Lattice complex - Revision history

BrianMcMahon: Tidied translations.

2017-11-14T17:51:47Z

Tidied translations.

BrianMcMahon: Style edits to align with printed edition

2017-05-15T14:45:34Z

Style edits to align with printed edition

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

2017-04-10T16:01:21Z

‎See also: 6th edition of ITA

MassimoNespolo: typo

2015-01-26T09:49:52Z

typo

MassimoNespolo: /* Symbols of invariant lattice complexes */ a couple of typos spotted by Gervais

2014-07-15T07:38:02Z

‎Symbols of invariant lattice complexes: a couple of typos spotted by Gervais

BrianMcMahon: Fixed some broken HTML in the table.

2012-02-06T15:49:25Z

Fixed some broken HTML in the table.

MassimoNespolo: /* See also */ link

2009-02-13T15:25:34Z

‎See also: link

MassimoNespolo: /* Classification */ typo and rearrangement

2009-02-13T15:23:16Z

‎Classification: typo and rearrangement

MassimoNespolo: /* Definition */ new section : classification

2009-02-13T15:22:47Z

‎Definition: new section : classification

MassimoNespolo: /* Definition */ link

2009-02-13T15:21:14Z

‎Definition: link

← Older revision		Revision as of 17:51, 14 November 2017
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−	<~~Font~~ color="red"> Gittercomplex</~~Font~~> (''Ge'').	+	<font color="red">Gittercomplex</font> (''Ge'').

