Bravais lattice

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In geometry and crystallography, a Bravais lattice 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

The 14 Bravais lattices are arrived at by combining one of the 7 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 7x6=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
	Image:Triclinic.png
monoclinic	P	C
	Image:Monoclinic.png	Image:Monoclinic-base-centered.png
orthorhombic	P	C	I	F
	Image:Orthorhombic.png	Image:Orthorhombic-base-centered.png	Image:Orthorhombic-body-centered.png	Image:Orthorhombic-face-centered.png
tetragonal	P	I
	Image:Tetragonal.png	Image:Tetragonal-body-centered.png
rhombohedral (trigonal)	P
	Image:Rhombohedral.png
hexagonal	P
	Image:Hexagonal.png
cubic	P	I	F
	Image:Cubic crystal shape.png	Image:Cubic-body-centered.png	Image:Cubic-face-centered.png

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

Crystal system	Volume
Triclinic	<math>abc \sqrt{1-Cos^2\alpha-Cos^2\beta-Cos^2\gamma+2Cos\alpha Cos\beta Cos\gamma}</math>
Monoclinic	<math>abc Sin\beta</math>
Orthorhombic	<math> abc </math>
Tetragonal	<math> a^2c </math>
Rhombohedral	<math> a^3 \sqrt{1-Cos^2\alpha-Cos^2\beta-Cos^2\gamma+2Cos\alpha Cos\beta Cos\gamma} </math>
Hexagonal	<math>\frac{\sqrt3 a^2c}{2}</math>
Cubic	<math> a^3</math>

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See also

• translational symmetry
• lattice (group)
• classification of latticesde:Bravais-Gitter

fr:Réseau de Bravais he:סריג ברווי nl:Bravaistralie ru:Решётка Браве