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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

\[m\overline{3}mI\]

See also

  • Section 8.2.3 of International Tables of Crystallography, Volume A
  • Sections 1.3.4 and 1.5.3 of International Tables of Crystallography, Volume B
  • Section 1.4 of International Tables of Crystallography, Volume C