@@ Line 6: / Line 6: @@
 A ''lattice complex'' is the set of all [[point configuration]]s that may be generated within one type of [[Wyckoff set]]. All [[Wyckoff position]]s, Wyckoff sets and types of Wyckoff sets that generate the same set of point configurations are assigned to the same lattice complex.
-Concretely, two Wyckoff positions are assigned to the same lattice complex if there is a suitable transformation that maps the point configurations of the two Wyckoff positions onto each other and if their space groups belong to the same [[crystal family]]. The 72 (in E<sup>2</sup>) or 1731 (in E<sup>3</sup>) Wyckoff positions are classified in 51 (E<sup>2</sup>) or 1128 (E<sup>3</sup>) types of Wyckoff sets. They are assigned to 30 (E<sup>2</sup>) or 402 (E<sup>3</sup>) lattice complexes.
+Concretely, two Wyckoff positions are assigned to the same lattice complex if there is a suitable transformation that maps the point configurations of the two Wyckoff positions onto each other and if their space groups belong to the same [[crystal family]]. The 72 (in ''E''<sup>2</sup>) or 1731 (in ''E''<sup>3</sup>) Wyckoff positions are classified in 51 (''E''<sup>2</sup>) or 1128 (''E''<sup>3</sup>) types of Wyckoff sets. They are assigned to 30 (''E''<sup>2</sup>) or 402 (''E''<sup>3</sup>) lattice complexes.
 The name ''lattice complex'' comes from the fact that an assemblage of points that are equivalent with respect to a group of symmetry operations including lattice translations can be visualized as a set of equivalent lattices.
@@ Line 27: / Line 27: @@
 <tr><td>B</td><td>000, <math>\frac{1}{2}</math>0<math>\frac{1}{2}</math></td><td>2</td></tr>
 <tr><td>C</td><td>000, <math>\frac{1}{2}</math><math>\frac{1}{2}</math>0</td><td>2</td></tr>
-<tr><td>D (Diamond structure)</td><td>000. <math>\frac{1}{4}</math><math>\frac{1}{4}</math><math>\frac{1}{4}</math>, 0<math>\frac{1}{2}</math><math>\frac{1}{2}</math>, <math>\frac{1}{2}</math>0,<math>\frac{1}{2}</math>, <math>\frac{1}{2}</math><math>\frac{1}{2}</math>0,<br><math>\frac{3}{4}</math><math>\frac{3}{4}</math><math>\frac{1}{4}</math>, <math>\frac{3}{4}</math><math>\frac{1}{4}</math><math>\frac{3}{4}</math>, <math>\frac{1}{4}</math><math>\frac{3}{4}</math><math>\frac{3}{4}</math></td><td>8</td></tr>
+<tr><td>D (Diamond structure)</td><td>000, <math>\frac{1}{4}</math><math>\frac{1}{4}</math><math>\frac{1}{4}</math>, 0<math>\frac{1}{2}</math><math>\frac{1}{2}</math>, <math>\frac{1}{2}</math>0,<math>\frac{1}{2}</math>, <math>\frac{1}{2}</math><math>\frac{1}{2}</math>0,<br><math>\frac{3}{4}</math><math>\frac{3}{4}</math><math>\frac{1}{4}</math>, <math>\frac{3}{4}</math><math>\frac{1}{4}</math><math>\frac{3}{4}</math>, <math>\frac{1}{4}</math><math>\frac{3}{4}</math><math>\frac{3}{4}</math></td><td>8</td></tr>
 <tr><td>E (Hexagonal close packing; Italian: Esagonale)</td><td><math>\frac{1}{3}</math><math>\frac{2}{3}</math><math>\frac{1}{4}</math>, <math>\frac{2}{3}</math><math>\frac{1}{3}</math><math>\frac{3}{4}</math></td><td>2</td></tr>
 <tr><td>F</td><td>000, 0<math>\frac{1}{2}</math><math>\frac{1}{2}</math>, <math>\frac{1}{2}</math>0<math>\frac{1}{2}</math>, <math>\frac{1}{2}</math><math>\frac{1}{2}</math>0</td><td>4</td></tr>
 <tr><td>G (Graphene)</td><td><math>\frac{1}{3}</math><math>\frac{2}{3}</math>0, <math>\frac{2}{3}</math><math>\frac{1}{3}</math>0 (hexagonal cell)</td><td>2</td></tr>
 <tr><td>I </td><td>000, <math>\frac{1}{2}</math><math>\frac{1}{2}</math><math>\frac{1}{2}</math></td><td>2</td></tr>
-<tr><td>J ("Jack")</td><td>0<math>\frac{1}{2}</math><math>\frac{1}{2}</math>, <math>\frac{1}{2}</math>0<math>\frac{1}{2}</math>, <math>\frac{1}{2}</math><math>\frac{1}{2}</math>0</td><td>3</td></tr>
+<tr><td>J ('Jack')</td><td>0<math>\frac{1}{2}</math><math>\frac{1}{2}</math>, <math>\frac{1}{2}</math>0<math>\frac{1}{2}</math>, <math>\frac{1}{2}</math><math>\frac{1}{2}</math>0</td><td>3</td></tr>
-<tr><td>J* = J+<math>\frac{1}{2}</math><math>\frac{1}{2}</math><math>\frac{1}{2}</math>J</td><td>0<math>\frac{1}{2}</math><math>\frac{1}{2}</math>, <math>\frac{1}{2}</math>0<math>\frac{1}{2}</math>, <math>\frac{1}{2}</math><math>\frac{1}{2}</math>0<br><math>\frac{1}{2}</math>00, 0<math>\frac{1}{2}</math>0, 00<math>\frac{1}{2}</math></td><td>6</td></tr>
+<tr><td>J* = J+<math>\frac{1}{2}</math><math>\frac{1}{2}</math><math>\frac{1}{2}</math>J</td><td>0<math>\frac{1}{2}</math><math>\frac{1}{2}</math>, <math>\frac{1}{2}</math>0<math>\frac{1}{2}</math>, <math>\frac{1}{2}</math><math>\frac{1}{2}</math>0,<br><math>\frac{1}{2}</math>00, 0<math>\frac{1}{2}</math>0, 00<math>\frac{1}{2}</math></td><td>6</td></tr>
 <tr><td>M</td><td>J in rhombohedral cell</td><td>3</td></tr>
-<tr><td>N (Kagome Net)</td><td><math>\frac{1}{2}</math>00, 0<math>\frac{1}{2}</math>0, <math>\frac{1}{2}</math><math>\frac{1}{2}</math>0 (hexagonal cell)</td><td>2</td></tr>
+<tr><td>N (Kagome net)</td><td><math>\frac{1}{2}</math>00, 0<math>\frac{1}{2}</math>0, <math>\frac{1}{2}</math><math>\frac{1}{2}</math>0 (hexagonal cell)</td><td>2</td></tr>
 <tr><td>P</td><td>000</td><td>1</td></tr>
 <tr><td><sup>+</sup>Q</td><td><math>\frac{1}{2}</math>00, 0<math>\frac{1}{2}</math><math>\frac{2}{3}</math>, <math>\frac{1}{2}</math><math>\frac{1}{2}</math><math>\frac{1}{3}</math> (hexagonal cell)</td><td>3</td></tr>
@@ Line 60: / Line 60: @@
 == Symbols of variant lattice complexes ==
-The symbol of a lattice complex with degrees of freedom may contain up to four pats:
+The symbol of a lattice complex with degrees of freedom may contain up to four parts:
 * a shift vector;
 * the distribution symmetry;
@@ Line 71: / Line 71: @@
 == See also ==
-*Chapter 3.4 of [http://it.iucr.org/A/ ''International Tables of Crystallography, Section A'', 6<sup>th</sup> edition]<br>
+*Chapter 3.4 of ''International Tables for Crystallography, Volume A'', 6th edition
 [[Category:Fundamental crystallography]]

