
\chapter{Space Groups}

\section{Introduction}

A point group is a collection of symmetry elements passing through a point, and therefore, necessarily does not include translations. {\em Space groups}, in contrast, include translations that are fractions of a repeat unit, for example, a $2_1$ axis which involves a rotation of 180$^\circ$ followed by a translation of $\frac{1}{2}$ of the repeat distance along the axis. Glide planes involve reflections followed by fractional translations. These translations are small, and hence do not manifest when the point group is determined from the external shape of well-formed crystals. However, they have consequences in structure determination and on other properties of crystals. There are  230 space groups which are made from combinations of the 32 crystallographic point groups with the 14 Bravais lattices. 

The relationships between the placing of atoms in the unit cell and the symmetry of the structure of lead titanate was described on page~\pageref{page:titanate}. The structure of PbTiO$_3$ is primitive tetragonal with point group $4mm$.  Since there are no screw axes or glide planes, the space group symbol for the titanate is $P4mm$, where the first symbol is to identify the lattice type as primitive. In this chapter, we will  consider a case where translational symmetry elements\footnote{other than the normal periodicity of a lattice} are present.

\section{Screw axes and glide planes}

A proper rotation axis designated $N$ brings a motif into coincidence after a rotation of $\frac{2\pi}{N}$. A screw axis, designated $N_p$, does the same but only when a translation of $\frac{p}{N}\times t$ follows the rotation by $\frac{2\pi}{N}$. Here $t$ is the distance between lattice points parallel to the axis. A screw diad ($2_1$) is illustrated in Fig.~\ref{fig:screw}, involving a rotation of 180$^\circ$ followed by a translation of $\frac{1}{2}t$ parallel to the axis. Other rotation axes following the same logic in terms of notation are: $3_1$,  $3_2$, $4_1$, $4_2$, $4_3$, $6_1$, $6_1$, $6_2$, $6_3$, $6_4$ and $6_5$.

\begin{figure}[!ht]
  \begin{center}
    \begin{minipage}[t]{0.45\linewidth}
\raisebox{-3.5cm}{\includegraphics[width=0.98\linewidth]{Space_groups/screw_2.eps}} 
    \end{minipage}\hfill
    \begin{minipage}[t]{0.5\linewidth}
  \caption{An illustration of the operation of a diad where the structure is restored by a rotation of 180$^\circ$, and a screw diad which involves the same rotation but then a translation along the axis by 0.5 (hence the notation $2_1$) of the repeat distance $t$. }
\label{fig:screw}
    \end{minipage}
  \end{center} \end{figure} 
 
 Glide planes are like mirror planes but a  reflection is followed by a translation in order to recover the symmetry.The terminology for glide planes is explained in Table~\ref{tab:notation}; the translations following reflection are parallel to the glide plane, the exception begin the diamond glide in a body-centred cubic structure.


\begin{table}[!ht]
\caption{Symbols for glide planes. `f.c.c.' and `b.c.c.' are abbreviations for face--centred cubic and body--centred cubic respectively.}
\begin{center}
\resizebox{0.99\textwidth}{!}{\begin{tabular}{lccc}\midrule
Glide character & Direction & Magnitude & Symbol \\\midrule 
 Axial glide & parallel to $a$, $b$ or $c$ axis & $\frac{1}{2}(a,~b,~{\rm or}~ c)$ & $a,~b,~{\rm or}~ c$ \\[6pt]
 Diagonal glide & parallel to face diagonal &  $\frac{1}{2}(a+c)$, $\frac{1}{2}(b+c)$ or $\frac{1}{2}(a+b)$ & $n$ \\[6pt]
 Diamond glide (f.c.c.) & parallel to face diagonal & $\frac{1}{4}(a+c)$, $\frac{1}{4}(b+c)$ or $\frac{1}{4}(a+b)$ & $d$ \\[6pt]
 Diamond glide (b.c.c.) & parallel to body diagonal & $\frac{1}{4}(a+b+c)$ & $d$ \\[6pt]\midrule
\end{tabular} }
\end{center}
\label{tab:notation}
\end{table}%


 \section{Cuprite} 
Consider now the structure of cuprite $({\rm Cu_2O})$ which is primitive cubic, with a motif of four copper atoms at 000,~ $\frac{1}{2}0\frac{1}{2}$,~ $0 \frac{1}{2}\frac{1}{2}$ and $\frac{1}{2}\frac{1}{2}0$, and two oxygen atoms at  $\frac{1}{4}\frac{1}{4}\frac{3}{4}$ and $\frac{3}{4}\frac{3}{4}\frac{1}{4}$ located at each lattice point. 
   
