Crystallographic orbit

Orbite cristallographique (Fr). Punktklage (Ge). Orbita cristallografica (It).

In mathematics, an orbit is a general group-theoretical term describing any set of objects that are mapped onto each other by the action of a group. In crystallography, the concept of orbit is used to indicate a point configuration in association with its generatig group.

Definition

From any point of the three-dimensional Euclidean space the symmetry operations of a given space group G generate an infinte set of points, called a crystallographic orbit. The space gorup G is called the generating space group of the orbit. Two crystallographic orbits are said configuration-equivalent if and only if their sets of points are identical.

Crystallographic orbits and site-symmetry groups

Each point of a crystallographic orbit defines uniquely a largest subgroup of G, which maps that point onto itself: its site-symmetry group. The site-symmetry groups belonging to different points out of the same crystallographic orbit are conjugate subgroups of G.

Crystallographic orbits and Wyckoff positions

Two crystallographic orbits of a space gorup G belong to the same Wyckoff position if and only if the site-symmetry groups of any two points from the first and the second orbit are conjugate subgroups of G.

Crystallographic orbits and Wyckoff sets

Two crystallographic orbits of a space gorup G belong to the same Wyckoff set if and only if the site-symmetry groups of any two points from the first and the second orbit are conjugate subgroups of the affine normalizer of G.

See also

• Chapter 8.3.2 of International Tables of Crystallography, Section A