In crystallography, a crystallographic point group is a set of symmetry operations, corresponding to one of the point groups in three dimensions, such that each operation (perhaps followed by a translation) would leave the structure of a crystal unchanged i.e. the same kinds of atoms would be placed in similar positions as before the transformation. For example, in many crystals in the cubic crystal system, a rotation of the unit cell by 90 degrees around an axis that is perpendicular to one of the faces of the cube is a symmetry operation that moves each atom to the location of another atom of the same kind, leaving the overall structure of the crystal unaffected.
In the classification of crystals, each point group defines a so-called (geometric) crystal class. There are infinitely many three-dimensional point groups. However, the crystallographic restriction on the general point groups results in there being only 32 crystallographic point groups. These 32 point groups are one-and-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.
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. For a periodic crystal (as opposed to a quasicrystal), the group must maintain the three-dimensional translational symmetry that defines crystallinity.
The point groups are named according to their component symmetries. There are several standard notations used by crystallographers, mineralogists, and physicists.
For the correspondence of the two systems below, see crystal system.
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.
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 (or cube), with (Oh) or without (O) improper operations (those that change handedness).
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.
Subgroup relations of the 32 crystallographic point groups (rows represent group orders from bottom to top as: 1,2,3,4,6,8,12,16,24, and 48.)
An abbreviated form of the Hermann–Mauguin notation commonly used for space groups also serves to describe crystallographic point groups. Group names are
Many of the crystallographic point groups share the same internal structure. For example, the point groups 1, 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:[2]
Convert all symmetry elements with translational components into their respective symmetry elements without translation symmetry. (Glide planes are converted into simple mirror planes; screw axes are converted into simple axes of rotation.)
Axes of rotation, rotoinversion axes, and mirror planes remain unchanged.
^"(International Tables) Abstract". Archived from the original on 2013-07-04. Retrieved 2011-11-25.
^Novak, I (1995-07-18). "Molecular isomorphism". European Journal of Physics. IOP Publishing. 16 (4): 151–153. Bibcode:1995EJPh...16..151N. doi:10.1088/0143-0807/16/4/001. ISSN 0143-0807. S2CID 250887121.
External links
Point-group symbols in International Tables for Crystallography (2006). Vol. A, ch. 12.1, pp. 818-820
Names and symbols of the 32 crystal classes in International Tables for Crystallography (2006). Vol. A, ch. 10.1, p. 794
January 06, 2023
crystallographic, point, group, crystallography, crystallographic, point, group, symmetry, operations, corresponding, point, groups, three, dimensions, such, that, each, operation, perhaps, followed, translation, would, leave, structure, crystal, unchanged, sa. In crystallography a crystallographic point group is a set of symmetry operations corresponding to one of the point groups in three dimensions such that each operation perhaps followed by a translation would leave the structure of a crystal unchanged i e the same kinds of atoms would be placed in similar positions as before the transformation For example in many crystals in the cubic crystal system a rotation of the unit cell by 90 degrees around an axis that is perpendicular to one of the faces of the cube is a symmetry operation that moves each atom to the location of another atom of the same kind leaving the overall structure of the crystal unaffected In the classification of crystals each point group defines a so called geometric crystal class There are infinitely many three dimensional point groups However the crystallographic restriction on the general point groups results in there being only 32 crystallographic point groups These 32 point groups are one and 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 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 For a periodic crystal as opposed to a quasicrystal the group must maintain the three dimensional translational symmetry that defines crystallinity Contents 1 Notation 1 1 Schoenflies notation 1 2 Hermann Mauguin notation 1 3 The correspondence between different notations 2 Isomorphisms 3 Deriving the crystallographic point group crystal class from the space group 4 See also 5 References 6 External linksNotation EditThe point groups are named according to their component symmetries There are several standard notations used by crystallographers mineralogists and physicists For the correspondence of the two systems below see crystal system Schoenflies notation Edit Main article Schoenflies notation Further information Point groups in three dimensions 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 or cube with Oh or