Solid State Chemistry and its Applications. Anthony R. West

bar comma ModifyingAbove 4 With bar"/> and , and the mirror plane, m (which is equivalent to ). These symmetry elements may occur either alone or in various possible combinations with each other to give a total of 32 possible crystallographic point groups. The method of drawing and labelling point groups used here is the same as that recommended in International Tables for X‐ray Crystallography, Vol 1. The symbols for the different point symmetry elements are given in Table 1.28. The thirty‐two point groups, classified according to their crystal system, are listed in Table 1.29 and shown diagrammatically in Appendix E.

       Table 1.28 Point symmetry elements

Symmetry element	Written symbol	Graphical symbol
Rotation axes	1	None
	2
	3
	4
	6
Inversion axes		None^a _____^b
Mirror plane	m	____

^a The inversion axis, , equivalent to a centre of symmetry, is represented as a small open circle, o, in space groups, but does not have a formal graphical representation in point groups, even though it is present in many point groups.

^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, mm2, mmm
Tetragonal	4, , 4/m, 422, 4mm, 2m, 4/mmm
Trigonal	3, , 32, 3m, m
Hexagonal	6, , 6/m, 622, 6mm, m2, 6/mmm
Cubic	23, m3, 432, m, m3m

      1.18.2 Stereographic projections and equivalent positions

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