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As noted earlier, any motif can be represented by a point called a node. Nodes, and the motifs they represent, can also be translated in three directions (t_a, t_b, and t_c) to produce three‐dimensional space point lattices and unit cells (Figure 4.10).

Unit cells are the three‐dimensional analogs of unit meshes. A unit cell is a parallelepiped whose edge lengths and volume are defined by the three unit translation vectors (t_a, t_b, and t_c). The unit cell is the smallest unit that contains all the information necessary to reproduce the mineral by three‐dimensional symmetry operations. Unit cells may be primitive (P), in which case they have nodes only at their corners and a total content of one node (=one motif). Non‐primitive cells are multiple because they contain extra nodes in one or more faces (A, B, C or F) or in their centers (I) and possess a total unit cell content of more than one node or motif.

Unit cells bear a systematic relationship to the coordination polyhedra and packing of atoms that characterize mineral structures, as illustrated by Figure 4.11.

In 1850, Bravais recognized that only 14 basic types of three‐dimensional translational point lattices exist; these are known as the 14 Bravais space point lattices (Klein and Hurlbut 1985) and define 14 basic types of unit cells. The 14 Bravais lattices are distinguished on the basis of (1) the magnitudes of the three unit translation vectors t_a, t_b, and t_c or more simply a, b, and c, (2) the angles (alpha, beta, and gamma) between them, where (α = b Λ c; β = c Λ a; γ = a Λ b), and (3) whether they are primitive lattices or some type of multiple lattice. Figure 4.12 illustrates the 14 Bravais space point lattices.

The translational symmetry of every mineral can be represented by one of the 14 basic types of unit cells. Each unit cell contains one or more nodes that represent motifs and contains all the information necessary to characterize chemical composition. Each unit cell also contains the rules according to which motifs are repeated by translation; the repeat distances, given by t_a = a, t_b = b, t_c = c, and directions, given by angles α, β, and γ. The 14 Bravais lattices can be grouped into crystal systems on the basis of the relative dimensions of the unit cell edges (a, b, and c) and the angles between them (α, β, and γ). These six (or seven if the hexagonal system is divided into trigonal and hexagonal) systems in which all minerals crystallize. These include the isometric (cubic), tetragonal, orthorhombic, monoclinic, triclinic, and hexagonal systems. The latter is subdivided into the hexagonal division or system and the trigonal (rhombohedral) division or system. Table 4.4 summarizes the characteristics of the Bravais lattices in the major crystal systems.

Table 4.3 The six crystal systems and 32 crystal classes, with their characteristic symmetry and crystal forms.

System	Crystal class	Class symmetry	Total symmetry
Isometric	Hexoctahedral		3A4, , 6A2, 9m
Hextetrahedral		, 4A3, 6m
Gyroidal	432	3A4, 4A3, 6A2
Diploidal		3A2, 3m,
Tetaroidal	23	3A2, 4A3
Tetragonal	Ditetragonal–dipyramidal	4/m2/m2/m	i, 1A4, 4A2, 5m
Tetragonal–scalenohedral		, 2A2, 2m
Ditetragonal–pyramidal	4mm	1A4, 4m
Tetragonal–trapezohedral	422	1A4, 4A2
Tetragonal–dipyramidal	4/m	i, 1A4, 1m
Tetragonal–disphenoidal
Tetragonal–pyramidal	4	1A4
Hexagonal(hexagonal)	Dihexagonal–dipyramidal	6/m2/m2/m	i, 1A6, 6A2, 7m
Ditrigonal–dipyramidal	6m2	1A6, 3A2, 3m
Dihexagonal–pyramidal	6mm	1A6, 6m
Hexagonal–trapezohedral	622	1A6, 6A2
Hexagonal–dipyramidal	6/m	i, 1A6, 1m
Trigonal–dipyramidal
Hexagonal–pyramidal	6	1A6
Hexagonal (rhombohedral or trigonal)	Hexagonal–scalenohedral		, 3A2, 3m
Ditrigonal–pyramidal	3m	1A3, 3m
Trigonal–trapezohedral	32	1A3, 3A2
Rhombohedral
Trigonal–pyramidal	3	1A3
Orthorhombic	Rhombic–dipyramidal	2/m2/m2/m	i, 3A2, 3m
Rhombic–pyramidal	mm2	1A2, 2m
Rhombic–disphenoidal	222	3A2
Monoclinic	Prismatic	2/m	i, 1A2, 1m
Sphenoidal	2	1A2
Domatic	m	1m
Triclinic	Pinacoidal		i
Pedial	1	None

Figure 4.11 Relationship between (a) atomic packing, (b) a unit cell, and (c) octahedral coordination polyhedra in halite (NaCl).

Source: Wenk and Bulakh (2016). © Cambridge University Press.

Figure 4.12 The 14 Bravais lattices and the six (or seven) crystal systems they represent.

Source: Courtesy of Steve Dutch.

Table 4.4 Major characteristics of Bravais lattice cells in the major crystal systems.

Crystal system	Unit cell edge lengths	Unit cell edge intersection angles	Bravais lattice types
Isometric (cubic)	(a = b = c) Preferred format for edges of equal length is (a1 = a2 = a3)	α = β = γ = 90°	Primitive (P) Body centered (I) Face centered (F)
Tetragonal	(a1 = a2 ≠ a3) or a = b ≠ c	α = β = γ = 90°	Primitive (P) Body centered (I)
Hexagonal (hexagonal)	(a1 = a2 ≠ c) (a = b ≠ c)	(α = β = 90o ≠ γ = 120°)	Primitive (P)
Hexagonal (trigonal or rhombohedral)	(a1 = a2 = a3)	α = β = γ ≠ 90°	Primitive (P)
Orthorhombic	a ≠ b ≠ c	(α = β = γ = 90°)	Primitive (P) Body centered (I) End centered (A, B, C) Face centered (F)
Monoclinic	a ≠ b ≠ c	(α = γ = 90° ≠ β)	Primitive (P) End centered (C)
Triclinic	a ≠ b ≠ c	(α, β, and γ ≠ 90°)	Primitive (P)