Z and Z'

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Definitions

In the description of a crystal structure, the letter Z characterises the number of (chemical) formula units contained in the unit cell. For 3D-periodic structures, this number is (almost) always an integer which is at least 1. (For the rare circumstances in which a non-integer value may be reported, see 'Non-integral values of Z below.)

When dealing with molecular structures, it is also convenient to define Z'. Z' indicates the number of formula units in the asymmetric unit. In other words, Z' is the value of Z divided by the multiplicity of the general position of the corresponding space group. Z' can be a fraction or a positive integer.

Examples

The cubic structure of NaCl belongs to space group [math]Fm\overline{3}m[/math]. The unit cell contains 4 Na and 4 Cl atoms. Consequently Z=4. for inorganic structures, Z' might be a fraction. For NaCl, Z' would be 4/196 or 1/48 indicating that each atom lies on highly symmetric special positions with site symmetry [math]m\overline{3}m[/math].

In molecular structures, Z' is often equal to one, but can be a rational fraction. If the chemical entity sits on or across crystallographic site symmetry, such as mirror planes, rotation axes and inversion centres or combinations thereof, Z' can be a rational fraction < 1 equal to the ratio Z over the multiplicity of the general position of the corresponding space group. When Z' is an integer n larger than 1, there are n symmetry-independent chemical entities in the asymmetric unit. These entities have identical chemical composition, atomic connectivity and stereochemistry. Such species may be conformers with very similar molecular conformations or different conformations of one or more groups within the molecule as a result of rotations about one or more bonds.

Remark

In some cases, there are however problems and issues to consider when estimating a value of Z' to a structure. The reference below, What is Z'?, lists a series of pitfalls in some complex structures illustrating the problems of assaining a proper value to Z'.

Non-integral values of Z

A crystallographic description of a 3D periodic structure expects an integer value of Z because of the symmetry properties of the determined space group. In certain cases of structural disorder, the repeating chemical unit may need to be described with relative proportions of atom species that do not conform to IUPAC standard nomenclature rules. To preserve the chemical identities of the molecular species by using IUPAC standard formulae, the IUCr Commission on Chemical Nomenclature (Brock, 2025) permits fractional values of Z to be reported where there is sufficient jutification for doing so.

Examples supplied in the cited report are: columns of n-mers ordered in 1D but disordered in 3D because of a near equivalence of intra­molecular distances between monomers and the intermolecular distances between n-mers; disorder at a single site of residues that cannot be distinguished crystallographically but are present in a fixed ratio; and the necessity to avoid fractional coefficients for molecules or ions in stoichiometric compounds (i.e. compounds in which the relative atomic proportions can be expressed as ratios of integers). The Commission Report should be consulted for further details.

Reference

Brock, C. P. (2025). Can Z, the number of formula units per unit cell, be a fraction? A report of the IUCr Commission on Crystallographic Nomenclature. Acta Cryst. A81, 405-408.

See also

• Unit cell
• Asymmetric unit
• What is Z'?
• Looking deeper into molecular structures with Z' > 1