# Difference between revisions of "Crystallographic orbit"

### From Online Dictionary of Crystallography

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− | + | <Font Color="blue"> Orbite cristallographique</Font> (''Fr''). <Font Color="red"> Punktklage </Font>(''Ge''). <Font color="black"> Orbita cristallografica </Font>(''It''). | |

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+ | 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. | ||

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+ | == 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. | ||

+ | |||

+ | == 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|site-symmetry group]]. The site-symmetry groups belonging to different points out of the ''same'' crystallographic orbit are conjugate subgroups of ''G''. | ||

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+ | == 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|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|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'' | ||

+ | |||

+ | [[Category:Fundamental crystallography]] |

## Revision as of 10:35, 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.

## 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*