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.

A Wyckoff letter is the conventional alphabetic label identifying a Wyckoff position within the tables for a space group. It is normally written beside the crystallographic site multiplicity, as in $1a$. Letters begin with $a$ for a position with maximal site-symmetry group order and proceed toward the general position. The letter is a label, not a coordinate or a chemical-species symbol.^[ref]

For example, the space group $P1$ has only translations. Every point has a trivial site-symmetry group, and the general position is labeled $1a$: one equivalent position per primitive unit cell, with free coordinates $(x,y,z)$.^[ref]

An original model makes the distinction from atom counting clear. Place one species at $(0.1,0.2,0.3)$ and another at $(0.4,0.1,0.7)$ in a generic $P1$ crystal structure. Both sites are described by $1a$, but they define different crystallographic orbits. The notation does not say that the entire structure contains only one atom per cell.

The space group, its setting and coordinate convention must accompany the letter. The same letter in another group need not denote the same symmetry or coordinates. Nor does matching multiplicity by itself identify the Wyckoff position, since distinct positions can have equal multiplicity.