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:

${\displaystyle \mathbf {R} =n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3}}$

(1)

where the n_i are any integers and a_i 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.

In crystallography, the Bravais lattice concept of an infinite array of discrete points is expanded using the concept of a unit cell which includes the space in between the discrete lattice points as well as any atoms in that space. There are two main types of unit cells: primitive unit cells and non-primitive unit cells.

A primitive unit cell for a given Bravais lattice can be chosen in more than one way (each way having a different shape), but each way will have the same volume and each way will have the property that a one-to-one correspondence can be established between the primitive unit cells and the discrete lattice points. The obvious primitive cell to associate with a particular choice of primitive vectors is the parallelepiped formed by them. That is, the set of all points r of the form:

${\displaystyle \mathbf {r} =x_{1}\mathbf {a} _{1}+x_{2}\mathbf {a} _{2}+x_{3}\mathbf {a} _{3}\qquad {\text{where }}0\leq x_{i}<1}$

(2)

Using the parallelepiped defined by the primitive vectors as the unit cell has the disadvantage in some cases of not clearly revealing the full symmetry of the lattice. One solution to this is to use the Wigner-Seitz primitive cell (consisting of all points in space that are closer to the given lattice point then to any other lattice point) which does display the full symmetry of the lattice. Another solution is to use a non-primitive unit cell which does display the full symmetry of the lattice. The non-primitive unit cell volume will be an integer multiple of the primitive unit cell volume.

The unit cell, whether primitive or not, when replicated once for each discrete lattice point, must exactly fill the entire space with no overlap and no gaps.

The expanded Bravais lattice concept, including the unit cell, 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) occurring exactly once in each primitive unit cell. The basis may consist of atoms, molecules, or polymer strings of solid matter. Consequently, the crystal looks the same when viewed in any given direction from any equivalent points in two different unit cells (two points in two different unit cells of the same lattice are equivalent if they have the same relative position with respect to their individual unit cell boundaries).

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.