without O improper operations those that change handedness Due to the crystallographic restriction theorem n 1 2 3 4 or 6 in 2 or 3 dimensional space n 1 2 3 4 6Cn C1 C2 C3 C4 C6Cnv C1v C1h C2v C3v C4v C6vCnh C1h C2h C3h C4h C6hDn D1 C2 D2 D3 D4 D6Dnh D1h C2v D2h D3h D4h D6hDnd D1d C2h D2d D3d D4d D6dS2n S2 S4 S6 S8 S12D4d 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 Hermann Mauguin notation Edit Main article Hermann Mauguin notation Subgroup relations of the 32 crystallographic point groups rows represent group orders from bottom to top as 1 2 3 4 6 8 12 16 24 and 48 An abbreviated form of the Hermann Mauguin notation commonly used for space groups also serves to describe crystallographic point groups Group names are Crystal family Crystal system Group namesCubic 23 m3 432 4 3m m3 mHexagonal Hexagonal 6 6 6 m 622 6mm 6 m2 6 mmmTrigonal 3 3 32 3m 3 mTetragonal 4 4 4 m 422 4mm 4 2m 4 mmmOrthorhombic 222 mm2 mmmMonoclinic 2 2 m mTriclinic 1 1 The correspondence between different notations Edit Crystal family Crystal system Hermann Mauguin Shubnikov 1 Schoenflies Orbifold Coxeter Order full short Triclinic 1 1 1 displaystyle 1 C1 11 11 1 2 displaystyle tilde 2 Ci S2 2 2 2Monoclinic 2 2 2 displaystyle 2 C2 22 2 2m m m displaystyle m Cs C1h 22 m displaystyle tfrac 2 m 2 m 2 m displaystyle 2 m C2h 2 2 2 4Orthorhombic 222 222 2 2 displaystyle 2 2 D2 V 222 2 2 4mm2 mm2 2 m displaystyle 2 cdot m C2v 22 2 42 m 2 m 2 m displaystyle tfrac 2 m tfrac 2 m tfrac 2 m mmm m 2 m displaystyle m cdot 2 m D2h Vh 222 2 2 8Tetragonal 4 4 4 displaystyle 4 C4 44 4 44 4 4 displaystyle tilde 4 S4 2 2 4 44 m displaystyle tfrac 4 m 4 m 4 m displaystyle 4 m C4h 4 2 4 8422 422 4 2 displaystyle 4 2 D4 422 4 2 84mm 4mm 4 m displaystyle 4 cdot m C4v 44 4 84 2m 4 2m 4 m displaystyle tilde 4 cdot m D2d Vd 2 2 2 4 84 m 2 m 2 m displaystyle tfrac 4 m tfrac 2 m tfrac 2 m 4 mmm m 4 m displaystyle m cdot 4 m D4h 422 4 2 16Hexagonal Trigonal 3 3 3 displaystyle 3 C3 33 3 33 3 6 displaystyle tilde 6 C3i S6 3 2 6 632 32 3 2 displaystyle 3 2 D3 322 3 2 63m 3m 3 m displaystyle 3 cdot m C3v 33 3 63 2 m displaystyle tfrac 2 m 3 m 6 m displaystyle tilde 6 cdot m D3d 2 3 2 6 12Hexagonal 6 6 6 displaystyle 6 C6 66 6 66 6 3 m displaystyle 3 m C3h 3 2 3 66 m displaystyle tfrac 6 m 6 m 6 m displaystyle 6 m C6h 6 2 6 12622 622 6 2 displaystyle 6 2 D6 622 6 2 126mm 6mm 6 m displaystyle 6 cdot m C6v 66 6 126 m2 6 m2 m 3 m displaystyle m cdot 3 m D3h 322 3 2 126 m 2 m 2 m displaystyle tfrac 6 m tfrac 2 m tfrac 2 m 6 mmm m 6 m displaystyle m cdot 6 m D6h 622 6 2 24Cubic 23 23 3 2 displaystyle 3 2 T 332 3 3 122 m displaystyle tfrac 2 m 3 m3 6 2 displaystyle tilde 6 2 Th 3 2 3 4 24432 432 3 4 displaystyle 3 4 O 432 4 3 244 3m 4 3m 3 4 displaystyle 3 tilde 4 Td 332 3 3 244 m displaystyle tfrac 4 m 3 2 m displaystyle tfrac 2 m m3 m 6 4 displaystyle tilde 6 4 Oh 432 4 3 48Isomorphisms EditSee also Crystal structure Crystal systems Many of the crystallographic point groups share the same internal structure For example the point groups 1 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 2 Hermann Mauguin Schoenflies Order Abstract group1 C1 1 C1 G 1 1 displaystyle G 1 1 1 Ci S2 2 C2 G 2 1 displaystyle G 2 1 2 C2 2m Cs C1h 23 C3 3 C3 G 3 1 displaystyle G 3 1 4 C4 4 C4 G 4 1 displaystyle G 4 1 4 S4 42 m C2h 4 D2 C2 C2 G 4 2 displaystyle G 4 2 222 D2 V 4mm2 C2v 43 C3i S6 6 C6 G 6 1 displaystyle G 6 1 6 C6 66 C3h 632 D3 6 D3 G 6 2 displaystyle G 6 2 3m C3v 6mmm D2h Vh 8 D2 C2 G 8 3 displaystyle G 8 3 4 m C4h 8 C4 C2 G 8 2 displaystyle G 8 2 422 D4 8 D4 G 8 4 displaystyle G 8 4 4mm C4v 84 2m D2d Vd 86 m C6h 12 C6 C2 G 12 2 displaystyle G 12 2 23 T 12 A4 G 12 5 displaystyle G 12 5 3 m D3d 12 D6 G 12 3 displaystyle G 12 3 622 D6 126mm C6v 126 m2 D3h 124 mmm D4h 16 D4 C2 G 16 9 displaystyle G 16 9 6 mmm D6h 24 D6 C2 G 24 5 displaystyle G 24 5 m3 Th 24 A4 C2 G 24 10 displaystyle G 24 10 432 O 24 S4 G 24 7 displaystyle G 24 7 4 3m Td 24m3 m Oh 48 S4 C2 G 48 7 displaystyle G 48 7 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 Deriving the crystallographic point group crystal class from the space group EditLeave out the Bravais lattice type Convert all symmetry elements with translational components into their respective symmetry elements without translation symmetry Glide planes are converted into simple mirror planes screw axes are converted into simple axes of rotation Axes of rotation rotoinversion axes and mirror planes remain unchanged See also EditMolecular symmetry Point group Space group Point groups in three dimensions Crystal systemReferences Edit International Tables Abstract Archived from the original on 2013 07 04 Retrieved 2011 11 25 Novak I 1995 07 18 Molecular isomorphism European Journal of Physics IOP Publishing 16 4 151 153 Bibcode 1995EJPh 16 151N doi 10 1088 0143 0807 16 4 001 ISSN 0143 0807 S2CID 250887121 External links EditPoint group symbols in International Tables for Crystallography 2006 Vol A ch 12 1 pp 818 820 Names and symbols of the 32 crystal classes in International Tables for Crystallography 2006 Vol A ch 10 1 p 794 Pictorial overview of the 32 groups Retrieved from https en wikipedia org w index php title Crystallographic point group amp oldid 1118961702, wikipedia, wiki, book, books, library,
