# 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">Kristallographisches Orbit</font> (''Ge''). <font color="black">Orbita cristallografica</font>(''It''). <font color="purple">結晶軌道</font> (''Ja''). <font color="green">Órbita cristalográfica</font> (''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 | + | 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 == | == Definition == | ||

− | From any point of the three-dimensional Euclidean space the symmetry operations of a given space group ''G'' generate an | + | 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 configuration-equivalent if and only if their sets of points are identical. | + | 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 == | == Crystallographic orbits and site-symmetry groups == | ||

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== Crystallographic orbits and Wyckoff positions == | == Crystallographic orbits and Wyckoff positions == | ||

− | Two crystallographic orbits of a space | + | Two crystallographic orbits of a space group ''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 == | == Crystallographic orbits and Wyckoff sets == | ||

− | Two crystallographic orbits of a space | + | Two crystallographic orbits of a space group ''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 == | == 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. | + | 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 [[eigensymmetry|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 [[eigensymmetry|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 [[Form|face form]] joined to the [[point group]] that has generated the form and the [[Form|face form]] detached from its generating [[point group]]. | ||

== Types of crystallographic orbits == | == 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 | + | 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''. | 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 == | == See also == | ||

− | * Chapter | + | * Chapter 1.4.4.1 of ''International Tables for Crystallography, Volume A'', 6th edition |

[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] |

## Latest revision as of 17:35, 9 November 2017

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.

## Contents

## 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