Example of the
BCC unit cell.
The unit cell is the basic building block of a crystal lattice (whether an atomic crystal or a nanoscale superlattice). Crystalline materials have a periodic structure, with the unit cell being the minimal volume necessary to fully describe the repeating structure. There are a finite number of possible symmetries for the repeating unit cell.
A unit cell can be defined by three vectors that lie along the edges of the enclosing parallelepped. We denote the vectors as
,
, and
; alternately the unit cell can be described by the lengths of these vectors (
,
,
), and the angles between them:
, the angle between
and 
, the angle between
and 
, the angle between
and 
Mathematical description
Vectors
There are many ways to define the Cartesian basis for the unit cell in real-space. A typical definition is:

There are many mathematically equivalent ways to express a given definition. For instance, the vector
can also be written as (c.f. these notes and Trueblood et al. Acta Cryst 1996, A52, 770-781):

Relations



Volume

If a, b, and c are the parallelepiped edge lengths, and α, β, and γ are the internal angles between the edges, the volume is

The volume of a unit cell with all edge-length equal to unity is:

Angles
is the angle between
and 
is the angle between
and 
is the angle between
and 
Unit cell definition using parallelepiped with lengths
a,
b,
c and angles between the sides given by α,β,γ (from Wikipedia
fractional coordinates).
Reciprocal vectors
The repeating structure of a unit cell creates peaks in reciprocal space. In particular, we observe maxima (constructive interference) when:

Where
,
, and
are integers. We define reciprocal-space vectors:

And we can then express the momentum transfer (
) in terms of these reciprocal vectors:

Combining with the three Laue equations yields:

Where
is a vector that defines the position of Bragg reflection
for the reciprocal-lattice.
Examples
Cubic
Since
,
, and:

And in reciprocal-space:

So:

And:

Hexagonal
Since
and
,
, and:

And in reciprocal-space:

So:

See Also