Arithmetic crystal class

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

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	$2P$
		4	$2C$
	$m$	5	$mP$
		6	$mC$
	$2/m$	7	$2/m P$
		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