Earth Materials. John O'Brien
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Название: Earth Materials
Автор: John O'Brien
Издательство: John Wiley & Sons Limited
Жанр: География
isbn: 9781119512219
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4.3.3 Plane lattice groups
When 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.
4.4 THREE‐DIMENSIONAL MOTIFS AND LATTICES
Minerals are three‐dimensional Earth materials with three‐dimensional crystal lattices. The fundamental units of pattern in any three‐dimensional lattice are three‐dimensional motifs that can be classified according to their translation‐free symmetries. These three‐dimensional equivalents of the two‐dimensional plane point groups are called space point groups.
Space point groups can be represented by nodes. These nodes can be translated to produce three‐dimensional patterns of points called space lattices. Space lattices are the three‐dimensional equivalents of plane nets or meshes. By analogy with unit meshes or nets, we can recognize the smallest three‐dimensional units, called unit cells, which contain all the information necessary to produce the three‐dimensional space lattices. In this section, we will briefly describe the space point groups, after which we will introduce Bravais lattices, unit cells, and their relationship to the six (or seven) major crystal systems to which minerals belong.
Figure 4.10 A primitive unit cell and a long‐range space point lattice that results from its repetition by symmetry operations in three dimensions.
4.4.1 Space point groups
In minerals, the fundamental motifs are parts of clusters of three‐dimensional coordination polyhedra sufficient to establish the composition of the mineral. When these are repeated in three dimensions during mineral growth, they produce the long‐range order characteristic of crystalline substances (Figure 4.10). Like all fundamental units of pattern, these three‐dimensional motifs can be classified on the basis of their translation‐free symmetries.
Only 32 different three‐dimensional motif symmetries exist. These define 32 space point groups, each with unique space point group symmetry. In minerals, the 32 crystal classes – to one of which all minerals belong – correspond СКАЧАТЬ