Space group

Space group

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)).

Contents

History

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

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

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

Translations

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

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

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

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

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 30o.
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.

Classification systems for space groups

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.

(Crystallographic) space group types (230 in three dimensions). Two space groups, considered as subgroups of the group of affine transformations of space, have the same space group type if they are conjugate by an orientation-preserving affine transformation. In three dimensions, for 11 of the affine space groups, there is no orientation-preserving map from the group to its mirror image, so if one distinguishes groups from their mirror images these each split into two cases. So there are 54+11=65 space group types that preserve orientation.
Affine space group types (219 in three dimensions). Two space groups, considered as subgroups of the group of affine transformations of space, have the same affine space group type if they are conjugate under an affine transformation. The affine space group type is determined by the underlying abstract group of the space group. In three dimensions there are 54 affine space group types that preserve orientation.
Arithmetic crystal classes (73 in three dimensions). These are determined by the point group together with the action of the point group on the subgroup of translations. In other words the arithmetic crystal classes correspond to conjugacy classes of finite subgroup of the general linear group GLn(Z) over the integers. A space group is called symmorphic (or split) if there is a point such that all symmetries are the product of a symmetry fixing this point and a translation. Equivalently, a space group is symmorphic if it is a semidirect product of its point group with its translation subgroup. There are 73 symmorphic space groups, with exactly one in each arithmetic crystal class. There are also 157 nonsymmorphic space group types with varying numbers in the arithmetic crystal classes.
(geometric) Crystal classes (32 in three dimensions). The crystal class of a space group is determined by its point group: the quotient by the subgroup of translations, acting on the lattice. Two space groups are in the same crystal class if and only if their point groups, which are subgroups of GL2(Z), are conjugate in the larger group GL2(Q). Bravais flocks (14 in three dimensions). These are determined by the underlying Bravais lattice type.

These correspond to conjugacy classes of lattice point groups in GL2(Z), where the lattice point group is the group of symmetries of the underlying lattice that fix a point of the lattice, and contains the point group.
Crystal systems. (7 in three dimensions) Crystal systems are an ad hoc modification of the lattice systems to make them compatible with the classification according to point groups. They differ from crystal families in that the hexagonal crystal family is split into two subsets, called the trigonal and hexagonal crystal systems. The trigonal crystal system is larger than the rhombohedral lattice system, the hexagonal crystal system is smaller than the hexagonal lattice system, and the remaining crystal systems and lattice systems are the same. Lattice systems (7 in three dimensions). The lattice system of a space group is determined by the conjugacy class of the lattice point group (a subgroup of GL2(Z)) in the larger group GL2(Q). In three dimensions the lattice point group can have one of the 7 different orders 2, 4, 8, 12, 16, 24, or 48. The hexagonal crystal family is split into two subsets, called the rhombohedral and hexagonal lattice systems.
Crystal families (6 in three dimensions). The point group of a space group does not quite determine its lattice system, because occasionally two space groups with the same point group may be in different lattice systems. Crystal families are formed from lattice systems by merging the two lattice systems whenever this happens, so that the crystal family of a space group is determined by either its lattice system or its point group. In 3 dimensions the only two lattice families that get merged in this way are the hexagonal and rhombohedral lattice systems, which are combined into the hexagonal crystal family. The 6 crystal families in 3 dimensions are called triclinic, monoclinic, orthorhombal, tetragonal, hexagonal, and cubic. Crystal families are commonly used in popular books on crystals, where they are sometimes called crystal systems.

