Arithmetic crystal class - Revision history

ShuhaoCao: Add two-dimensional arithmetic crystal classes

2025-07-06T13:26:00Z

Add two-dimensional arithmetic crystal classes

GervaisChapuis at 13:48, 14 March 2025

2025-03-14T13:48:38Z

BrianMcMahon: Tidied translations and corected Spanish (U. Mueller)

2017-11-08T18:04:56Z

Tidied translations and corected Spanish (U. Mueller)

MassimoNespolo: another "2" was missing!

2017-07-03T10:54:00Z

another "2" was missing!

MassimoNespolo: typo ("P" missing)

2017-06-30T17:41:03Z

typo ("P" missing)

MassimoNespolo: typo ("2" missing)

2017-06-30T17:36:40Z

typo ("2" missing)

BrianMcMahon at 12:24, 20 May 2017

2017-05-20T12:24:18Z

BrianMcMahon: Style edits to align with printed edition

2017-05-12T12:22:53Z

Style edits to align with printed edition

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

2017-04-10T14:47:23Z

‎See also: 6th edition of ITA

MassimoNespolo: spaces

2016-12-17T20:47:07Z

spaces

@@ Line 7: / Line 7: @@
 The group-theoretical definition of the arithmetic crystal classes is given in Chapter 1.3.4.4.1 of ''International Tables for Crystallography, Volume A'', 6th edition.
 == List of arithmetic crystal classes in three dimensions ==
-<table border cellspacing=0 cellpadding=5 align=center>
+<table border cellspacing=0 cellpadding=5 style="text-align: center;">
- <caption align=top> '''Three-dimensional arithmetic crystal classes.''' </caption>
+  <caption>
- <tr align=center>
+    '''The three-dimensional arithmetic crystal classes'''
-     <th rowspan=3> Crystal systems</th>
+  </caption>
-     <th colspan=3> Crystal class</th>
+  <tr>
- </tr>
+     <th rowspan=3 style="text-align: center;"> Crystal system </th>
- <tr align=center>
+     <th colspan=3 style="text-align: center;"> Crystal class </th>
-     <th rowspan=1 > Geometric</th>
+  </tr>
-     <th colspan=2> Arithmetic</th>
+  <tr>
- </tr>
+     <th rowspan=2 style="text-align: center;"> Geometric </th>
- <tr align=center>
+     <th colspan=2 style="text-align: center;"> Arithmetic </th>
-    <th> </th>
+  </tr>
-     <th> Number </th>
+  <tr>
-     <th> Symbol </th>
+     <th style="text-align: center;"> Number </th>
    </tr>
   <tr align=center>

@@ Line 38: / Line 38: @@
   <tr align=center>
     <td rowspan=6>Monoclinic</td>
-    <td><math> 2 </math></td>
+    <td rowspan=2><math> 2 </math></td>
     <td> 3 </td>
     <td><math> 2P </math></td>
     </tr>
   <tr align=center>
      <td rowspan=2><math> m </math></td>
-   <td> 4 </td>
+    <td> 5 </td>
-   <td><math> 2C </math></td>
+    <td><math> mP </math></td>
 </tr>
-<tr align=center>
+ <tr align=center>
     <td> 6 </td>
     <td><math> mC </math></td>
 </tr>
 <tr align=center>
     <td> 7 </td>
     <td><math> 2/m P </math></td>

← Older revision		Revision as of 18:04, 8 November 2017
Line 1:		Line 1:
−	<~~Font Color~~="blue">Classe cristalline arithmétique</~~Font~~> (''Fr''). <~~Font Color~~="red">Arithmetische Kristallklasse</~~Font~~> (''Ge''). <~~Font Color~~="~~green~~">~~Clase~~ cristallina ~~aritmética~~</~~Font~~> (''Sp''). <~~Font~~ color="~~black~~">~~Classe cristallina aritmetica~~</~~Font~~> (''It''). <~~Font~~ color="~~purple~~">~~代数的結晶類~~</~~Font~~> (''Ja'').	+	<font color="blue">Classe cristalline arithmétique</font> (''Fr''). <font color="red">Arithmetische Kristallklasse</font> (''Ge''). <font color="black">Classe cristallina aritmetica</font> (''It''). <font color="purple">代数的結晶類</font> (''Ja''). <font color="green">Clase cristalina aritmética</font> (''Sp'').

