Editing Crystallographic point group

{{Short description|Classification system for crystals}}
In [[crystallography]], a '''crystallographic point group''' is a [[point groups in three dimensions|three-dimensional point group]] whose symmetry operations are compatible with a three-dimensional crystallographic [[Lattice (group)|lattice]]. According to the [[crystallographic restriction theorem|crystallographic restriction]] it may only contain one-, two-, three-, four- and sixfold rotations or rotoinversions. This reduces the number of crystallographic point groups to 32 (from an infinity of general point groups). These 32 groups are the same as the 32 types of morphological (external) crystalline symmetries derived in 1830 by [[Johann F. C. Hessel|Johann Friedrich Christian Hessel]] from a consideration of observed crystal forms. In 1867 [[Axel Gadolin]], who was unaware of the previous work of Hessel, found the crystallographic point groups independently using [[stereographic projection]] to represent the symmetry elements of the 32 groups.<ref name="Authier">{{cite book |last1=Authier |first1=André |title=Early days of X-ray crystallography |date=2015 |publisher=Oxford University Press |location=Oxford |isbn=9780198754053 |doi=10.1093/acprof:oso/9780199659845.003.0012 |chapter=12. The Birth and Rise of the Space-Lattice Concept |url=https://academic.oup.com/book/8011/chapter/153381347 |url-access=registration |access-date=24 December 2024}}</ref>{{rp|page=379}}

In the classification of crystals, to each [[space group]] is associated a crystallographic point group by "forgetting" the translational components of the symmetry operations, that is, by turning screw rotations into rotations, glide reflections into reflections and moving all symmetry elements into the origin. Each crystallographic point group defines the '''(geometric) crystal class''' of the crystal.

The point group of a crystal determines, among other things, the directional variation of physical properties that arise from its structure, including [[crystal optics|optical properties]] such as [[birefringence|birefringency]], or electro-optical features such as the [[Pockels effect]].

==Notation==

The point groups are named according to their component symmetries. There are several standard notations used by crystallographers, [[mineralogist]]s, and [[physicists]].

For the correspondence of the two systems below, see '''[[crystal system]]'''.

===Schoenflies notation===
{{main|Schoenflies notation}}
{{details|Point groups in three dimensions}}

In [[Arthur Moritz Schoenflies|Schoenflies]] notation, point groups are denoted by a letter symbol with a subscript. The symbols used in crystallography mean the following:

*''C<sub>n</sub>'' (for [[cyclic group|cyclic]]) indicates that the group has an ''n''-fold rotation axis. ''C<sub>nh</sub>'' is ''C<sub>n</sub>'' with the addition of a mirror (reflection) plane perpendicular to the [[axis of rotation]]. ''C<sub>nv</sub>'' is ''C<sub>n</sub>'' with the addition of n mirror planes parallel to the axis of rotation.
*''S<sub>2n</sub>'' (for ''Spiegel'', German for [[mirror]]) denotes a group with only a ''2n''-fold [[Improper rotation|rotation-reflection axis]].
*''D<sub>n</sub>'' (for [[dihedral group|dihedral]], or two-sided) indicates that the group has an ''n''-fold rotation axis plus ''n'' twofold axes perpendicular to that axis. ''D<sub>nh</sub>'' has, in addition, a mirror plane perpendicular to the ''n''-fold axis. ''D<sub>nd</sub>'' has, in addition to the elements of ''D<sub>n</sub>'', mirror planes parallel to the ''n''-fold axis.
*The letter ''T'' (for [[tetrahedron]]) indicates that the group has the symmetry of a tetrahedron. ''T<sub>d</sub>'' includes [[improper rotation]] operations, ''T'' excludes improper rotation operations, and ''T<sub>h</sub>'' is ''T'' with the addition of an inversion.
*The letter ''O'' (for [[octahedron]]) indicates that the group has the symmetry of an octahedron, with (''O<sub>h</sub>'') or without (''O'') improper operations (those that change handedness).

