# Wyckoff position

### From Online Dictionary of Crystallography

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Position de Wyckoff (*Fr*). Wyckoff punktlage (*Ge*). Posizione di Wyckoff (*It*). ワイコフ位置 (*Ja*)

## Definition

A **Wyckoff position** of a space group G consists of all points *X* for which the site-symmetry groups are conjugate subgroups of G.

Each Wyckoff positon of a space group is labelled by a letter which is called the *Wyckoff letter*.

The number of different Wyckoff position of each space group is finite, the maximal numbers being 9 for plane groups (realized in *p*2*mm*) and 27 for space groups (realized in *Pmmm*).

There is a total of 72 Wyckoff positions in plane groups and 1731 Wyckoff positions in space groups.

The transfer of Wyckoff positions from individual space groups to space-group types is not unique because Wyckoff positions with the same type of site-symmetry group may be exchanged in different space groups of the same type. This is no longer true when one makes use of Wyckoff sets.

## Wyckoff positions of point groups

By analogy to the Wyckoff positions of space groups, Wyckoff positions of point groups have been defined too: here the term "position" indicates the position of face poles (form face forms) or of points (for point forms) in the stereographic projection. Like in space groups, in point groups too each Wyckoff position is labelled by a Wyckoff letter.

## History

The term *Wykcoff position* takes its origin from the first English collection of equivalent positions in space groups, which appeared in *The analytical expression of the results of the theory of space groups* by Ralph W.G. Wyckoff, published by Carnegie Institution of Washington (1922; second edition, 1930), and which can be considered an ancestor of the *International Tables for Crystallography*.

## See also

- crystallographic orbit
- form
- lattice complex
- point configuration
- stabilizer
- Wyckoff set
- Sections 1.4.4.2 of the
*International Tables of Crystallography*, Volume A, 6^{th}edition