Incomplete gamma function: Difference between revisions
→Special values: also updated the explanation for the E_n function and updated the notation |
fixing some incorrect capitals and some punctuation errors |
||
| Line 1: | Line 1: | ||
[[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.]] | |||
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. | |||
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". | |||
[[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. | |||
The six crystal families are formed by combining the hexagonal and trigonal crystal systems into one hexagonal family, in order to eliminate this confusion. | |||
==Overview== | |||
[[Image:Hanksite.JPG|thumb|Hexagonal [[hanksite]] crystal, with three-fold c-axis symmetry]] | |||
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. | |||
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, | |||
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. | |||
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. | |||
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. | |||
The relation between three-dimensional crystal families, crystal systems, and lattice systems is shown in the following table: | |||
{|class="wikitable" cellpadding=0 style="margin: 1em auto; text-align: center;" | |||
|- | |||
![[Crystal family]] | |||
!Crystal system | |||
!Required symmetries of point group | |||
![[Crystallographic point group|Point groups]] | |||
![[Space group]]s | |||
![[Bravais lattice]]s | |||
![[Lattice system]] | |||
|- | |||
|colspan=2|[[Triclinic]] | |||
|None | |||
|2 | |||
|2 | |||
|1 | |||
|[[Triclinic]] | |||
|- | |||
|colspan=2|[[Monoclinic]] | |||
|1 twofold [[Rotational symmetry|axis of rotation]] or 1 [[Reflection symmetry|mirror plane]] | |||
|3 | |||
|13 | |||
|2 | |||
|[[Monoclinic]] | |||
|- | |||
|colspan=2|[[Orthorhombic]] | |||
| 3 twofold axes of rotation or 1 twofold axis of rotation and two mirror planes. | |||
|3 | |||
|59 | |||
|4 | |||
|[[Orthorhombic]] | |||
|- | |||
|colspan=2|[[Tetragonal]] | |||
| 1 fourfold axis of rotation | |||
|7 | |||
|68 | |||
|2 | |||
|[[Tetragonal]] | |||
|- | |||
|rowspan=3|[[Hexagonal crystal family|Hexagonal]] | |||
|rowspan=2|[[Trigonal]] | |||
|rowspan=2|1 threefold axis of rotation | |||
|rowspan=2|5 | |||
|7 | |||
|1 | |||
|[[Rhombohedral lattice system|Rhombohedral]] | |||
|- | |||
|18 | |||
|rowspan=2|1 | |||
|rowspan=2|[[Hexagonal lattice system|Hexagonal]] | |||
|- | |||
|[[hexagonal crystal system|Hexagonal]] | |||
|1 sixfold axis of rotation | |||
|7 | |||
|27 | |||
|- | |||
|colspan=2|[[cubic crystal system|Cubic]] | |||
|4 threefold axes of rotation | |||
|5 | |||
|36 | |||
|3 | |||
|[[cubic crystal system|Cubic]] | |||
|- bgcolor=#e0e0e0 | |||
|'''Total:''' 6 | |||
|7 | |||
| | |||
|32 | |||
|230 | |||
|14 | |||
|7 | |||
|} | |||
==Crystal classes== | |||
The 7 crystal systems consist of 32 crystal classes (corresponding to the 32 [[crystallographic point group]]s) as shown in the following table: | |||
{| class=wikitable | |||
|- | |||
! crystal family | |||
! crystal system | |||
! [[point group]] / crystal class | |||
! [[Schönflies notation|Schönflies]] | |||
! [[Hermann–Mauguin notation|Hermann-Mauguin]] | |||
! [[Orbifold notation|Orbifold]] | |||
! [[Coxeter notation|Coxeter]] | |||
! Point symmetry | |||
! [[Symmetry number|Order]] | |||
! [[Group_theory#Abstract_groups|Abstract group]] | |||
|- | |||
| rowspan=2 colspan=2| [[triclinic crystal system|triclinic]] | |||
| triclinic-pedial | |||
| C<sub>1</sub> | |||
| 1 | |||
| 11 | |||
| [ ]<sup>+</sup> | |||
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]] | |||
| 1 | |||
| trivial <math>\mathbb{Z}_1</math> | |||
|- | |||
| triclinic-pinacoidal | |||
| C<sub>i</sub> | |||
| {{overline|1}} | |||
| 1x | |||
| [2,1<sup>+</sup>] | |||
| [[centrosymmetric]] | |||
| 2 | |||
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_2</math> | |||
|- | |||
| rowspan=3 colspan=2 | [[monoclinic crystal system|monoclinic]] | |||
| monoclinic-sphenoidal | |||
| C<sub>2</sub> | |||
| 2 | |||
| 22 | |||
| [2,2]<sup>+</sup> | |||
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]] | |||
| 2 | |||
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_2</math> | |||
|- | |||
| monoclinic-domatic | |||
| C<sub>s</sub> | |||
| m | |||
| *11 | |||
| [ ] | |||
| [[Polar point group|polar]] | |||
| 2 | |||
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_2</math> | |||
|- | |||
| monoclinic-[[prism (geometry)|prismatic]] | |||
| C<sub>2h</sub> | |||
| 2/m | |||
| 2* | |||
| [2,2<sup>+</sup>] | |||
| [[centrosymmetric]] | |||
| 4 | |||
| [[Klein four-group|Klein four]] <math>\mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2</math> | |||
