Editing Bravais lattice (section)

{{short description|Geometry and crystallography point array}}
{{Use dmy dates|date=February 2023}}
[[File:Crystal systems.jpg|thumb|The seven lattice systems and their Bravais lattices in three dimensions]]

In [[geometry]] and [[crystallography]], a '''Bravais lattice''', named after {{harvs|txt|first=Auguste |last=Bravais|year=1850|authorlink=Auguste Bravais}},<ref>{{cite journal|last1 = Aroyo|first1 = Mois I.|first2 = Ulrich|last2 = Müller|first3 = Hans|last3 = Wondratschek|title = Historical Introduction|journal = International Tables for Crystallography|volume = A1|issue = 1.1|pages = 2–5|year = 2006|url = http://it.iucr.org/A1a/ch1o1v0001/sec1o1o1/|doi = 10.1107/97809553602060000537|access-date = 2008-04-21|url-status = dead|archive-url = https://archive.today/20130704032928/http://it.iucr.org/A1a/ch1o1v0001/sec1o1o1/|archive-date = 2013-07-04|citeseerx = 10.1.1.471.4170}}</ref> is an infinite array of [[discrete point]]s generated by a set of [[Translation operator (quantum mechanics)#Discrete translational symmetry|discrete translation]] operations described in three dimensional space by

: <math>\mathbf{R} = n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + n_3 \mathbf{a}_3,</math>

where the ''n<sub>i</sub>'' are any integers, and '''a'''<sub>''i''</sub> are ''primitive translation vectors'', or ''primitive vectors'', which lie in different directions (not necessarily mutually perpendicular) and span the lattice. The choice of primitive vectors for a given Bravais lattice is not unique. A fundamental aspect of any Bravais lattice is that, for any choice of direction, the lattice appears exactly the same from each of the discrete lattice points when looking in that chosen direction.

The Bravais lattice concept is used to formally define a ''crystalline arrangement'' and its (finite) frontiers. A [[crystal]] is made up of one or more atoms, called the ''basis'' or ''motif'', at each lattice point. The ''basis'' may consist of [[atom]]s, [[molecule]]s, or [[polymer]] strings of [[Solid|solid matter]], and the lattice provides the locations of the basis. 

Two Bravais lattices are often considered equivalent if they have isomorphic [[symmetry group]]s. In this sense, there are 5 possible Bravais lattices in 2-dimensional space and 14 possible Bravais lattices in 3-dimensional space. The 14 possible symmetry groups of Bravais lattices are 14 of the 230 [[space group]]s. In the context of the space group classification, the Bravais lattices are also called Bravais classes, Bravais arithmetic classes, or Bravais flocks.<ref>{{cite web |title=Bravais class |url=http://reference.iucr.org/dictionary/Bravais_class |website=Online Dictionary of Crystallography |publisher=IUCr |access-date=8 August 2019}}</ref>