From Wikipedia, the free encyclopedia
Classification system for crystals
In crystallography, a crystallographic point group is a three-dimensional point group whose symmetry operations are compatible with a three-dimensional crystallographic lattice. According to the crystallographic restriction it may only contain one-, two-, three-, four- and sixfold rotations or rotoinversions. This reduces the number of crystallographic point groups to 32 (from an infinity of general point groups). These 32 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.[1]: 379
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.
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.
Schoenflies notation[edit]In Schoenflies notation, point groups are denoted by a letter symbol with a subscript. The symbols used in crystallography mean the following:
Due to the crystallographic restriction theorem, n = 1, 2, 3, 4, or 6 in 2- or 3-dimensional space.
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 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]An abbreviated form of the Hermann–Mauguin notation commonly used for space groups also serves to describe crystallographic point groups. Group names are
The correspondence between different notations[edit] Crystal family Crystal system Hermann-Mauguin Shubnikov[2] Schoenflies Orbifold Coxeter Order (full) (short) Triclinic 1 1 1 {\displaystyle 1\ } C1 11 [ ]+ 1 1 1 2 ~ {\displaystyle {\tilde {2}}} Ci = S2 × [2+,2+] 2 Monoclinic 2 2 2 {\displaystyle 2\ } C2 22 [2]+ 2 m m m {\displaystyle m\ } Cs = C1h * [ ] 2 2 m {\displaystyle {\tfrac {2}{m}}} 2/m 2 : m {\displaystyle 2:m\ } C2h 2* [2,2+] 4 Orthorhombic 222 222 2 : 2 {\displaystyle 2:2\ } D2 = V 222 [2,2]+ 4 mm2 mm2 2 ⋅ m {\displaystyle 2\cdot m\ } C2v *22 [2] 4 2 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] 8 Tetragonal 4 4 4 {\displaystyle 4\ } C4 44 [4]+ 4 4 4 4 ~ {\displaystyle {\tilde {4}}} S4 2× [2+,4+] 4 4 m {\displaystyle {\tfrac {4}{m}}} 4/m 4 : m {\displaystyle 4:m\ } C4h 4* [2,4+] 8 422 422 4 : 2 {\displaystyle 4:2\ } D4 422 [4,2]+ 8 4mm 4mm 4 ⋅ m {\displaystyle 4\cdot m\ } C4v *44 [4] 8 42m 42m 4 ~ ⋅ m {\displaystyle {\tilde {4}}\cdot m} D2d = Vd 2*2 [2+,4] 8 4 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] 16 Hexagonal Trigonal 3 3 3 {\displaystyle 3\ } C3 33 [3]+ 3 3 3 6 ~ {\displaystyle {\tilde {6}}} C3i = S6 3× [2+,6+] 6 32 32 3 : 2 {\displaystyle 3:2\ } D3 322 [3,2]+ 6 3m 3m 3 ⋅ m {\displaystyle 3\cdot m\ } C3v *33 [3] 6 3 2 m {\displaystyle {\tfrac {2}{m}}} 3m 6 ~ ⋅ m {\displaystyle {\tilde {6}}\cdot m} D3d 2*3 [2+,6] 12 Hexagonal 6 6 6 {\displaystyle 6\ } C6 66 [6]+ 6 6 6 3 : m {\displaystyle 3:m\ } C3h 3* [2,3+] 6 6 m {\displaystyle {\tfrac {6}{m}}} 6/m 6 : m {\displaystyle 6:m\ } C6h 6* [2,6+] 12 622 622 6 : 2 {\displaystyle 6:2\ } D6 622 [6,2]+ 12 6mm 6mm 6 ⋅ m {\displaystyle 6\cdot m\ } C6v *66 [6] 12 6m2 6m2 m ⋅ 3 : m {\displaystyle m\cdot 3:m\ } D3h *322 [3,2] 12 6 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] 24 Cubic 23 23 3 / 2 {\displaystyle 3/2\ } T 332 [3,3]+ 12 2 m {\displaystyle {\tfrac {2}{m}}} 3 m3 6 ~ / 2 {\displaystyle {\tilde {6}}/2} Th 3*2 [3+,4] 24 432 432 3 / 4 {\displaystyle 3/4\ } O 432 [4,3]+ 24 43m 43m 3 / 4 ~ {\displaystyle 3/{\tilde {4}}} Td *332 [3,3] 24 4 m {\displaystyle {\tfrac {4}{m}}} 3 2 m {\displaystyle {\tfrac {2}{m}}} m3m 6 ~ / 4 {\displaystyle {\tilde {6}}/4} Oh *432 [4,3] 48Many 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:[3]
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[edit]RetroSearch is an open source project built by @garambo | Open a GitHub Issue
Search and Browse the WWW like it's 1997 | Search results from DuckDuckGo
HTML:
3.2
| Encoding:
UTF-8
| Version:
0.7.4