Unit cell

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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


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):



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:


  • 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.



Since , , and:

And in reciprocal-space:




Since and , , and:

And in reciprocal-space:


See Also