In
crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wor ...
, the hexagonal crystal family is one of the six
crystal families, which includes two crystal systems (hexagonal and trigonal) and two lattice systems (hexagonal and rhombohedral). While commonly confused, the trigonal crystal system and the rhombohedral lattice system are not equivalent (see section
crystal systems
In crystallography, a crystal system is a set of point groups (a group of geometric symmetries with at least one fixed point). A lattice system is a set of Bravais lattices. Space groups are classified into crystal systems according to their poin ...
below).
In particular, there are crystals that have trigonal symmetry but belong to the hexagonal lattice (such as α-
quartz
Quartz is a hard, crystalline mineral composed of silica (silicon dioxide). The atoms are linked in a continuous framework of SiO4 silicon-oxygen tetrahedra, with each oxygen being shared between two tetrahedra, giving an overall chemical form ...
).
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.
There are 52 space groups associated with it, which are exactly those whose
Bravais lattice
In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by
: \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n_ ...
is either hexagonal or rhombohedral.
__TOC__
Lattice systems
The hexagonal crystal family consists of two
lattice system
In crystallography, a crystal system is a set of point groups (a group of geometric symmetries with at least one fixed point). A lattice system is a set of Bravais lattices. Space groups are classified into crystal systems according to their poi ...
s: hexagonal and rhombohedral. Each lattice system consists of one
Bravais lattice
In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by
: \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n_ ...
.
![RhombohedralR](https://upload.wikimedia.org/wikipedia/commons/d/d2/RhombohedralR.svg)
In the hexagonal family, the crystal is conventionally described by a right
rhombic prism
Prism usually refers to:
* Prism (optics), a transparent optical component with flat surfaces that refract light
* Prism (geometry), a kind of polyhedron
Prism may also refer to:
Science and mathematics
* Prism (geology), a type of sedimentary ...
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 (, , ) and (, , ), whereas in the alternative reverse setting they are at the coordinates (,,) and (,,). 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. The unit cell is a
rhombohedron
In geometry, a rhombohedron (also called a rhombic hexahedron or, inaccurately, a rhomboid
Traditionally, in two-dimensional geometry, a rhomboid is a parallelogram in which adjacent sides are of unequal lengths and angles are non-right ang ...
(which gives the name for the rhombohedral lattice). This is a unit cell with parameters ''a'' = ''b'' = ''c''; ''α'' = ''β'' = ''γ'' ≠ 90°.
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 ''m'' symmetry of the crystal lattice.
The rhombohedral unit cell for the hexagonal Bravais lattice is the D-centered cell, consisting of two additional lattice points which occupy one body diagonal of the unit cell with coordinates (, , ) and (, , ). However, such a description is rarely used.
Crystal systems
The hexagonal crystal family consists of two
crystal system
In crystallography, a crystal system is a set of point groups (a group of geometric symmetries with at least one fixed point). A lattice system is a set of Bravais lattices. Space groups are classified into crystal systems according to their po ...
s: trigonal and hexagonal. A crystal system is a set of
point group
In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every p ...
s in which the point groups themselves and their corresponding
space group
In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it unchan ...
s are assigned to a
lattice system
In crystallography, a crystal system is a set of point groups (a group of geometric symmetries with at least one fixed point). A lattice system is a set of Bravais lattices. Space groups are classified into crystal systems according to their poi ...
(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.
Trigonal crystal 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.
Hexagonal crystal system
The 7
point groups
In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every p ...
(''crystal classes'') in this crystal system are listed below, followed by their representations in
Hermann–Mauguin or international notation and
Schoenflies notation The Schoenflies (or Schönflies) notation, named after the German mathematician Arthur Moritz Schoenflies, is a notation primarily used to specify point groups in three dimensions. Because a point group alone is completely adequate to describe the ...
, and
mineral
In geology and mineralogy, a mineral or mineral species is, broadly speaking, a solid chemical compound with a fairly well-defined chemical composition and a specific crystal structure that occurs naturally in pure form.John P. Rafferty, ed. ( ...
examples, if they exist.
[
]
Hexagonal close packed
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 (, , )) associated with each lattice point.
Multi-element structures
Compounds that consist of more than one element (e.g. binary compound
In materials chemistry, a binary phase or binary compound is a chemical compound containing two different elements. Some binary phase compounds are molecular, e.g. carbon tetrachloride (CCl4). More typically binary phase refers to extended soli ...
s) 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 site
In crystallography, interstitial sites, holes or voids are the empty space that exists between the packing of atoms (spheres) in the crystal structure.
