Crystals are solids with periodic structures. Many materials, both organic and inorganic, are crystalline and many of the properties of these materials are dependent on the crystal packing. It is therefore essential to understand crystal packing for material discovery.
Unit Cells
A unit cell is a tileable repeating unit of a crystal. Each crystal has multiple valid representations of a unit cell, but the irreducible smallest cell is termed the “primitive” cell.
To define a unit cell, two things must be known:
Lattice Parameters
The lattice parameters describe the shape and size of the repeating space. There are two common representations:
Scalar Cartesian lengths (a, b, c,) and angles (α, β, γ).
A matrix of lattice vectors:
1x, 1y, 1z
2x, 2y, 2z
3x, 3y, 3z
Atomic coordinates
The position of atom within the unit cell is represented by a 3×1 vector (x, y, z).
These coordinates may be Cartesian (absolute) or fractional with respect to the unit cell lattice, where the origin is (0, 0, 0) and the opposite corner is (1, 1, 1).
Supercells
A supercell is a cell which contains a whole number multiple of a smaller unit cell. They are usually represented as a 3×1 vector (X, Y, Z) of the unit cell where the value of each component in the vector is the number of repeating units of the unit cell in that direction.
Although a unit cell is sufficient to describe the complete geometry of a crystal, when modelling the interactions with a cell, consideration must be given to the periodicity of the cell. With plane-wave density-functional theory, the periodicity is intrinsic to the method, but for a molecular mechanics model, or a machine-learning interatomic potential, it may be convenient to model the periodicity by building a supercell and modelling the interactions between molecules/atoms in the unit cell with molecules/atoms in repeating images of the unit cell.
Asymmetric Units
An asymmetric unit is an irreducible unit of a unit cell, where each asymmetric unit is related by symmetry, but the molecules/atoms within an asymmetric unit are not necessarily symmetry related. In the same way that the unit cell is related to the total crystal by translation, the asymmetric unit is related to the unit cell by the symmetry operators of the space group.
Z
Z is the total number of formula units (depending on the context, this may be atoms or molecules) in the unit cell.
Z’
Z is the total number of formula units (depending on the context, this may be atoms or molecules) in the asymmetric unit. Z’ is equal to Z divided by the number of symmetry operators in the space group.
G
G is the total number of whole free units (molecules, or independent atoms) in the asymmetric unit. It is distinct from Z’ in that the value is independent of the number of formula units. G is always equal to or greater than Z’.
You may also encounter the closely related parameter, Z”. This parameter differs from G in that it does not require the free units to be wholly inside the asymmetric units, which may yield non-integer Z” values. While a perfectly valid description of geometry, it is rarely practical for model fractions of a molecule, and therefore G is a more natural parameters for modelers.
Space Groups
A space group defines the constraints on the lattice parameters and the symmetry operators in the unit cell. For a cell with known symmetry, it is memory efficient to store the atomic coordinates of only the asymmetric unit, and then either the space group, or the symmetry operators.
Read more about space groups on the symmetry page.
