Crystallographic point group

In crystallography, a crystallographic point group is a three-dimensional point group whose symmetry operations are compatible with a three-dimensional crystallographic lattice. According to the crystallographic restriction it may only contain one-, two-, three-, four- and sixfold rotations or rotoinversions. This reduces the number of crystallographic point groups to 32 (from an infinity of general point groups). These 32 groups are the same as the 32 types of morphological (external) crystalline symmetries derived in 1830 by Johann Friedrich Christian Hessel from a consideration of observed crystal forms. In 1867 Axel Gadolin, who was unaware of the previous work of Hessel, found the crystallographic point groups independently using stereographic projection to represent the symmetry elements of the 32 groups.{{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}}{{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 optical properties such as 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, mineralogists, and physicists.

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

=Schoenflies notation=

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

C_n (for cyclic) indicates that the group has an n-fold rotation axis. C_nh is C_n with the addition of a mirror (reflection) plane perpendicular to the axis of rotation. C_nv is C_n with the addition of n mirror planes parallel to the axis of rotation.
S_2n (for Spiegel, German for mirror) denotes a group with only a 2n-fold rotation-reflection axis.
D_n (for dihedral, or two-sided) indicates that the group has an n-fold rotation axis plus n twofold axes perpendicular to that axis. D_nh has, in addition, a mirror plane perpendicular to the n-fold axis. D_nd has, in addition to the elements of D_n, mirror planes parallel to the n-fold axis.
The letter T (for tetrahedron) indicates that the group has the symmetry of a tetrahedron. T_d includes improper rotation operations, T excludes improper rotation operations, and T_h 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_h) 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_n \| C₁ \| C₂ \| C₃ \| C₄ \| C₆
C_nv \| C_1v=C_1h \| C_2v \| C_3v \| C_4v \| C_6v
C_nh \| C_1h \| C_2h \| C_3h \| C_4h \| C_6h
D_n \| D₁=C₂ \| D₂ \| D₃ \| D₄ \| D₆
D_nh \| D_1h=C_2v \| D_2h \| D_3h \| D_4h \| D_6h
D_nd \| D_1d=C_2h \| D_2d \| D_3d \|style="background:silver"\| D_4d \|style="background:silver"\| D_6d
S_2n \| S₂ \| S₄ \| S₆ \|style="background:silver"\| S₈ \|style="background:silver"\| S₁₂

class="wikitable"

! 1

! 2

! 3

! 4

! 6

C_n

| C₁

| C₂

| C₃

| C₄

| C₆

C_nv

| C_1v=C_1h

| C_2v

| C_3v

| C_4v

| C_6v

C_nh

| C_1h

| C_2h

| C_3h

| C_4h

| C_6h

D_n

| D₁=C₂

| D₂

| D₃

| D₄

| D₆

D_nh

| D_1h=C_2v

| D_2h

| D_3h

| D_4h

| D_6h

D_nd

| D_1d=C_2h

| D_2d

| D_3d

|style="background:silver"| D_4d

|style="background:silver"| D_6d

S_2n

| S₂

| S₄

| S₆

|style="background:silver"| S₈

|style="background:silver"| S₁₂

D_4d and D_6d are actually forbidden because they contain improper rotations with n=8 and 12 respectively. The 27 point groups in the table plus T, T_d, T_h, O and O_h constitute 32 crystallographic point groups.

= Hermann–Mauguin notation=

An abbreviated form of the Hermann–Mauguin notation commonly used for space groups also serves to describe crystallographic point groups. Group names are

class=wikitable ! Crystal family ! Crystal system !colspan=7\|Group names
colspan=2\|Cubic \|23\|\| m{{overline\|3}}\|\| \|\| 432\|\| {{overline\|4}}3m\|\| m{{overline\|3}}m \|\|
rowspan=2\|Hexagonal !Hexagonal \|6\|\| {{overline\|6}}\|\| ⁶⁄_m\|\| 622\|\| 6mm\|\| {{overline\|6}}m2\|\| 6/mmm
Trigonal \|3\|\| {{overline\|3}}\|\| \|\| 32\|\| 3m\|\| {{overline\|3}}m \|\|
colspan=2\|Tetragonal \|4\|\|{{overline\|4}}\|\| ⁴⁄_m\|\| 422\|\| 4mm\|\| {{overline\|4}}2m\|\|4/mmm
colspan=2\|Orthorhombic \| \|\| \|\| \|\|222\|\| \|\| mm2\|\| mmm
colspan=2\|Monoclinic \|2\|\| \|\| ²⁄_m\|\| \|\| m\|\| \|\|
colspan=2\|Triclinic \|1\|\| {{overline\|1}} \|\| \|\| \|\| \|\| \|\|

