Solid State Chemistry and its Applications. Anthony R. West

two elements. The choice of order of the symmetry elements is again immaterial; any two out of the three, in combination, will generate the third element.

      Orthorhombic point group mmm, Fig. 1.55, contains, as essential symmetry elements, three mirror planes mutually perpendicular to each other; as a consequence of the mirror planes, three mutually perpendicular 2‐fold axes are generated. (Note: the reverse process does NOT occur; the three 2‐fold axes in 222 do not lead to the automatic generation of mirror planes) These symmetry elements are shown in (f) with the 8 equivalent positions in (e). In this case, all three mirror planes are essential for the point group mmm.

Only three orthorhombic point groups are possible. If other combinations of 2‐fold axes and mirror planes are considered, they will turn out to be equivalent to one of the three allowed point groups. For instance, the combination 22m can be shown to yield the same set of positions and symmetries as mmm. Of the three orthorhombic point groups, only mmm possesses a centre of symmetry (the equivalent positions created by a centre of symmetry are shown in Fig. 1.54(a) for the triclinic point group ). Each position in mmm has a centrosymmetrically related partner, but this is not the case for 222 and mm2. The relevance of centrosymmetry is discussed in Section 1.18.4.

       Figure 1.56 Equivalent positions in the point group 222. In step 1, a 2‐fold axis perpendicular to the plane of the paper is added. In step 2, a second 2‐fold axis, running horizontally in the plane of the paper is added. In step 3, a third 2‐fold axis, running vertically in the plane of the paper has been added but is created automatically by step 2.

      As a final example, consider the trigonal point group 32 which is characterised by a single 3‐fold axis with a perpendicular 2‐fold axis, Fig. 1.57. The 3‐fold axis is oriented perpendicular to the plane of the paper (b). There are three 2‐fold axes lying in the plane of the paper and at 60° to each other, but only one of these is independent. To demonstrate this and find the equivalent positions, start with position 1 and consider the effect of the 3‐fold axis (rotation by 120°). Positions 3 and 5 result (a). Then consider the effect of one of the 2‐fold axes, say XX′ in (b). This generates three new positions: 1 → 4, 3 → 2 and 5 → 6; also, two more 2‐fold axes YY′ and ZZ′ are automatically generated, e.g. YY′ relates positions 1 and 6, 2 and 5, 3 and 4.

Of the 32 crystallographic point groups, 27 are non‐cubic and we have looked at 9 of these. The remaining 18, Appendix E, can be treated along similar lines and should cause no problem for the reader. The main difficulty likely to be encountered concerns the orientation of the different symmetry elements in a point group. Some guidelines are as follows: (i) in monoclinic, hexagonal, trigonal and tetragonal point groups, the unique axis is shown perpendicular to the plane of the paper (stereogram); (ii) an oblique line, as in 4/mmm, indicates that, in this case, the 4‐fold axis has a mirror plane perpendicular to it: it would be clearer if the symbol were written as (4/m)mm; (iii) in tetragonal, trigonal and hexagonal point groups, the 2‐fold axes, as in 2m, are always in the plane perpendicular to the unique axis (i.e. perpendicular to ).

       Figure 1.57 Trigonal point group 32.

      The 5 cubic point groups are rather more complicated to work with as they are difficult to represent by simple 2D projections. This is because (i) there are so many symmetry elements present and more importantly, (ii) many are not perpendicular to each other. Whereas for non‐cubic point groups, the symmetry axes are either in the plane or perpendicular to the plane of the stereograms, this is not generally possible for cubic point groups and oblique projections are needed to represent the 3‐fold axes. No further discussion of cubic point groups, which are shown in Appendix E, is given.

      1.18.3 Point symmetry of molecules: general and special positions

The relationships between point symmetry and structure are best illustrated by examples of small molecules. The methylene dichloride molecule, CH₂Cl₂, Fig. 1.58, possesses (i) a single 2‐fold axis which bisects the H–С–H and Cl–С–Cl bond angles and (ii) two mirror planes (a). The 2‐fold axis is parallel to the line of intersection of the mirror planes. These symmetry elements may be represented as in (b), in which the 2‐fold axis is perpendicular to the plane of the paper and the mirror planes appear in projection as horizontal and vertical lines. From inspection of Fig. 1.55, it is seen that CH₂C1₂ belongs to point group mm2. However, the number of equivalent positions in mm2 is four (c), which does not appear to tally with the realities of the CH₂CI₂ molecule. Thus, if we place a H atom at one equivalent position, there should be only two possible equivalent positions in the molecule, as there are only two H atoms present. This apparent anomaly is resolved by letting the equivalent positions in Fig. 1.55(c) lie on the vertical mirror plane instead of to either side of the mirror. This yields the arrangement shown in Fig. 1.58(c), which now has only two equivalent positions. Thus, positions 1 and 2 in Fig. 1.55(c) become the single position A in Fig. 1.58(c). We can now distinguish between the general equivalent positions of a point group and the special equivalent positions; the latter arise when the general positions lie on a symmetry element such as a mirror plane or rotation axis. Thus, A and В in Fig. 1.58(c) are both special positions.

       Figure 1.58 The symmetry of the methylene dichloride molecule, CH2Cl2, point group mm2.

       Figure 1.59 The symmetry of the methyl chloride molecule, CH3Cl, point group 3m.

      As a second example, the point symmetry of the methyl chloride molecule, CH₃CI is shown in Fig. 1.59. The molecule possesses one 3‐fold axis along the direction of the С–Cl bond (a). It has no 2‐fold axes but has three mirror planes oriented at 60° to each other; one is shown in (b). The 3‐fold axis coincides with the line of intersection of the mirror planes. The symmetry elements are shown as a stereogram in (c) and by comparison with Appendix E, we see that the point group is 3m. The six general equivalent positions in 3m are given in (d). We again have the problem that there are more equivalent positions than possible atoms and this is overcome by allowing the general positions to lie on the mirror planes (e); the number of positions is thereby reduced to three.

      1.18.4 Centrosymmetric and non‐centrosymmetric point groups

      Of the 32 point groups, 21 do not possess a centre of symmetry. The absence of a centre of symmetry is an essential but not sufficient requirement for the presence in crystals of optical activity, pyroelectricity and piezoelectricity (Chapters