OD structure - Revision history

BrianMcMahon: Tidied translations.

2017-11-16T13:15:06Z

Tidied translations.

BrianMcMahon: Style edits to align with printed edition

2017-05-16T10:24:19Z

Style edits to align with printed edition

MassimoNespolo: German

2009-02-23T14:44:00Z

German

MassimoNespolo at 17:44, 22 February 2009

2009-02-22T17:44:40Z

MassimoNespolo: internal link

2009-01-06T11:23:39Z

internal link

MassimoNespolo at 20:43, 9 April 2008

2008-04-09T20:43:08Z

MassimoNespolo at 14:26, 27 February 2007

2007-02-27T14:26:53Z

MassimoNespolo at 11:54, 27 February 2007

2007-02-27T11:54:42Z

MassimoNespolo: small typo

2006-04-28T07:51:37Z

small typo

MassimoNespolo at 07:49, 28 April 2006

2006-04-28T07:49:57Z

New page

OD structures consist of slabs with their own symmetry, containing coincidence operations constituting a [[diperiodic group]] ([[layer group]]) only within individual slabs. For the entire structure these coincidence operations are only local (partial), i.e. they are valid only in a subspace of the crystal space. The ambiguity (= existence of more than one equivalent possibilities) in the stacking of slabs arises from the existence of this local symmetry, which does not appear in the [[space group]] of the structure. The resulting structure can be "ordered" (''periodic'') or "disordered" (''non-periodic''), depending on the sequence of local symmetry operations relating pairs of slabs. The set of all the operations valid in the whole crystal space constitutes a [[space group]]; by adding the set of all the operations valid in a subspace of it, one obtains a [[space groupoid]]

In the OD theory, a central role is played by the '''vicinity condition''' ('''VC'''), which states the geometrical equivalence of layer pairs. The vicinity condition consists of three parts:
*'''VC α''': VC layers are either geometrically equivalent or, if not, they are relatively few in kind
*'''VC β''': translation groups of all VC layers are either identical or they have a common subgroup
*'''VC γ''': equivalent sides of equivalent layers are faced by equivalent sides of adjacent layers so that the resulting pairs are equivalent.

If the position of a layer is uniquely defined by the position of the adjacent layers and by the VC, the resulting structure is fully ordered. If, on the other hand, more than one position is possible that obeys the VC, the resulting structure is an ‘'''OD structure''' and the layers are '''OD layers'''. VC structures may thus be either fully ordered structures or OD structures. All OD structures are [[polytypism|polytypic]]; the reverse may or may not be true. Equivalency depends on the choice of OD layers and also on the definition of [[polytypism]].

All OD structures, even of different substances, built according to the same symmetry principle, belong to an OD [[groupoid family]]: this is an abstract family, whose members are the groupoids describing the symmetry of the substances sharing the same symmetry principle. Then, the OD structures of the same substance built on the same structural principle belong to one and the same family: the members of a family are individual, real structures.

==See also==
Chapter 9.2 of ''International Tables of Crystallography, Volume C''

[[Category:Fundamental crystallography]]

← Older revision		Revision as of 13:15, 16 November 2017
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−	<font color="blue">Structure OD</font> (''Fr''). <font color="red">OD ~~struktur~~</font> (''Ge''). <font color="black">Struttura OD</font> (''It''). <font color="purple">OD構造</font> (''Ja'').	+	<font color="blue">Structure OD</font> (''Fr''). <font color="red">OD Struktur</font> (''Ge''). <font color="black">Struttura OD</font> (''It''). <font color="purple">OD構造</font> (''Ja'').

