# 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

## Contents

## Mathematical description

### Vectors

### 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

### 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: