Editing Crystal structure (section)

== Classification by symmetry ==
{{main|Crystal system}}
The defining property of a crystal is its inherent symmetry. Performing certain symmetry operations on the crystal lattice leaves it unchanged. All crystals have [[translational symmetry]] in three directions, but some have other symmetry elements as well. For example, rotating the crystal 180° about a certain axis may result in an atomic configuration that is identical to the original configuration; the crystal has twofold rotational symmetry about this axis. In addition to rotational symmetry, a crystal may have symmetry in the form of mirror planes, and also the so-called compound symmetries, which are a combination of translation and rotation or mirror symmetries. A full classification of a crystal is achieved when all inherent symmetries of the crystal are identified.<ref>{{cite book|last1=Ashcroft|first1=N.|author-link1=Neil Ashcroft|last2=Mermin|first2=D.|author-link2=David Mermin|date=1976|title=Solid State Physics|publisher=Brooks/Cole (Thomson Learning, Inc.)|chapter=Chapter 7|isbn=978-0030493461}}</ref>

=== Lattice systems ===
Lattice systems are a grouping of crystal structures according to the point groups of their lattice. All crystals fall into one of seven lattice systems. They are related to, but not the same as the seven [[crystal system]]s. 
{| class="wikitable skin-invert-image"
|+Overview of common lattice systems
!scope="col" rowspan=2| Crystal family
!scope="col" rowspan=2| Lattice system
!scope="col" rowspan=2| Point group <br />([[Schoenflies notation|Schönflies notation]])
!scope="col" colspan=4| 14 Bravais lattices
|-
!scope="col"| Primitive (P)
!scope="col"| Base-centered (S)
!scope="col"| Body-centered (I)
!scope="col"| Face-centered (F)
|- align=center
!scope="row" colspan=2| [[Triclinic]] (a)
| C{{sub|i}}
| [[File:Triclinic.svg|80px|Triclinic]]
aP
|
|
|
|- align=center
!scope="row" colspan=2| [[Monoclinic]] (m)
| C{{sub|2h}}
| [[File:Monoclinic.svg|80px|Monoclinic, simple]]
mP
| [[File:Base-centered monoclinic.svg|80px|Monoclinic, centered]]
mS
|
|
|- align=center
!scope="row" colspan=2| [[Orthorhombic]] (o)
| D{{sub|2h}}
| [[File:Orthorhombic.svg|80px|Orthorhombic, simple]]
oP
| [[File:Orthorhombic-base-centered.svg|80px|Orthorhombic, base-centered]]
oS
| [[File:Orthorhombic-body-centered.svg|80px|Orthorhombic, body-centered]]
oI
| [[File:Orthorhombic-face-centered.svg|80px|Orthorhombic, face-centered]]
oF
|- align=center
!scope="row" colspan=2| [[Tetragonal]] (t)
| D{{sub|4h}}
| [[File:Tetragonal.svg|80px|Tetragonal, simple]]
tP
|
| [[File:Tetragonal-body-centered.svg|80px|Tetragonal, body-centered]]
tI
|
|- align=center
!scope="row" rowspan=2| [[Hexagonal crystal family|Hexagonal]] (h)
! Rhombohedral
| D{{sub|3d}}
| [[File:Rhombohedral.svg|80px|Rhombohedral]]
hR
|
|
|
|- align=center
!scope="row"| Hexagonal
| D{{sub|6h}}
| [[File:Hexagonal latticeFRONT.svg|80px|Hexagonal]]
hP
|
|
|
|- align=center
!scope="row" colspan=2| [[Cubic crystal system|Cubic]] (c)
| O{{sub|h}}
| [[File:Cubic.svg|80px|Cubic, simple]]
cP
|
| [[File:Cubic-body-centered.svg|80px|Cubic, body-centered]]
cI
| [[File:Cubic-face-centered.svg|80px|Cubic, face-centered]]
cF
|}
{{clear}}
The most symmetric, the [[cubic (crystal system)|cubic]] or isometric system, has the symmetry of a [[Cube (geometry)|cube]], that is, it exhibits four threefold rotational axes oriented at 109.5° (the [[tetrahedral angle]]) with respect to each other. These threefold axes lie along the body diagonals of the cube. The other six lattice systems, are [[Hexagonal lattice system|hexagonal]], [[tetragonal]], [[rhombohedral lattice system|rhombohedral]] (often confused with the [[trigonal crystal system]]), [[orthorhombic]], [[monoclinic]] and [[triclinic]].

==== Bravais lattices ====
[[Bravais lattice]]s, also referred to as ''space lattices'', describe the geometric arrangement of the lattice points,<ref name="Physics 1991"/> and therefore the translational symmetry of the crystal. The three dimensions of space afford 14 distinct Bravais lattices describing the translational symmetry. All crystalline materials recognized today, not including [[quasicrystal]]s, fit in one of these arrangements. The fourteen three-dimensional lattices, classified by lattice system, are shown above.

