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.
Contents

Overview 1

Crystal classes 2

Lattice systems 3

Crystal systems in fourdimensional space 4

See also 5

Notes 6

References 7

External links 8
Overview
Hexagonal
hanksite crystal, with threefold caxis symmetry
A lattice system is generally identified as a set of lattices with the same shape according to the relative lengths of the cell edges (a, b, c) and the angles between them (α, β, γ). Each lattice is assigned to one of the following classifications (lattice types) based on the positions of the lattice points within the cell: primitive (P), bodycentered (I), facecentered (F), basecentered (A, B, or C), and rhombohedral (R). The 14 unique combinations of lattice systems and lattice types are collectively known as the Bravais lattices. Associated with each lattice system is a set of point groups, sometimes called lattice point groups, which are subgroups of the arithmetic crystal classes. In total there are seven lattice systems: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic.
A crystal system is a set of point groups in which the point groups themselves and their corresponding space groups are assigned to the same 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. In total there are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic.
A crystal family is determined by lattices and point groups. It is formed by combining crystal systems which have space groups assigned to the same lattice system. In three dimensions, the crystal families are identical to the crystal systems except the hexagonal and trigonal crystal systems are combined into one hexagonal crystal family. In total there are six crystal families: triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, and cubic.
Spaces with less than three dimensions have the same number of crystal systems, crystal families, and lattice systems. In zero and onedimensional space, there is one crystal system. In twodimensional space, there are four crystal systems: oblique, rectangular, square, and hexagonal.
The relation between threedimensional crystal families, crystal systems, and lattice systems is shown in the following table:
Caution: There is no "trigonal" lattice system. To avoid confusion of terminology, don't use the term "trigonal lattice"; use the definition that "trigonal lattice" = "hexagonal lattice" ≠ "rhombohedral lattice".
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

HermannMauguin

Orbifold

Coxeter

Point symmetry

Order

Abstract group

triclinic

triclinicpedial

C_{1}

1

11

[ ]^{+}

enantiomorphic polar

1

trivial \mathbb{Z}_1

triclinicpinacoidal

C_{i}

1

1x

[2,1^{+}]

centrosymmetric

2

cyclic \mathbb{Z}_2

monoclinic

monoclinicsphenoidal

C_{2}

2

22

[2,2]^{+}

enantiomorphic polar

2

cyclic \mathbb{Z}_2

monoclinicdomatic

C_{s}

m

*11

[ ]

polar

2

cyclic \mathbb{Z}_2

monoclinicprismatic

C_{2h}

2/m

2*

[2,2^{+}]

centrosymmetric

4

Klein four \mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2

orthorhombic

orthorhombicsphenoidal

D_{2}

222

222

[2,2]^{+}

enantiomorphic

4

Klein four \mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2

orthorhombicpyramidal

C_{2v}

mm2

*22

[2]

polar

4

Klein four \mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2

orthorhombicbipyramidal

D_{2h}

mmm

*222

[2,2]

centrosymmetric

8

\mathbb{V}\times\mathbb{Z}_2

tetragonal

tetragonalpyramidal

C_{4}

4

44

[4]^{+}

enantiomorphic polar

4

cyclic \mathbb{Z}_4

tetragonaldisphenoidal

S_{4}

4

2x

[2^{+},2]

noncentrosymmetric

4

cyclic \mathbb{Z}_4

tetragonaldipyramidal

C_{4h}

4/m

4*

[2,4^{+}]

centrosymmetric

8

\mathbb{Z}_4\times\mathbb{Z}_2

tetragonaltrapezoidal

D_{4}

422

422

[2,4]^{+}

enantiomorphic

8

dihedral \mathbb{D}_8 = \mathbb{Z}_4\rtimes\mathbb{Z}_2

ditetragonalpyramidal

C_{4v}

4mm

*44

[4]

polar

8

dihedral \mathbb{D}_8 = \mathbb{Z}_4\rtimes\mathbb{Z}_2

tetragonalscalenoidal

D_{2d}

42m or 4m2

2*2

[2^{+},4]

noncentrosymmetric

8

dihedral \mathbb{D}_8 = \mathbb{Z}_4\rtimes\mathbb{Z}_2

ditetragonaldipyramidal

D_{4h}

4/mmm

*422

[2,4]

centrosymmetric

16

\mathbb{D}_8\times\mathbb{Z}_2

hexagonal

trigonal

trigonalpyramidal

C_{3}

3

33

[3]^{+}

enantiomorphic polar

3

cyclic \mathbb{Z}_3

rhombohedral

S_{6} (C_{3i})

