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 .

Contents

• History
• Elements
• Elements fixing a point
• Translations
• Glide planes
• Screw axes
• General formula
• Chirality
• Combinations
• Notation
• Classification systems
• In other dimensions
• Bieberbach's theorems
• Classification in small dimensions
• Magnetic groups and time reversal
• Table of space groups in 2 dimensions (wallpaper groups)
• Table of space groups in 3 dimensions
• Derivation of the crystal class from the space group
• References
• External links

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).

History

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] WilliamBarlow  ( 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.

Elements

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.

Notation

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.

Classification systems

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,andHusonet 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 other dimensions

Magnetic groups and time reversal

Main article: Magnetic space group

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 space groups in 2 dimensions (wallpaper groups)

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

Table of space groups in 3 dimensions

№	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.

Derivation of the crystal class from the space group

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.

Related Research Articles

In crystallography, crystal structure is a description of ordered arrangement of atoms, ions, or molecules in a crystalline material. Ordered structures occur from intrinsic nature of constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter.

In mathematics, a frieze or frieze pattern is a two-dimensional design that repeats in one direction. The term is derived from architecture and decorative arts, where such repeating patterns are often used. Frieze patterns can be classified into seven types according to their symmetries. The set of symmetries of a frieze pattern is called a frieze group.

A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, especially in textiles, tiles, and wallpaper.

In geometry, a glide reflection or transflection is a geometric transformation that consists of a reflection across a hyperplane and a translation ("glide") in a direction parallel to that hyperplane, combined into a single transformation. Because the distances between points are not changed under glide reflection, it is a motion or isometry. When the context is the two-dimensional Euclidean plane, the hyperplane of reflection is a straight line called the glide line or glide axis. When the context is three-dimensional space, the hyperplane of reflection is a plane called the glide plane. The displacement vector of the translation is called the glide vector.

In crystallography, a crystal system is a set of point groups. A lattice system is a set of Bravais lattices. Space groups are classified into crystal systems according to their point groups, and into lattice systems according to their Bravais lattices. Crystal systems that have space groups assigned to a common lattice system are combined into a crystal family.

In crystallography, the monoclinic crystal system is one of the seven crystal systems. A crystal system is described by three vectors. In the monoclinic system, the crystal is described by vectors of unequal lengths, as in the orthorhombic system. They form a parallelogram prism. Hence two pairs of vectors are perpendicular, while the third pair makes an angle other than 90°.

In crystallography, the orthorhombic crystal system is one of the 7 crystal systems. Orthorhombic lattices result from stretching a cubic lattice along two of its orthogonal pairs by two different factors, resulting in a rectangular prism with a rectangular base (a by b) and height (c), such that a, b, and c are distinct. All three bases intersect at 90° angles, so the three lattice vectors remain mutually orthogonal.

In geometry and crystallography, a Bravais lattice, named after Auguste Bravais, is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by

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The Schoenfliesnotation, named after the German mathematician Arthur Moritz Schoenflies, is a notation primarily used to specify point groups in three dimensions. Because a point group alone is completely adequate to describe the symmetry of a molecule, the notation is often sufficient and commonly used for spectroscopy. However, in crystallography, there is additional translational symmetry, and point groups are not enough to describe the full symmetry of crystals, so the full space group is usually used instead. The naming of full space groups usually follows another common convention, the Hermann–Mauguin notation, also known as the international notation.

In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every point group in dimension d is then a subgroup of the orthogonal group O(d). Point groups are used to describe the symmetries of geometric figures and physical objects such as molecules.

In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(3) itself is a subgroup of the Euclidean group E(3) of all isometries.

In geometry, Hermann–Mauguin notation is used to represent the symmetry elements in point groups, plane groups and space groups. It is named after the German crystallographer Carl Hermann and the French mineralogist Charles-Victor Mauguin. This notation is sometimes called international notation, because it was adopted as standard by the International Tables For Crystallography since their first edition in 1935.

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In geometry, a parallelohedron is a polyhedron that can be translated without rotations in 3-dimensional Euclidean space to fill space with a honeycomb in which all copies of the polyhedron meet face-to-face. There are five types of parallelohedron, first identified by Evgraf Fedorov in 1885 in his studies of crystallographic systems: the cube, hexagonal prism, rhombic dodecahedron, elongated dodecahedron, and truncated octahedron.

Johann Friedrich Christian Hessel was a German physician and professor of mineralogy at the University of Marburg.

A line group is a mathematical way of describing symmetries associated with moving along a line. These symmetries include repeating along that line, making that line a one-dimensional lattice. However, line groups may have more than one dimension, and they may involve those dimensions in its isometries or symmetry transformations.

In solid state physics, the magnetic space groups, or Shubnikov groups, are the symmetry groups which classify the symmetries of a crystal both in space, and in a two-valued property such as electron spin. To represent such a property, each lattice point is colored black or white, and in addition to the usual three-dimensional symmetry operations, there is a so-called "antisymmetry" operation which turns all black lattice points white and all white lattice points black. Thus, the magnetic space groups serve as an extension to the crystallographic space groups which describe spatial symmetry alone.

References

External links

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
• 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)