three‐, four‐ and sixfold axes on a single initial pole, represented as a dot
Figure 2.3 The repetition of an object by a mirror plane, (a) and (b), and by a centre of symmetry, (c)
Figure 2.4 The operation of the twofold rotoinversion axis,
Figure 2.5 The operation of the various rotoinversion axes that can occur in crystals
The onefold inversion axis is a centre of symmetry. The operations of the other rotoinversion axes are explained in Figures 2.4 and 2.5. The twofold rotation–inversion axis shown in Figure 2.4 repeats an object by rotation through 180° (360°/2) to give the dotted circle, followed by inversion to give the full circle. Similarly, the threefold inversion axis involves rotation through 360°/3 = 120° coupled with an inversion. In general, an n‐fold rotoinversion axis involves rotation through an angle of 2π/n coupled with inversion through a centre. The rotation and inversion are both part of the operation of repetition and must not be considered as separate operations. The operation of the various rotoinversion axes that can occur in crystals on a single initial pole is shown in Figure 2.5, with the pole of the rotoinversion axis at the centre of the primitive circle. The following symbols are used: inversion monad, , symbol ○; inversion diad, , symbol ⋄; inversion triad, , symbol ; inversion tetrad, , symbol ; inversion hexad, , symbol . Inspection of Figures 2.2–2.4 shows that is identical to a centre of symmetry, is identical to a mirror plane normal to the inversion diad, is identical to a triad axis plus a centre of symmetry and is identical to a triad axis normal to a mirror plane (symbol 3/m, the sign ‘/m’ indicating a mirror plane normal to an axis of symmetry).2 Only is unique. The operation of repetition described by cannot be reproduced by any combination of a proper rotation axis and a mirror plane or a centre of symmetry.
The various different combinations of 1, 2, 3, 4 and 6 pure rotation axes and , , , and rotoinversion axes constitute the 32 crystallographic point groups or crystal classes. These 32 classes are grouped into systems according to the presence of defining symmetry elements (see Table 1.3). Stereograms of each of the 32 crystallographic point groups or crystal classes are given in Figure 2.6, following the current conventions of the International Tables for Crystallography [3]. With the exception of the two triclinic point groups, each point group is depicted by two stereograms. The first stereogram shows how a single initial pole is repeated by the operations of the point group and the second stereogram shows all of the symmetry elements present. The nomenclature for describing the crystal classes is as follows. X indicates a rotation axis and an inversion axis. X/m is a rotation axis normal to a mirror plane, Xm a rotation axis with a mirror plane parallel to it and X2 a rotation axis with a diad normal to it. X/mm indicates a rotation axis with a mirror plane normal to it and another parallel to it. is an inversion axis with a parallel plane of symmetry. A plane of symmetry is an alternative description of a mirror plane.
Figure 2.6 Stereograms of the poles of equivalent general directions and of the symmetry operations of each of the 32 crystallographic point groups. The z‐axis is normal to the paper. In all the centrosymmetric classes, positions of centres of symmetry (inversion monads) lying within the primitive of the stereogram are shown.
Source: Taken from the International Tables for X‐ray Crystallography, Vol. 1 [4] and adapted to conform to current notation for the two centrosymmetric point groups in the cubic crystal system.
We shall describe each of the classes in Sections 2.2–2.8. A derivation of the 32 classes follows by noting from Sections 1.5 and 1.6 that the rotation axes consistent with translational symmetry are 1, 2, 3, 4 and 6. Individually, these give in total five crystal classes.
three‐, four‐ and sixfold axes on a single initial pole, represented as a dot