@@ Line 1: / Line 1: @@
-<font color="blue">Structure OD</font> (''Fr''); <font color="red">OD struktur</font> (''Ge''); <font color="black">Struttura OD</font> (''It''); <font color="purple">OD構造</font> (''Ja'').
+<font color="blue">Structure OD</font> (''Fr''). <font color="red">OD struktur</font> (''Ge''). <font color="black">Struttura OD</font> (''It''). <font color="purple">OD構造</font> (''Ja'').
-OD structures consist of slabs with their own symmetry, containing coincidence operations constituting a [[subperiodic group|diperiodic group]] (layer group) only within individual slabs. For the entire structure these coincidence operations are only local (partial), i.e. they are valid only in a subspace of the crystal space. The ambiguity (= existence of more than one equivalent possibilities) in the stacking of slabs arises from the existence of this local symmetry, which does not appear in the [[space group]] of the structure. The resulting structure can be "ordered" (''periodic'') or "disordered" (''non-periodic''), depending on the sequence of local symmetry operations relating pairs of slabs. The set of all the operations valid in the whole crystal space constitutes a [[space group]]; by adding the set of all the operations valid in a subspace of it, one obtains a space [[groupoid]].
+OD structures consist of slabs with their own symmetry, containing coincidence operations constituting a [[subperiodic group|diperiodic group]] (layer group) only within individual slabs. For the entire structure these coincidence operations are only local (partial), ''i.e.'' they are valid only in a subspace of the crystal space. The ambiguity (= existence of more than one equivalent possibilities) in the stacking of slabs arises from the existence of this local symmetry, which does not appear in the [[space group]] of the structure. The resulting structure can be 'ordered' (''periodic'') or 'disordered' (''non-periodic''), depending on the sequence of local symmetry operations relating pairs of slabs. The set of all the operations valid in the whole crystal space constitutes a [[space group]]; by adding the set of all the operations valid in a subspace of it, one obtains a space [[groupoid]].
 In the OD theory, a central role is played by the '''vicinity condition''' ('''VC'''), which states the geometrical equivalence of layer pairs. The vicinity condition consists of three parts:
-*'''VC &alpha;''': VC layers are either geometrically equivalent or, if not, they are relatively few in kind
+*'''VC''' &alpha;: VC layers are either geometrically equivalent or, if not, they are relatively few in kind.
-*'''VC &beta;''': translation groups of all VC layers are either identical or they have a common subgroup
+*'''VC''' &beta;: translation groups of all VC layers are either identical or they have a common subgroup.
-*'''VC &gamma;''': equivalent sides of equivalent layers are faced by equivalent sides of adjacent layers so that the resulting pairs are equivalent.
+*'''VC''' &gamma;: equivalent sides of equivalent layers are faced by equivalent sides of adjacent layers so that the resulting pairs are equivalent.
 If the position of a layer is uniquely defined by the position of the adjacent layers and by the VC, the resulting structure is fully ordered. If, on the other hand, more than one position is possible that obeys the VC, the resulting structure is an '''OD structure''' and the layers are '''OD layers'''. VC structures may thus be either fully ordered structures or OD structures. All OD structures are [[polytypism|polytypic]]; the reverse may or may not be true. Equivalency depends on the choice of OD layers and also on the definition of [[polytypism]].
 ==The notion of family==
-All OD structures, even of different substances, built according to the same symmetry principle, belong to an '''OD groupoid family'''. The notion OD groupoid family may be compared with the notion space-group type: the infinite space groups are classified in a finite number of space-group types. Similarly, the infinite OD groupoids are classified in a finite number of OD groupoid families. For OD structures of one kind of layers,  there are 400 OD  groupoid families
+All OD structures, even of different substances, built according to the same symmetry principle, belong to an '''OD groupoid family'''. The notion OD groupoid family may be compared with the notion space-group type: the infinite space groups are classified in a finite number of space-group types. Similarly, the infinite OD groupoids are classified in a finite number of OD groupoid families. For OD structures of one kind of layers,  there are 400 OD  groupoid families.
-The OD groupoid family is an abstract family, whose members are the groupoids describing the symmetry of the substances sharing the same symmetry principle. Moving from the abstract to the concrete level, the OD structures of the same substance built on the same structural principle - differing thus for their stacking mode - belong to one and the same family: the members of a family are individual, real structures.
+The OD groupoid family is an abstract family, whose members are the groupoids describing the symmetry of the substances sharing the same symmetry principle. Moving from the abstract to the concrete level, the OD structures of the same substance built on the same structural principle - differing thus in their stacking mode - belong to one and the same family: the members of a family are individual, real structures.
 ==See also==
-* Chapter 9.2 of ''[http://it.iucr.org/C/ International Tables for Crystallography, Volume C]''
+*Chapter 9.2 of ''[http://it.iucr.org/C/ International Tables for Crystallography, Volume C]''
 [[Category:Fundamental crystallography]]

← Older revision		Revision as of 14:44, 23 February 2009
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−	<font color="blue">Structure OD</font> (''Fr''); <font color="black">Struttura OD</font> (''It''); <font color="purple">OD構造</font> (''Ja'').	+	<font color="blue">Structure OD</font> (''Fr''); <font color="red">OD struktur</font> (''Ge''); <font color="black">Struttura OD</font> (''It''); <font color="purple">OD構造</font> (''Ja'').