The holes are easy to see if you try to pack circles together; no matter how close you get ...
s of the others.
Wurtzite structure
The wurtzite crystal structure is referred to by the Strukturbericht designation In crystallography, a Strukturbericht designation or Strukturbericht type is a system of detailed crystal structure classification by analogy to another known structure. The designations were intended to be comprehensive but are mainly used as suppl ...
B4 and the Pearson symbol
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:
* Diamond structure ...
hP4. The corresponding space group
In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it unchan ...
is No. 186 (in International Union of Crystallography classification) or P63mc (in Hermann–Mauguin notation
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 (who introduced it in 1928) and the French mineralogis ...
). The Hermann-Mauguin symbols in P63mc can be read as follows:
* 63.. : a six fold screw rotation around the c-axis
* .m. : a mirror plane with normal
* ..c : glide plane in the c-directions with normal .
Among the compounds that can take the wurtzite structure are wurtzite
Wurtzite is a zinc and iron sulfide mineral with the chemical formula , a less frequently encountered Polymorphism (materials science), structural polymorph form of sphalerite. The iron content is variable up to eight percent.Palache, Charles, Har ...
itself (ZnS ZNS or ZnS may refer to:
* Zinc sulfide or zinc sulphide (ZnS), a chemical compound
* ZNS-TV, a national radio and TV broadcaster operated by the state-owned Broadcasting Corporation of The Bahamas
** ZNS-1
ZNS-1 (branded as ''Radio Bahamas'') i ...
with up to 8% iron
Iron () is a chemical element with symbol Fe (from la, ferrum) and atomic number 26. It is a metal that belongs to the first transition series and group 8 of the periodic table. It is, by mass, the most common element on Earth, right in f ...
instead of zinc
Zinc is a chemical element with the symbol Zn and atomic number 30. Zinc is a slightly brittle metal at room temperature and has a shiny-greyish appearance when oxidation is removed. It is the first element in group 12 (IIB) of the periodi ...
), silver iodide
Silver iodide is an inorganic compound with the formula Ag I. The compound is a bright yellow solid, but samples almost always contain impurities of metallic silver that give a gray coloration. The silver contamination arises because AgI is hig ...
(AgI), zinc oxide
Zinc oxide is an inorganic compound with the formula . It is a white powder that is insoluble in water. ZnO is used as an additive in numerous materials and products including cosmetics, food supplements, rubbers, plastics, ceramics, glass, cemen ...
(ZnO), cadmium sulfide
Cadmium sulfide is the inorganic compound with the formula CdS. Cadmium sulfide is a yellow solid.Egon Wiberg, Arnold Frederick Holleman (2001''Inorganic Chemistry'' Elsevier It occurs in nature with two different crystal structures as the rare m ...
(CdS), cadmium selenide
Cadmium selenide is an inorganic compound with the formula Cadmium, CdSelenide, Se. It is a black to red-black solid that is classified as a II-VI semiconductor of the n-type semiconductor, n-type. Much of the current research on this compound i ...
(CdSe), silicon carbide
Silicon carbide (SiC), also known as carborundum (), is a hard chemical compound containing silicon and carbon. A semiconductor, it occurs in nature as the extremely rare mineral moissanite, but has been mass-produced as a powder and crystal sin ...
(α-SiC), gallium nitride
Gallium nitride () is a binary III/ V direct bandgap semiconductor commonly used in blue light-emitting diodes since the 1990s. The compound is a very hard material that has a Wurtzite crystal structure. Its wide band gap of 3.4 eV affords it ...
(GaN), aluminium nitride
Aluminium nitride ( Al N) is a solid nitride of aluminium. It has a high thermal conductivity of up to 321 W/(m·K) and is an electrical insulator. Its wurtzite phase (w-AlN) has a band gap of ~6 eV at room temperature and has a potenti ...
(AlN), boron nitride
Boron nitride is a thermally and chemically resistant refractory compound of boron and nitrogen with the chemical formula BN. It exists in various crystalline forms that are isoelectronic to a similarly structured carbon lattice. The hexagonal ...
(w-BN) and other semiconductors
A semiconductor is a material which has an electrical resistivity and conductivity, electrical conductivity value falling between that of a electrical conductor, conductor, such as copper, and an insulator (electricity), insulator, such as glas ...