class=wikitable

! Crystal family

! Crystal system

!colspan=7|Group names

colspan=2|Cubic

|23|| m{{overline|3}}|| || 432|| {{overline|4}}3m|| m{{overline|3}}m ||

rowspan=2|Hexagonal

!Hexagonal

|6|| {{overline|6}}|| ⁶⁄_m|| 622|| 6mm|| {{overline|6}}m2|| 6/mmm

Trigonal

|3|| {{overline|3}}|| || 32|| 3m|| {{overline|3}}m ||

colspan=2|Tetragonal

|4||{{overline|4}}|| ⁴⁄_m|| 422|| 4mm|| {{overline|4}}2m||4/mmm

colspan=2|Orthorhombic

| || || ||222|| || mm2|| mmm

colspan=2|Monoclinic

|2|| || ²⁄_m|| || m|| ||

colspan=2|Triclinic

|1|| {{overline|1}} || || || || ||

=The correspondence between different notations=

rowspan=2\|Crystal family !rowspan=2\|Crystal system !colspan=2\|Hermann-Mauguin !rowspan=2\|Shubnikov{{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 }} !rowspan=2\|Schoenflies !rowspan=2\|Orbifold !rowspan=2\|Coxeter !rowspan=2\|Order
class="wikitable"
align=center !(full) !(short)
align=center ! rowspan="2" colspan="2"\|Triclinic	1	1	$1\$	C₁	11	[ ]⁺	1
align=center \| {{overline\|1}}	{{overline\|1}}	$\tilde{2}$	C_i = S₂	×	[2⁺,2⁺]	2
align=center !rowspan="3" colspan="2"\| Monoclinic	2	2	$2\$	C₂	22	[2]⁺	2
align=center \| m	m	$m\$	C_s = C_1h	*	[ ]	2
align=center \| $\tfrac{2}{m}$	2/m	$2:m\$	C_2h	2*	[2,2⁺]	4
align=center !rowspan="3" colspan="2"\| Orthorhombic	222	222	$2:2\$	D₂ = V	222	[2,2]⁺	4
align=center \| mm2	mm2	$2 \cdot m\$	C_2v	*22	[2]	4
align=center \| $\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	mmm	$m \cdot 2:m\$	D_2h = V_h	*222	[2,2]	8
align=center ! rowspan="7" colspan="2"\|Tetragonal	4	4	$4\$	C₄	44	[4]⁺	4
align=center \| {{overline\|4}}	{{overline\|4}}	$\tilde{4}$	S₄	2×	[2⁺,4⁺]	4
align=center \| $\tfrac{4}{m}$	4/m	$4:m\$	C_4h	4*	[2,4⁺]	8
align=center \|422	422	$4:2\$	D₄	422	[4,2]⁺	8
align=center \|4mm	4mm	$4 \cdot m\$	C_4v	*44	[4]	8
align=center \| {{overline\|4}}2m	{{overline\|4}}2m	$\tilde{4}\cdot m$	D_2d = V_d	2*2	[2⁺,4]	8
align=center \| $\tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}$	4/mmm	$m \cdot 4:m\$	D_4h	*422	[4,2]	16
align=center !rowspan="12"\|Hexagonal !rowspan="5"\|Trigonal	3	3	$3\$	C₃	33	[3]⁺	3
align=center \|{{overline\|3}}	{{overline\|3}}	$\tilde{6}$	C_3i = S₆	3×	[2⁺,6⁺]	6
align=center \| 32	32	$3:2\$	D₃	322	[3,2]⁺	6
align=center \| 3m	3m	$3 \cdot m\$	C_3v	*33	[3]	6
align=center \| {{overline\|3}} $\tfrac{2}{m}$	{{overline\|3}}m	$\tilde{6}\cdot m$	D_3d	2*3	[2⁺,6]	12
align=center ! rowspan="7"\|Hexagonal	6	6	$6\$	C₆	66	[6]⁺	6
align=center \| {{overline\|6}}	{{overline\|6}}	$3:m\$	C_3h	3*	[2,3⁺]	6
align=center \| $\tfrac{6}{m}$	6/m	$6:m\$	C_6h	6*	[2,6⁺]	12
align=center \| 622	622	$6:2\$	D₆	622	[6,2]⁺	12
align=center \| 6mm	6mm	$6 \cdot m\$	C_6v	*66	[6]	12
align=center \| {{overline\|6}}m2	{{overline\|6}}m2	$m \cdot 3:m\$	D_3h	*322	[3,2]	12
align=center \| $\tfrac{6}{m}\tfrac{2}{m}\tfrac{2}{m}$	6/mmm	$m \cdot 6:m\$	D_6h	*622	[6,2]	24
align=center !rowspan="5" colspan="2"\|Cubic	23	23	$3/2\$	T	332	[3,3]⁺	12
align=center \| $\tfrac{2}{m}$ {{overline\|3}}	m{{overline\|3}}	$\tilde{6}/2$	T_h	3*2	[3⁺,4]	24
align=center \| 432	432	$3/4\$	O	432	[4,3]⁺	24
align=center \| {{overline\|4}}3m	{{overline\|4}}3m	$3/\tilde{4}$	T_d	*332	[3,3]	24
align=center \| $\tfrac{4}{m}$ {{overline\|3}} $\tfrac{2}{m}$	m{{overline\|3}}m	$\tilde{6}/4$	O_h	*432	[4,3]	48

