Space group

In mathematics, physics and chemistry, a space group is the symmetry group of a repeating pattern in space, usually in three dimensions.^[1] The elements of a space group (its symmetry operations) are the rigid transformations of the pattern that leave it unchanged. In three dimensions, space groups are classified into 219 distinct types, or 230 types if chiral copies are considered distinct. Space groups are discrete cocompact groups of isometries of an oriented Euclidean space in any number of dimensions. In dimensions other than 3, they are sometimes called Bieberbach groups.

In crystallography, space groups are also called the crystallographic or Fedorov groups, and represent a description of the symmetry of the crystal. A definitive source regarding 3-dimensional space groups is the International Tables for Crystallography Hahn (2002).

Space groups in 2 dimensions are the 17 wallpaper groups which have been known for several centuries, though the proof that the list was complete was only given in 1891, after the much more difficult classification of space groups had largely been completed.^[2]

In 1879 the German mathematician Leonhard Sohncke listed the 65 space groups (called Sohncke groups) whose elements preserve the chirality.^[3] More accurately, he listed 66 groups, but both the Russian mathematician and crystallographer Evgraf Fedorov and the German mathematician Arthur Moritz Schoenflies noticed that two of them were really the same. The space groups in three dimensions were first enumerated in 1891 by Fedorov^[4] (whose list had two omissions (I43d and Fdd2) and one duplication (Fmm2)), and shortly afterwards in 1891 were independently enumerated by Schönflies^[5] (whose list had four omissions (I43d, Pc, Cc, ?) and one duplication (P42₁m)). The correct list of 230 space groups was found by 1892 during correspondence between Fedorov and Schönflies.^[6] William Barlow (1894) later enumerated the groups with a different method, but omitted four groups (Fdd2, I42d, P42₁d, and P42₁c) even though he already had the correct list of 230 groups from Fedorov and Schönflies; the common claim that Barlow was unaware of their work is incorrect.^{[citation needed]} Burckhardt (1967) describes the history of the discovery of the space groups in detail.

The space groups in three dimensions are made from combinations of the 32 crystallographic point groups with the 14 Bravais lattices, each of the latter belonging to one of 7 lattice systems. What this means is that the action of any element of a given space group can be expressed as the action of an element of the appropriate point group followed optionally by a translation. A space group is thus some combination of the translational symmetry of a unit cell (including lattice centering), the point group symmetry operations of reflection, rotation and improper rotation (also called rotoinversion), and the screw axis and glide plane symmetry operations. The combination of all these symmetry operations results in a total of 230 different space groups describing all possible crystal symmetries.

The number of replicates of the asymmetric unit in a unit cell is thus the number of lattice points in the cell times the order of the point group. This ranges from 1 in the case of space group P1 to 192 for a space group like Fm3m, the NaCl structure.

The elements of the space group fixing a point of space are the identity element, reflections, rotations and improper rotations, including inversion points.

The translations form a normal abelian subgroup of rank 3, called the Bravais lattice (so named after French physicist Auguste Bravais). There are 14 possible types of Bravais lattice. The quotient of the space group by the Bravais lattice is a finite group which is one of the 32 possible point groups.

A glide plane is a reflection in a plane, followed by a translation parallel with that plane. This is noted by ${\displaystyle a}$, ${\displaystyle b}$, or ${\displaystyle c}$, depending on which axis the glide is along. There is also the ${\displaystyle n}$ glide, which is a glide along the half of a diagonal of a face, and the ${\displaystyle d}$ glide, which is a fourth of the way along either a face or space diagonal of the unit cell. The latter is called the diamond glide plane as it features in the diamond structure. In 17 space groups, due to the centering of the cell, the glides occur in two perpendicular directions simultaneously, i.e. the same glide plane can be called b or c, a or b, a or c. For example, group Abm2 could be also called Acm2, group Ccca could be called Cccb. In 1992, it was suggested to use symbol e for such planes. The symbols for five space groups have been modified:

A screw axis is a rotation about an axis, followed by a translation along the direction of the axis. These are noted by a number, n, to describe the degree of rotation, where the number is how many operations must be applied to complete a full rotation (e.g., 3 would mean a rotation one third of the way around the axis each time). The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. So, 2₁ is a twofold rotation followed by a translation of 1/2 of the lattice vector.

