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| [[File:Carbon lattice diamond.png|thumb|The [[Diamond cubic|diamond crystal structure]] belongs to the face-centered [[Cubic crystal system|cubic lattice]], with a repeated 2-atom pattern.]]
| | My name is Sabina (38 years old) and my hobbies are Lapidary and Tai Chi.<br><br>My blog [http://www.zthta.com/site/index.php/component/k2/item/9-?????-?????-???-???????-???????-??????-???/9-?????-?????-???-???????-???????-??????-???.html?start=8480 Fifa 15 coin generator] |
| In [[crystallography]], the terms '''crystal system''', '''crystal family''', and '''lattice system''' each refer to one of several classes of [[space group]]s, [[Bravais lattice|lattice]]s, [[point group]]s, or [[crystal]]s. Informally, two crystals tend to be in the same crystal system if they have similar symmetries, though there are many exceptions to this.
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| Crystal systems, crystal families, and lattice systems are similar but slightly different, and there is widespread confusion between them: in particular the [[trigonal crystal system]] is often confused with the [[rhombohedral lattice system]], and the term "crystal system" is sometimes used to mean "lattice system" or "crystal family".
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| [[Space group]]s and crystals are divided into 7 crystal systems according to their [[point group]]s, and into 7 lattice systems according to their [[Bravais lattice]]s. Five of the crystal systems are essentially the same as five of the lattice systems, but the hexagonal and trigonal crystal systems differ from the hexagonal and rhombohedral lattice systems.
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| The six crystal families are formed by combining the hexagonal and trigonal crystal systems into one hexagonal family, in order to eliminate this confusion.
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| ==Overview==
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| [[Image:Hanksite.JPG|thumb|Hexagonal [[hanksite]] crystal, with three-fold c-axis symmetry]]
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| A '''lattice system''' is a class of lattices with the same point group. In three dimensions there are seven lattice systems: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic. The lattice system of a crystal or space group is determined by its lattice but not always by its point group.
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| A '''crystal system''' is a class of point groups. Two point groups are placed in the same crystal system if the sets of possible lattice systems of their space groups are the same. For many point groups there is only one possible lattice system,
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| and in these cases the crystal system corresponds to a lattice system and is given the same name. However, for the five point groups in the trigonal crystal class there are two possible lattice systems for their point groups: rhombohedral or hexagonal. In three dimensions there are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. The crystal system of a crystal or space group is determined by its point group but not always by its lattice.
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| A ''' crystal family''' also consists of point groups and is formed by combining crystal systems whenever two crystal systems have space groups with the same lattice. In three dimensions a crystal family is almost the same as a crystal system (or lattice system), except that the hexagonal and trigonal crystal systems are combined into one hexagonal family. In three dimensions there are six crystal families: triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, and cubic. The crystal family of a crystal or space group is determined by either its point group or its lattice, and crystal families are the smallest collections of point groups with this property.
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| In dimensions less than three there is no essential difference between crystal systems, crystal families, and lattice systems. There are 1 in dimension 0, 1 in dimension 1, and 4 in dimension 2, called oblique, rectangular, square, and hexagonal.
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| The relation between three-dimensional crystal families, crystal systems, and lattice systems is shown in the following table:
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| {|class="wikitable" cellpadding=0 style="margin: 1em auto; text-align: center;"
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| |-
| |
| ![[Crystal family]]
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| !Crystal system
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| !Required symmetries of point group
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| ![[Crystallographic point group|Point groups]]
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| ![[Space group]]s
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| ![[Bravais lattice]]s
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| ![[Lattice system]]
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| |-
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| |colspan=2|[[Triclinic]]
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| |None
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| |2
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| |2
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| |1
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| |[[Triclinic]]
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| |-
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| |colspan=2|[[Monoclinic]]
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| |1 twofold [[Rotational symmetry|axis of rotation]] or 1 [[Reflection symmetry|mirror plane]]
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| |3
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| |13
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| |2
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| |[[Monoclinic]]
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| |-
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| |colspan=2|[[Orthorhombic]]
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| | 3 twofold axes of rotation or 1 twofold axis of rotation and two mirror planes.