← Older revision		Revision as of 11:23, 6 January 2009
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−	OD structures consist of slabs with their own symmetry, containing coincidence operations constituting a [[diperiodic group]] (layer group) only within individual slabs. For the entire structure these coincidence operations are only local (partial), i.e. they are valid only in a subspace of the crystal space. The ambiguity (= existence of more than one equivalent possibilities) in the stacking of slabs arises from the existence of this local symmetry, which does not appear in the [[space group]] of the structure. The resulting structure can be "ordered" (''periodic'') or "disordered" (''non-periodic''), depending on the sequence of local symmetry operations relating pairs of slabs. The set of all the operations valid in the whole crystal space constitutes a [[space group]]; by adding the set of all the operations valid in a subspace of it, one obtains a space [[groupoid]].	+	OD structures consist of slabs with their own symmetry, containing coincidence operations constituting a [[subperiodic group\|diperiodic group]] (layer group) only within individual slabs. For the entire structure these coincidence operations are only local (partial), i.e. they are valid only in a subspace of the crystal space. The ambiguity (= existence of more than one equivalent possibilities) in the stacking of slabs arises from the existence of this local symmetry, which does not appear in the [[space group]] of the structure. The resulting structure can be "ordered" (''periodic'') or "disordered" (''non-periodic''), depending on the sequence of local symmetry operations relating pairs of slabs. The set of all the operations valid in the whole crystal space constitutes a [[space group]]; by adding the set of all the operations valid in a subspace of it, one obtains a space [[groupoid]].

	In the OD theory, a central role is played by the '''vicinity condition''' ('''VC'''), which states the geometrical equivalence of layer pairs. The vicinity condition consists of three parts:		In the OD theory, a central role is played by the '''vicinity condition''' ('''VC'''), which states the geometrical equivalence of layer pairs. The vicinity condition consists of three parts:

← Older revision		Revision as of 20:43, 9 April 2008
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−	OD structures consist of slabs with their own symmetry, containing coincidence operations constituting a [[diperiodic group]] ([[layer group]]) only within individual slabs. For the entire structure these coincidence operations are only local (partial), i.e. they are valid only in a subspace of the crystal space. The ambiguity (= existence of more than one equivalent possibilities) in the stacking of slabs arises from the existence of this local symmetry, which does not appear in the [[space group]] of the structure. The resulting structure can be "ordered" (''periodic'') or "disordered" (''non-periodic''), depending on the sequence of local symmetry operations relating pairs of slabs. The set of all the operations valid in the whole crystal space constitutes a [[space group]]; by adding the set of all the operations valid in a subspace of it, one obtains a space [[groupoid]].	+	OD structures consist of slabs with their own symmetry, containing coincidence operations constituting a [[diperiodic group]] (layer group) only within individual slabs. For the entire structure these coincidence operations are only local (partial), i.e. they are valid only in a subspace of the crystal space. The ambiguity (= existence of more than one equivalent possibilities) in the stacking of slabs arises from the existence of this local symmetry, which does not appear in the [[space group]] of the structure. The resulting structure can be "ordered" (''periodic'') or "disordered" (''non-periodic''), depending on the sequence of local symmetry operations relating pairs of slabs. The set of all the operations valid in the whole crystal space constitutes a [[space group]]; by adding the set of all the operations valid in a subspace of it, one obtains a space [[groupoid]].

	In the OD theory, a central role is played by the '''vicinity condition''' ('''VC'''), which states the geometrical equivalence of layer pairs. The vicinity condition consists of three parts:		In the OD theory, a central role is played by the '''vicinity condition''' ('''VC'''), which states the geometrical equivalence of layer pairs. The vicinity condition consists of three parts:

