In crystallography, a crystallographic point group is a point group whose symmetry operations are compatible with the translational symmetry of crystallographic lattices. According to the crystallographic restriction theorem, point groups may only contain one-, two-, three-, four- and sixfold rotational symmetries. This reduces the number of crystallographic point groups to 10 in 2-dimensional space and 32 in 3-dimensional space.
In the classification of crystals, to each space group is associated a crystallographic point group by "forgetting" the translational components of the symmetry operations, that is, by turning screw rotations into rotations, glide reflections into reflections and moving all symmetry elements into the origin. Each crystallographic point group defines the (geometric) crystal class of the crystal.
The point group of a crystal determines, among other things, the directional variation of physical properties that arise from its structure, including optical properties such as birefringency, or electro-optical features such as the Pockels effect.
Graphical representations
It is common to display point groups graphically to develop an intuitive understanding of their symmetry. Typically stereographic projections are used as they preserve angular relations. Two types of projections can be made. The first, shown here, is a projection of the symmetry elements, to display their angular relations with respect to one another. In this case, each symmetry element is represented by a symbol shown in the table below. Thin lines are used to demarcate the sphere of the stereographic projection, and axes of rotation or rotoinversion where they do not intersect with a mirror plane. Mirror planes are represented by bold lines. The second type of projection, is of a general point, and all the additional points generated from that initial point using the symmetry elements of the point group. +Symmetry Elementgraphical representation2346=i=mMirror planes out of page are given by bold lines
In two dimensions
Hermann–Mauguin notation
An abbreviated form of the Hermann–Mauguin notation commonly used for wallpaper groups also serves to describe crystallographic point groups. In 2-dimensional space, group names are
Schoenflies notation
In Schoenflies notation, point groups are denoted by a letter symbol with a subscript. The symbols used in crystallography mean the following: Cn (for cyclic) indicates that the group has an n-fold rotation axis. Dn (for dihedral, or two-sided) indicates that the group has an n-fold rotation axis plus n twofold axes perpendicular to that axis.
n 1 2 3 4 6 Cn C1 C2 C3 C4 C6 Dn D1 D2 D3 D4 D6
In three dimensions
The 32 point groups are the same as the 32 types of morphological (external) crystalline symmetries derived in 1830 by Johann Friedrich Christian Hessel from a consideration of observed crystal forms. In 1867 Axel Gadolin, who was unaware of the previous work of Hessel, found the crystallographic point groups independently using stereographic projection to represent the symmetry elements of the 32 groups.
Crystal family Crystal system Point group / Crystal class Hermann–Mauguin Point symmetry Order Abstract group triclinic pedial 1 enantiomorphic polar 1 trivial pinacoidal centrosymmetric 2 cyclic monoclinic sphenoidal 2 enantiomorphic polar 2 cyclic domatic m polar 2 cyclic prismatic 2/m centrosymmetric 4 Klein four orthorhombic rhombic-disphenoidal 222 enantiomorphic 4 Klein four rhombic-pyramidal mm2 polar 4 Klein four rhombic-dipyramidal mmm (2/m 2/m 2/m) centrosymmetric 8 tetragonal tetragonal-pyramidal 4 enantiomorphic polar 4 cyclic tetragonal-disphenoidal non-centrosymmetric 4 cyclic tetragonal-dipyramidal 4/m centrosymmetric 8 tetragonal-trapezohedral 422 enantiomorphic 8 dihedral ditetragonal-pyramidal 4mm polar 8 dihedral tetragonal-scalenohedral 2m non-centrosymmetric 8 dihedral ditetragonal-dipyramidal 4/mmm (4/m 2/m 2/m) centrosymmetric 16 hexagonal trigonal trigonal-pyramidal 3 enantiomorphic polar 3 cyclic rhombohedral centrosymmetric 6 cyclic trigonal-trapezohedral 32 enantiomorphic 6 dihedral