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. | ||
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| + | == 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''. | ||
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| + | == 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''. | ||
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| + | == See also == | ||
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| + | * Chapter 8.3.2 of ''International Tables of Crystallography, Section A'' | ||
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| + | [[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