the cubic crystal class with the first symbol $m$ parallel to the $z$ axis (page~\pageref{page:pg}) and the mandatory symbol `3' to follow since the defining symmetry of a cubic system is four triads, in this case inversion triads. It is evident from the plotting of a general pole and its symmetry related poles shown in Fig.~\ref{fig:3m}c that there is a centre of symmetry. Pyrite belongs to the cubic class with point group $m3$. Fig.~\ref{fig:3m}d shows a projection of the crystal structure of pyrite, which is primitive cubic. The structure can be compared with the stereogram showing the point group symmetry in Fig.~\ref{fig:3m}c. Bearing in mind that point groups do not include translations, the glide planes and screw axes indicated in Fig.~\ref{fig:3m}d become mirrors and diads respectively.


\label{page:titanate} Crystals which lack a centre of symmetry can be piezoelectric, i.e. they develop a dipole on being deformed and will change shape on the application of an electrical field. Under normal conditions the crystal has positive and negative electrical charges which are symmetrically distributed, leaving the crystal in a neutral state when homogeneously deformed. The application of a stress causes charge asymmetry, and the development of a voltage across the crystal.  PbTiO$_3$ is such a substance (Fig.~\ref{fig:titanate}) where for example, the titanium atom is not located at exactly $\frac{1}{2},\frac{1}{2},\frac{1}{2}$. The structure is primitive tetragonal, with point group $4mm$.\footnote{i.e. by the convention quoted on page~\pageref{page:pg}, the $z$ direction is assigned to the four-fold rotation axis followed by $m$ for the mirrors with normals parallel to the $x$ and $y$ axes, and the final symbol for the tertiary mirrors with normals between the $x$ and $y$ axes.}
 
The stereogram illustrated in Fig.~\ref{fig:titanate}b shows that the structure has no centre of symmetry. Secondly, if an atom is placed at a general position (marked $a$) then there will be seven other identical atoms in the unit cell, inconsistent with the structure illustrated in Fig.~\ref{fig:titanate}a. An atom placed on a tetrad (marked $c$) will generate just one atom per unit cell; thus, there is just one Ti  at $\frac{1}{2},\frac{1}{2},0.572$. The oxygens at $\frac{1}{2},\frac{1}{2},0$ and $\frac{1}{2},\frac{1}{2},1$ also lie on a tetrad, and since they are shared between two cells, there is only one such oxygen in the unit cell. An atom, such as the oxygen at $0,\frac{1}{2},\frac{1}{2}$, placed on the mirror plane as illustrated by the poles marked $b$, would generate two such atoms in the cell (the four on the vertical faces are each shared by two cells).


This example shows how the placing of an atom within a unit cell must be consistent with the symmetry elements of the cell because there must be identical atoms at all symmetry related positions.
 

\begin{figure}[htp]
\centering
\subfigure(a){
\includegraphics[width=0.45\textwidth]{Stereograms/titanate.eps}}
\subfigure(b){
\includegraphics[width=0.45\textwidth]{Stereograms/4mm.eps}}
      \caption{(a) Lead titanate (PbTiO$_3$) is tetragonal with $a=0.3904$\,nm and $c=0.4152\,$nm. The Pb  is located at coordinates (0,0,0.116), the Ti  at ($\frac{1}{2},\frac{1}{2},0.572$), an oxygen atom  at $(0,\frac{1}{2},\frac{1}{2})$ and another oxygen  at $(\frac{1}{2},\frac{1}{2},0)$. The structure is primitive tetragonal with point group $4mm$. (b) Stereogram showing the symmetry operations associated with the point group $4mm$. }  \label{fig:titanate}
\end{figure}

There are many other consequences of whether or not a crystal has a centre of symmetry. The free energy of a small volume element containing a one-dimensional
composition gradient varies as follows:
 \begin{equation} g = g\{\overline c\} + \kappa_1 {{d c}\over {d z}} + \kappa_2
{{d^2 c}\over {dz^2}} + \kappa_3 \biggl({{d c}\over {d z}}\biggr)^2 
\end{equation}
where $c$ is the concentration, $g\{\overline c\}$ is the free energy of a homogeneous solution with composition $\overline c$,  $z$ is the distance and $\kappa_i$ are specific functions of the dependence of free energy on the gradients of concentration as described elsewhere.  In this, $\kappa_1$ is zero for a centrosymmetric crystal since the
free energy must then be invariant to a change in the sign of the
coordinate $z$. 

\section{Summary}

A circle on a sphere projects as a circle on the stereographic projection; angular relationships are therefore preserved. The method is therefore well suited to  look at angles between planes or directions in a crystal. The symmetry of the crystal in its three dimensions becomes easy to visualise by plotting the symmetry elements in their correct angular orientations on a stereographic projection. The placing of atoms inside the unit cell during attempts to solve structure must clearly be consistent with the symmetry determined, for example, from the external form of well-formed crystals. 

The stereographic projection can also be used quantitatively, by superimposing a Wulff net to directly measure angles between a pair of poles.  The net consists of projections of great and small circles, rather like the longitudes and latitudes on a globe representing the earth, The shortest distance between to points on the surface of a  sphere is when the two points are located on an arc of a circle which has the same diameter as the sphere. When using a Wulff net to measure the angle, it therefore is necessary to locate the two poles on a great circle.

We have seen that some quite elementary considerations of symmetry can indicate the occurrence of phenomena   such as piezoelectricity, or polar materials. There is much more to be explored in the chapters that follow.



