Bravais lattice


Bravais lattice

In geometry and crystallography, a Bravais lattice, named after Auguste Bravais, [cite journal | last = Aroyo | first = Mois I. | coauthors = Ulrich Müller and Hans Wondratschek | title = Historical Introduction | journal = International Tables for Crystallography | volume = A1 | issue = 1.1 | pages = 2–5 | publisher = Springer | date = 2006 | url = http://it.iucr.org/A1a/ch1o1v0001/sec1o1o1/ | doi = 10.1107/97809553602060000537 | accessdate =2008-04-21 ] 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.

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 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.

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

Bravais lattices in 2D

In two dimensions, there are five Bravais lattices. They are oblique, rectangular, centered rectangular, hexagonal, and square. [cite book |last=Kittel |first=Charles |title=Introduction to Solid State Physics |origyear=1953 |url= http://www.wiley.com/WileyCDA/WileyTitle/productCd-047141526X.html |accessdate=2008-04-21 |edition=Seventh Edition |year=1996 |publisher=John Wiley & Sons |location=New York |isbn=0-471-11181-3 |pages=10 |chapter=Chapter 1]

Bravais lattices in 4D

In four dimensions, there are 52 Bravais lattices. Of these, 21 are primitive and 31 are centered. [cite journal |author=Mackay AL and Pawley GS |title=Bravais Lattices in Four-dimensional Space |journal=Acta. cryst. |volume=16 |pages=11–19 |year=1963 |doi=10.1107/S0365110X63000037]

ee also

*translational symmetry
*lattice (group)
*classification of lattices
*Miller Index

References

External links

* A witty [http://www.haverford.edu/physics-astro/songs/bravais.htm musical mnemonic aid] written and performed by Walter Fox Smith


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Look at other dictionaries:

  • Bravais lattice — noun a 3 dimensional geometric arrangement of the atoms or molecules or ions composing a crystal • Syn: ↑space lattice, ↑crystal lattice • Hypernyms: ↑lattice * * * ˈbraˌvā ; brəˈvā , braˈ noun Usage: usually capitalized B …   Useful english dictionary

  • Bravais lattice — Bravė gardelė statusas T sritis Standartizacija ir metrologija apibrėžtis Erdvinės kristalo gardelės tipas, sudarytas iš dėsningai išsidėsčiusių atomų ar jonų – mazgų, kurie lygiagrečiu poslinkiu gali būti sutapatinti vienas su kitu. atitikmenys …   Penkiakalbis aiškinamasis metrologijos terminų žodynas

  • Bravais lattice — Bravė gardelė statusas T sritis fizika atitikmenys: angl. Bravais lattice vok. Bravais Gitter, n rus. решётка Бравэ, f pranc. réseau de Bravais, m …   Fizikos terminų žodynas

  • Bravais lattice — Bravė gardelė statusas T sritis chemija apibrėžtis Erdvinės kristalo gardelės tipas. atitikmenys: angl. Bravais lattice rus. решетка Браве …   Chemijos terminų aiškinamasis žodynas

  • Bravais lattice — /brav ay, breuh vay / Crystall. lattice (def. 4). [named after Auguste Bravais (d. 1863), French physicist] * * * ▪ crystallography       any of 14 possible three dimensional configurations of points used to describe the orderly arrangement of… …   Universalium

  • Bravais, Auguste — ▪ French physicist born Aug. 23, 1811, Annonay, Fr. died March 30, 1863, Le Chesnay       French physicist best remembered for his work on the lattice theory of crystals; Bravais lattices (Bravais lattice) are named for him.       Bravais… …   Universalium

  • Lattice plane — In crystallography, a lattice plane of a given Bravais lattice is a plane (or family of parallel planes) whose intersections with the lattice (or any crystalline structure of that lattice) are periodic (i.e. are described by 2d Bravais lattices) …   Wikipedia

  • Bravais-Gitter — Bravė gardelė statusas T sritis Standartizacija ir metrologija apibrėžtis Erdvinės kristalo gardelės tipas, sudarytas iš dėsningai išsidėsčiusių atomų ar jonų – mazgų, kurie lygiagrečiu poslinkiu gali būti sutapatinti vienas su kitu. atitikmenys …   Penkiakalbis aiškinamasis metrologijos terminų žodynas

  • Bravais-Gitter — Bravė gardelė statusas T sritis fizika atitikmenys: angl. Bravais lattice vok. Bravais Gitter, n rus. решётка Бравэ, f pranc. réseau de Bravais, m …   Fizikos terminų žodynas

  • lattice — latticelike, adj. /lat is/, n., v., latticed, latticing. n. 1. a structure of crossed wooden or metal strips usually arranged to form a diagonal pattern of open spaces between the strips. 2. a window, gate, or the like consisting of such a… …   Universalium


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