Due to the [[crystallographic restriction theorem]], ''n'' = 1, 2, 3, 4, or 6 in 2- or 3-dimensional space.

{| class="wikitable"
|-
! n
! 1
! 2
! 3
! 4
! 6
|-
| ''C<sub>n</sub>''
| ''C<sub>1</sub>''
| ''C<sub>2</sub>''
| ''C<sub>3</sub>''
| ''C<sub>4</sub>''
| ''C<sub>6</sub>''
|-
| ''C<sub>nv</sub>''
| ''C<sub>1v</sub>''=''C<sub>1h</sub>''
| ''C<sub>2v</sub>''
| ''C<sub>3v</sub>''
| ''C<sub>4v</sub>''
| ''C<sub>6v</sub>''
|-
| ''C<sub>nh</sub>''
| ''C<sub>1h</sub>''
| ''C<sub>2h</sub>''
| ''C<sub>3h</sub>''
| ''C<sub>4h</sub>''
| ''C<sub>6h</sub>''
|-
| ''D<sub>n</sub>''
| ''D<sub>1</sub>''=''C<sub>2</sub>''
| ''D<sub>2</sub>''
| ''D<sub>3</sub>''
| ''D<sub>4</sub>''
| ''D<sub>6</sub>''
|-
| ''D<sub>nh</sub>''
| ''D<sub>1h</sub>''=''C<sub>2v</sub>''
| ''D<sub>2h</sub>''
| ''D<sub>3h</sub>''
| ''D<sub>4h</sub>''
| ''D<sub>6h</sub>''
|-
| ''D<sub>nd</sub>''
| ''D<sub>1d</sub>''=''C<sub>2h</sub>''
| ''D<sub>2d</sub>''
| ''D<sub>3d</sub>''
|style="background:silver"| ''D<sub>4d</sub>''
|style="background:silver"| ''D<sub>6d</sub>''
|-
| ''S<sub>2n</sub>''
| ''S<sub>2</sub>''
| ''S<sub>4</sub>''
| ''S<sub>6</sub>''
|style="background:silver"|  ''S<sub>8</sub>''
|style="background:silver"|  ''S<sub>12</sub>''
|}
''D<sub>4d</sub>'' and ''D<sub>6d</sub>'' are actually forbidden because they contain [[improper rotation]]s with n=8 and 12 respectively. The 27 point groups in the table plus ''T'', ''T<sub>d</sub>'', ''T<sub>h</sub>'', ''O'' and ''O<sub>h</sub>'' constitute 32 crystallographic point groups.

=== Hermann–Mauguin notation===
{{main|Hermann–Mauguin notation}}

An abbreviated form of the [[Hermann–Mauguin notation]] commonly used for [[space group]]s also serves to describe crystallographic point groups. Group names are
{| class=wikitable
! Crystal family
! Crystal system
!colspan=7|Group names
|-
!colspan=2|[[cubic crystal system|Cubic]]
|23|| m{{overline|3}}|| || 432|| {{overline|4}}3m|| m{{overline|3}}m ||
|-
!rowspan=2|[[hexagonal crystal family|Hexagonal]]
!Hexagonal
|6|| {{overline|6}}|| <sup>6</sup>⁄<sub>m</sub>|| 622|| 6mm|| {{overline|6}}m2|| 6/mmm
|-
!Trigonal
|3|| {{overline|3}}|| || 32|| 3m|| {{overline|3}}m || 
|-
!colspan=2|[[tetragonal crystal system|Tetragonal]]
|4||{{overline|4}}|| <sup>4</sup>⁄<sub>m</sub>|| 422|| 4mm|| {{overline|4}}2m||4/mmm
|-
!colspan=2|[[orthorhombic crystal system|Orthorhombic]]
| || || ||222|| || mm2|| mmm
|-
!colspan=2|[[monoclinic crystal system|Monoclinic]]
|2|| || <sup>2</sup>⁄<sub>m</sub>|| || m|| || 
|-
!colspan=2|[[triclinic crystal system|Triclinic]]
|1|| {{overline|1}} || || || || ||
|}
{{clear}}