|- | |||
| rowspan=3 colspan=2| [[orthorhombic crystal system|orthorhombic]] | |||
| orthorhombic-sphenoidal | |||
| D<sub>2</sub> | |||
| 222 | |||
| 222 | |||
| [2,2]<sup>+</sup> | |||
| [[Chirality (chemistry)|enantiomorphic]] | |||
| 4 | |||
| [[Klein four-group|Klein four]] <math>\mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2</math> | |||
|- | |||
| orthorhombic-[[Pyramid (geometry)|pyramidal]] | |||
| C<sub>2v</sub> | |||
| mm2 | |||
| *22 | |||
| [2] | |||
| [[Polar point group|polar]] | |||
| 4 | |||
| [[Klein four-group|Klein four]] <math>\mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2</math> | |||
|- | |||
| orthorhombic-[[bipyramid]]al | |||
| D<sub>2h</sub> | |||
| mmm | |||
| *222 | |||
| [2,2] | |||
| [[centrosymmetric]] | |||
| 8 | |||
| <math>\mathbb{V}\times\mathbb{Z}_2</math> | |||
|- | |||
| rowspan=7 colspan=2| [[tetragonal crystal system|tetragonal]] | |||
| tetragonal-pyramidal | |||
| C<sub>4</sub> | |||
| 4 | |||
| 44 | |||
| [4]<sup>+</sup> | |||
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]] | |||
| 4 | |||
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_4</math> | |||
|- | |||
| tetragonal-disphenoidal | |||
| S<sub>4</sub> | |||
| {{overline|4}} | |||
| 2x | |||
| [2<sup>+</sup>,2] | |||
| [[non-centrosymmetric]] | |||
| 4 | |||
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_4</math> | |||
|- | |||
| tetragonal-dipyramidal | |||
| C<sub>4h</sub> | |||
| 4/m | |||
| 4* | |||
| [2,4<sup>+</sup>] | |||
| [[centrosymmetric]] | |||
| 8 | |||
| <math>\mathbb{Z}_4\times\mathbb{Z}_2</math> | |||
|- | |||
| tetragonal-trapezoidal | |||
| D<sub>4</sub> | |||
| 422 | |||
| 422 | |||
| [2,4]<sup>+</sup> | |||
| [[Chirality (chemistry)|enantiomorphic]] | |||
| 8 | |||
| [[Dihedral group|dihedral]] <math>\mathbb{D}_8 = \mathbb{Z}_4\rtimes\mathbb{Z}_2</math> | |||
|- | |||
| ditetragonal-pyramidal | |||
| C<sub>4v</sub> | |||
| 4mm | |||
| *44 | |||
| [4] | |||
| [[Polar point group|polar]] | |||
| 8 | |||
| [[Dihedral group|dihedral]] <math>\mathbb{D}_8 = \mathbb{Z}_4\rtimes\mathbb{Z}_2</math> | |||
|- | |||
| tetragonal-scalenoidal | |||
| D<sub>2d</sub> | |||
| {{overline|4}}2m or {{overline|4}}m2 | |||
| 2*2 | |||
| [2<sup>+</sup>,4] | |||
| [[non-centrosymmetric]] | |||
| 8 | |||
| [[Dihedral group|dihedral]] <math>\mathbb{D}_8 = \mathbb{Z}_4\rtimes\mathbb{Z}_2</math> | |||
|- | |||
| ditetragonal-dipyramidal | |||
| D<sub>4h</sub> | |||
| 4/mmm | |||
| *422 | |||
| [2,4] | |||
| [[centrosymmetric]] | |||
| 16 | |||
| <math>\mathbb{D}_8\times\mathbb{Z}_2</math> | |||
|- | |||
| rowspan=12|[[hexagonal crystal family|hexagonal]] || rowspan=5 | [[trigonal crystal system|trigonal]] | |||
| trigonal-pyramidal | |||
| C<sub>3</sub> | |||
| 3 | |||
| 33 | |||
| [3]<sup>+</sup> | |||
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]] | |||
| 3 | |||
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_3</math> | |||
|- | |||
| rhombohedral | |||
| S<sub>6</sub> (C<sub>3i</sub>) | |||
| {{overline|3}} | |||
| 3x | |||
| [2<sup>+</sup>,3<sup>+</sup>] | |||
| [[centrosymmetric]] | |||
| 6 | |||
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_6 = \mathbb{Z}_3\times\mathbb{Z}_2</math> | |||
|- | |||
| trigonal-trapezoidal | |||
| D<sub>3</sub> | |||
| 32 or 321 or 312 | |||
| 322 | |||
| [3,2]<sup>+</sup> | |||
| [[Chirality (chemistry)|enantiomorphic]] | |||
| 6 | |||
| [[Dihedral group|dihedral]] <math>\mathbb{D}_6 = \mathbb{Z}_3\rtimes\mathbb{Z}_2</math> | |||
|- | |||
| ditrigonal-pyramidal | |||
| C<sub>3v</sub> | |||
| 3m or 3m1 or 31m | |||
| *33 | |||
| [3] | |||
| [[Polar point group|polar]] | |||
| 6 | |||
| [[Dihedral group|dihedral]] <math>\mathbb{D}_6 = \mathbb{Z}_3\rtimes\mathbb{Z}_2</math> | |||
|- | |||
| ditrigonal-scalahedral | |||
| D<sub>3d</sub> | |||
| {{overline|3}}m or {{overline|3}}m1 or {{overline|3}}1m | |||
| 2*3 | |||
| [2<sup>+</sup>,6] | |||
| [[centrosymmetric]] | |||
| 12 | |||
| [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math> | |||
|- | |||
| rowspan=7 | [[Hexagonal crystal system|hexagonal]] | |||
| hexagonal-pyramidal | |||
| C<sub>6</sub> | |||
| 6 | |||
| 66 | |||
| [6]<sup>+</sup> | |||
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]] | |||
| 6 | |||
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_6 = \mathbb{Z}_3\times\mathbb{Z}_2</math> | |||
|- | |||
| trigonal-dipyramidal | |||
| C<sub>3h</sub> | |||
| {{overline|6}} | |||
| 3* | |||
| [2,3<sup>+</sup>] | |||
| [[non-centrosymmetric]] | |||
| 6 | |||
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_6 = \mathbb{Z}_3\times\mathbb{Z}_2</math> | |||
|- | |||
| hexagonal-dipyramidal | |||
| C<sub>6h</sub> | |||