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| |3
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| |59
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| |4
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| |[[Orthorhombic]]
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| |-
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| |colspan=2|[[Tetragonal]]
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| | 1 fourfold axis of rotation
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| |7
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| |68
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| |2
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| |[[Tetragonal]]
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| |-
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| |rowspan=3|[[Hexagonal crystal family|Hexagonal]]
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| |rowspan=2|[[Trigonal]]
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| |rowspan=2|1 threefold axis of rotation
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| |rowspan=2|5
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| |7
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| |1
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| |[[Rhombohedral lattice system|Rhombohedral]]
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| |-
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| |18
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| |rowspan=2|1
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| |rowspan=2|[[Hexagonal lattice system|Hexagonal]]
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| |-
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| |[[hexagonal crystal system|Hexagonal]]
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| |1 sixfold axis of rotation
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| |7
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| |27
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| |-
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| |colspan=2|[[cubic crystal system|Cubic]]
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| |4 threefold axes of rotation
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| |5
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| |36
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| |3
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| |[[cubic crystal system|Cubic]]
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| |- bgcolor=#e0e0e0
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| |'''Total:''' 6
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| |7
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| |32
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| |230
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| |14
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| |7
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| |}
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| ==Crystal classes==
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| The 7 crystal systems consist of 32 crystal classes (corresponding to the 32 [[crystallographic point group]]s) as shown in the following table:
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| {| class=wikitable
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| |-
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| ! crystal family
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| ! crystal system
| |
| ! [[point group]] / crystal class
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| ! [[Schönflies notation|Schönflies]]
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| ! [[Hermann–Mauguin notation|Hermann-Mauguin]]
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| ! [[Orbifold notation|Orbifold]]
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| ! [[Coxeter notation|Coxeter]]
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| ! Point symmetry
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| ! [[Symmetry number|Order]]
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| ! [[Group_theory#Abstract_groups|Abstract group]]
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| |-
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| | rowspan=2 colspan=2| [[triclinic crystal system|triclinic]]
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| | triclinic-pedial
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| | C<sub>1</sub>
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| | 1
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| | 11
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| | [ ]<sup>+</sup>
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| | [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
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| | 1
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| | trivial <math>\mathbb{Z}_1</math>
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| |-
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| | triclinic-pinacoidal
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| | C<sub>i</sub>
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| | {{overline|1}}
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| | 1x
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| | [2,1<sup>+</sup>]
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| | [[centrosymmetric]]
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| | 2
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| | [[Cyclic group|cyclic]] <math>\mathbb{Z}_2</math>
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| |-
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| | rowspan=3 colspan=2 | [[monoclinic crystal system|monoclinic]]
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| | monoclinic-sphenoidal
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| | C<sub>2</sub>
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| | 2
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| | 22
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| | [2,2]<sup>+</sup>
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| | [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
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| | 2
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| | [[Cyclic group|cyclic]] <math>\mathbb{Z}_2</math>
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| |-
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| | monoclinic-domatic
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| | C<sub>s</sub>
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| | m
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| | *11
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| | [ ]
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| | [[Polar point group|polar]]
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| | 2
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| | [[Cyclic group|cyclic]] <math>\mathbb{Z}_2</math>
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| |-
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| | monoclinic-[[prism (geometry)|prismatic]]
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| | C<sub>2h</sub>
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| | 2/m
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| | 2*
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| | [2,2<sup>+</sup>]
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| | [[centrosymmetric]]
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| | 4
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| | [[Klein four-group|Klein four]] <math>\mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2</math>
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| |-
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| | rowspan=3 colspan=2| [[orthorhombic crystal system|orthorhombic]]
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| | orthorhombic-sphenoidal
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| | D<sub>2</sub>
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| | 222
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| | 222
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| | [2,2]<sup>+</sup>
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| | [[Chirality (chemistry)|enantiomorphic]]
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| | 4
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| | [[Klein four-group|Klein four]] <math>\mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2</math>
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| |-
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| | orthorhombic-[[Pyramid (geometry)|pyramidal]]
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| | C<sub>2v</sub>
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| | mm2
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| | *22
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| | [2]
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| | [[Polar point group|polar]]
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| | 4
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| | [[Klein four-group|Klein four]] <math>\mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2</math>
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| |-
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| | orthorhombic-[[bipyramid]]al