. 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
In chemistry, the empirical formula of a chemical compound is the simplest whole number ratio of atoms present in a compound. A simple example of this concept is that the empirical formula of sulfur monoxide, or SO, would simply be SO, as is th ...
to denote the wurtzite crystal structure, as in w-BN
Boron nitride is a thermally and chemically resistant refractory compound of boron and nitrogen with the chemical formula BN. It exists in various Polymorphism (materials science), crystalline forms that are isoelectronic to a similarly structure ...
.
Each of the two individual atom types forms a sublattice which is hexagonal close-packed
In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement (or lattice). Carl Friedrich Gauss proved that the highest average density – that is, the greatest fraction of space occu ...
(HCP-type). When viewed all together, the atomic positions are the same as in lonsdaleite
Lonsdaleite (named in honour of Kathleen Lonsdale), also called hexagonal diamond in reference to the crystal structure, is an allotrope of carbon with a hexagonal lattice, as opposed to the cubical lattice of conventional diamond. It is found i ...
(hexagonal diamond
Diamond is a Allotropes of carbon, solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. Another solid form of carbon known as graphite is the Chemical stability, chemically stable form of car ...
). 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
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
). Due to this, wurtzite crystals can (and generally do) have properties such as piezoelectricity
Piezoelectricity (, ) is the electric charge that accumulates in certain solid materials—such as crystals, certain ceramics, and biological matter such as bone, DNA, and various proteins—in response to applied mechanical stress. The word ''p ...
and pyroelectricity
Pyroelectricity (from the two Greek words ''pyr'' meaning fire, and electricity) is a property of certain crystals which are naturally electrically polarized and as a result contain large electric fields. Pyroelectricity can be described as the a ...
, which centrosymmetric crystals lack.
Nickel arsenide structure
The nickel arsenide structure consists of two interpenetrating sublattices: a primitive hexagonal nickel sublattice and a hexagonal close-packed
In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement (or lattice). Carl Friedrich Gauss proved that the highest average density – that is, the greatest fraction of space occu ...
arsenic sublattice. Each nickel atom is octahedrally coordinated to six arsenic atoms, while each arsenic atom is trigonal prismatically coordinated to six nickel atoms. 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 chalcogen
The chalcogens (ore forming) ( ) are the chemical elements in group 16 of the periodic table. This group is also known as the oxygen family. Group 16 consists of the elements oxygen (O), sulfur (S), selenium (Se), tellurium (Te), and the radioact ...
ides, arsenide
In chemistry, an arsenide is a compound of arsenic with a less electronegative element or elements. Many metals form binary compounds containing arsenic, and these are called arsenides. They exist with many stoichiometries, and in this respect ars ...
s, antimonides Antimonides (sometimes called stibnides) are Chemical compound, compounds of antimony with more electropositive elements. The antimonide ion is Sb3−.
Reduction of antimony by alkali metals or by other methods leads to alkali metal antimonides of ...
and bismuth
Bismuth is a chemical element with the Symbol (chemistry), symbol Bi and atomic number 83. It is a post-transition metal and one of the pnictogens, with chemical properties resembling its lighter group 15 siblings arsenic and antimony. Elemental ...
ides of transition metal
In chemistry, a transition metal (or transition element) is a chemical element in the d-block of the periodic table (groups 3 to 12), though the elements of group 12 (and less often group 3) are sometimes excluded. They are the elements that can ...
s.
The following are the members of the nickeline group:[http://www.mindat.org/min-2901.html Mindat.org]
* Achavalite:
*Breithauptite
Breithauptite is a nickel antimonide mineral with the simple formula NiSb. Breithauptite is a metallic opaque copper-red mineral crystallizing in the hexagonal - dihexagonal dipyramidal crystal system. It is typically massive to reniform in habit, ...
:
*Freboldite:
*Kotulskite:
*Langistite:
*Nickeline:
*Sobolevskite:
*Sudburyite:
In two dimensions
There is only one hexagonal Bravais lattice in two dimensions: the hexagonal lattice.
See also
*Close-packing
In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement (or lattice). Carl Friedrich Gauss proved that the highest average density – that is, the greatest fraction of space occu ...
*Crystal structure
In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystal, crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric pat ...
References
External links
*
Mineralogy database
{{Crystal systems
Crystallography