3

3x

[2^{+},3^{+}]

centrosymmetric

6

cyclic \mathbb{Z}_6 = \mathbb{Z}_3\times\mathbb{Z}_2

trigonaltrapezoidal

D_{3}

32 or 321 or 312

322

[3,2]^{+}

enantiomorphic

6

dihedral \mathbb{D}_6 = \mathbb{Z}_3\rtimes\mathbb{Z}_2

ditrigonalpyramidal

C_{3v}

3m or 3m1 or 31m

*33

[3]

polar

6

dihedral \mathbb{D}_6 = \mathbb{Z}_3\rtimes\mathbb{Z}_2

ditrigonalscalahedral

D_{3d}

3m or 3m1 or 31m

2*3

[2^{+},6]

centrosymmetric

12

dihedral \mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2

hexagonal

hexagonalpyramidal

C_{6}

6

66

[6]^{+}

enantiomorphic polar

6

cyclic \mathbb{Z}_6 = \mathbb{Z}_3\times\mathbb{Z}_2

trigonaldipyramidal

C_{3h}

6

3*

[2,3^{+}]

noncentrosymmetric

6

cyclic \mathbb{Z}_6 = \mathbb{Z}_3\times\mathbb{Z}_2

hexagonaldipyramidal

C_{6h}

6/m

6*

[2,6^{+}]

centrosymmetric

12

\mathbb{Z}_6\times\mathbb{Z}_2

hexagonaltrapezoidal

D_{6}

622

622

[2,6]^{+}

enantiomorphic

12

dihedral \mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2

dihexagonalpyramidal

C_{6v}

6mm

*66

[6]

polar

12

dihedral \mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2

ditrigonaldipyramidal

D_{3h}

6m2 or 62m

*322

[2,3]

noncentrosymmetric

12

dihedral \mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2

dihexagonaldipyramidal

D_{6h}

6/mmm

*622

[2,6]

centrosymmetric

24

\mathbb{D}_{12}\times\mathbb{Z}_2

cubic

tetrahedral

T

23

332

[3,3]^{+}

enantiomorphic

12

alternating \mathbb{A}_4

hextetrahedral

T_{d}

43m

*332

[3,3]

noncentrosymmetric

24

symmetric \mathbb{S}_4

diploidal

T_{h}

m3

3*2

[3^{+},4]

centrosymmetric

24

\mathbb{A}_4\times\mathbb{Z}_2

gyroidal

O

432

432

[4,3]^{+}

enantiomorphic

24

symmetric \mathbb{S}_4

hexoctahedral

O_{h}

m3m

*432

[4,3]

centrosymmetric

48

\mathbb{S}_4\times\mathbb{Z}_2

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 noncentrosymmetric. Still, even for noncentrosymmetric case, inverted structure in some cases can be rotated to align with the original structure. This is the case of noncentrosymmetric 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 (meaning a line without an arrow) 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 noncentrosymmetric structures. There should also not be a mirror plane or 2fold 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.
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