The crystal structure consists of the same group of atoms, the ''basis'', positioned around each and every lattice point. This group of atoms therefore repeats indefinitely in three dimensions according to the arrangement of one of the Bravais lattices. The characteristic rotation and mirror symmetries of the unit cell is described by its [[crystallographic point group]].

===Crystal systems===
{{see also|Crystallographic point group#Isomorphisms}}

A crystal system is a set of point groups in which the point groups themselves and their corresponding space groups are assigned to a lattice system. Of the 32 point groups that exist in three dimensions, most are assigned to only one lattice system, in which case the crystal system and lattice system both have the same name. However, five point groups are assigned to two lattice systems, rhombohedral and hexagonal, because both lattice systems exhibit threefold rotational symmetry. These point groups are assigned to the trigonal crystal system.

{| class=wikitable
|+Overview of crystal systems
|-
!scope="col"| Crystal family
!scope="col"| Crystal system
!scope="col"| [[Point group]] / Crystal class
!scope="col"| [[Schönflies notation|Schönflies]]
!scope="col"| Point symmetry
!scope="col"| [[Symmetry number|Order]]
!scope="col"| [[Group theory#Abstract groups|Abstract group]]
|-
!scope="row" rowspan=2 colspan=2| [[triclinic crystal system|triclinic]]
| pedial	
| C{{sub|1}}
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 1
| trivial <math>\mathbb{Z}_1</math>
|-
| pinacoidal
| C{{sub|i}} (S{{sub|2}})
| [[centrosymmetric]]
| 2
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_2</math>
|-
!scope="row" rowspan=3 colspan=2 | [[monoclinic crystal system|monoclinic]]
| sphenoidal
| C{{sub|2}}
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 2
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_2</math>
|-
| domatic
| C{{sub|s}} (C{{sub|1h}})
| [[Polar point group|polar]]
| 2
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_2</math>
|-
| [[prism (geometry)|prismatic]]
| C{{sub|2h}}	
| [[centrosymmetric]]
| 4
| [[Klein four-group|Klein four]] <math>\mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2</math>
|-
!scope="row" rowspan=3 colspan=2| [[orthorhombic crystal system|orthorhombic]]
| rhombic-disphenoidal
| D{{sub|2}} (V)
| [[Chirality (chemistry)|enantiomorphic]]
| 4
| [[Klein four-group|Klein four]] <math>\mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2</math>
|-
| rhombic-[[Pyramid (geometry)|pyramidal]]
| C{{sub|2v}}	
| [[Polar point group|polar]]
| 4
| [[Klein four-group|Klein four]] <math>\mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2</math>
|-
| rhombic-[[dipyramid]]al
| D{{sub|2h}} (V{{sub|h}})	
| [[centrosymmetric]]
| 8
| <math>\mathbb{V}\times\mathbb{Z}_2</math>
|-
!scope="row" rowspan=7 colspan=2| [[tetragonal crystal system|tetragonal]]
| tetragonal-pyramidal	
| C{{sub|4}}
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 4
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_4</math>
|-
| tetragonal-disphenoidal
| S{{sub|4}}	
| [[non-centrosymmetric]]
| 4
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_4</math>
|-
| tetragonal-dipyramidal
| C{{sub|4h}}	
| [[centrosymmetric]]
| 8
| <math>\mathbb{Z}_4\times\mathbb{Z}_2</math>
|-
| tetragonal-trapezohedral
| D{{sub|4}}	
| [[Chirality (chemistry)|enantiomorphic]]
| 8
| [[Dihedral group|dihedral]] <math>\mathbb{D}_8 = \mathbb{Z}_4\rtimes\mathbb{Z}_2</math>
|-
| ditetragonal-pyramidal
| C{{sub|4v}}	
| [[Polar point group|polar]]
| 8
| [[Dihedral group|dihedral]] <math>\mathbb{D}_8 = \mathbb{Z}_4\rtimes\mathbb{Z}_2</math>
|-
| tetragonal-scalenohedral
| D{{sub|2d}} (V{{sub|d}})
| [[non-centrosymmetric]]
| 8
| [[Dihedral group|dihedral]] <math>\mathbb{D}_8 = \mathbb{Z}_4\rtimes\mathbb{Z}_2</math>
|-
| ditetragonal-dipyramidal
| D{{sub|4h}}	
| [[centrosymmetric]]
| 16
| <math>\mathbb{D}_8\times\mathbb{Z}_2</math>
|-
!scope="row" rowspan=12|[[hexagonal crystal family|hexagonal]] !! rowspan=5 | trigonal 
| trigonal-pyramidal	
| C{{sub|3}}
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 3
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_3</math>
|-
| rhombohedral
| C{{sub|3i}} (S{{sub|6}})
| [[centrosymmetric]]
| 6
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_6 = \mathbb{Z}_3\times\mathbb{Z}_2</math>
|-
| trigonal-trapezohedral
| D{{sub|3}}	
| [[Chirality (chemistry)|enantiomorphic]]
| 6
| [[Dihedral group|dihedral]] <math>\mathbb{D}_6 = \mathbb{Z}_3\rtimes\mathbb{Z}_2</math>
|-
| ditrigonal-pyramidal
| C{{sub|3v}}	
| [[Polar point group|polar]]
| 6
| [[Dihedral group|dihedral]] <math>\mathbb{D}_6 = \mathbb{Z}_3\rtimes\mathbb{Z}_2</math>
|-
| ditrigonal-scalenohedral
| D{{sub|3d}}	
| [[centrosymmetric]]
| 12
| [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math>
|-
!scope="row" rowspan=7 | hexagonal
| hexagonal-pyramidal	
| C{{sub|6}}
| [[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}}
| [[non-centrosymmetric]]
| 6
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_6 = \mathbb{Z}_3\times\mathbb{Z}_2</math>
|-
| hexagonal-dipyramidal
| C{{sub|6h}}	
| [[centrosymmetric]]
| 12
| <math>\mathbb{Z}_6\times\mathbb{Z}_2</math>
|-
| hexagonal-trapezohedral
| D{{sub|6}}	
| [[Chirality (chemistry)|enantiomorphic]]
| 12
| [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math>
|-
| dihexagonal-pyramidal
| C{{sub|6v}}	
| [[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}}	
| [[non-centrosymmetric]]
| 12
| [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math>
|-
| dihexagonal-dipyramidal
| D{{sub|6h}}	
| [[centrosymmetric]]
| 24
| <math>\mathbb{D}_{12}\times\mathbb{Z}_2</math> 
|-
!scope="row" rowspan=5 colspan=2 | [[cubic crystal system|cubic]]
| tetartoidal
| T 
| [[Chirality (chemistry)|enantiomorphic]]
| 12
| [[alternating group|alternating]] <math>\mathbb{A}_4</math>
|-
| diploidal
| T{{sub|h}}	
| [[centrosymmetric]]
| 24
| <math>\mathbb{A}_4\times\mathbb{Z}_2</math> 
|-
| gyroidal
| O
| [[Chirality (chemistry)|enantiomorphic]]
| 24
| [[symmetric group|symmetric]] <math>\mathbb{S}_4</math>
|-
| hextetrahedral
| T{{sub|d}}	
| [[non-centrosymmetric]]
| 24
| [[symmetric group|symmetric]] <math>\mathbb{S}_4</math>
|-
| hexoctahedral
| O{{sub|h}}	
| [[centrosymmetric]]
| 48
| <math>\mathbb{S}_4\times\mathbb{Z}_2</math> 
|}
{{clear}}
In total there are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic.

