Bravais lattice

A Bravais lattice is defined as an infinite set of points generated by integer linear combinations of a set of primitive translation vectors a, b, and c. Formally, the lattice is the set {R = n₁a + n₂b + n₃c} for integers n_i. This definition ensures that the lattice is closed under addition and subtraction, and that every point has the same coordination environment. The primitive vectors are not unique; any set that generates the same lattice is valid, but the choice affects the shape of the unit cell. The lattice is a mathematical abstraction, distinct from the crystal structure, which includes a basis of atoms attached to each lattice point.

In three dimensions, the 14 Bravais lattices are classified by their point group symmetry and centering type. They fall into seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral (trigonal), hexagonal, and cubic. Each system allows one or more lattice types: primitive (P), body-centered (I), face-centered (F), and base-centered (C). For example, the cubic system includes simple cubic (P), body-centered cubic (I), and face-centered cubic (F), but not base-centered cubic because it would reduce to a simpler tetragonal lattice. The rhombohedral system has only a primitive lattice, while the hexagonal system has only a primitive lattice with a special angle. The classification was derived by Auguste Bravais in 1850, building on earlier work by Moritz Ludwig Frankenheim, who had proposed 15 types; Bravais corrected this to 14 by recognizing that two of Frankenheim's types were equivalent.

In two dimensions, there are exactly five Bravais lattices: oblique, rectangular, centered rectangular, square, and hexagonal. These are used to describe surfaces and two-dimensional materials such as graphene, which has a hexagonal lattice. In higher dimensions, the number of Bravais lattices grows rapidly; for example, there are 64 in four dimensions and 189 in five dimensions, as classified by crystallographers. These higher-dimensional lattices are relevant in the study of quasicrystals and in the context of lattice theory in mathematics. The concept of a Bravais lattice also extends to any dimension, where it is defined as a discrete subgroup of Euclidean space that spans the space.

One lesser-known fact is that the Bravais lattice classification is not limited to crystals; it also applies to magnetic structures, where the lattice points represent magnetic moments, and to photonic crystals, where the periodic variation of refractive index follows a Bravais lattice. Another subtlety is that the choice of primitive vectors is not unique, and the Wigner-Seitz cell, a primitive cell with full point symmetry, is often used to visualize the lattice. The concept of a reciprocal lattice, fundamental in diffraction theory, is built on the Bravais lattice, and the Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice. Additionally, the 14 Bravais lattices are sometimes called the 14 space lattices, and they are distinct from the 230 space groups, which include additional symmetry operations like screw axes and glide planes. The work of Bravais was partly inspired by the earlier observations of René Just Haüy on the cleavage of crystals, and the classification has been extended to include non-Bravais lattices with a basis, which are used to describe complex crystal structures like diamond and zincblende.

Primitive vectors

The three non-coplanar translation vectors that generate a Bravais lattice through integer combinations.

Unit cell

A parallelepiped formed by the primitive vectors, which tiles space by translation.

Crystal system

One of seven categories of lattices based on their point group symmetry.

Centering

The addition of lattice points at the center of faces or body of a unit cell, leading to P, I, F, or C lattices.

Reciprocal lattice

The Fourier transform of a Bravais lattice, used in diffraction analysis.