\mathbf{R} = n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + n_3 \mathbf{a}_3,
where n_{1}, n_{2}, and n_{3} are integers and a_{1}, a_{2}, and a_{3} are three noncoplanar 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 fourdimensional space
The fourdimensional unit cell is defined by four edge lengths (a, b, c, d) and six interaxial angles (\alpha, \beta, \gamma, \delta, \epsilon, \zeta). The following conditions for the lattice parameters define 23 crystal families:
1 Hexaclinic: a\ne b \ne c \ne d, \alpha \ne \beta \ne \gamma \ne \delta \ne \epsilon \ne \zeta \ne 90 ^\circ
2 Triclinic: a\ne b \ne c \ne d, \alpha \ne \beta \ne \gamma \ne 90 ^\circ, \delta = \epsilon = \zeta = 90 ^\circ
3 Diclinic: a\ne b \ne c \ne d, \alpha \ne 90 ^\circ, \beta = \gamma = \delta = \epsilon = 90 ^\circ, \zeta \ne 90 ^\circ
4 Monoclinic: a\ne b \ne c \ne d, \alpha \ne 90 ^\circ, \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ
5 Orthogonal: a\ne b \ne c \ne d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ
6 Tetragonal Monoclinic: a\ne b = c \ne d, \alpha \ne 90 ^\circ, \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ
7 Hexagonal Monoclinic: a\ne b = c \ne d, \alpha \ne 90 ^\circ, \beta = \gamma = \delta = \epsilon = 90 ^\circ, \zeta = 120 ^\circ
8 Ditetragonal Diclinic: a = d \ne b = c, \alpha = \zeta = 90 ^\circ, \beta = \epsilon \ne 90 ^\circ, \gamma \ne 90 ^\circ, \delta = 180 ^\circ  \gamma
9 Ditrigonal (Dihexagonal) Diclinic: 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
10 Tetragonal Orthogonal: a\ne b = c \ne d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ
11 Hexagonal Orthogonal: a\ne b = c \ne d, \alpha = \beta = \gamma = \delta = \epsilon = 90 ^\circ, \zeta = 120 ^\circ
12 Ditetragonal Monoclinic: a = d \ne b = c, \alpha = \gamma = \delta = \zeta = 90 ^\circ, \beta = \epsilon \ne 90 ^\circ
13 Ditrigonal (Dihexagonal) Monoclinic: a = d \ne b = c, \alpha = \zeta = 120 ^\circ, \beta = \epsilon \ne 90 ^\circ, \gamma = \delta \ne 90 ^\circ, cos \gamma = \tfrac{1}{2} cos \beta
14 Ditetragonal Orthogonal: a = d \ne b = c, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ
15 Hexagonal Tetragonal: a = d \ne b = c, \alpha = \beta = \gamma = \delta = \epsilon = 90 ^\circ, \zeta = 120 ^\circ
16 Dihexagonal Orthogonal: a = d \ne b = c, \alpha = \zeta = 120 ^\circ, \beta = \gamma = \delta = \epsilon = 90 ^\circ,
17 Cubic Orthogonal: a = b = c \ne d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ
18 Octagonal: a = b = c = d, \alpha = \gamma = \zeta \ne 90 ^\circ, \beta = \epsilon = 90 ^\circ, \delta = 180 ^\circ  \alpha
19 Decagonal: a = b = c = d, \alpha = \gamma = \zeta \ne \beta = \delta = \epsilon, cos \beta = 0.5  cos \alpha
20 Dodecagonal: a = b = c = d, \alpha = \zeta = 90 ^\circ, \beta = \epsilon = 120 ^\circ, \gamma = \delta \ne 90 ^\circ
21 Diisohexagonal Orthogonal: a = b = c = d, \alpha = \zeta = 120 ^\circ, \beta = \gamma = \delta = \epsilon = 90 ^\circ
22 Icosagonal (Icosahedral): a = b = c = d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta, cos \alpha = \tfrac{1}{4}
23 Hypercubic: a = b = c = d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ
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 fourdimensional 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 threedimensional 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 threedimensional space groups P3_{1} and P3_{2}, P4_{1}22 and P4_{3}22. Starting from fourdimensional 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

Nonaxial 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

CryptoDitetragonal 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

CryptoDitrigonal 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

Diisohexagonal Orthogonal

Simple Diisohexagonal Orthogonal

29

9 (+2)

19 (+5)

1

Diisohexagonal Orthogonal RR

19 (+3)

1

Diisohexagonal Orthogonal P

Complex Diisohexagonal 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
Notes

^ Howard D. Flack (2003). "Chiral and Achiral Crystal Structures". Helvetica Chimica Acta 86: 905–921.

^ E. Koch , W. Fischer , U. Müller , in ‘International Tables for Crystallography, Vol. A, SpaceGroup Symmetry’, 5th edn., Ed. T. Hahn, Kluwer Academic Publishers, Dordrecht, 2002, Chapt. 10, p. 804.

^ ^{a} ^{b} E. J. W. Whittaker, An atlas of hyperstereograms of the fourdimensional crystal classes. Clarendon Press (Oxford Oxfordshire and New York) 1985.

^ ^{a} ^{b} H. Brown, R. Bülow, J. Neubüser, H. Wondratschek and H. Zassenhaus, Crystallographic Groups of FourDimensional Space. Wiley, NY, 1978.
References

Hahn, Theo, ed. (2002). International Tables for Crystallography, Volume A: Space Group Symmetry A (5th ed.). Berlin, New York:
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
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