Z and Z'

From Online Dictionary of Crystallography


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 always an integer which is at least 1.

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. uncertainties


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.


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'.

See also