The 230  3-Dimensional Space Groups

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The majority of the table is reference material.

Space Groups

The number of permutations of Bravais lattices with rotation and screw axes, mirror and glide planes, plus points of inversion is finite: there are only 230 unique combinations for three-dimensional symmetry, and these combinations are known as the 230 space groups. Use of powder diffraction for structural studies does not require a knowledge of their derivation, nor does it require you to memorize a list of the 230 combinations. However, you do need to understand some of the properties of space groups. In order to make life simple, the space groups can be classified according to certain symmetry types.

The table at the bottom of this page lists the 230 three-dimensional space groups used by crystallographers to describe the symmetry of their crystal structures. The space groups are numbered from 1 to 230 and are classified here according to the 7 crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic.

Within each crystal system, the space groups can be ordered by Laue class, crystal class (e.g. 2 < m < 2/m) and, finally, lattice centring (e.g. P < A,B,C < F < I ), as shown in the table below. Further ordering is based largely on the symmetry elements present: In general, rotation axes come before screw axes and mirror planes before glide planes. Note that space groups are not listed here in exact order of their standard space group number, though the order is very similar.

Notation

Each one of the 230 three-dimensional space groups is unique; however, they are frequently specified by means of space group symbols that are not unique to a particular space group. The reason for this is very simple: while each space group symmetry is unique, the choice of vectors that defines a unit cell for that symmetry is not unique. It was mentioned earlier, for example, that the unit cell for the centred monoclinic Bravais lattice is conventionally described as C-face centred with the unique symmetry axis direction being parallel to b. Space group symbols all begin with the lattice type as the first character of the symbol, e.g. C2. With a different choice of labelling for the unit-cell axes, the same space group (number 5) could easily be called A2; other possible symbols are I2, B2, or even F2. You should therefore be aware that the symbols used in the table below refer to so-called standard settings (and hence standard symbols) for the space groups.

Space groups possessing a point of inversion are termed centrosymmetric; these are shown in the table in red. Some space groups have no symmetry element that can change the handedness of an object; these are termed enantiomorphic space groups and are shown in magenta.

Crystal System Laue
Class
Crystal
Class
Lattice
Centring
230  3-Dimensional Space Groups
Triclinic -1 1 P P1
-1 P-1
Monoclinic 2/m 2 P P2, P21
C C2
m P Pm, Pc
C Cm, Cc
2/m P P2/m, P21m, P2/c, P21c
C C2/m, C2/c
Orthorhombic mmm 222 P P222, P2221, P21212, P212121
C C222, C2221
F F222
I I222, I212121
mm2 P Pmm2, Pmc21, Pcc2, Pma2, Pca21, Pnc2,
Pmn21, Pba2, Pna21, Pnn2
C or A Cmm2, Cmc21, Ccc2,
Amm2, Abm2, Ama2, Aba2
F Fmm2, Fdd2
I Imm2, Iba2, Ima2
mmm P Pmmm, Pnnn, Pccm, Pban, Pmma, Pnna,
Pmna, Pcca, Pbam, Pccn, Pbcm, Pnnm,
Pmmn, Pbcn, Pbca, Pnma
C Cmmm, Cmcm, Cmca, Cccm, Cmma, Ccca
F Fmmm, Fddd
I Immm, Ibam, Ibcm, Imma
Tetragonal 4/m 4 P P4, P41, P42, P43
I I4, I41
-4 P P-4
I I-4
4/m P P4/m, P42/m, P4/n, P42/n
I I4/m, I41/a
4/mmm 422 P P422, P4212, P4122, P41212, P4222,
P42212, P4322, P43212
I I422, I4212
4mm P P4mm, P4bm, P42cm, P42nm, P4cc,
P4nc, P42mc, P42bc
I I4mm, I4cm, I41md, I41cd
-42m P P-42m, P-42c, P-421m, P-421c
I I-42m, I-42d
-4m2 P P-4m2, P-4c2, P-4b2, P-4n2
I I-4m2, I-4c2
4/mmm P 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/mcm, P42/nmc, P42/ncm
I I4/mmm, I4/mcm, I41/amd, I41/acd
Trigonal -33 P P3, P31, P32
R R3
-3 P P-3
R R-3
-3m312 P P312, P3112, P3212
321 P321, P3121, P3221
R R32
31m P P31m, P31c
3m1 P3m1, P3c1
R R3m, R3c
-31m P P-31m, P-31c
-3m1 P-3m1, P-3c1
R R-3m, R-3c
Hexagonal 6/m 6 P P6, P61, P62, P63, P64, P65
-6 P-6
6/m P6/m, P63/m
6/mmm 622 P622, P6122, P6222, P6322, P6422, P6522
6mm P6mm, P6cc, P63cm, P63mc
-6m2 P-6m2, P-6c2
-62m P-62m, P62c
6/mmm P6/mmm, P6/mcc, P63/mcm, P63/mmc
Cubic m-3 23 P P23, P213
F F23
I I23, I213
m-3 P Pm-3, Pn-3, Pa-3
F Fm-3, Fd-3
I Im-3, Ia-3
m-3m 432 P P432, P4232, P4332, P4132
F F432, F4132
I I432, I4132
-43m P P-43m, P-43n
F F-43m, F-43c
I I-43m, I-43d
m-3m P Pm-3m, Pn-3n, Pm-3n, Pn-3m
F Fm-3m, Fm-3m, Fd-3m, Fd-3c
I Im-3m, Ia-3d

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