Bravais Lattices in 3 Dimensions
The 14 Bravais lattices in 3 dimensions are obtained by coupling one of the 7 lattice systems (or axial systems) with one of the lattice centerings. Each Bravais lattice refers to a distinct lattice type.
The lattice centerings are:
- Primitive (P): lattice points on the cell corners only.
- Body (I): one additional lattice point at the center of the cell.
- Face (F): one additional lattice point at center of each of the faces of the cell.
- Base (A, B or C): one additional lattice point at the center of each of one pair of the cell faces.
Not all combinations of the crystal systems and lattice centerings are needed to describe the possible lattices. There are in total 7 × 6 = 42 combinations, but it can be shown that several of these are in fact equivalent to each other. For example, the monoclinic I lattice can be described by a monoclinic C lattice by different choice of crystal axes. Similarly, all A- or B-centred lattices can be described either by a C- or P-centering. This reduces the number of combinations to 14 conventional Bravais lattices, shown in the table below.
| The 7 lattice systems | The 14 Bravais lattices | |||
| Triclinic | P | |||
| Monoclinic | P | C | ||
| Orthorhombic | P | C | I | F |
| Tetragonal | P | I | ||
| Rhombohedral |
P | |||
| Hexagonal | P | |||
| Cubic |
P (pcc) | I (bcc) | F (fcc) | |
The volume of the unit cell can be calculated by evaluating a · b × c where a, b, and c are the lattice vectors. The volumes of the Bravais lattices are given below:
| Lattice system | Volume | |||
| Triclinic | ||||
| Monoclinic | ||||
| Orthorhombic | ||||
| Tetragonal | ||||
| Rhombohedral | ||||
| Hexagonal | ||||
| Cubic | ||||
Centred Unit Cells :
| Crystal System | Possible Variations | Axial Distances (edge lengths) | Axial Angles | Examples |
|---|---|---|---|---|
| Cubic | Primitive, Body-centred, Face-centred | a = b = c | α = β = γ = 90° | NaCl, Zinc Blende, Cu |
| Tetragonal | Primitive, Body-centred | a = b ≠ c | α = β = γ = 90° | White tin, SnO2, TiO2, CaSO4 |
| Orthorhombic | Primitive, Body-centred, Face-centred, Base-centred | a ≠ b ≠ c | α = β = γ = 90° | Rhombic sulphur, KNO3, BaSO4 |
| Hexagonal | Primitive | a = b ≠ c | α = β = 90°, γ = 120° | Graphite, ZnO, CdS |
| Rhombohedral | Primitive | a = b = c | α = β = γ ≠ 90° | Calcite (CaCO3, Cinnabar (HgS) |
| Monoclinic | Primitive, Base-centred | a ≠ b ≠ c | α = γ = 90°, β ≠ 90° | Monoclinic sulphur, Na2SO4.10H2O |
| Triclinic | Primitive | a ≠ b ≠ c | α ≠ β ≠ γ ≠ 90° | K2Cr2O7, CuSO4.5H2O, H3BO3 |
Read more about this topic: Bravais Lattice
Famous quotes containing the word dimensions:
“The truth is that a Pigmy and a Patagonian, a Mouse and a Mammoth, derive their dimensions from the same nutritive juices.... [A]ll the manna of heaven would never raise the Mouse to the bulk of the Mammoth.”
—Thomas Jefferson (1743–1826)