===The correspondence between different notations===

{| class="wikitable"
|-
!rowspan=2|Crystal family
!rowspan=2|[[Crystal system]] 
!colspan=2|[[Hermann-Mauguin notation|Hermann-Mauguin]]
!rowspan=2|Shubnikov<ref>{{cite web |url=http://it.iucr.org/Ab/ch12o1v0001/sec12o1o3/ |title=(International Tables) Abstract |access-date=2011-11-25 |url-status=dead |archive-url=https://archive.today/20130704042455/http://it.iucr.org/Ab/ch12o1v0001/sec12o1o3/ |archive-date=2013-07-04 }}</ref> 
!rowspan=2|[[Schoenflies notation|Schoenflies]]
!rowspan=2|[[Orbifold notation|Orbifold]]
!rowspan=2|[[Coxeter notation|Coxeter]] 
!rowspan=2|Order
|- align=center
!(full)
!(short)
|- align=center
! rowspan="2" colspan="2"|[[triclinic crystal system|Triclinic]]
|| 1 || 1 ||  <math>1\ </math>||''C<sub>1</sub>'' || 11 || [&nbsp;]<sup>+</sup> || 1
|- align=center
| {{overline|1}} || {{overline|1}} || <math>\tilde{2}</math> ||''C<sub>i</sub> = S<sub>2</sub>'' || × || [2<sup>+</sup>,2<sup>+</sup>] ||2
|- align=center
!rowspan="3" colspan="2"| [[monoclinic crystal system|Monoclinic]]
|| 2 || 2 || <math>2\ </math> ||''C<sub>2</sub>'' || 22 || [2]<sup>+</sup> || 2
|- align=center
| m || m || <math>m\ </math> ||''C<sub>s</sub> = C<sub>1h</sub>'' || * || [&nbsp;] || 2
|- align=center
| <math>\tfrac{2}{m}</math> || 2/m || <math>2:m\ </math> || ''C<sub>2h</sub>'' ||  2* || [2,2<sup>+</sup>] || 4
|- align=center
!rowspan="3" colspan="2"| [[orthorhombic crystal system|Orthorhombic]]
||  222 ||222 ||<math>2:2\ </math> ||''D<sub>2</sub> = V'' || 222 || [2,2]<sup>+</sup> || 4
|- align=center
| mm2 || mm2 || <math>2 \cdot m\ </math> ||''C<sub>2v</sub>'' || *22 || [2] || 4
|- align=center
| <math>\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}</math> || mmm || <math>m \cdot 2:m\ </math> ||''D<sub>2h</sub>'' = ''V<sub>h</sub>'' ||  *222 || [2,2] || 8
|- align=center
! rowspan="7" colspan="2"|[[tetragonal crystal system|Tetragonal]]
||  4 || 4 || <math>4\ </math> ||''C<sub>4</sub>'' ||  44 || [4]<sup>+</sup> || 4
|- align=center
| {{overline|4}} ||{{overline|4}} || <math>\tilde{4}</math> 
|| ''S<sub>4</sub>'' || 2× || [2<sup>+</sup>,4<sup>+</sup>] ||4
|- align=center
|  <math>\tfrac{4}{m}</math> || 4/m || <math>4:m\ </math>|| ''C<sub>4h</sub>'' ||  4* || [2,4<sup>+</sup>] || 8
|- align=center
|422 || 422 || <math>4:2\ </math> || ''D<sub>4</sub>'' || 422 || [4,2]<sup>+</sup> || 8
|- align=center
|4mm || 4mm ||<math>4 \cdot m\ </math> || ''C<sub>4v</sub>'' ||  *44 || [4] || 8
|- align=center
