Difference between revisions of "Wyckoff position"
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| − | < | + | <font color="blue">Position de Wyckoff</font> (''Fr''). <font color="red">Punktlage, Wyckoff-Position</font> (''Ge''). <font color="black">Posizione di Wyckoff</font> (''It''). <font color="purple">ワイコフ位置</font> (''Ja''). <font color="green">Posición de Wyckoff</font> (''Sp''). |
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== Definition == | == Definition == | ||
| − | A '''Wyckoff position''' of a [[space group]] G consists of all points ''X'' for which the [[site symmetry|site-symmetry groups]] are [[conjugacy class|conjugate]] subgroups of G. | + | A '''Wyckoff position''' of a [[space group]] ''G'' consists of all points ''X'' for which the [[site symmetry|site-symmetry groups]] are [[conjugacy class|conjugate]] subgroups of ''G''. |
Each Wyckoff positon of a [[space group]] is labelled by a letter which is called the ''Wyckoff letter''. | Each Wyckoff positon of a [[space group]] is labelled by a letter which is called the ''Wyckoff letter''. | ||
| − | The number of different Wyckoff | + | The number of different Wyckoff positions of each [[space group]] is finite, the maximal numbers being 9 for plane groups (realized in ''p''2''mm'') and 27 for space groups (realized in ''Pmmm''). |
There is a total of 72 Wyckoff positions in plane groups and 1731 Wyckoff positions in space groups. | There is a total of 72 Wyckoff positions in plane groups and 1731 Wyckoff positions in space groups. | ||
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==Wyckoff positions of point groups== | ==Wyckoff positions of point groups== | ||
| − | By analogy to the Wyckoff positions of [[space group]]s, Wyckoff positions of [[point group]]s have been defined too: here the term | + | By analogy to the Wyckoff positions of [[space group]]s, Wyckoff positions of [[point group]]s have been defined too: here the term 'position' indicates the position of face poles (for face forms) or of points (for point forms) in the [[stereographic projection]]. Like in space groups, in point groups, too, each Wyckoff position is labelled by a Wyckoff letter. |
==History== | ==History== | ||
| − | The term '' | + | The term ''Wyckoff position'' takes its origin from the first English collection of equivalent positions in space groups, which appeared in ''The analytical expression of the results of the theory of space groups'' by Ralph W. G. Wyckoff, published by Carnegie Institution of Washington (1922; second edition, 1930), and which can be considered an ancestor of the ''International Tables for Crystallography''. |
== See also == | == See also == | ||
| − | *[[ | + | *[[Crystallographic orbit]] |
| − | *[[ | + | *[[Form]] |
| − | *[[ | + | *[[Lattice complex]] |
| − | *[[ | + | *[[Point configuration]] |
| − | *[[ | + | *[[Stabilizer]] |
*[[Wyckoff set]] | *[[Wyckoff set]] | ||
| − | * | + | *Chapter 1.4.4.2 of ''International Tables for Crystallography, Volume A'', 6th edition |
[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] | ||
Latest revision as of 14:52, 20 November 2017
Position de Wyckoff (Fr). Punktlage, Wyckoff-Position (Ge). Posizione di Wyckoff (It). ワイコフ位置 (Ja). Posición de Wyckoff (Sp).
Definition
A Wyckoff position of a space group G consists of all points X for which the site-symmetry groups are conjugate subgroups of G.
Each Wyckoff positon of a space group is labelled by a letter which is called the Wyckoff letter.
The number of different Wyckoff positions of each space group is finite, the maximal numbers being 9 for plane groups (realized in p2mm) and 27 for space groups (realized in Pmmm).
There is a total of 72 Wyckoff positions in plane groups and 1731 Wyckoff positions in space groups.
The transfer of Wyckoff positions from individual space groups to space-group types is not unique because Wyckoff positions with the same type of site-symmetry group may be exchanged in different space groups of the same type. This is no longer true when one makes use of Wyckoff sets.
Wyckoff positions of point groups
By analogy to the Wyckoff positions of space groups, Wyckoff positions of point groups have been defined too: here the term 'position' indicates the position of face poles (for face forms) or of points (for point forms) in the stereographic projection. Like in space groups, in point groups, too, each Wyckoff position is labelled by a Wyckoff letter.
History
The term Wyckoff position takes its origin from the first English collection of equivalent positions in space groups, which appeared in The analytical expression of the results of the theory of space groups by Ralph W. G. Wyckoff, published by Carnegie Institution of Washington (1922; second edition, 1930), and which can be considered an ancestor of the International Tables for Crystallography.
See also
- Crystallographic orbit
- Form
- Lattice complex
- Point configuration
- Stabilizer
- Wyckoff set
- Chapter 1.4.4.2 of International Tables for Crystallography, Volume A, 6th edition