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| | D<sub>2h</sub>
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| | mmm
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| | *222
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| | [2,2]
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| | [[centrosymmetric]]
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| | 8
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| | <math>\mathbb{V}\times\mathbb{Z}_2</math>
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| |-
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| | rowspan=7 colspan=2| [[tetragonal crystal system|tetragonal]]
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| | tetragonal-pyramidal
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| | C<sub>4</sub>
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| | 4
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| | 44
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| | [4]<sup>+</sup>
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| | [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
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| | 4
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| | [[Cyclic group|cyclic]] <math>\mathbb{Z}_4</math>
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| |-
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| | tetragonal-disphenoidal
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| | S<sub>4</sub>
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| | {{overline|4}}
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| | 2x
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| | [2<sup>+</sup>,2]
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| | [[non-centrosymmetric]]
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| | 4
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| | [[Cyclic group|cyclic]] <math>\mathbb{Z}_4</math>
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| |-
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| | tetragonal-dipyramidal
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| | C<sub>4h</sub>
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| | 4/m
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| | 4*
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| | [2,4<sup>+</sup>]
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| | [[centrosymmetric]]
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| | 8
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| | <math>\mathbb{Z}_4\times\mathbb{Z}_2</math>
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| |-
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| | tetragonal-trapezoidal
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| | D<sub>4</sub>
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| | 422
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| | 422
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| | [2,4]<sup>+</sup>
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| | [[Chirality (chemistry)|enantiomorphic]]
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| | 8
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| | [[Dihedral group|dihedral]] <math>\mathbb{D}_8 = \mathbb{Z}_4\rtimes\mathbb{Z}_2</math>
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| |-
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| | ditetragonal-pyramidal
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| | C<sub>4v</sub>
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| | 4mm
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| | *44
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| | [4]
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| | [[Polar point group|polar]]
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| | 8
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| | [[Dihedral group|dihedral]] <math>\mathbb{D}_8 = \mathbb{Z}_4\rtimes\mathbb{Z}_2</math>
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| |-
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| | tetragonal-scalenoidal
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| | D<sub>2d</sub>
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| | {{overline|4}}2m or {{overline|4}}m2
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| | 2*2
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| | [2<sup>+</sup>,4]
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| | [[non-centrosymmetric]]
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| | 8
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| | [[Dihedral group|dihedral]] <math>\mathbb{D}_8 = \mathbb{Z}_4\rtimes\mathbb{Z}_2</math>
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| |-
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| | ditetragonal-dipyramidal
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| | D<sub>4h</sub>
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| | 4/mmm
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| | *422
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| | [2,4]
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| | [[centrosymmetric]]
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| | 16
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| | <math>\mathbb{D}_8\times\mathbb{Z}_2</math>
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| |-
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| | rowspan=12|[[hexagonal crystal family|hexagonal]] || rowspan=5 | [[trigonal crystal system|trigonal]]
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| | trigonal-pyramidal
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| | C<sub>3</sub>
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| | 3
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| | 33
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| | [3]<sup>+</sup>
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| | [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
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| | 3
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| | [[Cyclic group|cyclic]] <math>\mathbb{Z}_3</math>
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| |-
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| | rhombohedral
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| | S<sub>6</sub> (C<sub>3i</sub>)
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| | {{overline|3}}
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| | 3x
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| | [2<sup>+</sup>,3<sup>+</sup>]
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| | [[centrosymmetric]]
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| | 6
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| | [[Cyclic group|cyclic]] <math>\mathbb{Z}_6 = \mathbb{Z}_3\times\mathbb{Z}_2</math>
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| |-
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| | trigonal-trapezoidal
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| | D<sub>3</sub>
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| | 32 or 321 or 312
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| | 322
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| | [3,2]<sup>+</sup>
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| | [[Chirality (chemistry)|enantiomorphic]]
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| | 6
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| | [[Dihedral group|dihedral]] <math>\mathbb{D}_6 = \mathbb{Z}_3\rtimes\mathbb{Z}_2</math>
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| |-
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| | ditrigonal-pyramidal
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| | C<sub>3v</sub>
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| | 3m or 3m1 or 31m
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| | *33
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| | [3]
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| | [[Polar point group|polar]]
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| | 6
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| | [[Dihedral group|dihedral]] <math>\mathbb{D}_6 = \mathbb{Z}_3\rtimes\mathbb{Z}_2</math>
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| |-
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| | ditrigonal-scalahedral
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| | D<sub>3d</sub>
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| | {{overline|3}}m or {{overline|3}}m1 or {{overline|3}}1m
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| | 2*3
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| | [2<sup>+</sup>,6]
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| | [[centrosymmetric]]
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| | 12
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| | [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math>
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| |-
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| | rowspan=7 | [[Hexagonal crystal system|hexagonal]]
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| | hexagonal-pyramidal
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| | C<sub>6</sub>
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| | 6
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| | 66