@@ Line 71: / Line 71: @@
 == See also ==
+*Chapter 3.4 of [http://it.iucr.org/A/ ''International Tables of Crystallography, Section A'', 6<sup>th</sup> edition]<br>
 [[Category:Fundamental crystallography]]

@@ Line 6: / Line 6: @@
 A ''lattice complex'' is the set of all [[point configuration]]s that may be generated within one type of [[Wyckoff set]]. All [[Wyckoff position]]s, Wyckoff sets and types of Wyckoff sets that generate the same set of point configurations are assigned to the same lattice complex.
-Concretely, two Wyckoff positions are assigned to the same lattice complex if there is a suitable trasformation that maps the point configurations of the two Wyckoff positions onto each other and if their space groups belong to the same [[crystal family]]. The 72 (in E<sup>2</sup>) or 1731 (in E<sup>3</sup>) Wyckoff positions are classified in 51 (E<sup>2</sup>) or 1128 (E<sup>3</sup>) types of Wyckoff sets. They are assigned to 30 (E<sup>2</sup>) or 402 (E<sup>3</sup>) lattice complexes.
+Concretely, two Wyckoff positions are assigned to the same lattice complex if there is a suitable transformation that maps the point configurations of the two Wyckoff positions onto each other and if their space groups belong to the same [[crystal family]]. The 72 (in E<sup>2</sup>) or 1731 (in E<sup>3</sup>) Wyckoff positions are classified in 51 (E<sup>2</sup>) or 1128 (E<sup>3</sup>) types of Wyckoff sets. They are assigned to 30 (E<sup>2</sup>) or 402 (E<sup>3</sup>) lattice complexes.
 The name ''lattice complex'' comes from the fact that an assemblage of points that are equivalent with respect to a group of symmetry operations including lattice translations can be visualized as a set of equivalent lattices.

@@ Line 45: / Line 45: @@
 <tr><td>T</td><td><math>\frac{1}{8}</math><math>\frac{1}{8}</math><math>\frac{1}{8}</math>, <math>\frac{3}{8}</math><math>\frac{3}{8}</math><math>\frac{1}{8}</math>, <math>\frac{1}{8}</math><math>\frac{5}{8}</math><math>\frac{5}{8}</math>, <math>\frac{7}{8}</math><math>\frac{7}{8}</math><math>\frac{1}{8}</math>, <math>\frac{7}{8}</math><math>\frac{5}{8}</math><math>\frac{3}{8}</math> + permutations</td><td>16</td></tr>
 <tr><td><sup>+</sup>V</td><td><math>\frac{1}{4}</math><math>\frac{1}{8}</math>0, <math>\frac{3}{4}</math><math>\frac{3}{8}</math>0, <math>\frac{3}{4}</math><math>\frac{5}{8}</math><math>\frac{1}{2}</math>, <math>\frac{7}{8}</math><math>\frac{1}{2}</math><math>\frac{1}{4}</math> + permutations</td><td>12</td></tr>
-<tr><td>'V</td><td>V·-1</td><td>12</td></tr>
+<tr><td><sup>-</sup>V</td><td><sup>+</sup>V·-1</td><td>12</td></tr>
-<tr><td>V*</td><td>V + 'V</td><td>24</td></tr>
+<tr><td>V*</td><td><sup>+</sup>V + <sup>-</sup>V</td><td>24</td></tr>
 <tr><td>W</td><td><math>\frac{1}{2}</math><math>\frac{1}{4}</math>0, 0<math>\frac{1}{2}</math><math>\frac{3}{4}</math> + permutations</td><td>6</td></tr>
 <tr><td>W'</td><td><math>\frac{1}{2}</math><math>\frac{1}{2}</math><math>\frac{1}{2}</math>W</td><td>6</td></tr>

@@ Line 54: / Line 54: @@
 <tr><td><sup>-</sup>Y</td><td><sup>+</sup>Y·-1</td><td>4</td></tr>
 <tr><td><sup>-</sup>Y'</td><td><math>\frac{1}{2}</math><math>\frac{1}{2}</math><math>\frac{1}{2}</math><sup>-</sup>Y</td><td>4</td></tr>
-<tr><td><sup>+</sup>Y*</td><td><sup>+</sup>Y+<sup>+</sup>Y'<td>8</td></td></tr>
+<tr><td><sup>+</sup>Y*</td><td><sup>+</sup>Y+<sup>+</sup>Y'<td>8</td></tr>
-<tr><td><sup>-</sup>Y*</td><td><sup>-</sup>Y+<sup>-</sup>Y'<td>8</td></td></tr>
+<tr><td><sup>-</sup>Y*</td><td><sup>-</sup>Y+<sup>-</sup>Y'<td>8</td></tr>
-<tr><td>Y**</td><td><sup>+</sup>Y*+<sup>-</sup>Y*<td>16</td></td></tr>
+<tr><td>Y**</td><td><sup>+</sup>Y*+<sup>-</sup>Y*<td>16</td></tr>
 </table>

