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As discussed earlier (see Compatibility of Rotations and Lattices), the seven 3-d crystal systems generate seven distinct unit cell shapes. Of these, six are designated as primitive lattices, which are symbolized “P”. One of the trigonal lattices receives another symbol. Valid centering options of the six primitive cells include body- (“I”), all face- (“F”), and base- (one-face; “A”, “B” or “C”) centering. The symbols for the various base-centered lattices indicate which sets of parallel faces of the primitive cell are centered: “A” corresponds to bc-faces at (b+c)/2; “B” corresponds to ac-faces at (a+c)/2; “C” corresponds to ab-faces at (a+b)/2.
| Unit Cell: |  |  |  |  |  |  |
|---|---|---|---|---|---|---|
| Lattice Type: | Primitive | Base-centered | Face-centered | Body-centered | ||
| Symbol | P | A | B | C | F | I |
| # Points: | 1 | 2 | 2 | 2 | 4 | 2 |
| Points: | [000] | [000], [0½½] | [000], [½0½] | [000], [½½0] | [000], [0½½], [½0½], [½½0] | [000], [0½½] |
As these diagrams illustrate, the base- and body-centered unit cells contain 2 lattice points, whereas all face-centered cells contain 4 lattice points. These are geometrical descriptions of centered lattices, but there are also arithmetical descriptions which make use of modular arithmetic using the lattice points:
- P-lattices = \(\left\{ \lbrack m\ n\ p\rbrack:m,n,p{\text\ may\ be}\ \mathbf{even}\ \text{or}\ \mathbf{odd}\ \text{integers} \right\}\)
- C-lattices = \(\left\{ \left\lbrack \frac{m}{2}{\ \frac{n}{2}\ }\frac{p}{2} \right\rbrack:m\ \text{and}\ n\ \text{are\ both}\ \mathbf{even}\ \text{or\ both}\ \mathbf{odd}\ \text{integers,}p\ \text{must\ be\ an}\ \mathbf{even}\ \text{integer} \right\}\)
- I-lattices = \(\left\{ \left\lbrack \frac{m}{2}{\ \frac{n}{2}\ }\frac{p}{2} \right\rbrack:m,\ n,\ \text{and}\ p\ \text{are\ all}\ \mathbf{even}\ \text{or\ all}\ \mathbf{odd}\ \text{integers} \right\}\)
- F-lattices = \(\left\{ \left\lbrack \frac{m}{2}{\ \frac{n}{2}\ }\frac{p}{2} \right\rbrack:m + n + p\ \text{must\ be\ an}\ \mathbf{even}\ \text{integer} \right\}\)
In this arithmetical description, the centered lattices are described by sets of lattice points based on half-integers with specific restrictions.
The trigonal unit cell with three equal sides and interior angles creates the rhombohedral lattice, symbolized by “R” and viewed in two perspectives to the right. The unit cell shape describes a primitive cell with 1 lattice point (green shaded region). The sum of the three primitive lattice vectors a + b + c is parallel to the C3 axis. Two other vectors, a ‒ b and b ‒ c, are perpendicular to the C3 axis and the angle between them is 120°. Therefore, the standard rhombohedral unit cell is a′ = a ‒ b, b′ = b ‒ c, c′ = a + b + c, so that a′ = b′ ≠ c′ and α′ = β′ = 90º, γ′ = 120º and contains 3 lattice points.
By considering centering in 3-d lattices, there are 14 distinct 3-d Bravais lattices, which are listed below. For each crystal class, not all centering options are possible because any centering must maintain the rotational symmetry of the system.
| Crystal System | Lattice Symmetry | Primitive Unit Cell Shape | Lattice Type | |
|---|---|---|---|---|
| (1) | Triclinic | \mathcal{C}_{i} | a ≠ b ≠ c; α ≠ β ≠ γ | P |
| (2) | Monoclinic | \mathcal{C}_{2v} – one C_{2} axis | a ≠ b ≠ c; α = γ = 90°, β ≠ 90° | P |
| (3) | a1 = a2 ≠ a3; α1 = α2 = 90°, α3 ≠ 90° | C | ||
| (4) | Orthorhombic | \mathcal{D}_{2h} – three ⊥ C_{2} axes | a ≠ b ≠ c; α = β = γ = 90° | P |
| (5) | a1 = a2 = a3; α1 ≠ α2 ≠ α3 | I | ||
| (6) | a1 ≠ a2 ≠ a3; α1 ≠ α2 ≠ α3 | F | ||
| (7) | a1 = a2 ≠ a3; α1 = α2 = 90°, α3 ≠ 90° | C | ||
| (8) | Tetragonal | \mathcal{D}_{4h} – one C_{4} axis | a = b ≠ c; α = β = γ = 90° | P |
| (9) | a1 = a2 ≠ a3; α1 = α2 ≠ α3 | I | ||
| (10) | Cubic | \mathcal{O}_{h} – four C_{3} axes | a = b = c; α = β = γ = 90° | P |
| (11) | a1 = a2 = a3; α1 = α2 = α3 = 109.5° | I | ||
| (12) | a1 = a2 = a3; α1 = α2 = α3 = 60° | F | ||
| (13) | Hexagonal | \mathcal{D}_{6h} – one C_{6} axis | a = b ≠ c; α = β = 90°, γ = 120° | P |
| Trigonal | \mathcal{D}_{3d} – one C_{3} axis | a = b ≠ c; α = β = 90°, γ = 120° | P | |
| (14) | a1 = a2 = a3; α1 = α2 = α3 ≠ 90° | R |
The cubic system allows just primitive (P), all face- (F) and body-centered (I) lattices. Base-centering destroys cubic rotational symmetry and is not a valid centering option for this system. For the face- and body-centered lattices, the corresponding primitive cells are indicated by red edges in the figure below. Each of the primitive cells is a rhombohedral primitive cell (\(a_{1} = a_{2} = a_{3}\) and \(\alpha_{1} = \alpha_{2} = \alpha_{3})\) but with special interior angles: the P-lattice cell has 90° interior angles; the F-lattice primitive cell has 60° interior angles; and the I-lattice primitive cell has 109.5° interior angles. Moreover, the volume of the F-lattice primitive cell is ¼ the volume of the cubic cell; the volume of the I-lattice primitive cell is ½ the volume of the cubic cell. In general, the volume of a centered cell with N lattice points is N times the volume of the corresponding primitive cell.