The general formula for the action of an element of a space group is

y = M.x + D

where M is its matrix, D is its vector, and where the element transforms point x into point y. In general, D = D (lattice) + D(M), where D(M) is a unique function of M that is zero for M being the identity. The matrices M form a point group that is a basis of the space group; the lattice must be symmetric under that point group, but the crystal structure itself may not be symmetric under that point group as applied to any particular point (that is, without a translation). For example, the diamond cubic structure does not have any point where the cubic point group applies.

The lattice dimension can be less than the overall dimension, resulting in a "subperiodic" space group. For (overall dimension, lattice dimension):

• (1,1): One-dimensional line groups
• (2,1): Two-dimensional line groups: frieze groups
• (2,2): Wallpaper groups
• (3,1): Three-dimensional line groups; with the 3D crystallographic point groups, the rod groups
• (3,2): Layer groups
• (3,3): The space groups discussed in this article

The 65 "Sohncke" space groups, not containing any mirrors, inversion points, improper rotations or glide planes, yield chiral crystals, not identical to their mirror image; whereas space groups that do include at least one of those give achiral crystals. Achiral molecules sometimes form chiral crystals, but chiral molecules always form chiral crystals, in one of the space groups that permit this.

Among the 65 Sohncke groups are 22 that come in 11 enantiomorphic pairs.

Only certain combinations of symmetry elements are possible in a space group. Translations are always present, and the space group P1 has only translations and the identity element. The presence of mirrors implies glide planes as well, and the presence of rotation axes implies screw axes as well, but the converses are not true. An inversion and a mirror implies two-fold screw axes, and so on.

There are at least ten methods of naming space groups. Some of these methods can assign several different names to the same space group, so altogether there are many thousands of different names.

Number

The International Union of Crystallography publishes tables of all space group types, and assigns each a unique number from 1 to 230. The numbering is arbitrary, except that groups with the same crystal system or point group are given consecutive numbers.

Hall notation^[7]

Space group notation with an explicit origin. Rotation, translation and axis-direction symbols are clearly separated and inversion centers are explicitly defined. The construction and format of the notation make it particularly suited to computer generation of symmetry information. For example, group number 3 has three Hall symbols: P 2y (P 1 2 1), P 2 (P 1 1 2), P 2x (P 2 1 1).

Schönflies notation

The space groups with given point group are numbered by 1, 2, 3, ... (in the same order as their international number) and this number is added as a superscript to the Schönflies symbol for the point group. For example, groups numbers 3 to 5 whose point group is C₂ have Schönflies symbols C¹
₂, C²
₂, C³
₂.

Coxeter notation

Spatial and point symmetry groups, represented as modifications of the pure reflectional Coxeter groups.

Geometric notation^[9]

A geometric algebra notation.

There are (at least) 10 different ways to classify space groups into classes. The relations between some of these are described in the following table. Each classification system is a refinement of the ones below it. To understand an explanation given here it may be necessary to understand the next one down.

Conway, Delgado Friedrichs, and Huson et al. (2001) gave another classification of the space groups, called a fibrifold notation, according to the fibrifold structures on the corresponding orbifold. They divided the 219 affine space groups into reducible and irreducible groups. The reducible groups fall into 17 classes corresponding to the 17 wallpaper groups, and the remaining 35 irreducible groups are the same as the cubic groups and are classified separately.

In n dimensions, an affine space group, or Bieberbach group, is a discrete subgroup of isometries of n-dimensional Euclidean space with a compact fundamental domain. Bieberbach (1911, 1912) proved that the subgroup of translations of any such group contains n linearly independent translations, and is a free abelian subgroup of finite index, and is also the unique maximal normal abelian subgroup. He also showed that in any dimension n there are only a finite number of possibilities for the isomorphism class of the underlying group of a space group, and moreover the action of the group on Euclidean space is unique up to conjugation by affine transformations. This answers part of Hilbert's eighteenth problem. Zassenhaus (1948) showed that conversely any group that is the extension^{[when defined as?]} of Zⁿ by a finite group acting faithfully is an affine space group. Combining these results shows that classifying space groups in n dimensions up to conjugation by affine transformations is essentially the same as classifying isomorphism classes for groups that are extensions of Zⁿ by a finite group acting faithfully.

It is essential in Bieberbach's theorems to assume that the group acts as isometries; the theorems do not generalize to discrete cocompact groups of affine transformations of Euclidean space. A counter-example is given by the 3-dimensional Heisenberg group of the integers acting by translations on the Heisenberg group of the reals, identified with 3-dimensional Euclidean space. This is a discrete cocompact group of affine transformations of space, but does not contain a subgroup Z³.

This table gives the number of space group types in small dimensions, including the numbers of various classes of space group. The numbers of enantiomorphic pairs are given in parentheses.

In addition to crystallographic space groups there are also magnetic space groups (also called two-color (black and white) crystallographic groups or Shubnikov groups). These symmetries contain an element known as time reversal. They treat time as an additional dimension, and the group elements can include time reversal as reflection in it. They are of importance in magnetic structures that contain ordered unpaired spins, i.e. ferro-, ferri- or antiferromagnetic structures as studied by neutron diffraction. The time reversal element flips a magnetic spin while leaving all other structure the same and it can be combined with a number of other symmetry elements. Including time reversal there are 1651 magnetic space groups in 3D (Kim 1999, p.428). It has also been possible to construct magnetic versions for other overall and lattice dimensions (Daniel Litvin's papers, (Litvin 2008), (Litvin 2005)). Frieze groups are magnetic 1D line groups and layer groups are magnetic wallpaper groups, and the axial 3D point groups are magnetic 2D point groups. Number of original and magnetic groups by (overall, lattice) dimension:(Palistrant 2012)(Souvignier 2006)

Overall dimension	Lattice dimension	Ordinary groups			Magnetic groups
		Name	Symbol	Count	Symbol	Count
0	0	Zero-dimensional symmetry group	${\displaystyle G_{0}}$	1	${\displaystyle G_{0}^{1}}$	2
1	0	One-dimensional point groups	${\displaystyle G_{10}}$	2	${\displaystyle G_{10}^{1}}$	5
	1	One-dimensional discrete symmetry groups	${\displaystyle G_{1}}$	2	${\displaystyle G_{1}^{1}}$	7
2	0	Two-dimensional point groups	${\displaystyle G_{20}}$	10	${\displaystyle G_{20}^{1}}$	31
	1	Frieze groups	${\displaystyle G_{21}}$	7	${\displaystyle G_{21}^{1}}$	31
	2	Wallpaper groups	${\displaystyle G_{2}}$	17	${\displaystyle G_{2}^{1}}$	80
3	0	Three-dimensional point groups	${\displaystyle G_{30}}$	32	${\displaystyle G_{30}^{1}}$	122
	1	Rod groups	${\displaystyle G_{31}}$	75	${\displaystyle G_{31}^{1}}$	394
	2	Layer groups	${\displaystyle G_{32}}$	80	${\displaystyle G_{32}^{1}}$	528
	3	Three-dimensional space groups	${\displaystyle G_{3}}$	230	${\displaystyle G_{3}^{1}}$	1651
4	0	Four-dimensional point groups	${\displaystyle G_{40}}$	271	${\displaystyle G_{40}^{1}}$	1202
	1		${\displaystyle G_{41}}$	343
	2		${\displaystyle G_{42}}$	1091
	3		${\displaystyle G_{43}}$	1594
	4	Four-dimensional discrete symmetry groups	${\displaystyle G_{4}}$	4894	${\displaystyle G_{4}^{1}}$	62227

Table of the wallpaper groups using the classification of the 2-dimensional space groups:

Crystal system, Bravais lattice	Geometric class, point group					Arithmetic class	Wallpaper groups (cell diagram)
	Int'l	Schön.	Orbifold	Cox.	Ord.
Oblique	1	C1	(1)	[ ]+	1	None	p1 (1)
	2	C2	(22)	[2]+	2	None	p2 (2222)
Rectangular	m	D1	(*)	[ ]	2	Along	pm (**)		pg (××)
	2mm	D2	(*22)	[2]	4	Along	pmm (*2222)		pmg (22*)
Centered rectangular	m	D1	(*)	[ ]	2	Between	cm (*×)
	2mm	D2	(*22)	[2]	4	Between	cmm (2*22)		pgg (22×)
Square	4	C4	(44)	[4]+	4	None	p4 (442)
	4mm	D4	(*44)	[4]	8	Both	p4m (*442)		p4g (4*2)
Hexagonal	3	C3	(33)	[3]+	3	None	p3 (333)
	3m	D3	(*33)	[3]	6	Between	p3m1 (*333)		p31m (3*3)
	6	C6	(66)	[6]+	6	None	p6 (632)
	6mm	D6	(*66)	[6]	12	Both	p6m (*632)

For each geometric class, the possible arithmetic classes are

• None: no reflection lines
• Along: reflection lines along lattice directions
• Between: reflection lines halfway in between lattice directions
• Both: reflection lines both along and between lattice directions

№	Crystal system, (count), Bravais lattice	Point group					Space groups (international short symbol)
		Int'l	Schön.	Orbifold	Cox.	Ord.
1	Triclinic (2)	1	C1	11	[ ]+	1	P1
2		1	Ci	1×	[2+,2+]	2	P1
3–5	Monoclinic (13)	2	C2	22	[2]+	2	P2, P21 C2
6–9		m	Cs	*11	[ ]	2	Pm, Pc Cm, Cc
10–15		2/m	C2h	2*	[2,2+]	4	P2/m, P21/m C2/m, P2/c, P21/c C2/c
16–24	Orthorhombic (59)	222	D2	222	[2,2]+	4	P222, P2221, P21212, P212121, C2221, C222, F222, I222, I212121
25–46		mm2	C2v	*22	[2]	4	Pmm2, Pmc21, Pcc2, Pma2, Pca21, Pnc2, Pmn21, Pba2, Pna21, Pnn2 Cmm2, Cmc21, Ccc2, Amm2, Aem2, Ama2, Aea2 Fmm2, Fdd2 Imm2, Iba2, Ima2
47–74		mmm	D2h	*222	[2,2]	8	Pmmm, Pnnn, Pccm, Pban, Pmma, Pnna, Pmna, Pcca, Pbam, Pccn, Pbcm, Pnnm, Pmmn, Pbcn, Pbca, Pnma Cmcm, Cmce, Cmmm, Cccm, Cmme, Ccce Fmmm, Fddd Immm, Ibam, Ibca, Imma
75–80	Tetragonal (68)	4	C4	44	[4]+	4	P4, P41, P42, P43, I4, I41
81–82		4	S4	2×	[2+,4+]	4	P4, I4
83–88		4/m	C4h	4*	[2,4+]	8	P4/m, P42/m, P4/n, P42/n I4/m, I41/a
89–98		422	D4	224	[2,4]+	8	P422, P4212, P4122, P41212, P4222, P42212, P4322, P43212 I422, I4122
99–110		4mm	C4v	*44	[4]	8	P4mm, P4bm, P42cm, P42nm, P4cc, P4nc, P42mc, P42bc I4mm, I4cm, I41md, I41cd
111–122		42m	D2d	2*2	[2+,4]	8	P42m, P42c, P421m, P421c, P4m2, P4c2, P4b2, P4n2 I4m2, I4c2, I42m, I42d
123–142		4/mmm	D4h	*224	[2,4]	16	P4/mmm, P4/mcc, P4/nbm, P4/nnc, P4/mbm, P4/mnc, P4/nmm, P4/ncc, P42/mmc, P42/mcm, P42/nbc, P42/nnm, P42/mbc, P42/mnm, P42/nmc, P42/ncm I4/mmm, I4/mcm, I41/amd, I41/acd
143–146	Trigonal (25)	3	C3	33	[3]+	3	P3, P31, P32 R3
147–148		3	S6	3×	[2+,6+]	6	P3, R3
149–155		32	D3	223	[2,3]+	6	P312, P321, P3112, P3121, P3212, P3221 R32
156–161		3m	C3v	*33	[3]	6	P3m1, P31m, P3c1, P31c R3m, R3c
162–167		3m	D3d	2*3	[2+,6]	12	P31m, P31c, P3m1, P3c1 R3m, R3c
168–173	Hexagonal (27)	6	C6	66	[6]+	6	P6, P61, P65, P62, P64, P63
174		6	C3h	3*	[2,3+]	6	P6
175–176		6/m	C6h	6*	[2,6+]	12	P6/m, P63/m
177–182		622	D6	226	[2,6]+	12	P622, P6122, P6522, P6222, P6422, P6322
183–186		6mm	C6v	*66	[6]	12	P6mm, P6cc, P63cm, P63mc
187–190		6m2	D3h	*223	[2,3]	12	P6m2, P6c2, P62m, P62c
191–194		6/mmm	D6h	*226	[2,6]	24	P6/mmm, P6/mcc, P63/mcm, P63/mmc
195–199	Cubic (36)	23	T	332	[3,3]+	12	P23, F23, I23 P213, I213
200–206		m3	Th	3*2	[3+,4]	24	Pm3, Pn3, Fm3, Fd3, Im3, Pa3, Ia3
207–214		432	O	432	[3,4]+	24	P432, P4232 F432, F4132 I432 P4332, P4132, I4132
215–220		43m	Td	*332	[3,3]	24	P43m, F43m, I43m P43n, F43c, I43d
221–230		m3m	Oh	*432	[3,4]	48	Pm3m, Pn3n, Pm3n, Pn3m Fm3m, Fm3c, Fd3m, Fd3c Im3m, Ia3d

