Earth Materials. John O'Brien

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Earth Materials - John  O'Brien


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      Square unit meshes (Figure 4.9a) are primitive and have equal unit translation vectors at 90° angles to each other (p, ta = tb, γ = 90°). Primitive rectangular unit meshes (Figure 4.9b) differ in that, although the unit translation vectors intersect at right angles, they are of unequal lengths (p, ta ≠ tb, γ = 90°). Diamond unit meshes have equal unit translation vectors that intersect at angles other than 60°, 90° or 120°. Diamond lattices can be produced and represented by primitive diamond unit meshes (p, ta = tb, γ ≠ 60°, 90° or 120°). They can also be produced by the translation of centered rectangular unit meshes (Figure 4.9c) in which the two unit mesh sides are unequal, the angle between them is 90°, and there is a second node in the center of the mesh (c, ta ≠ tb, γ = 90°). In a centered rectangular mesh there is a total content of two nodes = two motifs. If one looks closely, one may see evidence for glide reflection in the centered rectangular mesh and/or the larger diamond lattice. The hexagonal unit mesh (Figure 4.9d) is a special form of the primitive diamond mesh because, although the unit translation vectors are equal, the angles between them are 60° and 120° (p, ta = tb, γ = 120°). Rotation through 120o generates three such unit meshes which combine to produce a larger pattern with hexagonal symmetry. Oblique unit meshes (Figure 4.9e) are primitive and are characterized by unequal unit translation vectors that intersect at angles that are not 90°, 60° or 120° (p, ta ≠ tb, γ ≠ 90°, 60° or 120°) and produce the least regular, least symmetrical two‐dimensional lattices. The arrays of nodes on planes within minerals always correspond to one of these basic patterns.

      4.3.3 Plane lattice groups

Schematic illustration of 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.

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.

      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.

      4.4.1 Space point groups


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