# Difference between revisions of "Crystallographic orbit"

### From Online Dictionary of Crystallography

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== Definition == | == 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. | 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 == | == Crystallographic orbits and site-symmetry groups == |

## Revision as of 10:44, 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*.

## See also

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