Arithmetic crystal class

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, 2CmPmC, 2/mP, 2/mC (note that in the space group symbol the order is inversed: P2, C2, etc...). In some cases, the centering 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 centering 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 Section 8.2.3 of International Tables of Crystallography, Volume A.

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 $\overline{1}$ 2 $\overline{1}P$ 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 mm 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 $\overline{4}$ 24 $\overline{4}P$ 25 $\overline{4}I$ 4 / m 26 4 / mP 27 4 / mI 422 28 422P 29 422I 4mm 30 4mmP 31 4mmI $\overline{4}m$ 32 $\overline{4}2mP$ 33 $\overline{4}m2P$ 34 $\overline{4}m2I$ 35 $\overline{4}2mI$ 4 / mmm 36 4 / mmmP 37 4 / mmmI Trigonal 3 38 3P 39 3R $\overline{3}$ 40 $\overline{3}P$ 41 $\overline{3}R$ 32 42 312P 43 321P 44 32R 3m 45 3m1P 46 31mP 47 3mR $\overline{3}m$ 48 $\overline{3}1mP$ 49 $\overline{3}m1P$ 50 $\overline{3}mR$ Hexagonal 6 51 6P $\overline{6}$ 52 $\overline{6}P$ 6 / m 53 6 / mP 622 54 622P 6mm 55 6mmP $\overline{6}m$ 56 $\overline{6}2mP$ 57 $\overline{6}m2P$ 6 / mmm 58 6 / mmm Cubic 23 59 23P 60 23F 61 23I $m\overline{3}$ 62 $m\overline{3}P$ 63 $m\overline{3}F$ 64 $m\overline{3}I$ 432 65 432P 66 432F 67 432I $\overline{4}3m$ 68 $\overline{4}3mP$ 69 $\overline{4}3mF$ 70 $\overline{4}3mI$ $m\overline{3}m$ 71 $m\overline{3}mP$ 72 $m\overline{3}mF$ 73 $m\overline{3}mI$