| {{overline|4}}2m || {{overline|4}}2m || <math>\tilde{4}\cdot m</math> || ''D<sub>2d</sub>'' = ''V<sub>d</sub>''||  2*2 || [2<sup>+</sup>,4] || 8
|- align=center
| <math>\tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}</math> || 4/mmm || <math>m \cdot 4:m\ </math> || ''D<sub>4h</sub>'' || *422 || [4,2] || 16
|- align=center
!rowspan="12"|[[hexagonal crystal family|Hexagonal]]
!rowspan="5"|Trigonal
|| 3 || 3 || <math>3\ </math> || ''C<sub>3</sub>'' ||  33 || [3]<sup>+</sup> || 3
|- align=center
|{{overline|3}} || {{overline|3}} ||<math>\tilde{6}</math> || ''C<sub>3i</sub> = S<sub>6</sub>'' ||  3× || [2<sup>+</sup>,6<sup>+</sup>] ||6
|- align=center
| 32 || 32 || <math>3:2\ </math> || ''D<sub>3</sub>'' ||  322 || [3,2]<sup>+</sup> || 6
|- align=center
| 3m || 3m || <math>3 \cdot m\ </math> || ''C<sub>3v</sub>'' ||  *33 || [3] || 6
|- align=center
| {{overline|3}}<math>\tfrac{2}{m}</math> ||{{overline|3}}m || <math>\tilde{6}\cdot m</math> || ''D<sub>3d</sub>'' || 2*3 || [2<sup>+</sup>,6] || 12
|- align=center
! rowspan="7"|Hexagonal
||6 || 6 || <math>6\ </math> || ''C<sub>6</sub>'' ||  66 || [6]<sup>+</sup> || 6
|- align=center
| {{overline|6}} || {{overline|6}} || <math>3:m\ </math> || ''C<sub>3h</sub>'' ||  3* || [2,3<sup>+</sup>] || 6
|- align=center
| <math>\tfrac{6}{m}</math> || 6/m || <math>6:m\ </math> || ''C<sub>6h</sub>'' ||  6* || [2,6<sup>+</sup>] || 12
|- align=center
| 622 || 622 || <math>6:2\ </math> || ''D<sub>6</sub>'' ||  622 || [6,2]<sup>+</sup> || 12
|- align=center
| 6mm || 6mm ||<math>6 \cdot m\ </math> || ''C<sub>6v</sub>'' ||  *66 || [6] || 12
|- align=center
| {{overline|6}}m2 || {{overline|6}}m2 || <math>m \cdot 3:m\ </math> || ''D<sub>3h</sub>'' ||  *322 || [3,2] || 12
|- align=center
| <math>\tfrac{6}{m}\tfrac{2}{m}\tfrac{2}{m}</math> || 6/mmm || <math>m \cdot 6:m\ </math> || ''D<sub>6h</sub>'' ||  *622 || [6,2] || 24
|- align=center
!rowspan="5" colspan="2"|[[cubic crystal system|Cubic]]
|| 23 || 23 || <math>3/2\ </math> || ''T'' ||  332 || [3,3]<sup>+</sup> || 12
|- align=center
| <math>\tfrac{2}{m}</math>{{overline|3}} || m{{overline|3}} || <math>\tilde{6}/2</math> || ''T<sub>h</sub>'' ||  3*2 || [3<sup>+</sup>,4] || 24
|- align=center
| 432 || 432 ||  <math>3/4\ </math> || ''O''  ||  432 || [4,3]<sup>+</sup> || 24
|- align=center
| {{overline|4}}3m || {{overline|4}}3m || <math>3/\tilde{4}</math> || ''T<sub>d</sub>'' ||  *332 || [3,3] || 24
|- align=center
| <math>\tfrac{4}{m}</math>{{overline|3}}<math>\tfrac{2}{m}</math> || m{{overline|3}}m || <math>\tilde{6}/4</math> || ''O<sub>h</sub>'' ||  *432 || [4,3] || 48
|}

