# Difference between revisions of "Crystallographic orbit"

### From Online Dictionary of Crystallography

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Two crystallographic orbits of a space gorup ''G'' belong to the same [[Wyckoff set]] if and only if | Two crystallographic orbits of a space gorup ''G'' belong to the same [[Wyckoff set]] if and only if | ||

the [[Site symmetry|site-symmetry groups]] of any two points from the first and the second orbit are conjugate subgroups of the affine [[normalizer]] of ''G''. | the [[Site symmetry|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. | ||

== See also == | == See also == |

## Revision as of 10:45, 22 February 2007

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.

## Contents

## 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*.

## 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.

## See also

- Chapter 8.3.2 of
*International Tables of Crystallography, Section A*