| 6/m | |||
| 6* | |||
| [2,6<sup>+</sup>] | |||
| [[centrosymmetric]] | |||
| 12 | |||
| <math>\mathbb{Z}_6\times\mathbb{Z}_2</math> | |||
|- | |||
| hexagonal-trapezoidal | |||
| D<sub>6</sub> | |||
| 622 | |||
| 622 | |||
| [2,6]<sup>+</sup> | |||
| [[Chirality (chemistry)|enantiomorphic]] | |||
| 12 | |||
| [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math> | |||
|- | |||
| dihexagonal-pyramidal | |||
| C<sub>6v</sub> | |||
| 6mm | |||
| *66 | |||
| [6] | |||
| [[Polar point group|polar]] | |||
| 12 | |||
| [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math> | |||
|- | |||
| ditrigonal-dipyramidal | |||
| D<sub>3h</sub> | |||
| {{overline|6}}m2 or {{overline|6}}2m | |||
| *322 | |||
| [2,3] | |||
| [[non-centrosymmetric]] | |||
| 12 | |||
| [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math> | |||
|- | |||
| dihexagonal-dipyramidal | |||
| D<sub>6h</sub> | |||
| 6/mmm | |||
| *622 | |||
| [2,6] | |||
| [[centrosymmetric]] | |||
| 24 | |||
| <math>\mathbb{D}_{12}\times\mathbb{Z}_2</math> | |||
|- | |||
| rowspan=5 colspan=2 | [[cubic crystal system|cubic]] | |||
| tetrahedral | |||
| T || 23 | |||
| 332 | |||
| [3,3]<sup>+</sup> | |||
| [[Chirality (chemistry)|enantiomorphic]] | |||
| 12 | |||
| [[alternating group|alternating]] <math>\mathbb{A}_4</math> | |||
|- | |||
| hextetrahedral | |||
| T<sub>d</sub> | |||
| {{overline|4}}3m | |||
| *332 | |||
| [3,3] | |||
| [[non-centrosymmetric]] | |||
| 24 | |||
| [[symmetric group|symmetric]] <math>\mathbb{S}_4</math> | |||
|- | |||
| diploidal | |||
| T<sub>h</sub> | |||
| m{{overline|3}} | |||
| 3*2 | |||
| [3<sup>+</sup>,4] | |||
| [[centrosymmetric]] | |||
| 24 | |||
| <math>\mathbb{A}_4\times\mathbb{Z}_2</math> | |||
|- | |||
| gyroidal | |||
| O | |||
| 432 | |||
| 432 | |||
| [4,3]<sup>+</sup> | |||
| [[Chirality (chemistry)|enantiomorphic]] | |||
| 24 | |||
| [[symmetric group|symmetric]] <math>\mathbb{S}_4</math> | |||
|- | |||
| hexoctahedral | |||
| O<sub>h</sub> | |||
| m{{overline|3}}m | |||
| *432 | |||
| [4,3] | |||
| [[centrosymmetric]] | |||
| 48 | |||
| <math>\mathbb{S}_4\times\mathbb{Z}_2</math> | |||
|} | |||
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> | |||
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. | |||
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]]). | |||
<!--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. --> | |||
==Lattice systems == | |||
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 | |||
|- | |||
|| [[Image:Tetragonal.svg|80px|Tetragonal, simple]] | |||
|| [[Image:Tetragonal-body-centered.svg|80px|Tetragonal, body-centered]] | |||
|- | |||
|align=center| [[rhombohedral lattice system|rhombohedral]] <br>([[trigonal trapezohedron]]) | |||
| [[Image:Rhombohedral.svg|80px|Rhombohedral]] | |||
|- | |||
|align=center| [[Hexagonal crystal system|hexagonal]] (centered regular [[hexagon]]) | |||
| [[Image:Hexagonal lattice.svg|80px|Hexagonal]] | |||
|- | |||
|rowspan=2 align=center| [[Cubic crystal system|cubic]]<br>(isometric; [[cube]]) | |||
|align=center| simple | |||
|align=center| body-centered | |||
|align=center| face-centered | |||
|- | |||
|| [[Image:Cubic.svg|80px|Cubic, simple]] | |||
| [[Image:Cubic-body-centered.svg|80px|Cubic, body-centered]] | |||
| [[Image:Cubic-face-centered.svg|80px|Cubic, face-centered]] | |||
|} | |||
{{-}} | |||
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. | |||
Such symmetry groups consist of translations by vectors of the form | |||
:<math>\mathbf{R} = n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + n_3 \mathbf{a}_3,</math> | |||
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''. | |||
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. | |||
All crystalline materials must, by definition fit in one of these arrangements (not including [[quasicrystal]]s). | |||
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. | |||
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-->. | |||
==Crystal systems in four-dimensional space== | |||
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: | |||
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> | |||
2 Triclinic: <math>a\ne b \ne c \ne d, \alpha \ne \beta \ne \gamma \ne 90 ^\circ, \delta = \epsilon = \zeta = 90 ^\circ</math> | |||
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> | |||
4 Monoclinic: <math>a\ne b \ne c \ne d, \alpha \ne 90 ^\circ, \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math> | |||
5 Orthogonal: <math>a\ne b \ne c \ne d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math> | |||
6 Tetragonal Monoclinic: <math>a\ne b = c \ne d, \alpha \ne 90 ^\circ, \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math> | |||
