Wyckoff set - Revision history

BrianMcMahon: Added German and Spanish translations (U. Mueller)

2017-11-20T14:53:05Z

Added German and Spanish translations (U. Mueller)

BrianMcMahon: Style edits to align with printed edition

2017-05-17T18:28:23Z

Style edits to align with printed edition

MassimoNespolo: /* See also */ ITA 6th edition

2017-04-11T17:03:19Z

‎See also: ITA 6th edition

MassimoNespolo: /* Types of Wyckoff sets */ link

2016-11-24T13:44:39Z

‎Types of Wyckoff sets: link

MassimoNespolo: /* Types of Wyckoff sets */

2009-02-06T16:15:16Z

‎Types of Wyckoff sets

MassimoNespolo: /* Definition */

2009-02-06T16:13:43Z

‎Definition

MassimoNespolo: /* See also */ internal links

2009-02-06T15:05:28Z

‎See also: internal links

MassimoNespolo at 18:25, 9 February 2007

2007-02-09T18:25:28Z

New page

== Definition ==

A ''Wyckoff set'' with respect to a space group ''G'' is the set of all points ''X'' for which the [[site symmetry|site-symmetry groups]] are conjugate subgroups of the [[normalizer]] ''N'' of ''G'' in the group of all [[affine mapping]]s.

Any [[Wyckoff position]] of ''G'' is transformed onto itself by all elements of ''G'', but not necessarily by the elements of ''N''. Any Wyckoff set of ''G'' is instead transformed onto itself also by those elements of ''N'' that are contained in ''G''.

== See also ==

Section 8.3.2 of ''International Tables of Crystallography, Section A''<br>

[[Category:Fundamental crystallography]]

← Older revision		Revision as of 14:53, 20 November 2017
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		+	<font color="red">Wyckoff-Satz</font> (''Ge''). <font color="green">Conjunto de Wyckoff</font> (''Sp'').
		+
	== Definition ==		== Definition ==

@@ Line 6: / Line 6: @@
 == Types of Wyckoff sets ==
-[[Space group]]s are infinite in number but are classified in 219 affine types, or 230 crystallographic types. Two space groups are of the same type if they bear the same [[Hermann-Mauguin symbols|Hermann-Mauguin symbol]], ''i''.''e''. if the differ in their translation [[subgroup]] (their cell parameters).
+[[Space group]]s are infinite in number but are classified in 219 affine types, or 230 crystallographic types. Two space groups are of the same type if they bear the same [[Hermann-Mauguin symbols|Hermann-Mauguin symbol]], ''i.e.'' if they differ in their translation [[subgroup]] (their cell parameters).
-If one wants to transfer Wyckoff positions from individual space groups to space-group types, he meets the difficulty that the labelling of the Wyckoff positions by Wyckoff letters (Wyckoff notation) is not unique. For example, in the space groups of type ''P''-1 the eight classes of centres of inversion bring a different Wyckoff letter depending on the choice of the origin and on the permutations of the basis vectors. These eight positions belong however to the same Wyckoff set.
+The transfer of Wyckoff positions from individual space groups to space-group types meets the difficulty that the labelling of the Wyckoff positions by Wyckoff letters (Wyckoff notation) is not unique. For example, in the space groups of type <math>P\bar 1</math> the eight classes of centres of inversion bring a different Wyckoff letter depending on the choice of the origin and on the permutations of the basis vectors. These eight positions belong however to the same Wyckoff set.
 Different space groups of the same space-group type have corresponding Wyckoff sets, and one can deﬁne types of Wyckoff sets, consisting of individual Wyckoff sets, in the same way as types of space groups consist of individual space groups.
 <blockquote>
-Let the space groups ''G'' and ''G''&prime; belong to the same space-group type. The Wyckoff sets K of ''G'' and K&prime; of ''G''&prime; belong to the  same  type  of  Wyckoff  sets  if  the  afﬁne  mappings  which transform ''G'' onto ''G''&prime; also transform K onto K&prime;.
+Let the space groups ''G'' and ''G''&prime; belong to the same space-group type. The Wyckoff sets ''K'' of ''G'' and ''K''&prime; of ''G''&prime; belong to the  same  type  of  Wyckoff  sets  if  the  afﬁne  mappings  which transform ''G'' onto ''G''&prime; also transform ''K'' onto ''K''&prime;.
 </blockquote>
@@ Line 19: / Line 19: @@
 == See also ==
-*[[crystallographic orbit]]
+*[[Crystallographic orbit]]
-*[[lattice complex]]
+*[[Lattice complex]]
-*[[point configuration]]
+*[[Point configuration]]
-*[[stabilizer]]
+*[[Stabilizer]]
 *[[Wyckoff position]]
-*Section 1.4.4.3 of ''International Tables of Crystallography, Section A'', 6<sup>th</sup> edition
+*Chapter 1.4.4.3 of ''International Tables for Crystallography, Section A'', 6th edition
 [[Category:Fundamental crystallography]]

