Anthony R. West - Solid State Chemistry and its Applications

b The inversion axis, does not have a separate graphical symbol other than that of the mirror plane equivalent to it.

Table 1.29 The thirty‐two point groups

Crystal system	Point group
Triclinic	1,
Monoclinic	2, m , 2/ m
Orthorhombic	222, mm 2, mmm
Tetragonal	4, , 4/ m , 422, 4mm , 2 m , 4/ mmm
Trigonal	3, , 32, 3 m , m
Hexagonal	6, , 6/ m , 622, 6 mm , m 2, 6/ mmm
Cubic	23, m 3, 432, m , m 3 m

Point groups are represented graphically as stereographic projections . These are used a lot, especially in geology and mineralogy, to represent, in 3D, the directions in crystals and to show the relative orientations of crystal faces. To construct a stereographic projection, the different symmetry elements in a point group are encapsulated within a sphere, which becomes a circle in the projection. Usually, one of the rotation or inversion axes of the point group is arranged to be perpendicular to the plane of the circle and passes through its centre. Each point group is represented by two diagrams: the right hand one shows the symmetry elements; the left hand one shows the equivalent positions that are generated by the symmetry operations.

A simple point group that has only one symmetry element is the monoclinic point group 2, which consists of a single twofold rotation axis. It is shown as a stereographic projection in Fig. 1.52(b). The lens‐shaped symbol in the centre of the circle represents the 2‐fold rotation axis perpendicular to the plane of the circle and which passes through the centre of the sphere that the projection represents. The thin vertical line is a construction line; its significance can be seen in the companion drawing, (a) which shows the equivalent positions generated by the 2‐fold rotational symmetry. We know that, if an object possesses a 2‐fold rotation axis, it can be rotated by 180º about that axis to arrive at a position that is indistinguishable from the original position. This indistinguishable position or identical orientation is known, crystallographically, as an equivalent position and therefore, we can say that a 2‐fold rotation axis generates two equivalent positions. In (a), if our original position is shown as the small dot symbol, 1, that is above the plane of the stereographic projection, the 2‐fold axis generates an equivalent position at 2 which is also above the plane. On continuing with the rotation operation by a further 180º, position 2 moves around the circle to arrive back at starting position 1; hence, 1 and 2 are the two equivalent positions in this point group.

In Fig. 1.53(c, d), the trigonal point group, 3 is shown. This has a threefold axis perpendicular to the plane of the stereographic projection (d) and is represented by the solid triangle. Three equivalent positions are generated by the operation steps that involve rotation by 120°, either clockwise or anticlockwise, as shown in (c). Point groups 4 and 6 use the same principles as described for point groups 2 and 3 and are illustrated in Appendix E.

Figure 1.53 The point groups (a, b) 2, (c, d) 3, and (e–h) m.

Figure 1.54 The point groups (a) and (b, c) .

Monoclinic point group m is shown in Fig. 1.53(e, f) and in another orientation in (g, h). This has a single mirror plane which lies in the plane of the projection in (f) and is represented as a thick circle. Equivalent positions are generated by reflection across the mirror plane and hence, our starting position 1, above the plane generates an equivalent position 2, represented by an open circle, directly underneath the plane (e). In (h) the same point group, m , is shown but oriented vertically and perpendicular to the plane of the circle. It is represented by the thick line that bisects the projection that is shown; in this orientation, the equivalent positions are either side of the mirror and both are shown above the plane, (g); equally, they could both be below the plane.

The centre of symmetry in the point group, is shown in Fig. 1.54(a). It does not have a symbolic representation and hence, only one diagram showing the equivalent positions is given. Recall that a centre of symmetry represents two identical positions that are equidistant from the centre of an object and lie on a straight line passing through its centre. This generates the two equivalent positions shown in (a): 1 lies above the plane; 2 lies on a straight line that passes through the centre, to an equal distance the other side and therefore, is below the plane.

The inversion axes, spoken as: bar n, i.e. , , and are combined symmetry operations involving the rotation component, and inversion through the centre, as in . The trigonal point group is shown in Fig. 1.54(b, c). The symbol for , in (c), is a solid triangle with an open circular centre. Six equivalent positions (b) are generated, 1–6, before finally returning to starting position 1. Point groups , and are shown in Appendix E; readers may like to confirm for themselves that point group is equivalent to a mirror plane perpendicular to the 2‐fold inversion axis.