Isomorphisms

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 C₂. All isomorphic groups are of the same order, but not all groups of the same order are isomorphic. The point groups which are isomorphic are shown in the following table:{{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 }}

Hermann–Mauguin !Schoenflies !Order !colspan=2\|Abstract group
class="wikitable"
align=center	1	C₁	1	C₁	$G_1^1$
align=center \| {{overline\|1}}	C_i = S₂	2	rowspan="3"\| C₂	rowspan=3\| $G_2^1$
align=center	2	C₂	2
align=center \| m	C_s = C_1h	2
align=center	3	C₃	3	C₃	$G_3^1$
align=center	4	C₄	4	rowspan="2"\| C₄	rowspan=2\| $G_4^1$
align=center \| {{overline\|4}}	S₄	4
align=center \| 2/m	C_2h	4	rowspan="3" \| D₂ = C₂ × C₂	rowspan=3\| $G_4^2$
align=center	222	D₂ = V	4
align=center \| mm2	C_2v	4
align=center \|{{overline\|3}}	C_3i = S₆	6	rowspan="3"\|C₆	rowspan=3\| $G_6^1$
align=center	6	C₆	6
align=center \| {{overline\|6}}	C_3h	6
align=center \| 32	D₃	6	rowspan="2"\| D₃	rowspan=2\| $G_6^2$
align=center \| 3m	C_3v	6
align=center \| mmm	D_2h = V_h	8	D₂ × C₂	$G_8^3$
align=center \| 4/m	C_4h	8	C₄ × C₂	$G_8^2$
align=center \|422	D₄	8	rowspan="3"\| D₄	rowspan=3\| $G_8^4$
align=center \| 4mm	C_4v	8
align=center \| {{overline\|4}}2m	D_2d = V_d	8
align=center \| 6/m	C_6h	12	C₆ × C₂	$G_{12}^2$
align=center	23	T	12	A₄	$G_{12}^5$
align=center \| {{overline\|3}}m	D_3d	12	rowspan="4" \| D₆	rowspan=4\| $G_{12}^3$
align=center \| 622	D₆	12
align=center \| 6mm	C_6v	12
align=center \| {{overline\|6}}m2	D_3h	12
align=center \| 4/mmm	D_4h	16	D₄ × C₂	$G_{16}^9$
align=center \| 6/mmm	D_6h	24	D₆ × C₂	$G_{24}^5$
align=center \| m{{overline\|3}}	T_h	24	A₄ × C₂	$G_{24}^{10}$
align=center \| 432	O	24	rowspan="2"\| S₄	rowspan=2\| $G_{24}^{7}$
align=center \| {{overline\|4}}3m	T_d	24
align=center \| m{{overline\|3}}m	O_h	48	S₄ × C₂	$G_{48}^7$

This table makes use of cyclic groups (C₁, C₂, C₃, C₄, C₆), dihedral groups (D₂, D₃, D₄, D₆), one of the alternating groups (A₄), and one of the symmetric groups (S₄). Here the symbol " × " indicates a direct product.

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

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, rotoinversion axes, and mirror planes remain unchanged.

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]

Category:Symmetry

Category:Crystallography

Category:Discrete groups