@@ Line 83: / Line 83: @@
 </tr>
 <tr align=center>
-     <td rowspan=6><math> mm </math></td>
+     <td rowspan=6><math> mm2 </math></td>
     <td> 13 </td>
     <td><math> mm2P </math></td>

@@ Line 291: / Line 291: @@
     <td><math> 6/mmm </math></td>
     <td> 58 </td>
-    <td><math> 6/mmm </math></td>
+    <td><math> 6/mmmP </math></td>
     </tr>

@@ Line 170: / Line 170: @@
     </tr>
 <tr align=center>
-    <td rowspan=4><math> {\bar 4}m </math></td>
+    <td rowspan=4><math> {\bar 4}m2 </math></td>
     <td> 32 </td>
     <td><math> {\bar 4}2mP </math></td>
@@ Line 280: / Line 280: @@
     </tr>
 <tr align=center>
-    <td rowspan=2><math> {\bar 6}m </math></td>
+    <td rowspan=2><math> {\bar 6}m2 </math></td>
     <td> 56 </td>
     <td><math> {\bar 6}2mP </math></td>

@@ Line 6: / Line 6: @@
 The '''arithmetic crystal classes''' are obtained in an elementary fashion by combining the [[geometric crystal class]]es and the corresponding types of Bravais lattices. For instance, in the monoclinic system, there are three geometric crystal classes, 2, ''m'' and 2/''m'', and two types of Bravais lattices, ''P'' and ''C''. There are therefore six monoclinic arithmetic crystal classes. Their symbols are obtained by juxtaposing the symbol of the geometric class and that of the Bravais lattice, in that order: 2''P'', 2''C'', ''mP'', ''mC'', 2/''mP'', 2/''mC'' (note that in the space group symbol the order is inversed: ''P''2, ''C''2, ''etc.''). In some cases, the centring vectors of the Bravais lattice and some symmetry elements of the crystal class may or may not be parallel; for instance, in the geometric crystal class ''mm'' with the Bravais lattice ''C'', the centring vector and the two-fold axis may be perpendicular or coplanar, giving rise to two different arithmetic crystal classes, ''mm''2''C'' and 2''mmC'' (or ''mm''2''A'', since it is usual to orient the two-fold axis parallel to ''c''), respectively. There are 13 two-dimensional arithmetic crystal classes and 73 three-dimensional arithmetic crystal classes that are listed in the attached [[Arithmetic crystal class| table]]. Space groups belonging to the same geometric crystal class and with the same type of Bravais lattice belong to the same arithmetic crystal class; these are therefore in one to one correspondence with the [[symmorphic space groups]].
-The group-theoretical definition of the arithmetic crystal classes is given in Chapter 8.2.3 of ''International Tables for Crystallography, Volume A''.
+The group-theoretical definition of the arithmetic crystal classes is given in Chapter 1.3.4.4.1 of ''International Tables for Crystallography, Volume A'', 6th edition.
 == List of arithmetic crystal classes in three dimensions ==