7 Hexagonal Monoclinic: <math>a\ne b = c \ne d, \alpha \ne 90 ^\circ, \beta = \gamma = \delta = \epsilon = 90 ^\circ, \zeta = 120 ^\circ</math> | |||
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> | |||
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> | |||
10 Tetragonal Orthogonal: <math>a\ne b = c \ne d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math> | |||
11 Hexagonal Orthogonal: <math>a\ne b = c \ne d, \alpha = \beta = \gamma = \delta = \epsilon = 90 ^\circ, \zeta = 120 ^\circ</math> | |||
12 Ditetragonal Monoclinic: <math>a = d \ne b = c, \alpha = \gamma = \delta = \zeta = 90 ^\circ, \beta = \epsilon \ne 90 ^\circ</math> | |||
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> | |||
14 Ditetragonal Orthogonal: <math>a = d \ne b = c, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math> | |||
15 Hexagonal Tetragonal: <math>a = d \ne b = c, \alpha = \beta = \gamma = \delta = \epsilon = 90 ^\circ, \zeta = 120 ^\circ</math> | |||
16 Dihexagonal Orthogonal: <math>a = d \ne b = c, \alpha = \zeta = 120 ^\circ, \beta = \gamma = \delta = \epsilon = 90 ^\circ, </math> | |||
17 Cubic Orthogonal: <math>a = b = c \ne d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math> | |||
18 Octagonal: <math>a = b = c = d, \alpha = \gamma = \zeta \ne 90 ^\circ, \beta = \epsilon = 90 ^\circ, \delta = 180 ^\circ - \alpha</math> | |||
19 Decagonal: <math>a = b = c = d, \alpha = \gamma = \zeta \ne \beta = \delta = \epsilon, cos \beta = -0.5 - cos \alpha</math> | |||
20 Dodecagonal: <math>a = b = c = d, \alpha = \zeta = 90 ^\circ, \beta = \epsilon = 120 ^\circ, \gamma = \delta \ne 90 ^\circ</math> | |||
21 Di-isohexagonal Orthogonal: <math>a = b = c = d, \alpha = \zeta = 120 ^\circ, \beta = \gamma = \delta = \epsilon = 90 ^\circ</math> | |||
22 Icosagonal (Icosahedral): <math>a = b = c = d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta, cos \alpha = -\color{Black}\tfrac{1}{4}</math> | |||
23 Hypercubic: <math>a = b = c = d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ</math> | |||
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. | |||
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 | |||
!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 | |||
|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 | |||
|- | |||
| VI || colspan=2| Tetragonal Monoclinic || 7 | |||
|7 | |||
|88 | |||
|2 | |||
|Tetragonal Monoclinic P, I | |||
|- | |||
|rowspan=3| VII ||rowspan=3| Hexagonal Monoclinic | |||
|rowspan=2|Trigonal Monoclinic ||rowspan=2| 8 | |||
|rowspan=2|5 | |||
|9 | |||
|1 | |||
|Hexagonal Monoclinic R | |||
|- | |||
|15 | |||
|rowspan=2|1 | |||
|rowspan=2|Hexagonal Monoclinic P | |||
|- | |||
|Hexagonal Monoclinic || 9 | |||
|7 | |||
|25 | |||
|- | |||
| VIII || colspan=2| Ditetragonal Diclinic* ||10 | |||
|1 (+1) | |||
|1 (+1) | |||
|1 (+1) | |||
|Ditetragonal Diclinic P* | |||
|- | |||
|IX || colspan=2| Ditrigonal Diclinic* ||11 | |||
|2 (+2) | |||
|2 (+2) | |||
|1 (+1) | |||
|Ditrigonal Diclinic P* | |||
|- | |||
|rowspan=3| X ||rowspan=3| Tetragonal Orthogonal | |||
|rowspan=2|Inverse Tetragonal Orthogonal ||rowspan=2| 12 | |||
|rowspan=2|5 | |||
|7 | |||
|1 | |||
|Tetragonal Orthogonal KG | |||
|- | |||
|351 | |||
|rowspan=2|5 | |||
|rowspan=2|Tetragonal Orthogonal P, S, I, Z, G | |||
|- | |||
|Proper Tetragonal Orthogonal || 13 | |||
|10 | |||
|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 | |||
|- | |||
| XII || colspan=2| Ditetragonal Monoclinic* || 16 | |||
|1 (+1) | |||
|6 (+6) | |||
|3 (+3) | |||
|Ditetragonal Monoclinic P*, S*, D* | |||
|- | |||
| XIII || colspan=2| Ditrigonal Monoclinic* || 17 | |||
|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 | |||
|rowspan=2|5 | |||
|10 | |||
|1 | |||
|Ditetragonal Orthogonal D | |||
|- | |||
|165 (+2) | |||
|rowspan=2|2 | |||
|rowspan=2|Ditetragonal Orthogonal P, Z | |||
|- | |||
|Ditetragonal Orthogonal ||19 | |||
|6 | |||
|127 | |||
|- | |||
|XV ||colspan=2| Hexagonal Tetragonal || 20 | |||
|22 | |||
|108 | |||
|1 | |||
|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 | |||
|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]] | |||
*[[Crystal structure]] | |||
*[[Space group#Classification systems for space groups|List of the 230 crystallographic 3D space groups]] | |||
*[[Polar point group]] | |||
==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 --> | |||
<div class="references-small" > | |||
<references/> | |||
</div> | |||
==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] | |||
{{Crystal systems}} | |||
{{Mineral identification}} | |||
[[Category:Symmetry]] | |||
[[Category:Euclidean geometry]] | |||
[[Category:Crystallography]] | |||
[[Category:Morphology]] | |||