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

Bieberbach's theorems

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

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

Dimension Lattice types (sequence A004030 in OEIS) Crystallographic point groups (sequence A004028 in OEIS) Crystallographic space group types (sequence A006227 in OEIS) Affine space group types (sequence A004029 in OEIS) Classification
0 1 1 1 1 Trivial group
1 1 2 2 2 One is the group of integers and the other is the infinite dihedral group; see symmetry groups in one dimension
2 5 10 17 17 these 2D space groups are also called wallpaper groups or plane groups.
3 14 32 230 219 In 3D there are 230 crystallographic space group types, which reduces to 219 affine space group types because of some types being different from their mirror image; these are said to differ by "enantiomorphous character" (e.g. P3112 and P3212). Usually "space group" refers to 3D. They were enumerated independently by Barlow (1894), Fedorov (1891) and Schönflies (1891).
4 64 227 4894 4783 The 4895 4-dimensional groups were enumerated by Harold Brown, Rolf Bülow, and Joachim Neubüser et al. (1978).

Neubüser, Souvignier & Wondratschek (2002) corrected the number of enantiomorphic groups from 112 to 111, so total number of groups is 4783+111=4894. There are 44 enantiomorphic point groups in 4-dimensional space. If we consider enantiomorphic groups as different, the total number of point groups is 227+44=271
5 189 955 222018 Plesken & Schulz (2000) enumerated the ones of dimension 5
6 841 7104 28934974 28927922 Plesken & Schulz (2000) enumerated the ones of dimension 6.
Initially published number of 826 Lattice types in Plesken & Hanrath (1984) was corrected to 841 in Opgenorth, Plesken & Schulz (1998).
See also Janssen et al. (2002)

Double groups and time reversal

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

Table of space groups in 3 dimensions

# Crystal system Point group Space groups (international short symbol)
Intl Schönflies
1 Triclinic (2) 1 C1 P1
2 1 Ci P1
3-5 Monoclinic (13) 2 C2 P2, P21, C2
6-9 m Cs Pm, Pc, Cm, Cc
10-15 2/m C2h P2/m, P21/m, C2/m, P2/c, P21/c, C2/c
16-24 Orthorhombic (59) 222 D2 P222, P2221, P21212, P212121, C2221, C222, F222, I222, I212121
25-46 mm2 C2v 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 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 P4, P41, P42, P43, I4, I41
81-82 4 S4 P4, I4
83-88 4/m C4h P4/m, P42/m, P4/n, P42/n, I4/m, I41/a
89-98 422 D4 P422, P4212, P4122, P41212, P4222, P42212, P4322, P43212, I422, I4122
99-110 4mm C4v P4mm, P4bm, P42cm, P42nm, P4cc, P4nc, P42mc, P42bc, I4mm, I4cm, I41md, I41cd
111-122 42m D2d P42m, P42c, P421m, P421c, P4m2, P4c2, P4b2, P4n2, I4m2, I4c2, I42m, I42d
123-142 4/mmm D4h 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 P3, P31, P32, R3
147-148 3 S6 P3, R3
149-155 32 D3 P312, P321, P3112, P3121, P3212, P3221, R32
156-161 3m C3v P3m1, P31m, P3c1, P31c, R3m, R3c
162-167 3m D3d P31m, P31c, P3m1, P3c1, R3m, R3c,  
168-173 Hexagonal (27) 6 C6 P6, P61, P65, P62, P64, P63
174 6 C3h P6
175-176 6/m C6h P6/m, P63/m
177-182 622 D6 P622, P6122, P6522, P6222, P6422, P6322
183-186 6mm C6v P6mm, P6cc, P63cm, P63mc
187-190 6m2 D3h P6m2, P6c2, P62m, P62c
191-194 6/mmm D6h P6/mmm, P6/mcc, P63/mcm, P63/mmc
195-199 Cubic (36) 23 T P23, F23, I23, P213, I213
200-206 m3 Th Pm3, Pn3, Fm3, Fd3, Im3, Pa3, Ia3
207-214 432 O P432, P4232, F432, F4132, I432, P4332, P4132, I4132
215-220 43m Td P43m, F43m, I43m, P43n, F43c, I43d
221-230 m3m Oh 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 five space groups, all in the orthorhombic system and with a 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

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