A Bravais lattice is a mathematical abstraction with application to the study of crystalline solids. A Bravais lattice is an infinite set of points in space with positions such that at every point the arrangement of the surrounding points looks exactly the same. More precisely, a Bravais lattice is the set all of points with position vectors R that can be written as

R = n1a1 + n2a2 + n3a3,

where a1, a2, and a3 are three non-coplanar vectors and {n1, n2, n3} can take on any set of integer values.

It is easiest to visualize 2-d Bravais lattices. The simplest is the following square lattice (which is actually infinitely extended):

```
X      X      X      X      X      X      X

X      X      X      X      X      X      X

X      X      X      X      X      X      X

X      X      X      X      X      X      X

X      X      X      X      X      X      X

```

Notice that if this lattice were infinitely extended, the surroundings would look identical at every point X. The square lattice has 90 degree rotational symmetry. A fundamentally different 2-d Bravais lattice is the following rectangular lattice, with only 180 degree rotational symmetry:

```
X       X       X       X       X        X

X       X       X       X       X        X

X       X       X       X       X        X

X       X       X       X       X        X

```

One more important illustrative example is the following:

```
X       X       X       X       X        X

X       X       X       X       X       X

X       X       X       X       X        X

X       X       X       X       X       X

X       X       X       X       X        X

X       X       X       X       X       X

```

This new lattice could be called a "centered" rectangular lattice since it is just a rectangular lattice with additional points at the centers of the rectangles. Notice that centering the rectangular lattice creates another Bravais lattice. It is a fundamentally different lattice--rotating this lattice doesn't yield the simple rectangular lattice. Now imagine "squeezing" the centered rectangular lattice until it becomes a centered square lattice. Upon rotation by 45 degrees the centered square lattice would just be the simple square, uncentered square lattice with a smaller "lattice constant"! Thus the "centered square lattice" is not a new 2-d Bravais lattice--it is a redundancy.

Note: Two Bravais lattices with defining basis vectors {an} and {bn} are considered "equivalent" if an = c UtbnU for some constant c and for all n, where U is a unitary matrix that corresponds to a rotation operation. Would a mathematician verify that this is what I mean to say?

In three dimensions it turns out that there are only 14 nonequivalent Bravais lattices. This is fairly difficult to prove and to visualize. A. Bravais was the first person to correctly count the number of 3-d lattices in 1845.

The 14 Bravais lattices can be grouped into seven crystal systems based on their rotational symmetries. The seven crystal systems are the following: cubic, tetragonal, orthorhombic, monoclinic, triclinic, trigonal, and hexagonal. The reference listed below describes these systems in detail. Some of the crystal systems contain more than one Bravais lattice. For instance, the simple cubic, the face-centered cubic, and the body-centered cubic (each of these has a separate writeup) lattices are different Bravais lattices that share the same cubic rotational symmetries.

If a single atom were placed at every point in a Bravais lattice, a crystal with all the symmetries of the Bravais lattice would be formed. However, very often in a crystal a combination of atoms called a basis exists at every Bravais lattice point. Silicon and carbon, for instance, crystallize into the face-centered cubic lattice with a two-atom basis, creating the diamond structure. Thus real crystals usually aren't quite as symmetric as their Bravais lattices.

Reference: Solid State Physics by Ashcroft/Mermin

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