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| | [6]<sup>+</sup>
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| | [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
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| | 6
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| | [[Cyclic group|cyclic]] <math>\mathbb{Z}_6 = \mathbb{Z}_3\times\mathbb{Z}_2</math>
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| |-
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| | trigonal-dipyramidal
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| | C<sub>3h</sub>
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| | {{overline|6}}
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| | 3*
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| | [2,3<sup>+</sup>]
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| | [[non-centrosymmetric]]
| |
| | 6
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| | [[Cyclic group|cyclic]] <math>\mathbb{Z}_6 = \mathbb{Z}_3\times\mathbb{Z}_2</math>
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| |-
| |
| | hexagonal-dipyramidal
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| | C<sub>6h</sub>
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| | 6/m
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| | 6*
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| | [2,6<sup>+</sup>]
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| | [[centrosymmetric]]
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| | 12
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| | <math>\mathbb{Z}_6\times\mathbb{Z}_2</math>
| |
| |-
| |
| | hexagonal-trapezoidal
| |
| | D<sub>6</sub>
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| | 622
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| | 622
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| | [2,6]<sup>+</sup>
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| | [[Chirality (chemistry)|enantiomorphic]]
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| | 12
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| | [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math>
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| |-
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| | dihexagonal-pyramidal
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| | C<sub>6v</sub>
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| | 6mm
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| | *66
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| | [6]
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| | [[Polar point group|polar]]
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| | 12
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| | [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math>
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| |-
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| | ditrigonal-dipyramidal
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| | D<sub>3h</sub>
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| | {{overline|6}}m2 or {{overline|6}}2m
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| | *322
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| | [2,3]
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| | [[non-centrosymmetric]]
| |
| | 12
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| | [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math>
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| |-
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| | dihexagonal-dipyramidal
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| | D<sub>6h</sub>
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| | 6/mmm
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| | *622
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| | [2,6]
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| | [[centrosymmetric]]
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| | 24
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| | <math>\mathbb{D}_{12}\times\mathbb{Z}_2</math>
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| |-
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| | rowspan=5 colspan=2 | [[cubic crystal system|cubic]]
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| | tetrahedral
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| | T || 23
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| | 332
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| | [3,3]<sup>+</sup>
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| | [[Chirality (chemistry)|enantiomorphic]]
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| | 12
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| | [[alternating group|alternating]] <math>\mathbb{A}_4</math>
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| |-
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| | hextetrahedral
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| | T<sub>d</sub>
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| | {{overline|4}}3m
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| | *332
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| | [3,3]
| |
| | [[non-centrosymmetric]]
| |
| | 24
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| | [[symmetric group|symmetric]] <math>\mathbb{S}_4</math>
| |
| |-
| |
| | diploidal
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| | T<sub>h</sub>
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| | m{{overline|3}}
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| | 3*2
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| | [3<sup>+</sup>,4]
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| | [[centrosymmetric]]
| |
| | 24
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| | <math>\mathbb{A}_4\times\mathbb{Z}_2</math>
| |
| |-
| |
| | gyroidal
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| | O
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| | 432
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| | 432
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| | [4,3]<sup>+</sup>
| |
| | [[Chirality (chemistry)|enantiomorphic]]
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| | 24
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| | [[symmetric group|symmetric]] <math>\mathbb{S}_4</math>
| |
| |-
| |
| | hexoctahedral
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| | O<sub>h</sub>
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| | m{{overline|3}}m
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| | *432
| |
| | [4,3]
| |
| | [[centrosymmetric]]
| |
| | 48
| |
| | <math>\mathbb{S}_4\times\mathbb{Z}_2</math>
| |
| |}
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| Point symmetry can be thought of in the following fashion: consider the coordinates which make up the structure, and project 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 for non-centrosymmetric case, inverted structure in some cases can be rotated to align with the original structure. This is the case of non-centrosymmetric achiral structure. If the inverted structure cannot be rotated to align with the original structure, then the structure is chiral (enantiomorphic) and its symmetry group is '''''enantiomorphic'''''.<ref>{{cite journal|author=Howard D. Flack|year=2003|title=Chiral and Achiral Crystal Structures|journal=Helvetica Chimica Acta |volume=86|pages= 905–921|doi=10.1002/hlca.200390109}}</ref>
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| A direction is called polar if its two directional senses are geometrically or physically different. A polar symmetry direction of a crystal is called a polar axis.<ref>E. Koch , W. Fischer , U. Müller , in ‘International Tables for Crystallography, Vol. A, Space-Group Symmetry’, 5th edn., Ed. T. Hahn, Kluwer Academic Publishers, Dordrecht, 2002, Chapt. 10, p. 804.</ref> Groups containing a polar axis are called '''''[[polar point group|polar]]'''''. A polar crystal possess a "unique" axis (found in no other directions) such that some geometrical or physical property is different at the two ends of this axis. It may develop a [[Polarization density|dielectric polarization]], e.g. in [[Pyroelectricity|pyroelectric crystals]]. A polar axis can occur only in non-centrosymmetric structures. There should also not be a mirror plane or 2-fold axis perpendicular to the polar axis, because they will make both directions of the axis equivalent.
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| | |
| The [[crystal structure]]s of chiral biological molecules (such as [[protein]] structures) can only occur in the 11 [[Chirality (chemistry)|enantiomorphic]] point groups (biological molecules are usually [[Chirality (chemistry)|chiral]]).
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| <!--The protein assemblies themselves may have symmetries other than those given above, because they are not intrinsically restricted by the [[Crystallographic restriction theorem]]. For example the [[Rad52]] DNA binding protein has an 11-fold rotational symmetry (in human), however, it must form crystals in one of the 11 [[Chirality (chemistry)|enantiomorphic]] point groups given above. -->
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| | |
| ==Lattice systems ==
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| The distribution of the 14 Bravais lattice types into 7 lattice systems is given in the following table.
