Space group

In mathematics and physics, a space group is the symmetry group of a configuration in space, usually in three dimensions.[1] In three dimensions, there are 219 distinct types, or 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, 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 harder case of space groups had been done.

In 1879 Leonhard Sohncke listed the 65 space groups (sometimes called Sohncke space groups or chiral space groups) whose elements preserve the orientation. More accurately, he listed 66 groups, but Fedorov and Schönflies both noticed that two of them were really the same. The space groups in 3 dimensions were first enumerated by Fedorov (1891) (whose list had 2 omissions (I43d and Fdd2) and one duplication (Fmm2)), and shortly afterwards were independently enumerated by Schönflies (1891) (whose list had 4 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. 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 a myth. Burckhardt (1967) describes the history of the discovery of the space groups in detail.

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 different 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. Translation is defined as the face moves from one point to another point.

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):

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.

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.

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). Sometimes called Z-classes. 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.

Arithmetic crystal classes may be interpreted as different orientations of the point groups in the lattice, with the group elements' matrix components being constrained to have integer coefficients in lattice space. This is rather easy to picture in the two-dimensional, wallpaper group case. Some of the point groups have reflections, and the reflection lines can be along the lattice directions, halfway in between them, or both.

  • None: C1: p1; C2: p2; C3: p3; C4: p4; C6: p6
  • Along: D1: pm, pg; D2: pmm, pmg, pgg; D3: p31m
  • Between: D1: cm; D2: cmm; D3: p3m1
  • Both: D4: p4m, p4g; D6: p6m
(geometric) Crystal classes (32 in three dimensions). Sometimes called Q-classes. 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 GLn(Z), are conjugate in the larger group GLn(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 GLn(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 GLn(Z)) in the larger group GLn(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, including the numbers of various classes of space group. The numbers of enantiomorphic pairs are given in parentheses.

Dimensions Crystal families (sequence A004032 in OEIS) Crystal systems (sequence A004031 in OEIS) Bravais lattices (sequence A004030 in OEIS) Abstract crystallographic point groups (sequence A006226 in OEIS) Geometric crystal classes, Q-classes, crystallographic point groups (sequence A004028 in OEIS) Arithmetic crystal classes, Z-classes (sequence A004027 in OEIS) Affine space group types (sequence A004029 in OEIS) Crystallographic space group types (sequence A006227 in OEIS)
0a 1 1 1 1 1 1 1 1
1b 1 1 1 2 2 2 2 2
2c 4 4 5 9 10 13 17 17
3d 6 7 14 18 32 73 219 (+11) 230
4e 23 (+6) 33 (+7) 64 (+10) 118 227 (+44) 710 (+70) 4783 (+111) 4894
5f 32 59 189 239 955 6079 222018 (+79) 222097
6g 91 251 841 1594 7104 85311 (+30) 28927922 (+7052) 28934974

a - Trivial group
b - One is the group of integers and the other is the infinite dihedral group; see symmetry groups in one dimension
c - these 2D space groups are also called wallpaper groups or plane groups.
d - 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).
e - 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.
f - Plesken & Schulz (2000) enumerated the ones of dimension 5.
g - 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).

Magnetic 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:

Table of space groups in 2 dimensions (wallpaper groups)

Table of the wallpaper groups using the classification of the 3-dimensional space groups:

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

For each geometric class, the possible arithmetic classes are

Table of space groups in 3 dimensions

Further information: List of space groups
# Crystal system
(count)
Bravais lattice		 Point group Space groups (international short symbol)
Intl Schön.Orbifold notation Cox.Ord.
1 Triclinic
(2)
 1 C111[ ]+1 P1
2 1 Ci1×[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 C444[4]+4 P4, P41, P42, P43, I4, I41
81–82 4 S42×[2+,4+]4 P4, I4
83–88 4/m C4h4*[2,4+]8 P4/m, P42/m, P4/n, P42/n
I4/m, I41/a
89–98 422 D4224[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 D2d2*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 C333[3]+3 P3, P31, P32
R3
147–148 3 S63×[2+,6+]6 P3, R3
149–155 32 D3223[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 D3d2*3[2+,6]12 P31m, P31c, P3m1, P3c1
R3m, R3c
168–173 Hexagonal
(27)
 6 C666[6]+6 P6, P61, P65, P62, P64, P63
174 6 C3h3*[2,3+]6 P6
175–176 6/m C6h6*[2,6+]12 P6/m, P63/m
177–182 622 D6226[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 T332[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 O432[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, or C, standing for the principal, body centered, face centered, or C-face centered lattices.

References

  1. ↑ Hiller, Howard (1986). "Crystallography and cohomology of groups". Amer. Math. Monthly 93: 765–779. doi:10.2307/2322930.
  2. ↑ http://commons.wikimedia.org/wiki/Strukturbericht
  3. ↑ PDF The Crystallographic Space Groups in Geometric Algebra, David Hestenes and Jeremy Holt

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