Crystallography and Crystal Defects. Anthony Kelly

space (plane) group.

Table 2.4 Two‐dimensional lattices, point groups and space groups

System and lattice symbol	Point group	Space group symbols		Space group number
		Full	Short
Parallelogram (oblique) p (primitive)	1 2	p1 p211	p1 p2	1 2
Rectangular p and c (centred)	m	p1m1 p1g1 c1m1	pm pg cm	3 4 5
	2mm	p2mm p2mg p2gg c2mm	pmm pmg pgg cmm	6 7 8 9
Square p	4 4mm	p4 p4mm p4gm	p4 p4m p4g	10 11 12
Triequiangular (hexagonal) p	3 3m	p3 p3m1 p31m	p3 p3m1 p31m	13 14 15
	6 6mm	p6 p6mm	p6 p6m	16 17

      Note: The two distinct space groups p3m1 and p31m correspond to different orientations of the point group relative to the lattice. This does not lead to distinct groups in any other case.

      Below each of the diagrams in Figure 2.24, the equivalent general positions and special positions are also indicated. Special positions are positions located on at least one symmetry operator so that repetition of an initial point produces fewer equivalent positions than in the general case. The symmetry at each special position is also given.

      The group p1 is obtained by combining the parallelogram net and a onefold axis of rotational symmetry. There are no special positions in the cell. The group p2 arises by combining the parallelogram net and a diad. A mirror plane requires the rectangular net (Section 1.5) and if this net is combined with a single mirror, the space group pm, No. 3, results. Points and are in mirror relationship to one another. If the mirror in pm is replaced by a glide reflection line, the space group pg results. In No. 4 the glide reflection line g runs normal to the x‐axis. The centred rectangular lattice necessarily shows a glide reflection line as in cm, No. 5, but only one of these needs to be be present, corresponding to the plane point group m. Since, in this case, the net is multiply primitive, the motif associated with the lattice point at (,) is the same as that at (0, 0). Hence the coordinates of equivalent positions, if added to (,), give additional equivalent positions.

      If two mirror planes at right angles (point group 2mm) are combined with the rectangular lattice we get diads at the intersections of the mirrors as in pmm, No. 6. If one or both of the mirrors is replaced by g, the diads no longer lie at the intersections (see pmg, No. 7 and pgg, No. 8). The group cmm, No. 9, necessarily involves the presence of two sets of glide reflection lines, while p4, No. 10, denotes the square lattice and point group 4, which together necessarily involve the presence of diads. However, mirror planes are not required. If 4 lies at the intersection of two sets of mirrors we have p4m, No. 11, the diagonal glide reflection line necessarily being present. However, 4 can also lie at the intersection of two sets of glide reflection lines, in which case again two sets of mirrors arise but the mirrors intersect in diads, giving point group symmetry mm at these points (No. 12). The triequiangular net and point group 3 give the space group p3 (No. 13). If mirror planes are combined with the triad axis – the combination of the point group 3m and the triequiangular net – it is found that the mirrors can be arranged in two different ways with respect to the points of the net, yielding p31m and p3m1 (Nos. 14 and 15). With the hexagonal point group 6mm, which necessarily has two sets of mirrors, this duality does not arise and the two space groups are p6 and p6m (Nos. 16 and 17).

      Subtle differences in the nomenclature used here for the two‐dimensional space groups and that used in the various editions of the International Tables for Crystallography, such as choosing to describe space group No. 7 as p2mg rather than pmg, reflect the fact that for this two‐dimensional space group the presence of a mirror plane at right angle to a glide plane generates the diad. Similarly, in space group No. 17, the presence of one set of mirrors and the sixfold axes generates the second set of mirrors, and so the space group can be described either as p6m or p6mm (Table 2.4).

2.14 Nomenclature for Point Groups and Space Groups

The nomenclature we have introduced in this chapter to describe point groups and space groups conforms to a notation known as the Hermann–Mauguin notation arising from the work of Carl Hermann [14] and Charles‐Victor Mauguin [15] and used in the Internationale Tabellen zur Bestimmung von Kristallstrukturen, edited by Hermann and published in 1935. However, another notation for describing point group and space group symmetry is also in common use. This second notation, known as the Schoenflies notation, is named after Arthur Moritz Schoenflies [13]. This latter notation is used in particular in physical chemistry in connection with molecular symmetry and molecular spectroscopy [16], and it is also useful when considering group theoretical aspects of point groups and space groups [17]. The equivalence between the Hermann–Mauguin notation and the Schoenflies notation is discussed in detail by Hans Burzlaff and Helmuth Zimmermann in chapter 3.3 of [3]. The reason why the Hermann–Mauguin notation might be preferred over the Schoenflies notation is neatly summarized by Phillips [2]: ‘each space group symbol [in the Hermann–Mauguin notation] conveys all the essential information, whereas we must always have a catalogue at hand to follow an arbitrary serial numbering such as that of Schoenflies’ (p. 340).

2.15 Groups, Subgroups, and Supergroups

      It