[[Category:Mineralogy]] | |||
[[ru:Сингония]] | |||
Revision as of 22:07, 28 January 2014

In crystallography, the terms crystal system, crystal family, and lattice system each refer to one of several classes of space groups, lattices, point groups, or crystals. Informally, two crystals tend to be in the same crystal system if they have similar symmetries, though there are many exceptions to this.
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".
Space groups and crystals are divided into 7 crystal systems according to their point groups, and into 7 lattice systems according to their Bravais lattices. 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. The six crystal families are formed by combining the hexagonal and trigonal crystal systems into one hexagonal family, in order to eliminate this confusion.
Overview
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.
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, 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.
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.
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.
The relation between three-dimensional crystal families, crystal systems, and lattice systems is shown in the following table:
| Crystal family | Crystal system | Required symmetries of point group | Point groups | Space groups | Bravais lattices | Lattice system |
|---|---|---|---|---|---|---|
| Triclinic | None | 2 | 2 | 1 | Triclinic | |
| Monoclinic | 1 twofold axis of rotation or 1 mirror plane | 3 | 13 | 2 | Monoclinic | |
| Orthorhombic | 3 twofold axes of rotation or 1 twofold axis of rotation and two mirror planes. | 3 | 59 | 4 | Orthorhombic | |
| Tetragonal | 1 fourfold axis of rotation | 7 | 68 | 2 | Tetragonal | |
| Hexagonal | Trigonal | 1 threefold axis of rotation | 5 | 7 | 1 | Rhombohedral |
| 18 | 1 | Hexagonal | ||||
| Hexagonal | 1 sixfold axis of rotation | 7 | 27 | |||
| Cubic | 4 threefold axes of rotation | 5 | 36 | 3 | Cubic | |
| Total: 6 | 7 | 32 | 230 | 14 | 7 | |
Crystal classes
The 7 crystal systems consist of 32 crystal classes (corresponding to the 32 crystallographic point groups) as shown in the following table:
| crystal family | crystal system | point group / crystal class | Schönflies | Hermann-Mauguin | Orbifold | Coxeter | Point symmetry | Order | Abstract group |
|---|---|---|---|---|---|---|---|---|---|
| triclinic | triclinic-pedial | C1 | 1 | 11 | [ ]+ | enantiomorphic polar | 1 | trivial | |
| triclinic-pinacoidal | Ci | Template:Overline | 1x | [2,1+] | centrosymmetric | 2 | cyclic | ||
| monoclinic | monoclinic-sphenoidal | C2 | 2 | 22 | [2,2]+ | enantiomorphic polar | 2 | cyclic | |
| monoclinic-domatic | Cs | m | *11 | [ ] | polar | 2 | cyclic | ||
| monoclinic-prismatic | C2h | 2/m | 2* | [2,2+] | centrosymmetric | 4 | Klein four | ||
| orthorhombic | orthorhombic-sphenoidal | D2 | 222 | 222 | [2,2]+ | enantiomorphic | 4 | Klein four | |
| orthorhombic-pyramidal | C2v | mm2 | *22 | [2] | polar | 4 | Klein four | ||
| orthorhombic-bipyramidal | D2h | mmm | *222 | [2,2] | centrosymmetric | 8 | |||
| tetragonal | tetragonal-pyramidal | C4 | 4 | 44 | [4]+ | enantiomorphic polar | 4 | cyclic | |
| tetragonal-disphenoidal | S4 | Template:Overline | 2x | [2+,2] | non-centrosymmetric | 4 | cyclic | ||
| tetragonal-dipyramidal | C4h | 4/m | 4* | [2,4+] | centrosymmetric | 8 | |||
| tetragonal-trapezoidal | D4 | 422 | 422 | [2,4]+ | enantiomorphic | 8 | dihedral | ||
| ditetragonal-pyramidal | C4v | 4mm | *44 | [4] | polar | 8 | dihedral | ||
| tetragonal-scalenoidal | D2d | Template:Overline2m or Template:Overlinem2 | 2*2 | [2+,4] | non-centrosymmetric | 8 | dihedral | ||
| ditetragonal-dipyramidal | D4h | 4/mmm | *422 | [2,4] | centrosymmetric | 16 | |||
| hexagonal | trigonal | trigonal-pyramidal | C3 | 3 | 33 | [3]+ | enantiomorphic polar | 3 | cyclic |
| rhombohedral | S6 (C3i) | Template:Overline | 3x | [2+,3+] | centrosymmetric | 6 | cyclic | ||
| trigonal-trapezoidal | D3 | 32 or 321 or 312 | 322 | [3,2]+ | enantiomorphic | 6 | dihedral | ||
| ditrigonal-pyramidal | C3v | 3m or 3m1 or 31m | *33 | [3] | polar | 6 | dihedral | ||
| ditrigonal-scalahedral | D3d | Template:Overlinem or Template:Overlinem1 or Template:Overline1m | 2*3 | [2+,6] | centrosymmetric | 12 | dihedral | ||
| hexagonal | hexagonal-pyramidal | C6 | 6 | 66 | [6]+ | enantiomorphic polar | 6 | cyclic | |
| trigonal-dipyramidal | C3h | Template:Overline | 3* | [2,3+] | non-centrosymmetric | 6 | cyclic | ||
| hexagonal-dipyramidal | C6h | 6/m | 6* | [2,6+] | centrosymmetric | 12 | |||
| hexagonal-trapezoidal | D6 | 622 | 622 | [2,6]+ | enantiomorphic | 12 | dihedral | ||
| dihexagonal-pyramidal | C6v | 6mm | *66 | [6] | polar | 12 | dihedral | ||