| |
| | |
| {| class=wikitable
| |
| |align=center|'''The 7 lattice systems'''
| |
| |colspan=4 align=center| '''The 14 Bravais Lattices'''
| |
| |-
| |
| |colspan=1 align=center| [[triclinic crystal system|triclinic]] ([[parallelepiped]])
| |
| || [[Image:Triclinic.svg|80px|Triclinic]]
| |
| |-
| |
| |rowspan=2 align=center| [[monoclinic crystal system|monoclinic]] (right [[prism (geometry)|prism]] with [[parallelogram]] base; here seen from above)
| |
| |align=center| simple
| |
| |align=center| base-centered
| |
| |-
| |
| || [[Image:Monoclinic.svg|80px|Monoclinic, simple]]
| |
| || [[Image:Monoclinic-base-centered.svg|80px|Monoclinic, centered]]
| |
| |-
| |
| |rowspan=2 align=center| [[orthorhombic crystal system|orthorhombic]] ([[cuboid]])
| |
| |align=center| simple
| |
| |align=center| base-centered
| |
| |align=center| body-centered
| |
| |align=center| face-centered
| |
| |-
| |
| || [[Image:Orthorhombic.svg|80px|Orthohombic, simple]]
| |
| || [[Image:Orthorhombic-base-centered.svg|80px|Orthohombic, base-centered]]
| |
| || [[Image:Orthorhombic-body-centered.svg|80px|Orthohombic, body-centered]]
| |
| || [[Image:Orthorhombic-face-centered.svg|80px|Orthohombic, face-centered]]
| |
| |-
| |
| |rowspan=2 align=center| [[tetragonal crystal system|tetragonal]] (square [[cuboid]])
| |
| |align=center|simple
| |
| |align=center| body-centered
| |
| |-
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| || [[Image:Tetragonal.svg|80px|Tetragonal, simple]]
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| || [[Image:Tetragonal-body-centered.svg|80px|Tetragonal, body-centered]]
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| |-
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| |align=center| [[rhombohedral lattice system|rhombohedral]] <br>([[trigonal trapezohedron]])
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| | [[Image:Rhombohedral.svg|80px|Rhombohedral]]
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| |-
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| |align=center| [[Hexagonal crystal system|hexagonal]] (centered regular [[hexagon]])
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| | [[Image:Hexagonal lattice.svg|80px|Hexagonal]]
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| |-
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| |rowspan=2 align=center| [[Cubic crystal system|cubic]]<br>(isometric; [[cube]])
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| |align=center| simple
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| |align=center| body-centered
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| |align=center| face-centered
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| |-
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| || [[Image:Cubic.svg|80px|Cubic, simple]]
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| | [[Image:Cubic-body-centered.svg|80px|Cubic, body-centered]]
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| | [[Image:Cubic-face-centered.svg|80px|Cubic, face-centered]]
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| |}
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| {{-}}
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| In [[geometry]] and [[crystallography]], a '''Bravais lattice''' is a category of [[symmetry group]]s for [[translational symmetry]] in three directions, or correspondingly, a category of translation [[Lattice (group)|lattice]]s.
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| Such symmetry groups consist of translations by vectors of the form
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| :<math>\mathbf{R} = n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + n_3 \mathbf{a}_3,</math> | |
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| where ''n''<sub>1</sub>, ''n''<sub>2</sub>, and ''n''<sub>3</sub> are [[integer]]s and ''a''<sub>1</sub>, ''a''<sub>2</sub>, and ''a''<sub>3</sub> are three non-coplanar vectors, called ''primitive vectors''.
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| These lattices are classified by [[space group]] of the translation lattice itself; there are 14 Bravais lattices in three dimensions; each can apply in one lattice system only. They represent the maximum symmetry a structure with the translational symmetry concerned can have.
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| All crystalline materials must, by definition fit in one of these arrangements (not including [[quasicrystal]]s).
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| For convenience a Bravais lattice is depicted by a unit cell which is a factor 1, 2, 3 or 4 larger than the [[primitive cell]]. Depending on the symmetry of a crystal or other pattern, the [[fundamental domain]] is again smaller, up to a factor 48.
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| The Bravais lattices were studied by [[Moritz Ludwig Frankenheim]] (1801–1869), in 1842, who found that there were 15 Bravais lattices. This was corrected to 14 by [[Auguste Bravais|A. Bravais]] in 1848<!-- or 1849 or 1850, Britannica has two different years-->.
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| ==Crystal systems in four-dimensional space==
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| The four-dimensional unit cell is defined by four edge lengths (<math>a, b, c, d</math>) and six interaxial angles (<math>\alpha, \beta, \gamma, \delta, \epsilon, \zeta</math>). The following conditions for the lattice parameters define 23 crystal families:
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| 1 Hexaclinic: <math>a\ne b \ne c \ne d, \alpha \ne \beta \ne \gamma \ne \delta \ne \epsilon \ne \zeta \ne 90 ^\circ</math>
