Crystallographic orbit

Orbite cristallographique (Fr). Kristallographisches Orbit (Ge). Orbita cristallografica(It). 結晶軌道 (Ja). Órbita cristalográfica (Sp).

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 generating group.

Definition

From any point of the three-dimensional Euclidean space the symmetry operations of a given space group G generate an infinite set of points, called a crystallographic orbit. The space group G is called the generating space group of the orbit. Two crystallographic orbits are said to be 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 group 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 group 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.

Crystallographic orbits and point configurations

The concept of crystallographic orbit is closely related to that of point configuration, but differs from it by the fact that point configurations are detached from their generating space groups.

A set of points has an inherent symmetry, which corresponds to the group which has generated the set. However, this set of points may occur not only in its generating space group, but also in other space groups of different type. A set of points taken with its inherent symmetry but detached from its generating group is called a point configuration.

The relation between crystallographic orbits and point configurations in point space has a close analogy in vector space in the relation between the face form joined to the point group that has generated the form and the face form detached from its generating point group.

Types of crystallographic orbits

The generating space group of any crystallographic orbit may be compared with the inherent symmetry of its point configuration. If both groups coincide, the orbit is called a characteristic crystallographic orbit, otherwise it is called a non-characteristic crystallographic orbit. If the inherent symmetry group contains translations additional to those of the generating space group, the orbit is called an extraordinary crystallographic orbit.

See also

• Chapter 1.4.4.1 of International Tables for Crystallography, Volume A, 6th edition