ditrigonal-pyramidal 3m polar 6 dihedral ditrigonal-scalenohedral m ( 2/m) centrosymmetric 12 dihedral hexagonal hexagonal-pyramidal 6 enantiomorphic polar 6 cyclic trigonal-dipyramidal non-centrosymmetric 6 cyclic hexagonal-dipyramidal 6/m centrosymmetric 12 hexagonal-trapezohedral 622 enantiomorphic 12 dihedral dihexagonal-pyramidal 6mm polar 12 dihedral ditrigonal-dipyramidal m2 non-centrosymmetric 12 dihedral dihexagonal-dipyramidal 6/mmm (6/m 2/m 2/m) centrosymmetric 24 cubic tetartoidal 23 enantiomorphic 12 alternating diploidal m (2/m 3) centrosymmetric 24 gyroidal 432 enantiomorphic 24 symmetric hextetrahedral 3m non-centrosymmetric 24 symmetric hexoctahedral mm (4/m 3 2/m) centrosymmetric 48
The point symmetry of a structure can be further described as follows. Consider the points that make up the structure, and reflect them all through a single point, so that (x,y,z) becomes (−x,−y,−z). This is the 'inverted structure'. If the original structure and inverted structure are identical, then the structure is centrosymmetric. Otherwise it is non-centrosymmetric. Still, even in the non-centrosymmetric case, the inverted structure can in some cases be rotated to align with the original structure. This is a non-centrosymmetric achiral structure. If the inverted structure cannot be rotated to align with the original structure, then the structure is chiral or enantiomorphic and its symmetry group is enantiomorphic.
A direction (meaning a line without an arrow) is called polar if its two-directional senses are geometrically or physically different. A symmetry direction of a crystal that is polar is called a polar axis. Groups containing a polar axis are called polar. A polar crystal possesses a unique polar axis (more precisely, all polar axes are parallel). Some geometrical or physical property is different at the two ends of this axis: for example, there might develop a dielectric polarization as in pyroelectric crystals. A polar axis can occur only in non-centrosymmetric structures. There cannot be a mirror plane or twofold axis perpendicular to the polar axis, because they would make the two directions of the axis equivalent.
The crystal structures of chiral biological molecules (such as protein structures) can only occur in the 65 enantiomorphic space groups (biological molecules are usually chiral).
Hermann–Mauguin notation
An abbreviated form of the Hermann–Mauguin notation commonly used for space groups also serves to describe crystallographic point groups. In 3-dimensional space, group names are
Schoenflies notation
In Schoenflies notation, point groups are denoted by a letter symbol with a subscript. The symbols used in crystallography mean the following:
Cn (for cyclic) indicates that the group has an n-fold rotation axis. Cnh is Cn with the addition of a mirror (reflection) plane perpendicular to the axis of rotation. Cnv is Cn with the addition of n mirror planes parallel to the axis of rotation. S2n (for Spiegel, German for mirror) denotes a group with only a 2n-fold rotation-reflection axis. Dn (for dihedral, or two-sided) indicates that the group has an n-fold rotation axis plus n twofold axes perpendicular to that axis. Dnh has, in addition, a mirror plane perpendicular to the n-fold axis. Dnd has, in addition to the elements of Dn, mirror planes parallel to the n-fold axis. The letter T (for tetrahedron) indicates that the group has the symmetry of a tetrahedron. Td includes improper rotation operations, T excludes improper rotation operations, and Th is T with the addition of an inversion. The letter O (for octahedron) indicates that the group has the symmetry of an octahedron, with (Oh) or without (O) improper operations (those that change handedness).
n 1 2 3 4 6 Cn C1 C2 C3 C4 C6 Cnv C1v=C1h C2v C3v C4v C6v Cnh C1h C2h C3h C4h C6h Dn D1=C2 D2 D3 D4 D6 Dnh D1h=C2v D2h D3h D4h D6h Dnd D1d=C2h D2d D3d D4d D6d S2n S2 S4 S6 S8 S12 D4d and D6d are actually forbidden because they contain improper rotations with n=8 and 12 respectively. The 27 point groups in the table plus T, Td, Th, O and Oh constitute 32 crystallographic point groups.
