Bravais lattices
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In geometry and crystallography, a Bravais lattice, named after Auguste Bravais,[1] 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. Related to Bravais lattices are Crystallographic point groups of which there are 32 and Space groups of which there are 230.

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Development of the Bravais lattices

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

The lattice centerings are:

  • Primitive centering (P): lattice points on the cell corners only
  • Body centered (I): one additional lattice point at the centre of the cell
  • Face centered (F): one additional lattice point at centre 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 7times6 = 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.

The 7 Crystal systems The 14 Bravais lattices
triclinic P
Triclinic
monoclinic P C
Monoclinic, simple Monoclinic, centered
orthorhombic P C I F
Orthohombic, simple Orthohombic, base-centered Orthohombic, body-centered Orthohombic, face-centered
tetragonal P I
Tetragonal, simple Tetragonal, body-centered
rhombohedral
(trigonal)		 P
Rhombohedral
hexagonal A
Hexagonal
cubic
 P (pcc) I (bcc) F (fcc)
Cubic, simple Cubic, body-centered Cubic, face-centered


The volume of the unit cell can be calculated by evaluating \mathbf{a} \cdot \mathbf{b} \times \mathbf{c} where \mathbf{a}, \mathbf{b}, and \mathbf{c} are the lattice vectors. The volumes of the Bravais lattices are given below:

Crystal system Volume
Triclinic abc \sqrt{1-\cos^2\alpha-\cos^2\beta-\cos^2\gamma+2\cos\alpha \cos\beta \cos\gamma}
Monoclinic abcsinα
Orthorhombic abc
Tetragonal a2c
Rhombohedral a^3 \sqrt{1 - 3\cos^2\alpha + 2\cos^3\alpha}
Hexagonal \frac{3\sqrt{3\,}\, a^2c}{2}
Cubic a3


Bravais lattices in 2D

In two dimensions, there are five Bravais lattices. They are oblique, rectangular, centered rectangular, hexagonal, and square.[2]

The five fundamental two-dimensional Bravais lattices: 1 oblique, 2 rectangular, 3 centered rectangular, 4 hexagonal, and 5 square
The five fundamental two-dimensional Bravais lattices: 1 oblique, 2 rectangular, 3 centered rectangular, 4 hexagonal, and 5 square

Bravais lattices in 4D

In four dimensions, there are 52 Bravais lattices. Of these, 21 are primitive and 31 are centered.[3]

See also

References

  1. ^ Aroyo, Mois I.; Ulrich Müller and Hans Wondratschek (2006). "Historical Introduction". International Tables for Crystallography A1 (1.1): 2–5. Springer. doi:10.1107/97809553602060000537. Retrieved on 2008-04-21. 
  2. ^ Kittel, Charles [1953] (1996). "Chapter 1", Introduction to Solid State Physics, Seventh Edition (in English), New York: John Wiley & Sons, 10. ISBN 0-471-11181-3. Retrieved on 2008-04-21. 
  3. ^ Mackay AL and Pawley GS (1963). "Bravais Lattices in Four-dimensional Space". Acta. cryst. 16: 11–19. doi:10.1107/S0365110X63000037. 

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