Wyckoff set

Definition

A Wyckoff set with respect to a space group G is the set of all points X for which the site-symmetry groups are conjugate subgroups of the normalizer N of G in the group of all affine mappings.

Any Wyckoff position of G is transformed onto itself by all elements of G, but not necessarily by the elements of N. Any Wyckoff set of G is instead transformed onto itself also by those elements of N that are contained in G.

Types of Wyckoff sets

Space groups are infinite in number but are classified in 219 affine types, or 230 crystallographic types. Two space groups are of the same type if they bear the same Hermann-Mauguin symbol, i.e. if the differ in their translation subgroup (their cell parameters).

If one wants to transfer Wyckoff positions from individual space groups to space-group types, he meets the difficulty that the labelling of the Wyckoff positions by Wyckoff letters (Wyckoff notation) is not unique. For example, in the space groups of type P-1 the eight classes of centres of inversion bring a different Wyckoff letter depending on the choice of the origin and on the permutations of the basis vectors. These eight positions belong however to the same Wyckoff set.

Different space groups of the same space-group type have corresponding Wyckoff sets, and one can define types of Wyckoff sets, consisting of individual Wyckoff sets, in the same way as types of space groups consist of individual space groups.

Let the space groups G and G′ belong to the same space-group type. The Wyckoff sets K of G and K′ of G′ belong to the same type of Wyckoff sets if the affine mappings which transform G onto G′ also transform K onto K′.

There is a total of 51 types of Wyckoff sets in plane groups and 1128 types of Wyckoff sets in space groups.

See also

• crystallographic orbit
• lattice complex
• point configuration
• stabilizer
• Wyckoff position
• Section 8.3.2 of International Tables of Crystallography, Section A