← Older revision		Revision as of 14:26, 27 February 2007
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−	OD structures consist of slabs with their own symmetry, containing coincidence operations constituting a [[diperiodic group]] ([[layer group]]) only within individual slabs. For the entire structure these coincidence operations are only local (partial), i.e. they are valid only in a subspace of the crystal space. The ambiguity (= existence of more than one equivalent possibilities) in the stacking of slabs arises from the existence of this local symmetry, which does not appear in the [[space group]] of the structure. The resulting structure can be "ordered" (''periodic'') or "disordered" (''non-periodic''), depending on the sequence of local symmetry operations relating pairs of slabs. The set of all the operations valid in the whole crystal space constitutes a [[space group]]; by adding the set of all the operations valid in a subspace of it, one obtains a [[~~space~~ groupoid]]	+	OD structures consist of slabs with their own symmetry, containing coincidence operations constituting a [[diperiodic group]] ([[layer group]]) only within individual slabs. For the entire structure these coincidence operations are only local (partial), i.e. they are valid only in a subspace of the crystal space. The ambiguity (= existence of more than one equivalent possibilities) in the stacking of slabs arises from the existence of this local symmetry, which does not appear in the [[space group]] of the structure. The resulting structure can be "ordered" (''periodic'') or "disordered" (''non-periodic''), depending on the sequence of local symmetry operations relating pairs of slabs. The set of all the operations valid in the whole crystal space constitutes a [[space group]]; by adding the set of all the operations valid in a subspace of it, one obtains a space [[groupoid]].

	In the OD theory, a central role is played by the '''vicinity condition''' ('''VC'''), which states the geometrical equivalence of layer pairs. The vicinity condition consists of three parts:		In the OD theory, a central role is played by the '''vicinity condition''' ('''VC'''), which states the geometrical equivalence of layer pairs. The vicinity condition consists of three parts:

← Older revision		Revision as of 11:54, 27 February 2007
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		+	<font color="blue">Structure OD</font> (''Fr''); <font color="black">Struttura OD</font> (''It''); <font color="purple">OD構造</font> (''Ja'').
		+
		+
	OD structures consist of slabs with their own symmetry, containing coincidence operations constituting a [[diperiodic group]] ([[layer group]]) only within individual slabs. For the entire structure these coincidence operations are only local (partial), i.e. they are valid only in a subspace of the crystal space. The ambiguity (= existence of more than one equivalent possibilities) in the stacking of slabs arises from the existence of this local symmetry, which does not appear in the [[space group]] of the structure. The resulting structure can be "ordered" (''periodic'') or "disordered" (''non-periodic''), depending on the sequence of local symmetry operations relating pairs of slabs. The set of all the operations valid in the whole crystal space constitutes a [[space group]]; by adding the set of all the operations valid in a subspace of it, one obtains a [[space groupoid]]		OD structures consist of slabs with their own symmetry, containing coincidence operations constituting a [[diperiodic group]] ([[layer group]]) only within individual slabs. For the entire structure these coincidence operations are only local (partial), i.e. they are valid only in a subspace of the crystal space. The ambiguity (= existence of more than one equivalent possibilities) in the stacking of slabs arises from the existence of this local symmetry, which does not appear in the [[space group]] of the structure. The resulting structure can be "ordered" (''periodic'') or "disordered" (''non-periodic''), depending on the sequence of local symmetry operations relating pairs of slabs. The set of all the operations valid in the whole crystal space constitutes a [[space group]]; by adding the set of all the operations valid in a subspace of it, one obtains a [[space groupoid]]

@@ Line 6: / Line 6: @@
 *'''VC &gamma;''': equivalent sides of equivalent layers are faced by equivalent sides of adjacent layers so that the resulting pairs are equivalent.
-If the position of a layer is uniquely defined by the position of the adjacent layers and by the VC, the resulting structure is fully ordered. If, on the other hand, more than one position is possible that obeys the VC, the resulting structure is an ‘'''OD structure''' and the layers are '''OD layers'''. VC structures may thus be either fully ordered structures or OD structures. All OD structures are [[polytypism|polytypic]]; the reverse may or may not be true. Equivalency depends on the choice of OD layers and also on the definition of [[polytypism]].
+If the position of a layer is uniquely defined by the position of the adjacent layers and by the VC, the resulting structure is fully ordered. If, on the other hand, more than one position is possible that obeys the VC, the resulting structure is an '''OD structure''' and the layers are '''OD layers'''. VC structures may thus be either fully ordered structures or OD structures. All OD structures are [[polytypism|polytypic]]; the reverse may or may not be true. Equivalency depends on the choice of OD layers and also on the definition of [[polytypism]].
 All OD structures, even of different substances, built according to the same symmetry principle, belong to an OD [[groupoid family]]: this is an abstract family, whose members are the groupoids describing the symmetry of the substances sharing the same symmetry principle. Then, the OD structures of the same substance built on the same structural principle belong to one and the same family: the members of a family are individual, real structures.