In geometry and crystallography, a Bravais lattice, named after Auguste Bravais, is an infinite set of points generated by a set of discrete translation operations. A crystal is made up of one or more atoms (the basis) which is repeated at each lattice point. The crystal then looks the same when viewed from any of the lattice points. In all, there are 14 possible Bravais lattices that fill three-dimensional space.

Development of the Bravais lattices

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The 14 Bravais lattices are arrived at by combining one of the seven crystal systems (or axial systems) with one of the lattice centerings.

The lattice centerings are:

• Primitive centering (P): lattice points on the cell corners only
• Body centered (I): one additional lattice point at the center of the cell
• Face centered (F): one additional lattice point at center of each of the faces of the cell
• Centered on a single face (A, B or C centering): one additional lattice point at the center of one 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-centered 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.

Crystal system	Bravais lattices
triclinic	P
monoclinic	P	C
orthorhombic	P	C	I	F
tetragonal	P	I
rhombohedral (trigonal)	P
hexagonal	P
cubic	P	I	F

The volume of the unit cell can be calculated by evaluating $\backslash underline\; a.\backslash underline\; b\; \backslash times\; \backslash underline\; c$ where $\backslash underline\; a,\; \backslash underline\; b$, and $\backslash underline\; c$ are the lattice vectors, the volumes of the Bravais lattices are given below:

Crystal system	Volume
Triclinic	$abc\; \backslash sqrt\{1-\backslash cos^2\backslash alpha-\backslash cos^2\backslash beta-\backslash cos^2\backslash gamma+2\backslash cos\backslash alpha\; \backslash cos\backslash beta\; \backslash cos\backslash gamma\}$
Monoclinic	$abc\; \backslash sin\backslash beta$
Orthorhombic	$abc$
Tetragonal	$a^2c$
Rhombohedral	$a^3\; \backslash sqrt\{1-\backslash cos^2\backslash alpha-\backslash cos^2\backslash beta-\backslash cos^2\backslash gamma+2\backslash cos\backslash alpha\; \backslash cos\backslash beta\; \backslash cos\backslash gamma\}$
Hexagonal	$\backslash frac\{\backslash sqrt\{3\backslash ,\}\backslash ,\; a^2c\}\{2\}$
Cubic	$a^3$

See also

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• translational symmetry
• lattice (group)
• classification of lattices

Crystallography | Solid state physics

Bravais-Gitter | Réseau de Bravais | סריג ברווי | 브라베이 격자 | Bravaistralie | Решётка Браве