| ditrigonal-dipyramidal | D3h | Template:Overlinem2 or Template:Overline2m | *322 | [2,3] | non-centrosymmetric | 12 | dihedral | ||
| dihexagonal-dipyramidal | D6h | 6/mmm | *622 | [2,6] | centrosymmetric | 24 | |||
| cubic | tetrahedral | T | 23 | 332 | [3,3]+ | enantiomorphic | 12 | alternating | |
| hextetrahedral | Td | Template:Overline3m | *332 | [3,3] | non-centrosymmetric | 24 | symmetric | ||
| diploidal | Th | mTemplate:Overline | 3*2 | [3+,4] | centrosymmetric | 24 | |||
| gyroidal | O | 432 | 432 | [4,3]+ | enantiomorphic | 24 | symmetric | ||
| hexoctahedral | Oh | mTemplate:Overlinem | *432 | [4,3] | centrosymmetric | 48 | |||
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.[1]
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.[2] Groups containing a polar axis are called 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 dielectric polarization, e.g. in 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.
The crystal structures of chiral biological molecules (such as protein structures) can only occur in the 11 enantiomorphic point groups (biological molecules are usually chiral).
Lattice systems
The distribution of the 14 Bravais lattice types into 7 lattice systems is given in the following table.
| The 7 lattice systems | The 14 Bravais Lattices | |||
| triclinic (parallelepiped) | ||||
| monoclinic (right prism with parallelogram base; here seen from above) | simple | base-centered | ||
| orthorhombic (cuboid) | simple | base-centered | body-centered | face-centered |
| Orthohombic, face-centered | ||||
| tetragonal (square cuboid) | simple | body-centered | ||
| Tetragonal, simple | Tetragonal, body-centered | |||
| rhombohedral (trigonal trapezohedron) |
Rhombohedral | |||
| hexagonal (centered regular hexagon) | Hexagonal | |||
| cubic (isometric; cube) |
simple | body-centered | face-centered | |
| Cubic, simple | Cubic, body-centered | Cubic, face-centered | ||
Benefits of Residing in a Apartment or Landed property in Singapore Property New Launches & Project Showcase In Singapore Many residential Singapore property sales involve buying property in Singapore at new launches. These are often homes underneath building, being sold new by developers. New Launch Singapore Property, 28 Imperial Residences Coming To Geylang Lorong 26 The property market is slowing down, based on personal property transactions in May Cell Apps FREE Sign Up Log in Property Brokers Feedback
Individuals all wish to be seen having the identical foresight as the experts in property investment or the massive names in their own fields. Thus the discharge of these tales works to encourage different buyers to observe suit. Bartley Ridge is the most popular new launch in district 13. Irresistible pricing from $1,1xx psf. Bartley Ridge is a ninety nine-12 months leasehold new condominium at Mount Vernon road, good next to Bartley MRT station (CC12). If you want to get more Rehda Johor chairman Koh Moo Hing said potential property consumers in the two areas Http://Modern.Dowatch.Net/Profile/Mic31K/Created/Topics are now adopting a wait-and-see attitude. How can I get the ebrochure and flooring plans of the new launch projects ? The Existing Mortgage on your HDB District 13, Freehold condominium District 11, Freehold Cluster landed house Sea Horizon EC @ Pasir Ris
FindSgNewLaunch is the main Singapore Property web site - one of the best place to begin your actual estate search whether you might be an investor, shopping for for own use, or searching for a spot to lease. With detailed details about each property, together with maps and pictures. We deliver you probably the most complete choice out there. No. For brand spanking new Singapore property gross sales, you possibly can withdraw at any time earlier than booking the unit, without penalty. On the preview, the agent will let you recognize the exact worth for you to resolve whether or not to proceed or not. Solely when you resolve to proceed will the agent book the unit for you. Pending for Sale Licence Approval All Pending for Sale Licence Approval New launch FREEHOLD condominium @ Braddell New launch condominium combined growth at Yishun PROJECT TITLE
To not worry, we'll hold you in our VIP Precedence list for future new launch VIP Preview. We'll contact you to establish your wants and advocate related tasks, both new launch or resale properties that probably match your standards. In case you're looking for resale property, such as these few years old, or just got Short-term Occupation Permit (PRIME), you might click on here right here for fast search and submit your shortlisted listings to us, we'll check and call you for viewing.