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.8\textwidth]{Space_groups/cuprite2.eps}
\caption{Projection along the [001] direction of four unit cells of cuprite $({\rm Cu_2O})$. The fractional height along [001] are indicated, with unlabelled atoms at 0 and 1.}
\label{fig:cuprite}
\end{center}
\end{figure} 

Fig.~\ref{fig:cuprite} shows a projection of four such unit cells along the [001] direction, with fractional coordinates along that direction indicated. The environment of each of the six atoms within the unit cell is different, consistent with the cubic-P lattice. There are mirror planes parallel to $\{110\}$ -- the trace of only one of these is illustrated in Fig.~\ref{fig:cuprite} for the sake of clarity. Note also that there are $4_2$ axes located at $\frac{1}{4}\frac{1}{4}z$ and $\frac{3}{4}\frac{3}{4}z$ and their equivalent positions. A $4_2$ axis involves a rotation of 90$^\circ$ followed by a translation by $\frac{2}{4}$ of the repeat distance along that axis. The diagram also shows that the structure has a centre of symmetry at $\frac{1}{2}\,\frac{1}{2}\,\frac{1}{2}$.

The glide plane parallel to $\{100\}$ (Fig.~\ref{fig:cuprite}) involves a reflection followed by two orthogonal translations parallel to the plane by $\frac{1}{2}$ the lattice parameter. It is, therefore, an $n$-glide plane according to the notation in Table~\ref{tab:notation}.

Since cuprite is cubic, the defining symmetry is four triads. The structure clearly has a centre of symmetry at the body-centre, so the three fold axes which pass through this centre are in fact $\overline 3$ inversion triads ({\em cf.} Figs.~\ref{fig:bar2}d,e).  

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.5\textwidth]{Space_groups/cuprite_cell.eps}
\caption{The unit cell of cuprite.}
\label{fig:cuprite_2}
\end{center}
\end{figure} 

\section{Location of atoms in cuprite cell}

The number of atoms in the unit cell depends on where it is located, because symmetry demands that it will exist at all equivalent positions. The space group $Pn\overline 3 m$ reduces to the point group $m3m$ when translations are omitted. Fig.~\ref{fig:m3m}a shows that placing of an atom at a general location leads to 48 such atoms in the cell.  If located on a mirror plane, the number of positions becomes 24 since an atom on a mirror plane has no reflection from that plane Fig.~\ref{fig:m3m}b.

\begin{figure}[htp]
\centering
\subfigure(a){
\includegraphics[width=0.35\textwidth]{Space_groups/cuprite_48.eps}}\hfill
\subfigure(b){
\includegraphics[width=0.35\textwidth]{Space_groups/cuprite_24.eps}}\\
\subfigure(c){
\includegraphics[width=0.35\textwidth]{Space_groups/pn3m_1.eps}}\hfill
\subfigure(d){
\includegraphics[width=0.35\textwidth]{Space_groups/pn3m_2.eps}}
\caption{Stereograms consistent with the point group $m3m$, showing the number of equivalent positions when (a) an atom is placed at general location $x,\,y\,z$ and all equivalent positions;  (b) an atom is placed at general location $x,\,x,\,z$ and all equivalent positions. (c) Projection of the cuprite unit cell with the traces of glide planes marked as dashed lines and those of mirror planes as heavy lines.  Initial placing of atoms copper in black and oxygen in red. (d) New set of atoms generated by the operation of the glide planes. The relation between the original and new atoms is marked by blue lines.}
\label{fig:m3m}
\end{figure}

A copper atom located at 0,0,0 (Fig.~\ref{fig:m3m}c,d) has the point symmetry  $\overline 3m$. The $n$-glide plane marked by the horizontal line generates the atom at the opposite face-centre at a height $\frac{1}{2}$ and the other symmetry elements then generate atoms at all face-centres and corners of the cube. Therefore, four equivalent copper atoms in the unit cell.

