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4.3.3 Plane lattice groups
ОглавлениеWhen the ten plane point groups are combined with the five unit meshes in all ways that are compatible, a total of 17 plane lattice groups are recognized on the basis of the total symmetry of their plane lattices. Note that these symmetries involve translation‐free symmetry operations that include rotation and reflection, translation and compound symmetry operations such as glide reflection. Table 4.2 summarizes the 17 plane lattice groups and their symmetries. Primitive lattices are denoted by “P” and centered lattices by “C.” Axes of rotation for the entire pattern perpendicular to the plane are noted by 1, 2, 3, 4, and 6. Mirror planes perpendicular to the plane are denoted by “m”; glide planes perpendicular to the plane are denoted by “g.”
Figure 4.9 The five principal types of meshes or nets and their unit meshes (shaded gray): (a) square, (b) primitive rectangle, (c) diamond or centered rectangle, (d) hexagonal, (e) oblique.
Source: Nesse (2016). © Oxford University Press.
Table 4.2 The 17 plane lattice groups and the unique combination of point group and unit mesh that characterizes each.
| Lattice | Point group | Plane group |
|---|---|---|
| Oblique (P) | 1 | P1 |
| 2 | P2 | |
| Rectangular (P and C) | m | Pm |
| Pg | ||
| Cm | ||
| 2mm | P2mm | |
| P2mg | ||
| P2gg | ||
| C2mm | ||
| Square (P) | 4 | P4 |
| 4mm | P4mm | |
| P4gm | ||
| Hexagonal (P) (rhombohedral) | 3 | P3 |
| 3m | P3m1 | |
| P3lm | ||
| Hexagonal (P) (hexagonal) | 6 | P6 |
| 6mm | P6mm |
The details of plane lattice groups are well documented (see for example, Klein and Dutrow 2007), but are beyond the introductory material in this text.
