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Next: 4. Space Groups Up: Symmetry Previous: 2. Combination of Symmetry Elements in

3. Symmetry Elements in Arrays

If you look back at Fig. 1.1, you will see that it combines elements of symmetry with a repeating pattern. We call such a repeating pattern of motifs an array ; the smallest convenient parallelogram that can be repeated without change of orientation to produce the pattern is called the unit cell . A two-dimensional unit cell is outlined in Fig. 1.1; note that while the choice of origin is somewhat arbitrary the shape is not. (Check this by identifying other ways of defining the repeating unit.)

In a three-dimensional array, such as a crystal, the unit cell is a parallelipeped defined by intercepts a, b, c on three axes (x, y, z ) and the angles between them, $\alpha$, $\beta$, $\gamma$, as shown in Fig. 3.1. If the contents have no symmetry or a centre of symmetry only, the unit cell can have this quite general shape; more symmetrical cell contents restrict the values of the interaxial angles and the relative sizes of the edges in the manner given in Table 1. Thus a single direction of twofold symmetry (monoclinic system) makes two of the angles into right-angles but places no restrictions on the third or on the edge dimensions; at the other extreme, cubic symmetry produces a cell whose edges are all equal and whose angles are all 90$^{\circ}$ (i.e. a cube!).


 
Fig. 3.1. A generalised unit cell, showing the repeat distances, a , b , c along the x , y and z axes, and the interaxial angles $\alpha$, $\beta$, $\gamma$.
\begin{figure}
\includegraphics {fig3.1.ps}
\end{figure}

An array can have any of the symmetry elements that we have discussed already, including no symmetry at all (Fig. 3.2), and it can also have additional types of symmetry not possible in finite objects. Consider Fig. 3.3 which shows an aerial view of a boat rowed by eight crew; provided that they have been well coached they present a symmetrical appearance, but it is not one that can be exactly described by any of the symmetry elements introduced so far. It is obviously related to a mirror plane, but in Fig. 3.3 each rower is the mirror image of one rowing immediately in front or behind. Any figure is related to the next by moving one place along the boat and then reflecting across a mirror plane. A symmetry operation of this type is called -- very descriptively -- a glide plane . Because a glide plane combines the operation of reflection with that of translation it occurs only in extended arrays.

An analogous operation combining rotation and translation is called -- equally descriptively -- a screw axis . As an actual example of an object possessing this type of symmetry, a bolt is really better than a screw, since most screws taper to a point, but the action of driving a screw -- or using a corkscrew! -- illustrates very vividly the operation of a screw axis. Other familiar objects having screw axes are spiral staircases, springs, and some climbing plants. Formal examples are shown in Fig. 3.4, together with a formal representation of a glide plane.


 
Fig. 3.2. An array of repeating motifs: neither the motif nor the array contains any elements of symmetry.
\begin{figure}
\includegraphics {fig3.2.ps}
\end{figure}


 
Fig. 3.3. A stylised aerial view of a well coached `eight`, showing a translational symmetry operation: each rower is related to the next by a combination of translation and reflection.
\begin{figure}
\includegraphics {fig3.3.ps}
\end{figure}


 
Fig. 3.4. Translational symmetry elements. (a) A twofold screw axis, 21, shown perpendicular to the plane of the paper (left) and in the plane of the paper (right). Each half revolution is accompanied by a translation through half the repeat distance. (b) A fourfold screw axis, 41. (c) A glide plane. Translation from left to right across the page is accompanied by reflection through the plane of the paper.
\begin{figure}
\includegraphics {fig3.4.ps}
\end{figure}

The general symbol for a screw axis is Nn, where N is the order (2, 3, 4 or 6) of the axis, and n /N the translation distance expressed as a fraction of the repeat unit. Thus, 41, shown in Fig. 3.4b, means that the asymmetric unit moves $\frac{1}{4}$ of a repeat unit along the axis for each $\frac{1}{4}$ of a revolution about that axis. Glide planes are symbolised by a letter indicating the direction of the glide: the letters a, b and c mean that the direction of glide is parallel to the a, b and c axes, respectively, while n and d refer to glide planes in which the direction of glide is diagonally across a face of the unit cell or along a body diagonal.

A combination of parallel translational and non-translational symmetry elements produces an interesting effect on the way in which the pattern repeats. Consider Fig. 3.5. The pattern in 3.5a contains mirror planes: note that two mirror planes are associated with each repeat unit across the page. The pattern in 3.5b is based on a similar motif related by glide planes (g ); again two are associated with each repeat unit across the page. In 3.5c there are parallel mirror and glide planes, and as a result the grouping of motifs at the cente of the rectangular cell is identical with that at the corners. Such a pattern is called centred , while those of 3.5a and 3.5b are said to be primitive . It is always possible to define a smaller primitive cell for a centred pattern, such as the diamond-shaped cell outlined at the lower right of 3.5c. However, this is not normally done, partly because such a cell is a less convenient shape, but more importantly because its axes no longer bear the correct relationship to the symmetry elements of the pattern.


 
Fig. 3.5. Non-translational and translational symmetry elements and their combination. (a) Mirror planes, (b) glide planes, (c) parallel mirror and glide planes, producing a centred pattern. The dashed lines outline a possible (but inconvenient) primitive cell.
\begin{figure}
\includegraphics {fig3.5.ps}
\end{figure}


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Next: 4. Space Groups Up: Symmetry Previous: 2. Combination of Symmetry Elements in

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