Crystal system

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The diamond crystal structure belongs to the face-centered cubic lattice, with a repeated two-atom pattern. Carbon lattice diamond.png
The diamond crystal structure belongs to the face-centered cubic lattice, with a repeated two-atom pattern.

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 are in the same crystal system if they have similar symmetries, although there are many exceptions to this.

Contents

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 seven crystal systems according to their point groups, and into seven 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

Hexagonal hanksite crystal, with threefold c-axis symmetry Hanksite.JPG
Hexagonal hanksite crystal, with threefold c-axis symmetry

A lattice system is a class of lattices with the same set of lattice point groups, which are subgroups of the arithmetic crystal classes. The 14 Bravais lattices are grouped into seven lattice systems: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic.

In a crystal system, a set of point groups 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 both the crystal and lattice systems have the same name. However, five point groups are assigned to two lattice systems, rhombohedral and hexagonal, because both 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 a common lattice system. In three dimensions, the crystal families and systems are identical, except the hexagonal and trigonal crystal systems, which 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 one-dimensional space, there is one crystal system. In 2D space, there are four crystal systems: 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 (6)Crystal system (7)Required symmetries of point group Point groups Space groups Bravais lattices Lattice system
Triclinic None221 Triclinic
Monoclinic 1 twofold axis of rotation or 1 mirror plane 3132 monoclinic
Orthorhombic 3 twofold axes of rotation or 1 twofold axis of rotation and 2 mirror planes3594 Orthorhombic
Tetragonal 1 fourfold axis of rotation7682 Tetragonal
Hexagonal Trigonal1 threefold axis of rotation571Rhombohedral
181Hexagonal
Hexagonal1 sixfold axis of rotation727
Cubic 3 fourfold axes of rotation5363 Cubic
67Total32230147
Note: there is no "trigonal" lattice system. To avoid confusion of terminology, the term "trigonal lattice" is not used.

Crystal classes

The 7 crystal systems consist of 32 crystal classes (corresponding to the 32 crystallographic point groups) as shown in the following table below:

Crystal familyCrystal system Point group / Crystal class Schönflies Hermann–Mauguin Orbifold Coxeter Point symmetry Order Abstract group
triclinic pedialC1111[ ]+ enantiomorphic polar 1trivial
pinacoidalCi (S2)11x[2,1+] centrosymmetric 2 cyclic
monoclinic sphenoidalC2222[2,2]+ enantiomorphic polar 2 cyclic
domaticCs (C1h)m*11[ ] polar 2 cyclic
prismatic C2h2/m2*[2,2+] centrosymmetric 4 Klein four
orthorhombic rhombic-disphenoidalD2 (V)222222[2,2]+ enantiomorphic 4 Klein four
rhombic-pyramidal C2vmm2*22[2] polar 4 Klein four
rhombic-dipyramidal D2h (Vh)mmm*222[2,2] centrosymmetric 8
tetragonal tetragonal-pyramidalC4444[4]+ enantiomorphic polar 4 cyclic
tetragonal-disphenoidalS442x[2+,2] non-centrosymmetric 4 cyclic
tetragonal-dipyramidalC4h4/m4*[2,4+] centrosymmetric 8
tetragonal-trapezohedralD4422422[2,4]+ enantiomorphic 8 dihedral
ditetragonal-pyramidalC4v4mm*44[4] polar 8 dihedral
tetragonal-scalenohedralD2d (Vd)42m or 4m22*2[2+,4] non-centrosymmetric 8 dihedral
ditetragonal-dipyramidalD4h4/mmm*422[2,4] centrosymmetric 16
hexagonal trigonaltrigonal-pyramidalC3333[3]+ enantiomorphic polar 3 cyclic
rhombohedralC3i (S6)33x[2+,3+] centrosymmetric 6 cyclic
trigonal-trapezohedralD332 or 321 or 312322[3,2]+ enantiomorphic 6 dihedral
ditrigonal-pyramidalC3v3m or 3m1 or 31m*33[3] polar 6 dihedral
ditrigonal-scalenohedralD3d3m or 3m1 or 31m2*3[2+,6] centrosymmetric 12 dihedral
hexagonalhexagonal-pyramidalC6666[6]+ enantiomorphic polar 6 cyclic
trigonal-dipyramidalC3h63*[2,3+] non-centrosymmetric 6 cyclic
hexagonal-dipyramidalC6h6/m6*[2,6+] centrosymmetric 12
hexagonal-trapezohedralD6622622[2,6]+ enantiomorphic 12 dihedral
dihexagonal-pyramidalC6v6mm*66[6] polar 12 dihedral
ditrigonal-dipyramidalD3h6m2 or 62m*322[2,3] non-centrosymmetric 12 dihedral
dihexagonal-dipyramidalD6h6/mmm*622[2,6] centrosymmetric 24
cubic tetartoidalT23332[3,3]+ enantiomorphic 12 alternating
diploidalThm33*2[3+,4] centrosymmetric 24
gyroidalO432432[4,3]+ enantiomorphic 24 symmetric
hextetrahedral Td43m*332[3,3] non-centrosymmetric 24 symmetric
hexoctahedral Ohm3m*432[4,3] centrosymmetric 48

The point symmetry of a structure can be further described as follows. Consider the points that make up the structure, and reflect 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 in the non-centrosymmetric case, the inverted structure can in some cases be rotated to align with the original structure. This is a non-centrosymmetric achiral structure. If the inverted structure cannot be rotated to align with the original structure, then the structure is chiral or 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 symmetry direction of a crystal that is polar is called a polar axis. [2] Groups containing a polar axis are called polar . A polar crystal possesses a unique polar axis (more precisely, all polar axes are parallel). Some geometrical or physical property is different at the two ends of this axis: for example, there might develop a dielectric polarization as in pyroelectric crystals. A polar axis can occur only in non-centrosymmetric structures. There cannot be a mirror plane or twofold axis perpendicular to the polar axis, because they would make the two directions of the axis equivalent.

The crystal structures of chiral biological molecules (such as protein structures) can only occur in the 65 enantiomorphic space groups (biological molecules are usually chiral).

Bravais lattices

There are seven different kinds of crystal systems, and each kind of crystal system has four different kinds of centerings (primitive, base-centered, body-centered, face-centered). However, not all of the combinations are unique; some of the combinations are equivalent while other combinations are not possible due to symmetry reasons. This reduces the number of unique lattices to the 14 Bravais lattices.

The distribution of the 14 Bravais lattices into lattice systems and crystal families is given in the following table.

Crystal familyLattice system Schönflies 14 Bravais lattices
PrimitiveBase-centeredBody-centeredFace-centered
triclinic Ci Triclinic.svg
monoclinic C2h Monoclinic.svg Monoclinic-base-centered.svg
orthorhombic D2h Orthorhombic.svg Orthorhombic-base-centered.svg Orthorhombic-body-centered.svg Orthorhombic-face-centered.svg
tetragonal D4h Tetragonal.svg Tetragonal-body-centered.svg
hexagonal rhombohedralD3d Rhombohedral.svg
hexagonalD6h Hexagonal latticeFRONT.svg
cubic Oh Cubic.svg Cubic-body-centered.svg Cubic-face-centered.svg

In geometry and crystallography, a Bravais lattice is a category of translative symmetry groups (also known as lattices) in three directions.

Such symmetry groups consist of translations by vectors of the form

R = n1a1 + n2a2 + n3a3,

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 the space group of the lattice itself, viewed as a collection of points; there are 14 Bravais lattices in three dimensions; each belongs to one lattice system only. They[ clarification needed ] represent the maximum symmetry a structure with the given translational symmetry can have.

All crystalline materials (not including quasicrystals) must, by definition, fit into one of these arrangements.

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 in 1842, who found that there were 15 Bravais lattices. This was corrected to 14 by A. Bravais in 1848.