Correspondence between different notations
Crystal familyCrystal system Hermann-MauguinShubnikov SchoenfliesOrbifoldCoxeter Order(full)(short)Triclinic 1 1 C1 11 [ ]+ 1 Ci = S2 × [2+,2+] 2 Monoclinic 2 2 C2 22 + 2 m m Cs = C1h * [ ] 2 2/m C2h 2* [2,2+] 4 Orthorhombic 222 222 D2 = V 222 [2,2]+ 4 mm2 mm2 C2v *22 4 mmm D2h = Vh *222 [2,2] 8Tetragonal 4 4 C4 44 + 4 S4 2× [2+,4+] 4 4/m C4h 4* [2,4+] 8422 422 D4 422 [4,2]+ 84mm 4mm C4v *44 8 2m 2m D2d = Vd 2*2 [2+,4] 8 4/mmm D4h *422 [4,2] 16HexagonalTrigonal 3 3 C3 33 + 3 C3i = S6 3× [2+,6+] 6 32 32 D3 322 [3,2]+ 6 3m 3m C3v *33 6 m D3d 2*3 [2+,6] 12Hexagonal6 6 C6 66 + 6 C3h 3* [2,3+] 6 6/m C6h 6* [2,6+] 12 622 622 D6 622 [6,2]+ 12 6mm 6mm C6v *66 12 m2 m2 D3h *322 [3,2] 12 6/mmm D6h *622 [6,2] 24Cubic 23 23 T 332 [3,3]+ 12 m Th 3*2 [3+,4] 24 432 432 O 432 [4,3]+ 24 3m 3m Td *332 [3,3] 24 mm Oh *432 [4,3] 48
Isomorphisms Two groups are said to be isomorphic if there exists a bijective and homomorphic mapping between the two groups. That is to say that simply renaming the elements of one group with the elements of the second group in the right manner will give you the second group and visa versa. Many of the crystallographic point groups share the same internal structure in this sense. For example, the point groups , 2, and m contain different geometric symmetry operations, (inversion, rotation, and reflection, respectively) but all share the structure of the cyclic group C2. All isomorphic groups are of the same order, but not all groups of the same order are isomorphic. The point groups which are isomorphic are shown in the following table:
Hermann–MauguinSchoenfliesOrderAbstract group 1 C1 1 C1 Ci = S2 2 C2 2 C2 2 m Cs = C1h 2 3 C3 3 C3 4 C4 4 C4 S4 4 2/m C2h 4 D2 = C2 × C2 222 D2 = V 4 mm2 C2v 4 C3i = S6 6 C66 C6 6 C3h 6 32 D3 6 D3 3m C3v 6 mmm D2h = Vh 8 D2 × C2 4/m C4h 8 C4 × C2422 D4 8 D4 4mm C4v 8 2m D2d = Vd 8 6/m C6h 12 C6 × C2 23 T 12 A4 m D3d 12 D6 622 D6 12 6mm C6v 12 m2 D3h 12 4/mmm D4h 16 D4 × C2 6/mmm D6h 24 D6 × C2 m Th 24 A4 × C2 432 O 24 S4 3m Td 24 mm Oh 48 S4 × C2
This table makes use of cyclic groups (C1, C2, C3, C4, C6), dihedral groups (D2, D3, D4, D6), one of the alternating groups (A4), and one of the symmetric groups (S4). Here the symbol " × " indicates a direct product.
Symmetry in understanding crystal properties The symmetry of a material can have profound effects on what properties are 'allowed' to be displayed by that crystal. These influences are summarized in Von Neumann's principle and more generally by the Curie Law's. These laws state that the symmetry of a crystals physical properties must be at least as symmetric as the crystal itself. A common example given is in piezoelectricity and pyroelectricity. These properties generate an electric dipole in a crystal under strain or thermal changes. An electric dipole is directional and as such cannot exist in crystals with inversion symmetry. The converse, however, is not true. Crystals without inversion symmetry do not necessarily display piezo- or pyro-electricity. Symmetry in Polycrystals Polycrystals contain many small crystals of different orientations. In an idealized case, with sufficiently small crystals in a sufficiently large sample, every orientation is represented, leading to an approximately isotropic material. In effect, this allows a polycrystal to display properties of a higher symmetry than the individual crystals composing it. For example, an ideal polycrystal or pyroelectric crystallites will display electric dipoles in every direction, which cancel out, giving the polycrystal a net zero polarization.