 Wyckoff position

A Wyckoff position of a space group consists of all points whose site-symmetry groups are conjugate subgroups of that space group. It classifies sites by their symmetry relationship within the whole group. A Wyckoff letter labels each such class in the conventional tables; the label is accompanied by its crystallographic site multiplicity.^[ref]

A Wyckoff position need not specify one particular atom or one fixed coordinate triplet. A general position usually has free fractional parameters. Choosing those parameters produces a crystallographic orbit, the collection of equivalent sites generated from that choice by the space group. Different parameter choices can give distinct orbits belonging to the same Wyckoff position.

An original comparison uses $P\bar{1}$, with operations $\mathbf{r}\mapsto \pm \mathbf{r}+\mathbf{n}$, where $\mathbf{n}$ is a lattice vector. Generic positions form pairs $\{\mathbf{r},-\mathbf{r}\}$ modulo lattice translations and have trivial site-symmetry groups. Many different pairs therefore belong to the same general Wyckoff position.

The origin and the half-cell point $(1/2,0,0)$ instead each have an inversion-containing site-symmetry group of order2. Their groups are isomorphic but not conjugate within this space group: a group operation sends the origin to an integer lattice point, never to that half-cell point. They occupy different special Wyckoff positions. Group order or multiplicity alone therefore does not uniquely identify a Wyckoff position.

This classification helps constrain independent atomic coordinates in a crystal structure; it describes allowed symmetry, not which chemical species occupies a site or its fractional occupation.