In four-dimensional space

‌The four-dimensional unit cell is defined by four edge lengths (a, b, c, d) and six interaxial angles (α, β, γ, δ, ε, ζ). The following conditions for the lattice parameters define 23 crystal families

Crystal families in 4D space
No.FamilyEdge lengthsInteraxial angles
1Hexaclinicabcdαβγδεζ ≠ 90°
2Triclinicabcdαβγ ≠ 90°
δ = ε = ζ = 90°
3Diclinicabcdα ≠ 90°
β = γ = δ = ε = 90°
ζ ≠ 90°
4Monoclinicabcdα ≠ 90°
β = γ = δ = ε = ζ = 90°
5Orthogonalabcdα = β = γ = δ = ε = ζ = 90°
6Tetragonal monoclinicab = cdα ≠ 90°
β = γ = δ = ε = ζ = 90°
7Hexagonal monoclinicab = cdα ≠ 90°
β = γ = δ = ε = 90°
ζ = 120°
8Ditetragonal diclinica = db = cα = ζ = 90°
β = ε ≠ 90°
γ ≠ 90°
δ = 180° − γ
9Ditrigonal (dihexagonal) diclinica = db = cα = ζ = 120°
β = ε ≠ 90°
γδ ≠ 90°
cos δ = cos β − cos γ
10Tetragonal orthogonalab = cdα = β = γ = δ = ε = ζ = 90°
11Hexagonal orthogonalab = cdα = β = γ = δ = ε = 90°, ζ = 120°
12Ditetragonal monoclinica = db = cα = γ = δ = ζ = 90°
β = ε ≠ 90°
13Ditrigonal (dihexagonal) monoclinica = db = cα = ζ = 120°
β = ε ≠ 90°
γ = δ ≠ 90°
cos γ = −1/2cos β
14Ditetragonal orthogonala = db = cα = β = γ = δ = ε = ζ = 90°
15Hexagonal tetragonala = db = cα = β = γ = δ = ε = 90°
ζ = 120°
16Dihexagonal orthogonala = db = cα = ζ = 120°
β = γ = δ = ε = 90°
17Cubic orthogonala = b = cdα = β = γ = δ = ε = ζ = 90°
18Octagonala = b = c = dα = γ = ζ ≠ 90°
β = ε = 90°
δ = 180° − α
19Decagonala = b = c = dα = γ = ζβ = δ = ε
cos β = −1/2 − cos α
20Dodecagonala = b = c = dα = ζ = 90°
β = ε = 120°
γ = δ ≠ 90°
21Diisohexagonal orthogonala = b = c = dα = ζ = 120°
β = γ = δ = ε = 90°
22Icosagonal (icosahedral)a = b = c = dα = β = γ = δ = ε = ζ
cos α = −1/4
23Hypercubica = b = c = dα = β = γ = δ = ε = ζ = 90°

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 an asterisk. The number of enantiomorphic pairs are given in parentheses. Here the term "enantiomorphic" has a different meaning than in the table for three-dimensional crystal classes. The latter means, that enantiomorphic point groups describe chiral (enantiomorphic) structures. In the current table, "enantiomorphic" means that a group itself (considered as a 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.