An oxygen atom at $\frac{1}{4},\frac{1}{4},\frac{3}{4}$  has the point symmetry $\overline 4 3 m$. The operation of the $n$-glide plane leads to an oxygen atom at  $\frac{3}{4},\frac{3}{4},\frac{1}{4}$. All other symmetry operations have the same effect so there are only two oxygen atoms per cell, consistend with the chemical formula ${\rm Cu_2O}$; the density of the compound can be measured to verify this.  The results are summarise in Table~\ref{tab:table}. The International Union of Crystallography publishes much more comprehensive space group tables which list the multiplicity of coordinates as a function of symmetry at that coordinate, for all possible scenarios and for all space groups. 
\begin{table}[!ht]
\caption{Space group for cuprite, $Pn\overline 3m$. The origin of the cell is set on a copper atom.}
\begin{center}
\begin{tabular}{lccccc}\midrule
Point symmetry & Number & \multicolumn{4}{c}{Coordinates} \\\midrule
1 & 48 & $x,y,z$ & $\overline x,\overline y,z$&  $\overline x,y,\overline z$&  $\ldots$\\[6pt]
$m$ & 24 & $x,x,z$ & $\overline x,\overline x,z$ & $\overline x,x,\overline z$& $\ldots$\\[6pt]
$\vdots$ & & & & \\
$\vdots$ & & & & \\
$\overline 3m$ & 4 &0,0,0 & $\frac{1}{2},\frac{1}{2},0$ & $\frac{1}{2},0,\frac{1}{2}$ & $0,\frac{1}{2},\frac{1}{2}$ \\[6pt]
$\overline 4 3 m$ & 2& $\frac{1}{4},\frac{1}{4},\frac{3}{4}$ & $\frac{3}{4},\frac{3}{4},\frac{1}{4}$ & & \\[6pt]\midrule 
\end{tabular}
\end{center}
\label{tab:table}
\end{table}

\section{Shape of Precipitates}

Crystallography is one of the characteristics that can influence the shape of a precipitate, but its influence is usually overwhelming when the precipitation occurs in the solid state. In some circumstances, it is symmetry which determines the shape of precipitates which form inside a solid.

Suppose that $G_m$ and $G_p$ represent the symmetry groups of the matrix and precipitate respectively. The common symmetry (intersection group) between these two phases in the observed orientation relationship is defined as $H$. Given a planar interface between them, symmetry operations consistent with $H$ leads to other orientations of the interface until the particle is fully enclosed and its shape defined. Note that any operation of $H$ is a symmetry operation of the parent and product and hence each crystal orientation is brought into congruence so the orientation of the two crystals is not affected by $H$.


Aluminium has the space group $Fm\overline 3 m$ and the silver--rich precipitate $\Omega$ has the space group $Fmmm$. The orientation relation has been measured to be:
$$[001]_{\Omega}\parallel [111]_{\rm Al}$$   $$[100]_{\Omega}\parallel [10\overline 1]_{\rm Al}$$ 
Fig.~\ref{fig:common} shows the stereograms of the symmetry elements of the two crystals, in the correct relative orientation and it is evident the common point group symmetry is $2/m$, which is consistent with the observed plate shape of the $\Omega$ both with respect to the diad and the mirror plane. 


\begin{figure}[!ht]
\centering
\subfigure(a){
\includegraphics[width=0.25\textwidth]{Space_groups/m3m.eps}}
\subfigure(b){
\includegraphics[width=0.25\textwidth]{Space_groups/mmm.eps}}
\subfigure(c){
\includegraphics[width=0.3\textwidth]{Space_groups/omega_shape.eps}}
\caption{(a) Stereographic representation of the symmetry elements in the point group $m3m$ for aluminium. (b) Stereographic representation of the symmetry elements in the point group $mmm$ for $\Omega$. The symmetry elements shared by Al and $\Omega$ are highlighted in red. (c) The precipitate shape observed, corresponding to the shared symmetry group $2/m$.}
\label{fig:common}
\end{figure}

\section{Summary}

We have touched on the notion of space groups which represent all the symmetry elements of the crystal. The appreciation of space groups is essential in the solution of structures, transformations which lead to changes in symmetry, and the properties of crystals. A great deal of the work in identifying the number of symmetry related positions as a function of the space group has already been documented, with the information routinely available either from the tables published by the International Union of Crystallography or via the world wide web.