Oceanfront Suites, irresistible pricing for a 946 leasehold property with magnificent sea view. Dreaming of basking and feeling the warmth of pure sunlight is now just a click on away. Oceanfront Suites - Seaside residing no longer needs to remain an unattainable This Cambodia new launch, a mega development has also 762 residential models. Additionally located within this Oxley abroad property is a mega shopping center with 627 outlets and also up to 963 available workplace spaces and is surrounded by quite a few Embassy, resorts, Casinos and many vacationer relax space. Belysa EC @ Pasir Ris Esparina EC @ Sengkang Dell Launches World's first Gender-GEDI Female Entrepreneurship Index on 06/04/thirteen by Istanbul, Turkey. Paris Ris EC @ Paris Ris in search of indication of curiosity.
The developer should open a Venture Account with a financial institution or monetary establishment for every housing venture he undertakes, before he's issued with a Sale License (license to sell models in his development). All payments from buyers before completion of the challenge, and construction loans, go into the mission account. New launch rental LA FIESTA, an thrilling new condominium located along Sengkang Square / Compassvale Highway is a brief stroll to the bustling Sengkang City Centre the place the bus interchange, Sengkang MRT and LRT stations are located. Glorious location,Premium rental with Bayfront resort lifestyle theme and views ofwaterscape. Close to EC pricing - Worth for cash! Apr 02, 2013 Sengkang New Rental Launch, La Fiesta- Sengkang MRTstation at your gate.
As The Hillford property launch at Jalan Jurong Kechil may be very close to to beauty world mrt , the environment for the plot of land which belongs to World Class Land remains very upbeat as it is rather close to to Holland Village. Review now by visiting the brand new apartment pages on our website, each displaying complete particulars and the latest information of each new launch. You can even contact us directly to obtain quick & correct answers to all of your questions with high of the road service. An inevitable conclusion is that costs within the property market have just set new highs. The apparent connotation for potential buyers is to take motion now before prices bounce again. tract and points to his property line, marked by a big maple in a sea of Search SG Developersale.com
In geometry and crystallography, a Bravais lattice is a category of symmetry groups for translational symmetry in three directions, or correspondingly, a category of translation lattices.
Such symmetry groups consist of translations by vectors of the form
where n1, n2, and n3 are integers and a1, a2, and a3 are three non-coplanar vectors, called primitive vectors.
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.
All crystalline materials must, by definition fit in one of these arrangements (not including quasicrystals).
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.
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 A. Bravais in 1848.
Crystal systems in four-dimensional space
The four-dimensional unit cell is defined by four edge lengths () and six interaxial angles (). The following conditions for the lattice parameters define 23 crystal families:
9 Ditrigonal (Dihexagonal) Diclinic:
13 Ditrigonal (Dihexagonal) Monoclinic:
21 Di-isohexagonal Orthogonal:
The names here are given according to Whittaker.[3] They are almost the same as in Brown et al,[4] 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.
The relation between four-dimensional crystal families, crystal systems, and lattice systems is shown in the following table.[3][4] 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 P31 and P32, P4122 and P4322. Starting from four-dimensional space, point groups also can be enantiomorphic in this sense.