Crystal systems in 4D space
No. of
crystal family
Crystal familyCrystal systemNo. of
crystal system
Point groupsSpace groupsBravais latticesLattice system
IHexaclinic1221Hexaclinic P
IITriclinic23132Triclinic P, S
IIIDiclinic32123Diclinic P, S, D
IVMonoclinic442076Monoclinic P, S, S, I, D, F
VOrthogonalNon-axial orthogonal5221Orthogonal KU
1128Orthogonal P, S, I, Z, D, F, G, U
Axial orthogonal63887
VITetragonal monoclinic77882Tetragonal monoclinic P, I
VIIHexagonal monoclinicTrigonal monoclinic8591Hexagonal monoclinic R
151Hexagonal monoclinic P
Hexagonal monoclinic9725
VIIIDitetragonal diclinic*101 (+1)1 (+1)1 (+1)Ditetragonal diclinic P*
IXDitrigonal diclinic*112 (+2)2 (+2)1 (+1)Ditrigonal diclinic P*
XTetragonal orthogonalInverse tetragonal orthogonal12571Tetragonal orthogonal KG
3515Tetragonal orthogonal P, S, I, Z, G
Proper tetragonal orthogonal13101312
XIHexagonal orthogonalTrigonal orthogonal1410812Hexagonal orthogonal R, RS
1502Hexagonal orthogonal P, S
Hexagonal orthogonal1512240
XIIDitetragonal monoclinic*161 (+1)6 (+6)3 (+3)Ditetragonal monoclinic P*, S*, D*
XIIIDitrigonal monoclinic*172 (+2)5 (+5)2 (+2)Ditrigonal monoclinic P*, RR*
XIVDitetragonal orthogonalCrypto-ditetragonal orthogonal185101Ditetragonal orthogonal D
165 (+2)2Ditetragonal orthogonal P, Z
Ditetragonal orthogonal196127
XVHexagonal tetragonal20221081Hexagonal tetragonal P
XVIDihexagonal orthogonalCrypto-ditrigonal orthogonal*214 (+4)5 (+5)1 (+1)Dihexagonal orthogonal G*
5 (+5)1Dihexagonal orthogonal P
Dihexagonal orthogonal231120
Ditrigonal orthogonal221141
161Dihexagonal orthogonal RR
XVIICubic orthogonalSimple cubic orthogonal24591Cubic orthogonal KU
965Cubic orthogonal P, I, Z, F, U
Complex cubic orthogonal2511366
XVIIIOctagonal*262 (+2)3 (+3)1 (+1)Octagonal P*
XIXDecagonal27451Decagonal P
XXDodecagonal*282 (+2)2 (+2)1 (+1)Dodecagonal P*
XXIDiisohexagonal orthogonalSimple diisohexagonal orthogonal299 (+2)19 (+5)1Diisohexagonal orthogonal RR
19 (+3)1Diisohexagonal orthogonal P
Complex diisohexagonal orthogonal3013 (+8)15 (+9)
XXIIIcosagonal317202Icosagonal P, SN
XXIIIHypercubicOctagonal hypercubic3221 (+8)73 (+15)1Hypercubic P
107 (+28)1Hypercubic Z
Dodecagonal hypercubic3316 (+12)25 (+20)
Total23 (+6)33 (+7)227 (+44)4783 (+111)64 (+10)33 (+7)

See also

Related Research Articles

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Pleochroism optical phenomenon in which a substance appears to be different colors when observed at different angles, especially with polarized light

Pleochroism is an optical phenomenon in which a substance has different colors when observed at different angles, especially with polarized light.

Tetragonal crystal system Lattice point group

In crystallography, the tetragonal crystal system is one of the 7 crystal systems. Tetragonal crystal lattices result from stretching a cubic lattice along one of its lattice vectors, so that the cube becomes a rectangular prism with a square base and height.

Space group symmetry group of a configuration in space

In mathematics, physics and chemistry, a space group is the symmetry group of a configuration in space, usually in three dimensions. In three dimensions, there are 219 distinct types, or 230 if chiral copies are considered distinct. Space groups are also studied in dimensions other than 3 where they are sometimes called Bieberbach groups, and are discrete cocompact groups of isometries of an oriented Euclidean space.

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Monoclinic crystal system one of the 7 crystal systems in crystallography

In crystallography, the monoclinic crystal system is one of the seven crystal systems. A crystal system is described by three vectors. In the monoclinic system, the crystal is described by vectors of unequal lengths, as in the orthorhombic system. They form a rectangular prism with a parallelogram as its base. Hence two pairs of vectors are perpendicular, while the third pair makes an angle other than 90°.

In crystallography, the orthorhombic crystal system is one of the 7 crystal systems. Orthorhombic lattices result from stretching a cubic lattice along two of its orthogonal pairs by two different factors, resulting in a rectangular prism with a rectangular base and height (c), such that a, b, and c are distinct. All three bases intersect at 90° angles, so the three lattice vectors remain mutually orthogonal.

In geometry and crystallography, a Bravais lattice, named after Auguste Bravais (1850), is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by:

In crystallography, a crystallographic point group is a set of symmetry operations, corresponding to one of the point groups in three dimensions, such that each operation would leave the structure of a crystal unchanged i.e. the same kinds of atoms would be placed in similar positions as before the transformation. For example, in a primitive cubic crystal system, a rotation of the unit cell by 90 degree around an axis that is perpendicular to two parallel faces of the cube, intersecting at its center, is a symmetry operation that projects each atom to the location of one of its neighbor leaving the overall structure of the crystal unaffected.

Triclinic crystal system lattice point group

In crystallography, the tricliniccrystal system is one of the 7 crystal systems. A crystal system is described by three basis vectors. In the triclinic system, the crystal is described by vectors of unequal length, as in the orthorhombic system. In addition, the angles between these vectors must all be different and may not include 90°.