@@ Line 1: / Line 1: @@
-<Font Color="blue">Classe cristalline arithmétique</Font> (''Fr''); <Font Color="red">Arithmetische Kristallklasse</Font> (''Ge''); <Font Color="green">Clase cristallina aritmética</Font> (''Sp''); <Font color="black">Classe cristallina aritmetica</Font> (''It''); <Font color="purple">代数的結晶類</Font> (''Ja'').
+<Font Color="blue">Classe cristalline arithmétique</Font> (''Fr''). <Font Color="red">Arithmetische Kristallklasse</Font> (''Ge''). <Font Color="green">Clase cristallina aritmética</Font> (''Sp''). <Font color="black">Classe cristallina aritmetica</Font> (''It''). <Font color="purple">代数的結晶類</Font> (''Ja'').
 == Definition ==
-The '''arithmetic crystal classes''' are obtained in an elementary fashion by combining the [[geometric crystal class]]es and the corresponding types of Bravais lattices. For instance, in the monoclinic system, there are three geometric crystal classes, 2, ''m'' and 2/''m'', and two types of Bravais lattices, ''P'' and ''C''. There are therefore six monoclinic arithmetic crystal classes. Their symbols are obtained by juxtaposing the symbol of the geometric class and that of the Bravais lattice, in that order: 2''P'', 2''C'', ''mP'', ''mC'', 2/''mP'', 2/''mC'' (note that in the space group symbol the order is inversed: ''P''2, ''C''2, etc...). In some cases, the centring vectors of the Bravais lattice and some symmetry elements of the crystal class may or may not be parallel; for instance, in the geometric crystal class ''mm'' with the Bravais lattice ''C'', the centring vector and the two-fold axis may be perpendicular or coplanar, giving rise to two different arithmetic crystal classes, ''mm''2''C'' and 2''mmC'' (or ''mm''2''A'', since it is usual to orient the two-fold axis parallel to ''c''), respectively. There are 13 two-dimensional arithmetic crystal classes and 73 three-dimensional arithmetic crystal classes that are listed in the attached [[Arithmetic crystal class| table]]. Space groups belonging to the same geometric crystal class and with the same type of Bravais lattice belong to the same arithmetic crystal class; these are therefore in one to one correspondence with the [[symmorphic space groups]].
+The '''arithmetic crystal classes''' are obtained in an elementary fashion by combining the [[geometric crystal class]]es and the corresponding types of Bravais lattices. For instance, in the monoclinic system, there are three geometric crystal classes, 2, ''m'' and 2/''m'', and two types of Bravais lattices, ''P'' and ''C''. There are therefore six monoclinic arithmetic crystal classes. Their symbols are obtained by juxtaposing the symbol of the geometric class and that of the Bravais lattice, in that order: 2''P'', 2''C'', ''mP'', ''mC'', 2/''mP'', 2/''mC'' (note that in the space group symbol the order is inversed: ''P''2, ''C''2, ''etc.''). In some cases, the centring vectors of the Bravais lattice and some symmetry elements of the crystal class may or may not be parallel; for instance, in the geometric crystal class ''mm'' with the Bravais lattice ''C'', the centring vector and the two-fold axis may be perpendicular or coplanar, giving rise to two different arithmetic crystal classes, ''mm''2''C'' and 2''mmC'' (or ''mm''2''A'', since it is usual to orient the two-fold axis parallel to ''c''), respectively. There are 13 two-dimensional arithmetic crystal classes and 73 three-dimensional arithmetic crystal classes that are listed in the attached [[Arithmetic crystal class| table]]. Space groups belonging to the same geometric crystal class and with the same type of Bravais lattice belong to the same arithmetic crystal class; these are therefore in one to one correspondence with the [[symmorphic space groups]].
-The group-theoretical definition of the arithmetic crystal classes is given in Section 8.2.3 of ''International Tables of Crystallography, Volume A''.
+The group-theoretical definition of the arithmetic crystal classes is given in Chapter 8.2.3 of ''International Tables for Crystallography, Volume A''.
 == List of arithmetic crystal classes in three dimensions ==
@@ Line 366: / Line 366: @@
 == See also ==
 *[[Geometric crystal class]]
-*Section 1.3.4.4.1 of ''International Tables of Crystallography, Volume A'', 6<sup>th</sup> edition<br>
+*Chapter 1.3.4.4.1 of ''International Tables for Crystallography, Volume A'', 6th edition<br>
-*Sections 1.3.4 and 1.5.3 of ''International Tables of Crystallography, Volume B''<br>
+*Chapters 1.3.4 and 1.5.3 of ''International Tables for Crystallography, Volume B''<br>
-*Section 1.4 of ''International Tables of Crystallography, Volume C''<br>
+*Chapter 1.4 of ''International Tables for Crystallography, Volume C''<br>
 [[Category:Fundamental crystallography]]

← Older revision		Revision as of 20:47, 17 December 2016
Line 1:		Line 1:
−	<Font Color="blue"> Classe cristalline arithmétique </Font> (''Fr.''). <Font Color="red"> Arithmetische Kristallklasse </Font>(''Ge''). <Font Color="green">Clase cristallina aritmética</Font>(''Sp''). <Font color="black"> Classe cristallina aritmetica </Font>(''It''). <Font color="purple"> 代数的結晶類</Font> (''Ja'')	+	<Font Color="blue">Classe cristalline arithmétique</Font> (''Fr''); <Font Color="red">Arithmetische Kristallklasse</Font> (''Ge''); <Font Color="green">Clase cristallina aritmética</Font> (''Sp''); <Font color="black">Classe cristallina aritmetica</Font> (''It''); <Font color="purple">代数的結晶類</Font> (''Ja'').