@@ Line 19: / Line 19: @@
 == See also ==
 *[[crystallographic orbit]]
 *[[lattice complex]]
@@ Line 25: / Line 24: @@
 *[[stabilizer]]
 *[[Wyckoff position]]
-*Section 8.3.2 of ''International Tables of Crystallography, Section A''<br>
+*Section 1.4.4.3 of ''International Tables of Crystallography, Section A'', 6<sup>th</sup> edition
 [[Category:Fundamental crystallography]]

@@ Line 6: / Line 6: @@
 == Types of Wyckoff sets ==
-[[Space group]]s are infinite in number but are classified in 219 affine types, or 230 crystallographic types. Two space groups are of the same type if they bear the same Hermann-Mauguin symbol, ''i''.''e''. if the differ in their translation [[subgroup]] (their cell parameters).
+[[Space group]]s are infinite in number but are classified in 219 affine types, or 230 crystallographic types. Two space groups are of the same type if they bear the same [[Hermann-Mauguin symbols|Hermann-Mauguin symbol]], ''i''.''e''. if the differ in their translation [[subgroup]] (their cell parameters).
 If one wants to transfer Wyckoff positions from individual space groups to space-group types, he meets the difficulty that the labelling of the Wyckoff positions by Wyckoff letters (Wyckoff notation) is not unique. For example, in the space groups of type ''P''-1 the eight classes of centres of inversion bring a different Wyckoff letter depending on the choice of the origin and on the permutations of the basis vectors. These eight positions belong however to the same Wyckoff set.

@@ Line 15: / Line 15: @@
 Let the space groups ''G'' and ''G''&prime; belong to the same space-group type. The Wyckoff sets K of ''G'' and K&prime; of ''G''&prime; belong to the  same  type  of  Wyckoff  sets  if  the  afﬁne  mappings  which transform ''G'' onto ''G''&prime; also transform K onto K&prime;.
 </blockquote>
 == See also ==

← Older revision		Revision as of 16:13, 6 February 2009
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	Any [[Wyckoff position]] of ''G'' is transformed onto itself by all elements of ''G'', but not necessarily by the elements of ''N''. Any Wyckoff set of ''G'' is instead transformed onto itself also by those elements of ''N'' that are contained in ''G''.		Any [[Wyckoff position]] of ''G'' is transformed onto itself by all elements of ''G'', but not necessarily by the elements of ''N''. Any Wyckoff set of ''G'' is instead transformed onto itself also by those elements of ''N'' that are contained in ''G''.
		+
		+	== Types of Wyckoff sets ==
		+	[[Space group]]s are infinite in number but are classified in 219 affine types, or 230 crystallographic types. Two space groups are of the same type if they bear the same Hermann-Mauguin symbol, ''i''.''e''. if the differ in their translation [[subgroup]] (their cell parameters).
		+
		+	If one wants to transfer Wyckoff positions from individual space groups to space-group types, he meets the difficulty that the labelling of the Wyckoff positions by Wyckoff letters (Wyckoff notation) is not unique. For example, in the space groups of type ''P''-1 the eight classes of centres of inversion bring a different Wyckoff letter depending on the choice of the origin and on the permutations of the basis vectors. These eight positions belong however to the same Wyckoff set.
		+
		+	Different space groups of the same space-group type have corresponding Wyckoff sets, and one can deﬁne types of Wyckoff sets, consisting of individual Wyckoff sets, in the same way as types of space groups consist of individual space groups.
		+
		+	<blockquote>
		+	Let the space groups ''G'' and ''G''′ belong to the same space-group type. The Wyckoff sets K of ''G'' and K′ of ''G''′ belong to the same type of Wyckoff sets if the afﬁne mappings which transform ''G'' onto ''G''′ also transform K onto K′.
		+	</blockquote>

	== See also ==		== See also ==