| Crystal system | Trigonal | Hexagonal | |
|---|---|---|---|
| Lattice system |  Rhombohedral |  Hexagonal | |
| Example | .jpg) Dolomite (white) |  α-Quartz |  Beryl |
Incrystallography, thehexagonal crystal family is one of the sixcrystal families, which includes two crystal systems (hexagonal andtrigonal) and two lattice systems (hexagonal andrhombohedral). While commonly confused, the trigonal crystal system and the rhombohedral lattice system are not equivalent (see sectioncrystal systems below).[1] In particular, there are crystals that have trigonal symmetry but belong to the hexagonal lattice (such as α-quartz).
The hexagonal crystal family consists of the 12 point groups such that at least one of their space groups has the hexagonal lattice as underlying lattice, and is the union of the hexagonal crystal system and the trigonal crystal system.[2] There are 52 space groups associated with it, which are exactly those whoseBravais lattice is either hexagonal or rhombohedral.
The hexagonal crystal family consists of twolattice systems: hexagonal and rhombohedral. Each lattice system consists of oneBravais lattice.
| Bravais lattice | Hexagonal | Rhombohedral |
|---|---|---|
| Pearson symbol | hP | hR |
| Hexagonal unit cell |  |  |
| Rhombohedral unit cell |  | |
In the hexagonal family, the crystal is conventionally described by a rightrhombicprism unit cell with two equal axes (a bya), an included angle of 120° (γ) and a height (c, which can be different froma) perpendicular to the two base axes.
The hexagonal unit cell for the rhombohedral Bravais lattice is the R-centered cell, consisting of two additional lattice points which occupy one body diagonal of the unit cell. There are two ways to do this, which can be thought of as two notations which represent the same structure. In the usual so-called obverse setting, the additional lattice points are at coordinates (2⁄3,1⁄3,1⁄3) and (1⁄3,2⁄3,2⁄3), whereas in the alternative reverse setting they are at the coordinates (1⁄3,2⁄3,1⁄3) and (2⁄3,1⁄3,2⁄3).[3] In either case, there are 3 lattice points per unit cell in total and the lattice is non-primitive.
The Bravais lattices in the hexagonal crystal family can also be described by rhombohedral axes.[4] The unit cell is arhombohedron (which gives the name for the rhombohedral lattice). This is a unit cell with parametersa =b =c;α =β =γ ≠ 90°.[5] In practice, the hexagonal description is more commonly used because it is easier to deal with a coordinate system with two 90° angles. However, the rhombohedral axes are often shown (for the rhombohedral lattice) in textbooks because this cell reveals the3m symmetry of the crystal lattice.
The rhombohedral unit cell for the hexagonal Bravais lattice is the D-centered[1] cell, consisting of two additional lattice points which occupy one body diagonal of the unit cell with coordinates (1⁄3,1⁄3,1⁄3) and (2⁄3,2⁄3,2⁄3). However, such a description is rarely used.
| Crystal system | Required symmetries of point group | Point groups | Space groups | Bravais lattices | Lattice system |
|---|---|---|---|---|---|
| Trigonal | 1 threefold axis of rotation | 5 | 7 | 1 | Rhombohedral |
| 18 | 1 | Hexagonal | |||
| Hexagonal | 1 sixfold axis of rotation | 7 | 27 |
The hexagonal crystal family consists of twocrystal systems: trigonal and hexagonal. A crystal system is a set ofpoint groups in which the point groups themselves and their correspondingspace groups are assigned to alattice system (see table inCrystal system#Crystal classes).
The trigonal crystal system consists of the 5 point groups that have a single three-fold rotation axis, which includes space groups 143 to 167. These 5 point groups have 7 corresponding space groups (denoted by R) assigned to the rhombohedral lattice system and 18 corresponding space groups (denoted by P) assigned to the hexagonal lattice system. Hence, the trigonal crystal system is the only crystal system whose point groups have more than one lattice system associated with their space groups.
The hexagonal crystal system consists of the 7 point groups that have a single six-fold rotation axis. These 7 point groups have 27 space groups (168 to 194), all of which are assigned to the hexagonal lattice system.
