Crystal lattice

In geometry and crystallography, a Bravais lattice, named after Auguste Bravais (1850), is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by:





R

=

n

1




a


1


+

n

2




a


2


+

n

3




a


3




{\displaystyle \mathbf {R} =n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3}}
where the ni are any integers and ai are 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 will appear 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 a periodic arrangement of one or more atoms (the basis or motif) at each lattice point. The basis may consist of atoms, molecules, or polymer strings of solid matter.
Two Bravais lattices are often considered equivalent if they have isomorphic symmetry groups. In this sense, there are 14 possible Bravais lattices in three-dimensional space. The 14 possible symmetry groups of Bravais lattices are 14 of the 230 space groups. In the context of the space group classification, the Bravais lattices are also called Bravais classes, Bravais arithmetic classes, or Bravais flocks.

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