Hexagonal crystal family#Wurtzite structure

The hexagonal crystal family consists of two lattice systems: hexagonal and rhombohedral. Each lattice system consists of one Bravais lattice.

File:RhombohedralR.svg

In the hexagonal family, the crystal is conventionally described by a right rhombic prism unit cell with two equal axes (a by a), an included angle of 120° (γ) and a height (c, which can be different from a) 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}}).{{cite book|title=Mathematical Techniques in Crystallography and Materials Science|author=Edward Prince|year=2004|publisher=Springer Science & Business Media|page=41}} 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.{{cite web|url=http://img.chem.ucl.ac.uk/sgp/medium/sgp.htm|title=Medium-Resolution Space Group Diagrams and Tables|website=img.chem.ucl.ac.uk}} The unit cell is a rhombohedron (which gives the name for the rhombohedral lattice). This is a unit cell with parameters a = b = c; α = β = γ ≠ 90°.{{cite book|last1=Ashcroft|first1=Neil W.|last2=Mermin|first2=N. David|date=1976|title=Solid State Physics|edition=1st|page=[https://archive.org/details/solidstatephysic00ashc/page/119 119]|publisher=Holt, Rinehart and Winston |isbn=0-03-083993-9|url=https://archive.org/details/solidstatephysic00ashc/page/119}} 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 the {{overline|3}}m symmetry of the crystal lattice.

The rhombohedral unit cell for the hexagonal Bravais lattice is the D-centered{{r|hahn}} 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.

The hexagonal crystal family consists of two crystal systems: trigonal and hexagonal. A crystal system is a set of point groups in which the point groups themselves and their corresponding space groups are assigned to a lattice system (see table in Crystal 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.{{cite book|url=https://books.google.com/books?id=co07Qn9HzP0C&pg=PA62|page=62|title=A Field Guide to Rocks and Minerals|first1=Frederick H.|last1=Pough|first2=Roger Tory|last2=Peterson|publisher=Houghton Mifflin Harcourt|year=1998|isbn=0-395-91096-X}}{{cite book|last1=Hurlbut|first1=Cornelius S.|last2=Klein|first2=Cornelis|date=1985|title=Manual of Mineralogy|edition=20th|pages=[https://archive.org/details/manualofmineralo00klei/page/78 78–89]|publisher=Wiley |isbn=0-471-80580-7|url=https://archive.org/details/manualofmineralo00klei/page/78}}{{cite web|url=http://webmineral.com/crystall.shtml|title=Crystallography and Minerals Arranged by Crystal Form|website=Webmineral}}

The 7 point groups (crystal classes) in this crystal system are listed below, followed by their representations in Hermann–Mauguin or international notation and Schoenflies notation, and mineral examples, if they exist.{{cite web|url=http://webmineral.com/crystall.shtml |title=Crystallography |publisher=Webmineral.com |access-date=2014-08-03}}

The unit cell volume is given by a²c•sin(60°)

{{main|Close-packing of equal spheres}}

File:Hexagonal close packed.svg

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.{{Cite book |url=https://books.google.com/books?id=lTRRAAAAMAAJ |title=An introduction to mathematical crystallography |last=Jaswon |first=Maurice Aaron |date=1965-01-01 |publisher=American Elsevier Pub. Co. |language=en}}

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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 the interstitial sites of the others.

{{Category see also|Wurtzite structure type}}

File:Wurtzite cellGIF.gif

Image:Wurtzite-unit-cell-3D-balls.png{{Citation needed|date=August 2018}}]]

Image:Wurtzite polyhedra.png

The wurtzite crystal structure is referred to by the Strukturbericht designation B4 and the Pearson symbol hP4. The corresponding space group is No. 186 (in International Union of Crystallography classification) or P6₃mc (in Hermann–Mauguin notation). The Hermann-Mauguin symbols in P6₃mc can be read as follows:{{cite book |last1=Hitchcock |first1=Peter B |title=International tables for crystallography volume A. |date=1988 | url=https://www.springer.com/series/6995}}

• 6₃.. : a six fold screw rotation around the c-axis
• .m. : a mirror plane with normal {100}
• ..c : glide plane in the c-directions with normal {120}.

Among the compounds that can take the wurtzite structure are wurtzite itself (ZnS with up to 8% iron instead of zinc), 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 other semiconductors.{{Cite journal|last1=Togo|first1=Atsushi|last2=Chaput|first2=Laurent|last3=Tanaka|first3=Isao|date=2015-03-20|title=Distributions of phonon lifetimes in Brillouin zones|journal=Physical Review B|volume=91|issue=9|pages=094306|doi=10.1103/PhysRevB.91.094306|arxiv=1501.00691|bibcode=2015PhRvB..91i4306T |s2cid=118851924 }} 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 the empirical formula to denote the wurtzite crystal structure, as in w-BN.

Each of the two individual atom types forms a sublattice which is hexagonal close-packed (HCP-type). When viewed all together, the atomic positions are the same as in lonsdaleite (hexagonal diamond). Each atom is tetrahedrally coordinated. The structure can also be described as an HCP lattice of zinc with sulfur atoms occupying half of the tetrahedral voids or vice versa.

The wurtzite structure is non-centrosymmetric (i.e., lacks inversion symmetry). Due to this, wurtzite crystals can (and generally do) have properties such as piezoelectricity and pyroelectricity, which centrosymmetric crystals lack.{{citation needed|date=January 2012}}

{{Category see also|Nickel arsenide structure type}}

The nickel arsenide structure consists of two interpenetrating sublattices: a primitive hexagonal nickel sublattice and a hexagonal close-packed arsenic sublattice. Each nickel atom is octahedrally coordinated to six arsenic atoms, while each arsenic atom is trigonal prismatically coordinated to six nickel atoms.Inorganic Chemistry by Duward Shriver and Peter Atkins, 3rd Edition, W.H. Freeman and Company, 1999, pp.47,48. The structure can also be described as an HCP lattice of arsenic with nickel occupying each octahedral void.

Compounds adopting the NiAs structure are generally the chalcogenides, arsenides, antimonides and bismuthides of transition metals. {{Citation needed|date=November 2019}}

Image:Nickel-arsenide-3D-unit-cell.png

The following are the members of the nickeline group:http://www.mindat.org/min-2901.html Mindat.org

• Achavalite: {{chem2|FeSe}}
• Breithauptite: {{chem2|NiSb}}
• Freboldite: {{chem2|CoSe}}
• Kotulskite: {{chem2|Pd(Te,Bi)}}
• Langistite: {{chem2|(Co,Ni)As}}
• Nickeline: {{chem2|NiAs}}
• Sobolevskite: {{chem2|Pd(Bi,Te)}}
• Sudburyite: {{chem2|(Pd,Ni)Sb}}

{{main|Hexagonal lattice}}

There is only one hexagonal Bravais lattice in two dimensions: the hexagonal lattice.

• Close-packing
• Crystal structure

{{reflist}}

• {{Commonscatinline|Hexagonal lattices}}
• [http://webmineral.com/ Mineralogy database]

{{Crystal systems}}

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