In mathematics and geometry, a space group is a symmetry group, usually for three dimensions, that divides space into discrete repeatable domains.

In three dimensions, there are 219 unique types, or counted as 230 if chiral copies are considered distinct. Space groups are also studied in dimensions other than 3 where they are sometimes called Bieberbach groups, and are discrete cocompact groups of isometries of an oriented Euclidean space.

In crystallography, they 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[link]

Space groups in 2 dimensions are the 17 wallpaper groups which have been known for several centuries.

The space groups in 3 dimensions were first enumerated by Fyodorov (1891), and shortly afterwards were independently enumerated by Schönflies (1891) and Barlow (1894). These first enumerations all contained several minor mistakes, and the correct list of 230 space groups was found during correspondence between Fyodorov and Schönflies.

Elements of a space group[link]

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. This results in a space group being 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 unique space groups describing all possible crystal symmetries.

Elements fixing a point[link]

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

Translations[link]

The translations form a normal abelian subgroup of rank 3, called the Bravais lattice. 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.

Glide planes[link]

A glide plane is a reflection in a plane, followed by a translation parallel with that plane. This is noted by a, b or c, depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the 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.

Screw axes[link]

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.

General formula[link]

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.

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

Notation for space groups[link]

There are at least eight 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.

• International symbol or Hermann–Mauguin notation. The Hermann–Mauguin (or international) notation describes the lattice and some generators for the group. It has a shortened form called the international short symbol, which is the one most commonly used in crystallography, and usually consists of a set of four symbols. The first describes the centering of the Bravais lattice (P, A, B, C, I, R or F). The next three describe the most prominent symmetry operation visible when projected along one of the high symmetry directions of the crystal. These symbols are the same as used in point groups, with the addition of glide planes and screw axis, described above. By way of example, the space group of quartz is P3₁21, showing that it exhibits primitive centering of the motif (i.e., once per unit cell), with a threefold screw axis and a twofold rotation axis. Note that it does not explicitly contain the crystal system, although this is unique to each space group (in the case of P3₁21, it is trigonal).

In the international short symbol the first symbol (3₁ in this example) denotes the symmetry along the major axis (c-axis in trigonal cases), the second (2 in this case) along axes of secondary importance (a and b) and the third symbol the symmetry in another direction. In the trigonal case there also exists a space group P3₁12. In this space group the twofold axes are not along the a and b-axes but in a direction rotated by 30°.

The international symbols and international short symbols for some of the space groups were changed slightly between 1935 and 2002, so several space groups have 4 different international symbols in use.

• Hall notation. 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.

• Shubnikov symbol

• 2D:Orbifold notation and 3D:Fibrifold notation. As the name suggests, the orbifold notation describes the orbifold, given by the quotient of Euclidean space by the space group, rather than generators of the space group. It was introduced by Conway and Thurston, and is not used much outside mathematics. Some of the space groups have several different fibrifolds associated to them, so have several different fibrifold symbols.

• Coxeter notation – Spacial and point symmetry groups, represented as modications of the pure reflectional Coxeter groups.

Classification systems for space groups[link]

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.

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.

Space groups in other dimensions[link]

Bieberbach's theorems[link]

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 18th problem. Zassenhaus (1948) showed that conversely any group that is the extension 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.

Classification in small dimensions[link]

This table give the number of space group types in small dimensions.

Magnetic groups and time reversal[link]

In addition to crystallographic space groups there are also magnetic space groups (also called two-color (black and white) crystallographic 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:

• (1,1): 2, 7
• (2,1): 7, 31
• (2,2): 17, 80
• (3,1): 75, 394 (rod groups, not 3D line groups in general)
• (3,2): 80, 528
• (3,3): 230, 1651
• (4,4): 4894, 62227 (Souvignier 2006)

Table of space groups in 3 dimensions[link]

Note. An e plane is a double glide plane, one having glides in two different directions. They are found in seven orthorombic, 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, or C, standing for the principal, body centered, face centered, or C-face centered lattices.