Note: An e plane is a double glide plane, one having glides in two different directions. They are found in seven orthorhombic, five tetragonal and five cubic space groups, all with centered lattice. The use of the symbol e became official with Hahn (2002).

The lattice system can be found as follows. If the crystal system is not trigonal then the lattice system is of the same type. If the crystal system is trigonal, then the lattice system is hexagonal unless the space group is one of the seven in the rhombohedral lattice system consisting of the 7 trigonal space groups in the table above whose name begins with R. (The term rhombohedral system is also sometimes used as an alternative name for the whole trigonal system.) The hexagonal lattice system is larger than the hexagonal crystal system, and consists of the hexagonal crystal system together with the 18 groups of the trigonal crystal system other than the seven whose names begin with R.

The Bravais lattice of the space group is determined by the lattice system together with the initial letter of its name, which for the non-rhombohedral groups is P, I, F, A or C, standing for the principal, body centered, face centered, A-face centered or C-face centered lattices. There are seven rhombohedral space groups, with initial letter R.

1. Leave out the Bravais type
2. Convert all symmetry elements with translational components into their respective symmetry elements without translation symmetry (Glide planes are converted into simple mirror planes; Screw axes are converted into simple axes of rotation)
3. Axes of rotation, rotoinversion axes and mirror planes remain unchanged.

Wikimedia Commons has media related to Space groups.

• International Union of Crystallography
• Point Groups and Bravais Lattices Archived 2012-07-16 at the Wayback Machine
• [1] Bilbao Crystallographic Server
• Space Group Info (old)
• Space Group Info (new)
• Crystal Lattice Structures: Index by Space Group
• Full list of 230 crystallographic space groups
• Interactive 3D visualization of all 230 crystallographic space groups Archived 2021-04-18 at the Wayback Machine
• Huson, Daniel H. (1999), The Fibrifold Notation and Classification for 3D Space Groups (PDF)^{[permanent dead link‍]}
• The Geometry Center: 2.1 Formulas for Symmetries in Cartesian Coordinates (two dimensions)
• The Geometry Center: 10.1 Formulas for Symmetries in Cartesian Coordinates (three dimensions)