Arithmetic crystal class

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Classe cristalline arithmétique (Fr). Arithmetische Kristallklasse (Ge). Clase cristallina aritmética (Sp). Classe cristallina aritmetica (It). 代数的結晶類 (Ja).


Definition

The arithmetic crystal classes are obtained in an elementary fashion by combining the geometric crystal classes 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: 2P, 2C, mP, mC, 2/mP, 2/mC (note that in the space group symbol the order is inversed: P2, C2, 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, mm2C and 2mmC (or mm2A, 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 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 1.3.4.4.1 of International Tables for Crystallography, Volume A, 6th edition.

List of arithmetic crystal classes in three dimensions

Three-dimensional arithmetic crystal classes.
Crystal systems Crystal class
Geometric Arithmetic
Number Symbol
Triclinic 1 1 1P
 {\bar 1} 2  {\bar 1}P
Monoclinic 2 3 2P
m 4 2C
5 mP
2 / m 6 mC
7 2 / mP
8 2 / mC
Orthorhombic 222 9 222P
10 222C
11 222F
12 222I
mm2 13 mm2P
14 mm2C
15 2mmC
(mm2A)
16 mm2F
17 mm2I
mmm 18 mmmP
19 mmmC
20 mmmF
21 mmmI
Tetragonal 4 22 4P
23 4I
 {\bar 4} 24  {\bar 4}P
25  {\bar 4}I
4 / m 26 4 / mP
27 4 / mI
422 28 422P
29 422I
4mm 30 4mmP
31 4mmI
 {\bar 4}m2 32  {\bar 4}2mP
33  {\bar 4}m2P
34  {\bar 4}m2I
35  {\bar 4}2mI
4 / mmm 36 4 / mmmP
37 4 / mmmI
Trigonal 3 38 3P
39 3R
 {\bar 3} 40  {\bar 3}P
41  {\bar 3}R
32 42 312P
43 321P
44 32R
3m 45 3m1P
46 31mP
47 3mR
 {\bar 3}m 48  {\bar 3}1mP
49  {\bar 3}m1P
50  {\bar 3}mR
Hexagonal 6 51 6P
 {\bar 6} 52  {\bar 6}P
6 / m 53 6 / mP
622 54 622P
6mm 55 6mmP
 {\bar 6}m2 56  {\bar 6}2mP
57  {\bar 6}m2P
6 / mmm 58 6 / mmmP
Cubic 23 59 23P
60 23F
61 23I
 m{\bar 3} 62  m{\bar 3}P
63  m{\bar 3}F
64  m{\bar 3}I
432 65 432P
66 432F
67 432I
 {\bar 4}3m 68  {\bar 4}3m P
69  {\bar 4}3m F
70  {\bar 4}3m I
 m{\bar 3}m 71  m{\bar 3}mP
72  m{\bar 3}mF
73  m{\bar 3}mI

See also

  • Geometric crystal class
  • Chapter 1.3.4.4.1 of International Tables for Crystallography, Volume A, 6th edition
  • Chapters 1.3.4 and 1.5.3 of International Tables for Crystallography, Volume B
  • Chapter 1.4 of International Tables for Crystallography, Volume C