Cubic crystal system lattice point group

In crystallography, the cubiccrystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals.

Crystal twinning when two separate crystals share some of the same crystal lattice points in a symmetrical manner

Crystal twinning occurs when two separate crystals share some of the same crystal lattice points in a symmetrical manner. The result is an intergrowth of two separate crystals in a variety of specific configurations. The surface along which the lattice points are shared in twinned crystals is called a composition surface or twin plane.

Vauxite phosphate mineral

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Centrosymmetry

In crystallography, a point group which contains an inversion center as one of its symmetry elements is centrosymmetric. In such a point group, for every point in the unit cell there is an indistinguishable point. Such point groups are also said to have inversion symmetry. Point reflection is a similar term used in geometry. Crystals with an inversion center cannot display certain properties, such as the piezoelectric effect.

In geometry, Hermann–Mauguin notation is used to represent the symmetry elements in point groups, plane groups and space groups. It is named after the German crystallographer Carl Hermann and the French mineralogist Charles-Victor Mauguin. This notation is sometimes called international notation, because it was adopted as standard by the International Tables For Crystallography since their first edition in 1935.

The Pearson symbol, or Pearson notation, is used in crystallography as a means of describing a crystal structure, and was originated by W.B. Pearson. The symbol is made up of two letters followed by a number. For example:

Hexagonal crystal family lattice point group

In crystallography, the hexagonal crystal family is one of the six crystal families, which includes two crystal systems and two lattice systems.

Californium compounds any chemical compound having at least one californium atom

Few compounds of californium have been made and studied. The only californium ion that is stable in aqueous solutions is the californium(III) cation. The other two oxidation states are IV (strong oxidizing agents) and II (strong reducing agents). The element forms a water-soluble chloride, nitrate, perchlorate, and sulfate and is precipitated as a fluoride, oxalate or hydroxide. If problems of availability of the element could be overcome, then CfBr2 and CfI2 would likely be stable.

Berkelium compounds Any chemical compound having at least one berkelium atom

Berkelium forms a number of chemical compounds, where it normally exists in an oxidation state of +3 or +4, and behaves similarly to its lanthanide analogue, terbium. Like all actinides, berkelium easily dissolves in various aqueous inorganic acids, liberating gaseous hydrogen and converting into the trivalent oxidation state. This trivalent state is the most stable, especially in aqueous solutions, but tetravalent berkelium compounds are also known. The existence of divalent berkelium salts is uncertain and has only been reported in mixed lanthanum chloride-strontium chloride melts. Aqueous solutions of Bk3+ ions are green in most acids. The color of the Bk4+ ions is yellow in hydrochloric acid and orange-yellow in sulfuric acid. Berkelium does not react rapidly with oxygen at room temperature, possibly due to the formation of a protective oxide surface layer; however, it reacts with molten metals, hydrogen, halogens, chalcogens and pnictogens to form various binary compounds. Berkelium can also form several organometallic compounds.

Stibarsen native element mineral

Stibarsen or allemontite is a natural form of arsenic antimonide (AsSb) or antimony arsenide (SbAs). The name stibarsen is derived from Latin stibium (antimony) and arsenic, whereas allemonite refers to the locality Allemont in France where the mineral was discovered. It is found in veins at Allemont, Isère, France; Valtellina, Italy; and the Comstock Lode, Nevada; and in a lithium pegmatites at Varuträsk, Sweden. Stibarsen is often mixed with pure arsenic or antimony, and the original description in 1941 proposed to use stibarsen for AsSb and allemontite for the mixtures. Since 1982, the International Mineralogical Association considers stibarsen as the correct mineral name.

References

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  2. Hahn (2002) , p. 804
  3. 1 2 Whittaker, E. J. W. (1985). An Atlas of Hyperstereograms of the Four-Dimensional Crystal Classes. Oxford & New York: Clarendon Press.
  4. 1 2 Brown, H.; Bülow, R.; Neubüser, J.; Wondratschek, H.; Zassenhaus, H. (1978). Crystallographic Groups of Four-Dimensional Space. New York: Wiley.