Arithmetic crystal classes

Other languages

Classes cristallines arithmétiques (Fr). Arithmetische Kristall Klassen (Ge). Clases cristallinas aritméticas (Sp).

Definition

The arithmetic crystal classes are obtained in an elementary fashion by combining the geometrical crystal classes and the corrresponding 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...

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 $2/m P$
8 $2/m C$
Orthorhombic $222$ 9 $222 P$
10 $222 C$
11 $222 F$
12 $222 I$
$mm$ 13 $mm2 P$
14 $mm2 C$
15 $2mmC$
$(Amm2)$
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 $4mmI$
${\bar 4}m$ 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 $6mmP$
${\bar 6}m$ 56 ${\bar 6}2mP$
57 ${\bar 6}m2P$
$6/mmm$ 58 $6/mmm$
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 $m{\bar 3}mI$