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| 2 Triclinic: <math>a\ne b \ne c \ne d, \alpha \ne \beta \ne \gamma \ne 90 ^\circ, \delta = \epsilon = \zeta = 90 ^\circ</math>
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| 3 Diclinic: <math>a\ne b \ne c \ne d, \alpha \ne 90 ^\circ, \beta = \gamma = \delta = \epsilon = 90 ^\circ, \zeta \ne 90 ^\circ</math>
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| 4 Monoclinic: <math>a\ne b \ne c \ne d, \alpha \ne 90 ^\circ, \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math>
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| 5 Orthogonal: <math>a\ne b \ne c \ne d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math>
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| 6 Tetragonal Monoclinic: <math>a\ne b = c \ne d, \alpha \ne 90 ^\circ, \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math>
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| 7 Hexagonal Monoclinic: <math>a\ne b = c \ne d, \alpha \ne 90 ^\circ, \beta = \gamma = \delta = \epsilon = 90 ^\circ, \zeta = 120 ^\circ</math>
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| 8 Ditetragonal Diclinic: <math>a = d \ne b = c, \alpha = \zeta = 90 ^\circ, \beta = \epsilon \ne 90 ^\circ, \gamma \ne 90 ^\circ, \delta = 180 ^\circ - \gamma </math>
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| 9 Ditrigonal (Dihexagonal) Diclinic: <math>a = d \ne b = c, \alpha = \zeta = 120 ^\circ, \beta = \epsilon \ne 90 ^\circ, \gamma \ne \delta \ne 90 ^\circ, cos \delta = cos \beta - cos \gamma</math> | |
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| 10 Tetragonal Orthogonal: <math>a\ne b = c \ne d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math>
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| 11 Hexagonal Orthogonal: <math>a\ne b = c \ne d, \alpha = \beta = \gamma = \delta = \epsilon = 90 ^\circ, \zeta = 120 ^\circ</math>
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| 12 Ditetragonal Monoclinic: <math>a = d \ne b = c, \alpha = \gamma = \delta = \zeta = 90 ^\circ, \beta = \epsilon \ne 90 ^\circ</math>
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| 13 Ditrigonal (Dihexagonal) Monoclinic: <math>a = d \ne b = c, \alpha = \zeta = 120 ^\circ, \beta = \epsilon \ne 90 ^\circ, \gamma = \delta \ne 90 ^\circ, cos \gamma = -\color{Black}\tfrac{1}{2} cos \beta</math>
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| 14 Ditetragonal Orthogonal: <math>a = d \ne b = c, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math>
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| 15 Hexagonal Tetragonal: <math>a = d \ne b = c, \alpha = \beta = \gamma = \delta = \epsilon = 90 ^\circ, \zeta = 120 ^\circ</math>
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| 16 Dihexagonal Orthogonal: <math>a = d \ne b = c, \alpha = \zeta = 120 ^\circ, \beta = \gamma = \delta = \epsilon = 90 ^\circ, </math>
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| 17 Cubic Orthogonal: <math>a = b = c \ne d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math>
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| 18 Octagonal: <math>a = b = c = d, \alpha = \gamma = \zeta \ne 90 ^\circ, \beta = \epsilon = 90 ^\circ, \delta = 180 ^\circ - \alpha</math>
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| 19 Decagonal: <math>a = b = c = d, \alpha = \gamma = \zeta \ne \beta = \delta = \epsilon, cos \beta = -0.5 - cos \alpha</math>
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| 20 Dodecagonal: <math>a = b = c = d, \alpha = \zeta = 90 ^\circ, \beta = \epsilon = 120 ^\circ, \gamma = \delta \ne 90 ^\circ</math>
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| 21 Di-isohexagonal Orthogonal: <math>a = b = c = d, \alpha = \zeta = 120 ^\circ, \beta = \gamma = \delta = \epsilon = 90 ^\circ</math>
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| 22 Icosagonal (Icosahedral): <math>a = b = c = d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta, cos \alpha = -\color{Black}\tfrac{1}{4}</math>
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| | |
| 23 Hypercubic: <math>a = b = c = d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math>
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| | |
| The names here are given according to Whittaker.<ref name="Whittaker">E. J. W. Whittaker, An atlas of hyperstereograms of the four-dimensional crystal classes. Clarendon Press (Oxford Oxfordshire and New York) 1985.</ref> They are almost the same as in Brown ''et al'',<ref name="Brown">H. Brown, R. Bülow, J. Neubüser, H. Wondratschek and H. Zassenhaus, Crystallographic Groups of Four-Dimensional Space. Wiley, NY, 1978.</ref> with exception for names of the crystal families 9, 13, and 22. The names for these three families according to Brown ''et al'' are given in parenthesis.
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| | |
| The relation between four-dimensional crystal families, crystal systems, and lattice systems is shown in the following table.<ref name="Whittaker"/><ref name="Brown"/> Enantiomorphic systems are marked with asterisk. The number of enantiomorphic pairs are given in parentheses. Here the term "enantiomorphic" has different meaning than in table for three-dimensional crystal classes. The latter means, that enantiomorphic point groups describe chiral (enantiomorphic) structures. In the current table, "enantiomorphic" means, that group itself (considered as geometric object) is enantiomorphic, like enantiomorphic pairs of three-dimensional space groups P3<sub>1</sub> and P3<sub>2</sub>, P4<sub>1</sub>22 and P4<sub>3</sub>22. Starting from four-dimensional space, point groups also can be enantiomorphic in this sense.
