International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2015). Vol. A, ch. 1.4, pp. 48-49

Section 1.4.1.5. Hermann–Mauguin symbols of the plane groups

H. Wondratscheke

1.4.1.5. Hermann–Mauguin symbols of the plane groups

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The principles of the HM symbols for space groups are retained in the HM symbols for plane groups (also known as wallpaper groups). The rotation axes along c of three dimensions are replaced by rotation points in the ab plane; the possible orders of rotations are the same as in three-dimensional space: 2, 3, 4 and 6. The lattice (sometimes called net) of a plane group is spanned by the two basis vectors a and b, and is designated by a lower-case letter. The choice of a lattice basis, i.e. of a minimal cell, leads to a primitive lattice p, in addition a c-centred lattice is conventionally used. The nets are listed in Table 3.1.2.1[link] . The reflections and glide reflections through planes of the space groups are replaced by reflections and glide reflections through lines. Glide reflections are called g independent of the direction of the glide line. The arrangement of the constituents in the HM symbol is displayed in Table 1.4.1.2[link].

Table 1.4.1.2| top | pdf |
The structure of the Hermann–Mauguin symbols for the plane groups

The positions of the representative symmetry directions for the crystal systems are given. The lattice symbol and the maximal order of rotations around a point are followed by two positions for symmetry directions.

Crystal systemLattice(s) First positionSecond positionThird position
Oblique p   1 or 2
Rectangular p, c 1 or 2 a b
Tetragonal p 4 a ab
Hexagonal p 3 a 1
      or
    3 1 ab
    6 a ab

Short HM symbols are used only if there is at most one symmetry direction, e.g. p411 is replaced by p4 (no symmetry direction), p1m1 is replaced by pm (one symmetry direction) etc.

There are four crystal systems of plane groups, cf. Table 3.2.3.1[link] . The analogue of the triclinic crystal system is called oblique, the analogues of the monoclinic and orthorhombic crystal systems are rectangular. Both have rotations of order 2 at most. The presence of reflection or glide reflection lines in the rectangular crystal system allows one to choose a rectangular basis with one basis vector perpendicular to a symmetry line and one basis vector parallel to it. The square crystal system is analogous to the tetragonal crystal system for space groups by the occurrence of fourfold rotation points and a square net. Plane groups with threefold and sixfold rotation points are united in the hexagonal crystal system with a hexagonal net.

Plane groups occur as sections and projections of the space groups, cf. Section 1.4.5[link]. In order to maintain the relations to the space groups, the symmetry directions of the symmetry lines are determined by their normals, not by the directions of the lines themselves. This is important because the normal of the line, not the direction of the line itself, determines the position in the HM symbol.

  • (1) In oblique plane groups there is no symmetry direction: HM symbols are p1 or p2.

  • (2) Rectangular plane groups may have no rotations and then only one symmetry direction: p1m1 = pm, p1g1 = pg and c1m1 = cm. If there are twofold rotations, the HM symbol starts with p2 or c2, followed by the symmetry m or g first perpendicular to a and then perpendicular to b. The con­ventional HM symbol p2mg describes a plane group with a reflection line running perpendicular to a (parallel to b) and a glide-reflection line running from the back to the front (perpendicular to b and thus parallel to a). There are four plane-group types: p2mm, p2mg, p2gg and c2mm. The constituent `2' was sometimes omitted in older HM symbols.

  • (3) There is one square plane group with only rotations and no symmetry directions, the net is a square net: p411 = p4. The generating symmetry of symmetry directions perpendicular to a and ab are listed in the second and third positions: p4mm with reflection lines perpendicular to a and b and p4gm with glide lines in the same directions. Reflection lines and glide lines perpendicular to ab (and a + b) alternate.

  • (4) Five plane groups belong to the hexagonal crystal system. The trigonal and hexagonal plane groups p311 = p3 and p611 = p6 contain only rotations. In the other trigonal plane groups there is exactly one set of symmetry directions; its representative direction is either perpendicular to a (p3m1) or perpendicular to ab (p31m).

    The HM symbols p3m1 and p31m may be easily confused, although they are different. Apart from the different orientations of their symmetry directions, in a plane group of type p3m1, all rotation points lie on reflection lines, but in p31m not all of them do.

    The hexagonal plane group p6mm displays representative directions of mirror lines perpendicular to a and perpendicular to ab.








































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