Statistical mechanics, structure of matter Definition

Bravais lattice

Definition

Let $n \ge 1$ , in physics usually $n \in \set{1, 2, 3}$ . A Bravais lattice in $\R^n$ is a set of the form $\Lambda = \set{m_1 a_1 + \cdots + m_n a_n : m_1, \dots, m_n \in \Z},$ where $a_1, \dots, a_n$ are linearly independent vectors of $\R^n$ , called primitive vectors. Equivalently, it is a lattice in $\R^n$ : a discrete subgroup of $(\R^n, +)$ that spans $\R^n$ . The parallelepiped $\set{t_1 a_1 + \cdots + t_n a_n : 0 \le t_i < 1}$ is a primitive cell; its volume $\abs{\det(a_1, \dots, a_n)}$ does not depend on the choice of primitive vectors.

Remarks

Example: the face-centred cubic lattice is the Bravais lattice with primitive vectors $a_1 = \tfrac{a}{2}(0, 1, 1)$ , $a_2 = \tfrac{a}{2}(1, 0, 1)$ , $a_3 = \tfrac{a}{2}(1, 1, 0)$ for some $a > 0$ . These vectors are independent, since $\det(a_1, a_2, a_3) = a^3/4 \ne 0$ , so the primitive cell has volume $a^3/4$ .

Non-example: the vertices of the honeycomb tiling of the plane do not form a Bravais lattice. The set is not invariant under its own difference vectors. If $x$ and $y$ are neighbouring vertices, the translation by $y - x$ maps $x$ to a vertex but maps $y$ to the centre of a hexagon. The honeycomb is a triangular Bravais lattice with a basis of two points.

Bravais lattices are grouped into types by their point symmetry and centring. There are $5$ types in dimension $2$ and $14$ types in dimension $3$ .

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Lattice (group) concept
Bravais lattice definition

Tag	`32DNGJM4TM`
URI	`m2://32DNGJM4TM`
Kind	definition
Writing	W0 stub
MSC	82D25, 74E15, 11H06
Defines	Bravais lattice
Uses	Lattice (group)
Changed	3 November 2026

Bravais lattice

Definition

Remarks

Reading path

Further reading

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