| No. of Crystal family |
Crystal family | Crystal system | No. of Crystal system |
Point groups | Space groups | Bravais lattices | Lattice system |
|---|---|---|---|---|---|---|---|
| I | Hexaclinic | 1 | 2 | 2 | 1 | Hexaclinic P | |
| II | Triclinic | 2 | 3 | 13 | 2 | Triclinic P, S | |
| III | Diclinic | 3 | 2 | 12 | 3 | Diclinic P, S, D | |
| IV | Monoclinic | 4 | 4 | 207 | 6 | Monoclinic P, S, S, I, D, F | |
| V | Orthogonal | Non-axial Orthogonal | 5 | 2 | 2 | 1 | Orthogonal KU |
| 112 | 8 | Orthogonal P, S, I, Z, D, F, G, U | |||||
| Axial Orthogonal | 6 | 3 | 887 | ||||
| VI | Tetragonal Monoclinic | 7 | 7 | 88 | 2 | Tetragonal Monoclinic P, I | |
| VII | Hexagonal Monoclinic | Trigonal Monoclinic | 8 | 5 | 9 | 1 | Hexagonal Monoclinic R |
| 15 | 1 | Hexagonal Monoclinic P | |||||
| Hexagonal Monoclinic | 9 | 7 | 25 | ||||
| VIII | Ditetragonal Diclinic* | 10 | 1 (+1) | 1 (+1) | 1 (+1) | Ditetragonal Diclinic P* | |
| IX | Ditrigonal Diclinic* | 11 | 2 (+2) | 2 (+2) | 1 (+1) | Ditrigonal Diclinic P* | |
| X | Tetragonal Orthogonal | Inverse Tetragonal Orthogonal | 12 | 5 | 7 | 1 | Tetragonal Orthogonal KG |
| 351 | 5 | Tetragonal Orthogonal P, S, I, Z, G | |||||
| Proper Tetragonal Orthogonal | 13 | 10 | 1312 | ||||
| XI | Hexagonal Orthogonal | Trigonal Orthogonal | 14 | 10 | 81 | 2 | Hexagonal Orthogonal R, RS |
| 150 | 2 | Hexagonal Orthogonal P, S | |||||
| Hexagonal Orthogonal | 15 | 12 | 240 | ||||
| XII | Ditetragonal Monoclinic* | 16 | 1 (+1) | 6 (+6) | 3 (+3) | Ditetragonal Monoclinic P*, S*, D* | |
| XIII | Ditrigonal Monoclinic* | 17 | 2 (+2) | 5 (+5) | 2 (+2) | Ditrigonal Monoclinic P*, RR* | |
| XIV | Ditetragonal Orthogonal | Crypto-Ditetragonal Orthogonal | 18 | 5 | 10 | 1 | Ditetragonal Orthogonal D |
| 165 (+2) | 2 | Ditetragonal Orthogonal P, Z | |||||
| Ditetragonal Orthogonal | 19 | 6 | 127 | ||||
| XV | Hexagonal Tetragonal | 20 | 22 | 108 | 1 | Hexagonal Tetragonal P | |
| XVI | Dihexagonal Orthogonal | Crypto-Ditrigonal Orthogonal* | 21 | 4 (+4) | 5 (+5) | 1 (+1) | Dihexagonal Orthogonal G* |
| 5 (+5) | 1 | Dihexagonal Orthogonal P | |||||
| Dihexagonal Orthogonal | 23 | 11 | 20 | ||||
| Ditrigonal Orthogonal | 22 | 11 | 41 | ||||
| 16 | 1 | Dihexagonal Orthogonal RR | |||||
| XVII | Cubic Orthogonal | Simple Cubic Orthogonal | 24 | 5 | 9 | 1 | Cubic Orthogonal KU |
| 96 | 5 | Cubic Orthogonal P, I, Z, F, U | |||||
| Complex Cubic Orthogonal | 25 | 11 | 366 | ||||
| XVIII | Octagonal* | 26 | 2 (+2) | 3 (+3) | 1 (+1) | Octagonal P* | |
| XIX | Decagonal | 27 | 4 | 5 | 1 | Decagonal P | |
| XX | Dodecagonal* | 28 | 2 (+2) | 2 (+2) | 1 (+1) | Dodecagonal P* | |
| XXI | Di-isohexagonal Orthogonal | Simple Di-isohexagonal Orthogonal | 29 | 9 (+2) | 19 (+5) | 1 | Di-isohexagonal Orthogonal RR |
| 19 (+3) | 1 | Di-isohexagonal Orthogonal P | |||||
| Complex Di-isohexagonal Orthogonal | 30 | 13 (+8) | 15 (+9) | ||||
| XXII | Icosagonal | 31 | 7 | 20 | 2 | Icosagonal P, SN | |
| XXIII | Hypercubic | Octagonal Hypercubic | 32 | 21 (+8) | 73 (+15) | 1 | Hypercubic P |
| 107 (+28) | 1 | Hypercubic Z | |||||
| Dodecagonal Hypercubic | 33 | 16 (+12) | 25 (+20) | ||||
| Total: | 23 (+6) | 33 (+7) | 227 (+44) | 4783 (+111) | 64 (+10) | 33 (+7) | |
See also
- Crystal cluster
- Crystal structure
- List of the 230 crystallographic 3D space groups
- Polar point group
Notes
- ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ 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.
- ↑ 3.0 3.1 E. J. W. Whittaker, An atlas of hyperstereograms of the four-dimensional crystal classes. Clarendon Press (Oxford Oxfordshire and New York) 1985.
- ↑ 4.0 4.1 H. Brown, R. Bülow, J. Neubüser, H. Wondratschek and H. Zassenhaus, Crystallographic Groups of Four-Dimensional Space. Wiley, NY, 1978.
References
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
External links
- Overview of the 32 groups
- Mineral galleries – Symmetry
- all cubic crystal classes, forms and stereographic projections (interactive java applet)
- Crystal system at the Online Dictionary of Crystallography
- Crystal family at the Online Dictionary of Crystallography
- Lattice system at the Online Dictionary of Crystallography
- Conversion Primitive to Standard Conventional for VASP input files
- Learning Crystallography