Arithmetic crystal class

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...). 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. They do not contain glide or screw elements and are therefore in one to one correspondence with the symmorphic groups.

The group-theoretical definition of the arithmetic crystal classes is given in Section 8.2.3 of International Tables of Crystallography, Volume A.

Three-dimensional arithmetic crystal classes.

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

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