| |
| {|class="wikitable" cellpadding=4 cellspacing=0
| |
| |-align=center
| |
| !No. of <br />Crystal family
| |
| !Crystal family
| |
| !Crystal system
| |
| !No. of <br>Crystal system
| |
| !Point groups
| |
| !width=120|Space groups
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| !Bravais lattices
| |
| !Lattice system
| |
| |-
| |
| | I ||colspan=2| Hexaclinic|| 1
| |
| |2
| |
| |2
| |
| |1
| |
| |Hexaclinic P
| |
| |-
| |
| | II || colspan=2| Triclinic|| 2
| |
| |3
| |
| |13
| |
| |2
| |
| |Triclinic P, S
| |
| |-
| |
| | III ||colspan=2| Diclinic|| 3
| |
| |2
| |
| |12
| |
| |3
| |
| |Diclinic P, S, D
| |
| |-
| |
| |IV || colspan=2| Monoclinic|| 4
| |
| |4
| |
| |207
| |
| |6
| |
| |Monoclinic P, S, S, I, D, F
| |
| |-
| |
| |rowspan=3| V ||rowspan=3| Orthogonal
| |
| |rowspan=2|Non-axial Orthogonal|| rowspan=2| 5
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| |rowspan=2|2
| |
| |2
| |
| |1
| |
| |Orthogonal KU
| |
| |-
| |
| |112
| |
| |rowspan=2|8
| |
| |rowspan=2|Orthogonal P, S, I, Z, D, F, G, U
| |
| |-
| |
| |Axial Orthogonal|| 6
| |
| |3
| |
| |887
| |
| |-
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| | VI || colspan=2| Tetragonal Monoclinic || 7
| |
| |7
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| |88
| |
| |2
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| |Tetragonal Monoclinic P, I
| |
| |-
| |
| |rowspan=3| VII ||rowspan=3| Hexagonal Monoclinic
| |
| |rowspan=2|Trigonal Monoclinic ||rowspan=2| 8
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| |rowspan=2|5
| |
| |9
| |
| |1
| |
| |Hexagonal Monoclinic R
| |
| |-
| |
| |15
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| |rowspan=2|1
| |
| |rowspan=2|Hexagonal Monoclinic P
| |
| |-
| |
| |Hexagonal Monoclinic || 9
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| |7
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| |25
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| |-
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| | VIII || colspan=2| Ditetragonal Diclinic* ||10
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| |1 (+1)
| |
| |1 (+1)
| |
| |1 (+1)
| |
| |Ditetragonal Diclinic P*
| |
| |-
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| |IX || colspan=2| Ditrigonal Diclinic* ||11
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| |2 (+2)
| |
| |2 (+2)
| |
| |1 (+1)
| |
| |Ditrigonal Diclinic P*
| |
| |-
| |
| |rowspan=3| X ||rowspan=3| Tetragonal Orthogonal
| |
| |rowspan=2|Inverse Tetragonal Orthogonal ||rowspan=2| 12
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| |rowspan=2|5
| |
| |7
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| |1
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| |Tetragonal Orthogonal KG
| |
| |-
| |
| |351
| |
| |rowspan=2|5
| |
| |rowspan=2|Tetragonal Orthogonal P, S, I, Z, G
| |
| |-
| |
| |Proper Tetragonal Orthogonal || 13
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| |10
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| |1312
| |
| |-
| |
| |rowspan=3|XI ||rowspan=3| Hexagonal Orthogonal
| |
| |rowspan=2|Trigonal Orthogonal ||rowspan=2| 14
| |
| |rowspan=2|10
| |
| |81
| |
| |2
| |
| |Hexagonal Orthogonal R, RS
| |
| |-
| |
| |150
| |
| |rowspan=2|2
| |
| |rowspan=2|Hexagonal Orthogonal P, S
| |
| |-
| |
| |Hexagonal Orthogonal || 15
| |
| |12
| |
| |240
| |
| |-
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| | XII || colspan=2| Ditetragonal Monoclinic* || 16
| |
| |1 (+1)
| |
| |6 (+6)
| |
| |3 (+3)
| |
| |Ditetragonal Monoclinic P*, S*, D*
| |
| |-
| |
| | XIII || colspan=2| Ditrigonal Monoclinic* || 17
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| |2 (+2)
| |
| |5 (+5)
| |
| |2 (+2)
| |
| |Ditrigonal Monoclinic P*, RR*
| |
| |-
| |
| |rowspan=3| XIV ||rowspan=3| Ditetragonal Orthogonal
| |
| |rowspan=2|Crypto-Ditetragonal Orthogonal ||rowspan=2| 18
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| |rowspan=2|5
| |
| |10
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| |1
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| |Ditetragonal Orthogonal D
| |
| |-
| |
| |165 (+2)
| |
| |rowspan=2|2
| |
| |rowspan=2|Ditetragonal Orthogonal P, Z
| |
| |-
| |
| |Ditetragonal Orthogonal ||19
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| |6
| |
| |127
| |
| |-
| |
| |XV ||colspan=2| Hexagonal Tetragonal || 20
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| |22
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| |108
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| |1
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| |Hexagonal Tetragonal P
| |
| |-
| |
| |rowspan=5| XVI || rowspan=5| Dihexagonal Orthogonal
| |
| |rowspan=2| Crypto-Ditrigonal Orthogonal* || rowspan=2|21
| |
| |rowspan=2|4 (+4)
| |
| |5 (+5)
| |
| |1 (+1)
| |
| |Dihexagonal Orthogonal G*
| |
| |-
| |
| |5 (+5)
| |
| |rowspan=3|1
| |
| |rowspan=3|Dihexagonal Orthogonal P
| |
| |-
| |
| |Dihexagonal Orthogonal || 23
