Bravais Lattices
A Bravais lattice is one of exactly 14 distinct three-dimensional lattice types — infinite, periodic arrays of points in which every point has an identical surrounding environment — that together provide a complete geometric classification of every possible periodic crystal structure, first worked out mathematically by Auguste Bravais in 1850.
Curious
The Codex's crystal structure entry introduces the idea of a lattice — the repeating geometric skeleton underlying a crystal's atomic arrangement. This entry looks specifically at the complete classification of possible lattices: remarkably, despite the seemingly limitless variety of ways points could in principle be arranged periodically in three dimensions, mathematics restricts the possibilities to exactly 14 distinct types, called the Bravais lattices.
Each Bravais lattice is defined by the shape of its repeating unit cell (the smallest building block that, tiled in three dimensions, reproduces the whole lattice) and where additional lattice points, if any, sit within that cell — at the corners only, or also at the body centre, face centres, or specific edge centres.
Exploring
| Crystal system | Bravais lattices in this system | Example material |
|---|---|---|
| Cubic | Simple, body-centred, face-centred (3 lattices) | Table salt (face-centred), iron (body-centred) |
| Hexagonal | Simple hexagonal (1 lattice) | Graphite, ice |
| Tetragonal | Simple, body-centred (2 lattices) | Tin (white tin) |
| Others (orthorhombic, trigonal, monoclinic, triclinic) | 8 further lattices across 4 systems | Various minerals |
Deep Dive
Auguste Bravais's 1850 classification predates X-ray crystallography (which would not confirm atomic-scale crystal structures experimentally until 1912) by over 60 years — his work was purely a mathematical exercise in enumerating the distinct ways points could be arranged periodically in three-dimensional space consistent with translational symmetry, grouped by the rotational and reflective symmetries each arrangement could support. That his purely mathematical classification later proved to exactly match the geometric possibilities found in real crystals is a striking early example of mathematics anticipating physical reality.
The 7 crystal systems: The 14 Bravais lattices are grouped into 7 crystal systems based on the symmetry of their conventional unit cell: triclinic (least symmetric, all cell angles and edge lengths generally different), monoclinic, orthorhombic, tetragonal, trigonal (rhombohedral), hexagonal, and cubic (most symmetric, all edges equal, all angles 90°). Some systems support multiple distinct Bravais lattices (cubic has 3: simple, body-centred, face-centred; the added lattice points at the body or face centres represent genuinely distinct symmetry, not merely more atoms), while others support only one.
The Bravais lattice classification extends naturally to two dimensions (5 distinct planar lattices) and has been generalised mathematically to higher dimensions as well, though only the three-dimensional case has direct physical application to ordinary crystalline solids. The full space group classification of crystal symmetry (230 distinct space groups in three dimensions, combining the 14 Bravais lattices with additional point-group symmetries) provides an even more complete framework, used throughout modern crystallography and materials science for precisely specifying crystal structures.
Concept Relationship Map
Requires: crystal structure
Enables: reciprocal lattice [PENDING FULL ENTRY], Miller indices [PENDING FULL ENTRY]
Commonly confused with: treating a Bravais lattice as the full crystal structure, rather than the point pattern that must be combined with a basis
Sources
- Bravais, A. (1850). Journal de l'École Polytechnique, 19, 1–128.
- Kittel, C. (2004). Introduction to Solid State Physics, 8th ed. Wiley, Ch. 1.