The 5 point groups in this crystal system are listed below, with their international number and notation, their space groups in name and example crystals.[6][7][8]
| Space group no. | Point group | Type | Examples | Space groups | |||||
|---|---|---|---|---|---|---|---|---|---|
| Name[1] | Intl | Schoen. | Orb. | Cox. | Hexagonal | Rhombohedral | |||
| 143–146 | Trigonal pyramidal | 3 | C3 | 33 | [3]+ | enantiomorphicpolar | carlinite,jarosite | P3, P31, P32 | R3 |
| 147–148 | Rhombohedral | 3 | C3i (S6) | 3× | [2+,6+] | centrosymmetric | dolomite,ilmenite | P3 | R3 |
| 149–155 | Trigonal trapezohedral | 32 | D3 | 223 | [2,3]+ | enantiomorphic | abhurite, alpha-quartz (152, 154),cinnabar | P312, P321, P3112, P3121, P3212, P3221 | R32 |
| 156–161 | Ditrigonal pyramidal | 3m | C3v | *33 | [3] | polar | schorl,cerite,tourmaline,alunite,lithium tantalate | P3m1, P31m, P3c1, P31c | R3m, R3c |
| 162–167 | Ditrigonal scalenohedral | 3m | D3d | 2*3 | [2+,6] | centrosymmetric | antimony,hematite,corundum,calcite,bismuth | P31m, P31c, P3m1, P3c1 | R3m, R3c |
The 7point groups (crystal classes) in this crystal system are listed below, followed by their representations inHermann–Mauguin or international notation andSchoenflies notation, andmineral examples, if they exist.[2][9]
| Space group no. | Point group | Type | Examples | Space groups | ||||
|---|---|---|---|---|---|---|---|---|
| Name[1] | Intl | Schoen. | Orb. | Cox. | ||||
| 168–173 | Hexagonal pyramidal | 6 | C6 | 66 | [6]+ | enantiomorphicpolar | nepheline,cancrinite | P6, P61, P65, P62, P64, P63 |
| 174 | Trigonal dipyramidal | 6 | C3h | 3* | [2,3+] | cesanite,laurelite | P6 | |
| 175–176 | Hexagonal dipyramidal | 6/m | C6h | 6* | [2,6+] | centrosymmetric | apatite,vanadinite | P6/m, P63/m |
| 177–182 | Hexagonal trapezohedral | 622 | D6 | 226 | [2,6]+ | enantiomorphic | kalsilite, beta-quartz | P622, P6122, P6522, P6222, P6422, P6322 |
| 183–186 | Dihexagonal pyramidal | 6mm | C6v | *66 | [6] | polar | greenockite,wurtzite[10] | P6mm, P6cc, P63cm, P63mc |
| 187–190 | Ditrigonal dipyramidal | 6m2 | D3h | *223 | [2,3] | benitoite | P6m2, P6c2, P62m, P62c | |
| 191–194 | Dihexagonal dipyramidal | 6/mmm | D6h | *226 | [2,6] | centrosymmetric | beryl | P6/mmm, P6/mcc, P63/mcm, P63/mmc |
The unit cell volume is given bya2c•sin(60°)
Hexagonal close packed (hcp) is one of the two simple types of atomic packing with the highest density, the other being the face-centered cubic (fcc). However, unlike the fcc, it is not a Bravais lattice, as there are two nonequivalent sets of lattice points. Instead, it can be constructed from the hexagonal Bravais lattice by using a two-atom motif (the additional atom at about (2⁄3, 1⁄3, 1⁄2)) associated with each lattice point.[11]
Compounds that consist of more than one element (e.g.binary compounds) often have crystal structures based on the hexagonal crystal family. Some of the more common ones are listed here. These structures can be viewed as two or more interpenetrating sublattices where each sublattice occupies theinterstitial sites of the others.
The wurtzite crystal structure is referred to by theStrukturbericht designation B4 and thePearson symbol hP4. The correspondingspace group isNo. 186 (in International Union of Crystallography classification) orP63mc (inHermann–Mauguin notation). The Hermann-Mauguin symbols inP63mc can be read as follows:[13]
Among the compounds that can take the wurtzite structure arewurtzite itself (ZnS with up to 8%iron instead ofzinc),silver iodide (AgI),zinc oxide (ZnO),cadmium sulfide (CdS),cadmium selenide (CdSe),silicon carbide (α-SiC),gallium nitride (GaN),aluminium nitride (AlN),boron nitride (w-BN) and othersemiconductors.[14] In most of these compounds, wurtzite is not the favored form of the bulk crystal, but the structure can be favored in some nanocrystal forms of the material.
In materials with more than one crystal structure, the prefix "w-" is sometimes added to theempirical formula to denote the wurtzite crystal structure, as inw-BN.
Each of the two individual atom types forms a sublattice which ishexagonal close-packed (HCP-type). When viewed all together, the atomic positions are the same as inlonsdaleite (hexagonaldiamond). Each atom istetrahedrally coordinated. The structure can also be described as an HCP lattice of zinc with sulfur atoms occupying half of thetetrahedral voids or vice versa.
The wurtzite structure isnon-centrosymmetric (i.e., lacksinversion symmetry). Due to this, wurtzite crystals can (and generally do) have properties such aspiezoelectricity andpyroelectricity, which centrosymmetric crystals lack.[citation needed]
The nickel arsenide structure consists of two interpenetrating sublattices: a primitive hexagonal nickel sublattice and ahexagonal close-packed arsenic sublattice. Each nickel atom isoctahedrallycoordinated to six arsenic atoms, while each arsenic atom istrigonal prismatically coordinated to six nickel atoms.[15] The structure can also be described as an HCP lattice of arsenic with nickel occupying eachoctahedral void.
Compounds adopting the NiAs structure are generally thechalcogenides,arsenides,antimonides andbismuthides oftransition metals.[citation needed]
The following are the members of the nickeline group:[16]
There is only one hexagonal Bravais lattice in two dimensions: the hexagonal lattice.
| Bravais lattice | Hexagonal |
|---|---|
| Pearson symbol | hp |
| Unit cell |  |
Media related toHexagonal lattices at Wikimedia Commons| Seven 3D systems |
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| Four 2D systems | |
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