| |
| |11
| |
| |20
| |
| |-
| |
| |rowspan=2| Ditrigonal Orthogonal || rowspan=2| 22
| |
| |rowspan=2| 11
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| |41
| |
| |-
| |
| |16
| |
| |1
| |
| |Dihexagonal Orthogonal RR
| |
| |-
| |
| |rowspan=3| XVII ||rowspan=3| Cubic Orthogonal
| |
| |rowspan=2|Simple Cubic Orthogonal ||rowspan=2| 24
| |
| |rowspan=2|5
| |
| |9
| |
| |1
| |
| |Cubic Orthogonal KU
| |
| |-
| |
| |96
| |
| |rowspan=2|5
| |
| |rowspan=2|Cubic Orthogonal P, I, Z, F, U
| |
| |-
| |
| |Complex Cubic Orthogonal || 25
| |
| |11
| |
| |366
| |
| |-
| |
| | XVIII ||colspan=2| Octagonal* || 26
| |
| |2 (+2)
| |
| |3 (+3)
| |
| |1 (+1)
| |
| | Octagonal P*
| |
| |-
| |
| | XIX ||colspan=2| Decagonal || 27
| |
| |4
| |
| |5
| |
| |1
| |
| | Decagonal P
| |
| |-
| |
| | XX ||colspan=2| Dodecagonal* ||28
| |
| |2 (+2)
| |
| |2 (+2)
| |
| |1 (+1)
| |
| | Dodecagonal P*
| |
| |-
| |
| |rowspan=3| XXI ||rowspan=3| Di-isohexagonal Orthogonal
| |
| |rowspan=2| Simple Di-isohexagonal Orthogonal || rowspan=2| 29
| |
| |rowspan=2|9 (+2)
| |
| |19 (+5)
| |
| |1
| |
| |Di-isohexagonal Orthogonal RR
| |
| |-
| |
| |19 (+3)
| |
| |rowspan=2|1
| |
| |rowspan=2|Di-isohexagonal Orthogonal P
| |
| |-
| |
| |Complex Di-isohexagonal Orthogonal ||30
| |
| |13 (+8)
| |
| |15 (+9)
| |
| |-
| |
| |XXII ||colspan=2| Icosagonal|| 31
| |
| |7
| |
| |20
| |
| |2
| |
| | Icosagonal P, SN
| |
| |-
| |
| |rowspan=3| XXIII ||rowspan=3| Hypercubic
| |
| |rowspan=2| Octagonal Hypercubic||rowspan=2|32
| |
| |rowspan=2|21 (+8)
| |
| |73 (+15)
| |
| |1
| |
| |Hypercubic P
| |
| |-
| |
| |107 (+28)
| |
| |rowspan=2|1
| |
| |rowspan=2|Hypercubic Z
| |
| |-
| |
| |Dodecagonal Hypercubic|| 33
| |
| |16 (+12)
| |
| |25 (+20)
| |
| |- bgcolor=#e0e0e0
| |
| |'''Total:'''
| |
| |23 (+6)
| |
| |33 (+7)
| |
| |
| |
| |227 (+44)
| |
| |4783 (+111)
| |
| |64 (+10)
| |
| |33 (+7)
| |
| |}
| |
| | |
| ==See also==
| |
| *[[Crystal cluster]]
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| *[[Crystal structure]]
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| *[[Space group#Classification systems for space groups|List of the 230 crystallographic 3D space groups]]
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| *[[Polar point group]]
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| | |
| ==Notes==
| |
| <!--This article uses the Cite.php citation mechanism. If you would like more information on how to add references to this article, please see http://meta.wikimedia.org/wiki/Cite/Cite.php -->
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| <div class="references-small" >
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| <references/>
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| </div>
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| | |
| ==References==
| |
| *{{Cite book | editor1-last=Hahn | editor1-first=Theo | title=International Tables for Crystallography, Volume A: Space Group Symmetry | url=http://it.iucr.org/A/ | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=5th | isbn=978-0-7923-6590-7 | doi=10.1107/97809553602060000100 | year=2002 | volume=A}}
| |
| | |
| ==External links==
| |
| *[http://newton.ex.ac.uk/research/qsystems/people/goss/symmetry/Solids.html Overview of the 32 groups]
| |
| *[http://mineral.galleries.com/minerals/symmetry/symmetry.htm Mineral galleries – Symmetry]
| |
| *[http://www.ifg.uni-kiel.de/kubische_Formen all cubic crystal classes, forms and stereographic projections (interactive java applet)]
| |
| *[http://reference.iucr.org/dictionary/Crystal_system Crystal system] at the [http://reference.iucr.org/dictionary/Main_Page Online Dictionary of Crystallography]
| |
| *[http://reference.iucr.org/dictionary/Crystal_family Crystal family] at the [http://reference.iucr.org/dictionary/Main_Page Online Dictionary of Crystallography]
| |
| *[http://reference.iucr.org/dictionary/Lattice_system Lattice system] at the [http://reference.iucr.org/dictionary/Main_Page Online Dictionary of Crystallography]
| |
| *[http://materials.duke.edu/awrapper.html Conversion Primitive to Standard Conventional for VASP input files]
| |
| *[http://www.xtal.iqfr.csic.es/Cristalografia/index-en.html Learning Crystallography]
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| | |
| {{Crystal systems}}
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| {{Mineral identification}}
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| [[Category:Symmetry]]
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| [[Category:Euclidean geometry]]
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| [[Category:Crystallography]]
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| [[Category:Morphology]]
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| [[Category:Mineralogy]]
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| [[ru:Сингония]]
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