==== Point groups ====
The [[crystallographic point group]] or ''crystal class'' is the mathematical group comprising the symmetry operations that leave at least one point unmoved and that leave the appearance of the crystal structure unchanged. These symmetry operations include

*''Reflection'', which reflects the structure across a ''reflection plane''
*''Rotation'', which rotates the structure a specified portion of a circle about a ''rotation axis''
*''Inversion'', which changes the sign of the coordinate of each point with respect to a ''center of symmetry'' or ''inversion point''
*''[[Improper rotation]]'', which consists of a rotation about an axis followed by an inversion.

Rotation axes (proper and improper), reflection planes, and centers of symmetry are collectively called ''symmetry elements''. There are 32 possible crystal classes. Each one can be classified into one of the seven crystal systems.

=== Space groups ===
In addition to the operations of the point group, the [[space group]] of the crystal structure contains translational symmetry operations. These include:
*Pure ''translations'', which move a point along a vector
*''Screw axes'', which rotate a point around an axis while translating parallel to the axis.<ref name=Sands>
{{cite book |title=Introduction to Crystallography |author=Donald E. Sands |chapter-url=https://books.google.com/books?id=h_A5u5sczJoC&pg=PA71 |chapter=§4-2 Screw axes and §4-3 Glide planes |pages= 70–71 |isbn=978-0486678399 |publisher=Courier-Dover |year=1994 |edition=Reprint of WA Benjamin corrected 1975}} 
</ref>
*''Glide planes'', which reflect a point through a plane while translating it parallel to the plane.<ref name=Sands/>

There are 230 distinct space groups.