Lattice, unit cell, asymmetric unit

Lattice and Unit Cell

A lattice is a construct that divides space into regular, translationally periodic units. The unit lattice is defined by basis vectors (a, b, and c), that span the lattice. Combination of the unit lattice with molecular motifs generates the unit cell of the crystal. A crystal is a translationally periodic, finite assembly of unit lattices. Each identical unit cell contains the same number of identically arranged molecules. The molecules contained in each unit cell can be related by internal symmetry, generating a unit cell packed with multiple, symmetry equivalent instances or symmetry equivalent copies of the molecule.

The asymmetric unit (ASU) of the unit cell

The smallest object needed to generate the whole unit cell by applying the crystallographic operations is called the asymmetric unit (AU) of the unit cell. The ASU of a unit cell contains all the necessary information to generate the complete unit cell of a crystal structure by applying its symmetry operations to the asymmetric unit.

Examples:

The area (or volume in 3D) of the ASU for:
- a plane group p4: one-quarter of the unit cell
- a trigonal (p3) group: one-third of the trigonal unit cell area (2D) or volume (3D)
- an hexagonal (p6) group: one-sixth of the unit cell (half of the trigonal AU, and this is because in p6 there are additional 2-fold symmetry in the unit cell with respect to p3)