References[link]

• Barlow, W (1894), "Über die geometrischen Eigenschaften starrer Strukturen und ihre Anwendung auf Kristalle", Z. Kristallogr. 23: 1–63 
• Bieberbach, Ludwig (1911), "Über die Bewegungsgruppen der Euklidischen Räume", Mathematische Annalen 70 (3): 297–336, DOI:10.1007/BF01564500, ISSN 0025-5831 
• Bieberbach, Ludwig (1912), "Über die Bewegungsgruppen der Euklidischen Räume (Zweite Abhandlung.) Die Gruppen mit einem endlichen Fundamentalbereich", Mathematische Annalen 72 (3): 400–412, DOI:10.1007/BF01456724, ISSN 0025-5831 
• Brown, Harold; Bülow, Rolf; Neubüser, Joachim; Wondratschek, Hans; Zassenhaus, Hans (1978), Crystallographic groups of four-dimensional space, New York: Wiley-Interscience [John Wiley & Sons], ISBN 978-0-471-03095-9, MR0484179 
• Burckhardt, Johann Jakob (1947), Die Bewegungsgruppen der Kristallographie, Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften, 13, Verlag Birkhäuser, Basel, MR0020553 
• Conway, John Horton; Delgado Friedrichs, Olaf; Huson, Daniel H.; Thurston, William P. (2001), "On three-dimensional space groups", Beiträge zur Algebra und Geometrie. Contributions to Algebra and Geometry 42 (2): 475–507, ISSN 0138-4821, MR1865535, http://www.emis.de/journals/BAG/vol.42/no.2/17.html 
• Fedorov, E. S. (1891), "Symmetry of Regular Systems of Figures", Zap. Mineral. Obch. 28 (2): 1–146 
• Fedorov, E. S. (1971), Symmetry of crystals, ACA Monograph, 7, American Crystallographic Association 
• Hahn, Th. (2002), Hahn, Theo, ed., International Tables for Crystallography, Volume A: Space Group Symmetry, A (5th ed.), Berlin, New York: Springer-Verlag, DOI:10.1107/97809553602060000100, ISBN 978-0-7923-6590-7, http://it.iucr.org/A/ 
• Hall, S.R. (1981), "Space-Group Notation with an Explicit Origin", Acta Cryst. A37: 517–525 
• Janssen, T.; Birman, J.L.; Dénoyer, F.; Koptsik, V.A.; Verger-Gaugry, J.L.; Weigel, D.; Yamamoto, A.; Abrahams, S.C. et al. (2002), "Report of a Subcommittee on the Nomenclature of n-Dimensional Crystallography. II. Symbols for arithmetic crystal classes, Bravais classes and space groups", Acta Cryst. A 58 (Pt 6): 605–621, DOI:10.1107/S010876730201379X 
• Kim, Shoon K. (1999), Group theoretical methods and applications to molecules and crystals, Cambridge University Press, ISBN 978-0-521-64062-6, MR1713786 
• Litvin, D.B. (May 2008), "Tables of crystallographic properties of magnetic space groups", Acta Cryst. A 64 (Pt 3): 419–24, DOI:10.1107/S010876730800768X, PMID 18421131 
• Litvin, D.B. (May 2005), "Tables of properties of magnetic subperiodic groups", Acta Cryst. A 61 (Pt 3): 382–5, DOI:10.1107/S010876730500406X, PMID 15846043 
• Neubüser, J.; Souvignier, B.; Wondratschek, H. (2002), "Corrections to Crystallographic Groups of Four-Dimensional Space by Brown et al. (1978) [New York: Wiley and Sons]", Acta Cryst. A 58 (Pt 3): 301, DOI:10.1107/S0108767302001368 
• Opgenorth, J; Plesken, W; Schulz, T (1998), "Crystallographic Algorithms and Tables", Acta Cryst. A 54 (Pt 5): 517–531, DOI:10.1107/S010876739701547X 
• Plesken, Wilhelm; Hanrath, W (1984), "The lattices of six-dimensional space", Math. Comp. 43 (168): 573–587 
• Plesken, Wilhelm; Schulz, Tilman (2000), "Counting crystallographic groups in low dimensions", Experimental Mathematics 9 (3): 407–411, ISSN 1058-6458, MR1795312, http://projecteuclid.org/euclid.em/1045604675 
• Schönflies, Arthur Moritz (1891), "Theorie der Kristallstruktur", Gebr. Bornträger, Berlin. 
• Souvignier, Bernd (2006), "The four-dimensional magnetic point and space groups", Z. Kristallogr. 221: 77–82 
• Vinberg, E. (2001), "Crystallographic group", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4, http://www.encyclopediaofmath.org/index.php?title=C/c027190 
• Zassenhaus, Hans (1948), "Über einen Algorithmus zur Bestimmung der Raumgruppen", Commentarii Mathematici Helvetici 21: 117–141, DOI:10.1007/BF02568029, ISSN 0010-2571, MR0024424, http://www.digizeitschriften.de/index.php?id=166&ID=380406 

External links[link]

• International Union of Crystallography
• Point Groups and Bravais Lattices
• 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
• Huson, Daniel H. (1999), The Fibrifold Notation and Classification for 3D Space Groups, http://www-ab.informatik.uni-tuebingen.de/talks/pdfs/Fibrifolds-Princeton%201999.pdf 
• 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)