@@ Line 72: / Line 72: @@
 == See also ==
-Chapter 14 of ''International Tables of Crystallography, Section A''<br>
+Chapter 14 of [http://it.iucr.org/A/ ''International Tables of Crystallography, Section A'']<br>
 [[Category:Fundamental crystallography]]

@@ Line 11: / Line 11: @@
 ==Classification==
-If a first lattice complex formes a true subset of a second one, ''i''.''e''. if each point configuration of the first lattice complex also belongs to the second one, then the first one is called a '''limiting complex''' of the second one and the second complex is called a '''comprehensive complex''' of the first one.
+If a lattice complex can be generated in different space-group types, one of them stands out because its corresponding [[Wyckoff position]]s show the highest [[site symmetry]]. This is called the ''characteristic space-group type of the lattice complex''. The characteristic space-group type and the corresponding oriented [[site symmetry]] express the common symmetry properties of all [[point configuration]]s of a lattice complex. In the symbol of a lattice complex, however, instead of the [[site symmetry]], the Wyckoff letter of one of the [[Wyckoff position]]s with that [[site symmetry]] is given. This [[Wyckoff position]] is called the ''characteristic Wyckoff position of the lattice complex''.
-If a lattice complex can be generated in different space-group types, one of them stands out because its corresponding [[Wyckoff position]]s show the highest [[site symmetry]]. This is called the ''characteristic space-group type of the lattice complex''. The characteristic space-group type and the corresponding oriented [[site symmetry]] express the common symmetry properties of all [[point configuration]]s of a lattice complex. In the symbol of a lattice complex, however, instead of the [[site symmetry]], the Wyckoff letter of one of the [[Wyckoff position]]s with that [[site symmetry]] is given. This [[Wyckoff position]] is called the ''characteristic Wyckoff position of the lattice complex''.
+If a first lattice complex forms a true subset of a second one, ''i''.''e''. if each point configuration of the first lattice complex also belongs to the second one, then the first one is called a '''limiting complex''' of the second one and the second complex is called a '''comprehensive complex''' of the first one.
 Lattice complexes are called ''invariant'' if they can occupy a parameterless position in a space group. The points in an invariant lattice complex can be split into different assemblages of equivalent points, the sum of which constitute a ''variant'' lattice complex (also termed a ''lattice complex with degrees of freedom''). Variant lattice complexes are classified into ''univariant'', ''bivariant'' and ''trivariant'' according to the number of parameters that can be varied independently.

← Older revision		Revision as of 15:21, 13 February 2009
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	A ''lattice complex'' is the set of all [[point configuration]]s that may be generated within one type of [[Wyckoff set]]. All [[Wyckoff position]]s, Wyckoff sets and types of Wyckoff sets that generate the same set of point configurations are assigned to the same lattice complex.		A ''lattice complex'' is the set of all [[point configuration]]s that may be generated within one type of [[Wyckoff set]]. All [[Wyckoff position]]s, Wyckoff sets and types of Wyckoff sets that generate the same set of point configurations are assigned to the same lattice complex.

−	Concretely, two Wyckoff positions are assigned to the same lattice complex if there is a suitable trasformation that maps the point configurations of the two Wyckoff positions onto each other and if their space groups belong to the same crystal family. The 72 (in E<sup>2</sup>) or 1731 (in E<sup>3</sup>) Wyckoff positions are classified in 51 (E<sup>2</sup>) or 1128 (E<sup>3</sup>) types of Wyckoff sets. They are assigned to 30 (E<sup>2</sup>) or 402 (E<sup>3</sup>) lattice complexes.	+	Concretely, two Wyckoff positions are assigned to the same lattice complex if there is a suitable trasformation that maps the point configurations of the two Wyckoff positions onto each other and if their space groups belong to the same [[crystal family]]. The 72 (in E<sup>2</sup>) or 1731 (in E<sup>3</sup>) Wyckoff positions are classified in 51 (E<sup>2</sup>) or 1128 (E<sup>3</sup>) types of Wyckoff sets. They are assigned to 30 (E<sup>2</sup>) or 402 (E<sup>3</sup>) lattice complexes.