== Isomorphisms ==
{{See also|Crystal structure#Crystal systems}}

Many of the crystallographic point groups share the same internal structure. For example, the point groups {{overline|1}}, 2, and m contain different geometric symmetry operations, (inversion, rotation, and reflection, respectively) but all share the structure of the [[cyclic group]] C<sub>2</sub>. All [[Group isomorphism|isomorphic]] groups are of the same [[Order (group theory)|order]], but not all groups of the same order are isomorphic. The point groups which are isomorphic are shown in the following table:<ref>{{cite journal | last=Novak | first=I | title=Molecular isomorphism | journal=European Journal of Physics | publisher=IOP Publishing | volume=16 | issue=4 | date=1995-07-18 | issn=0143-0807 | doi=10.1088/0143-0807/16/4/001 | pages=151–153| bibcode=1995EJPh...16..151N | s2cid=250887121 }}</ref>

{| class="wikitable"
|-
![[Hermann–Mauguin notation|Hermann–Mauguin]]
![[Schoenflies notation|Schoenflies]]
![[Order (group theory)|Order]]
!colspan=2|[[List of small groups|Abstract group]]
|- align=center
|| 1 || ''C<sub>1</sub>'' || 1 || [[Trivial group|C<sub>1</sub>]] ||<math>G_1^1</math>
|- align=center
| {{overline|1}} ||''C<sub>i</sub> = S<sub>2</sub>'' ||2 || rowspan="3"| [[Cyclic group|C<sub>2</sub>]] ||rowspan=3|<math>G_2^1</math>
|- align=center
|| 2 ||''C<sub>2</sub>'' || 2
|- align=center
| m ||''C<sub>s</sub> = C<sub>1h</sub>'' || 2
|- align=center
|| 3 || ''C<sub>3</sub>'' || 3 || [[Cyclic group|C<sub>3</sub>]] ||<math>G_3^1</math>
|- align=center
|| 4 || ''C<sub>4</sub>'' || 4 ||rowspan="2"| [[Cyclic group|C<sub>4</sub>]] ||rowspan=2|<math>G_4^1</math>
|- align=center
| {{overline|4}} || ''S<sub>4</sub>'' || 4
|- align=center
| 2/m ||  ''C<sub>2h</sub>'' || 4 || rowspan="3" | [[Klein four-group|D<sub>2</sub>]] = C<sub>2</sub> × C<sub>2</sub> ||rowspan=3|<math>G_4^2</math>
|- align=center
||  222 ||''D<sub>2</sub> = V'' || 4
|- align=center
| mm2 ||''C<sub>2v</sub>'' ||  4
|- align=center
|{{overline|3}} ||''C<sub>3i</sub> = S<sub>6</sub>'' || 6 ||rowspan="3"|[[Cyclic group|C<sub>6</sub>]]||rowspan=3|<math>G_6^1</math>
|- align=center
||6 || ''C<sub>6</sub>'' || 6
|- align=center
| {{overline|6}} || ''C<sub>3h</sub>'' || 6
|- align=center
| 32 || ''D<sub>3</sub>'' || 6 || rowspan="2"| [[Dihedral group of order 6|D<sub>3</sub>]]||rowspan=2|<math>G_6^2</math>
|- align=center
| 3m || ''C<sub>3v</sub>'' || 6
|- align=center
| mmm ||''D<sub>2h</sub>'' = ''V<sub>h</sub>'' || 8 || D<sub>2</sub> × C<sub>2</sub>||<math>G_8^3</math>
|- align=center
|  4/m || ''C<sub>4h</sub>'' || 8 || C<sub>4</sub> × C<sub>2</sub>||<math>G_8^2</math>
|- align=center
|422 || ''D<sub>4</sub>'' || 8 || rowspan="3"| [[Dihedral group of order 8|D<sub>4</sub>]]||rowspan=3|<math>G_8^4</math>
|- align=center
| 4mm || ''C<sub>4v</sub>'' || 8
|- align=center
| {{overline|4}}2m || ''D<sub>2d</sub>'' = ''V<sub>d</sub>''|| 8
|- align=center
| 6/m || ''C<sub>6h</sub>'' || 12 || C<sub>6</sub> × C<sub>2</sub>||<math>G_{12}^2</math>
|- align=center
|| 23 || ''T'' || 12 || [[Alternating group|A<sub>4</sub>]]||<math>G_{12}^5</math>
|- align=center
| {{overline|3}}m || ''D<sub>3d</sub>'' || 12 || rowspan="4" | [[Dihedral group|D<sub>6</sub>]]||rowspan=4|<math>G_{12}^3</math>
|- align=center
| 622 || ''D<sub>6</sub>'' || 12
|- align=center
| 6mm || ''C<sub>6v</sub>'' || 12
|- align=center
| {{overline|6}}m2 || ''D<sub>3h</sub>'' || 12
|- align=center
| 4/mmm || ''D<sub>4h</sub>'' || 16 || D<sub>4</sub> × C<sub>2</sub>||<math>G_{16}^9</math>
|- align=center
| 6/mmm || ''D<sub>6h</sub>'' || 24 || D<sub>6</sub> × C<sub>2</sub>||<math>G_{24}^5</math>
|- align=center
| m{{overline|3}} || ''T<sub>h</sub>'' || 24 || A<sub>4</sub> × C<sub>2</sub>||<math>G_{24}^{10}</math>
|- align=center
| 432 || ''O''  ||  24 || rowspan="2"| [[Symmetric group|S<sub>4</sub>]]||rowspan=2|<math>G_{24}^{7}</math>
|- align=center
| {{overline|4}}3m || ''T<sub>d</sub>'' || 24
|- align=center
|  m{{overline|3}}m || ''O<sub>h</sub>'' || 48 || S<sub>4</sub> × C<sub>2</sub>||<math>G_{48}^7</math>
|}

