Bravais lattice

A Bravais lattice is an infinite set of discrete points laid out in space with a periodic arrangement, such that the arrangement and orientation appears the same everywhere in space.

An alternative definition in NN dimensions goes as follows: a Bravais lattice consists of all points R\mathbf{R} defined by

R=∑i=1Nniai\mathbf{R}=\sum_{i=1}^{N} n_{i}\mathbf{a}_{i}

where n1,…,nN∈Zn_{1},\ldots,n_{N}\in \mathbb{Z} and a1,…,aN\mathbf{a}_{1},\ldots,\mathbf{a}_{N} are a set of appropriate linearly independent vectors called primitive vectors. Each point in the Bravais lattice is then generated by a sequence of translations a1,…,aN\mathbf{a}_{1},\ldots,\mathbf{a}_{N} and n1,…,nNn_{1},\ldots,n_{N} represent the number of those translations to be done to reach a given point. Negative values for n1,…,nNn_{1},\ldots,n_{N} mean translations in the opposite direction.

A Bravais lattice is a mathematical construct that shows how space can be tessellated through translations only. This is why not all sets of linearly independent vectors make for primitive vectors: the shape that they define must fill space completely when translated. For instance, a cube is a valid shape, but a pyramid is not, since it can't fill space without using rotations. Moreover, for a given lattice, the set of primitive vectors is not unique. There are multiple — in fact infinitely many — sets of primitive vectors that can define any given lattice.

When we speak of the "shape" of the lattice, we are technically referring to its primitive unit cell. The primitive cell of a lattice is the space identified by the primitive vectors that, when translated, completely fills space. Each primitive cell must contain precisely one lattice point (equivalently: one reciprocal lattice point). Just like the primitive vectors, the primitive cell of a given lattice is not unique.

It's also possible to define nonprimitive unit cells by using a subset of the primitive vectors, provided that they also completely fill space. These are typically useful when the primitive cells themselves are complicated.

Due to the arbitrary nature of the primitive cell, the choice is mostly a matter of convention and practicality. The most common choice is known as the Wigner-Seitz cell, which is a region of space around a lattice point that is closer to that point than to any other point.

Each point in the lattice has a set of points closest to it, all at the same distance. These are called its nearest neighbors. The number of nearest neighbors is the same for all points, as it is a property of the lattice itself, and it is known as the coordination number of the lattice.

In 2D

Given the definitions of a Bravais lattice, there aren't infinite possible lattices that satisfy them. In fact, in 2D, there's only 5 possible lattices that can be considered Bravais lattices:

  • square,
  • rectangular,
  • centered rectangular (orthorhombic)
  • hexagonal
  • oblique (rhomboidal)

In 3D

In three dimensions, there are 14 possible Bravais lattices:

  • simple cubic (SC)
  • face-centered cubic (FCC)
  • body-centered cubic (BCC)
  • simple hexagonal
  • rhomboedric
  • simple tetragonal
  • body-centered tetragonal
  • orthorhombic
  • base-centered orthorhombic
  • face-centered orthorhombic
  • body-centered orthorhombic
  • simple monoclinic
  • base-centered monoclinic
  • triclinic

A nice visualization tool can be found at CrystalWalk.