This table makes use of [[cyclic group]]s (C<sub>1</sub>, C<sub>2</sub>, C<sub>3</sub>, C<sub>4</sub>, C<sub>6</sub>), [[dihedral group]]s (D<sub>2</sub>, D<sub>3</sub>, D<sub>4</sub>, D<sub>6</sub>), one of the [[alternating group]]s (A<sub>4</sub>), and one of the [[symmetric group]]s (S<sub>4</sub>). Here the symbol " × " indicates a [[Direct product of groups|direct product]].

==Deriving the crystallographic point group (crystal class) from the space group==

# Leave out the [[Bravais lattice]] type.
# Convert all symmetry elements with translational components into their respective symmetry elements without translation symmetry. (Glide planes are converted into simple mirror planes; screw axes are converted into simple axes of rotation.)
# Axes of rotation, [[Improper rotation|rotoinversion]] axes, and mirror planes remain unchanged.

==See also==
* [[Molecular symmetry]]
* [[Point group]]
* [[Space group]]
* [[Point groups in three dimensions]]
* [[Crystal system]]

== References ==
<references />

==External links==
*[https://archive.today/20130704033345/http://it.iucr.org/Ab/ch12o1v0001/ Point-group symbols in International Tables for Crystallography (2006). Vol. A, ch. 12.1, pp. 818-820]
*[https://archive.today/20130704032551/http://it.iucr.org/Ab/ch10o1v0001/table10o1o2o4/ Names and symbols of the 32 crystal classes in International Tables for Crystallography (2006). Vol. A, ch. 10.1, p. 794]
*[https://web.archive.org/web/20120204104121/http://newton.ex.ac.uk/research/qsystems/people/goss/symmetry/Solids.html Pictorial overview of the 32 groups]

{{Crystal systems}}

[[Category:Symmetry]]
[[Category:Crystallography]]
[[Category:Discrete groups]]