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Cubic crystal system

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(Redirected fromCubic system)
Crystallographic system where the unit cell is in the shape of a cube
A rock containing three crystals ofpyrite (FeS2). The crystal structure of pyrite is primitive cubic, and this is reflected in the cubic symmetry of its naturalcrystal facets.
A network model of a primitive cubic system
The primitive and cubic close-packed (also known as face-centered cubic) unit cells

Incrystallography, thecubic (orisometric)crystal system is acrystal system where theunit cell is in the shape of acube. This is one of the most common and simplest shapes found incrystals andminerals.

There are three main varieties of these crystals:

  • Primitive cubic (abbreviatedcP and alternatively calledsimple cubic)
  • Body-centered cubic (abbreviatedcI orbcc)
  • Face-centered cubic (abbreviatedcF orfcc)

Note: the termfcc is often used in synonym for thecubic close-packed orccp structure occurring in metals. However,fcc stands for a face-centered cubic Bravais lattice, which is not necessarily close-packed when a motif is set onto the lattice points. E.g. the diamond and the zincblende lattices arefcc but not close-packed.Each is subdivided into other variants listed below. Although theunit cells in these crystals are conventionally taken to be cubes, theprimitive unit cells often are not.

Bravais lattices

[edit]
Further information:Bravais lattice

The three Bravais latices in the cubic crystal system are:

Bravais latticePrimitive
cubic		Body-centered
cubic		Face-centered
cubic
Pearson symbolcPcIcF
Unit cell

The primitive cubic lattice (cP) consists of onelattice point on each corner of the cube; this means each simple cubic unit cell has in total one lattice point. Each atom at a lattice point is then shared equally between eight adjacent cubes, and the unit cell therefore contains in total one atom (1⁄8 × 8).[1]

The body-centered cubic lattice (cI) has one lattice point in the center of the unit cell in addition to the eight corner points. It has a net total of two lattice points per unit cell (1⁄8 × 8 + 1).[1]

The face-centered cubic lattice (cF) has lattice points on the faces of the cube, that each gives exactly one half contribution, in addition to the corner lattice points, giving a total of four lattice points per unit cell (1⁄8 × 8 from the corners plus1⁄2 × 6 from the faces).

The face-centered cubic lattice is closely related to thehexagonal close packed (hcp) system, where two systems differ only in the relative placements of their hexagonal layers. The[111] plane of a face-centered cubic lattice is a hexagonal grid.

Attempting to create a base-centered cubic lattice (i.e., putting an extra lattice point in the center of each horizontal face) results in a simpletetragonalBravais lattice.

Coordination number (CN) is the number of nearest neighbors of a central atom in the structure.[1] Each sphere in a cP lattice has coordination number 6, in a cI lattice 8, and in a cF lattice 12.

Atomic packing factor (APF) is the fraction of volume that is occupied by atoms. The cP lattice has an APF of about 0.524, the cI lattice an APF of about 0.680, and the cF lattice an APF of about 0.740.

Crystal classes

[edit]
Further information:Crystallographic point group

Theisometric crystal system class names, point groups (inSchönflies notation,Hermann–Mauguin notation,orbifold, andCoxeter notation), type, examples, international tables for crystallography space group number,[2] andspace groups are listed in the table below. There are a total 36 cubic space groups.

No.Point groupTypeExampleSpace groups
Name[3]Schön.IntlOrb.Cox.PrimitiveFace-centeredBody-centered
195–197TetartoidalT23332[3,3]+enantiomorphicUllmannite,Sodium chlorateP23F23I23
198–199P213I213
200–204DiploidalTh2/m3
(m3)		3*2[3+,4]centrosymmetricPyritePm3, Pn3Fm3, Fd3I3
205–206Pa3Ia3
207–211GyroidalO432432[3,4]+enantiomorphicPetziteP432, P4232F432, F4132I432
212–214P4332, P4132I4132
215–217HextetrahedralTd43m*332[3,3]SphaleriteP43mF43mI43m
218–220P43nF43cI43d
221–230HexoctahedralOh4/m32/m
(m3m)		*432[3,4]centrosymmetricGalena,HalitePm3m, Pn3n,Pm3n, Pn3mFm3m, Fm3c, Fd3m, Fd3cIm3m, Ia3d

Other terms for hexoctahedral are: normal class,holohedral, ditesseral central class,galena type.

Single element structures

[edit]
Visualisation of a diamond cubic unit cell: 1. Components of a unit cell, 2. One unit cell, 3. A lattice of 3 x 3 x 3 unit cells
See also:Periodic table (crystal structure)

As a rule, since atoms in a solid attract each other, the more tightly packed arrangements of atoms tend to be more common. (Loosely packed arrangements do occur, though, for example if theorbital hybridization demands certainbond angles.) Accordingly, the primitive cubic structure, with especially low atomic packing factor, is rare in nature, but is found inpolonium.[4][5] Thebcc andfcc, with their higher densities, are both quite common in nature. Examples ofbcc includeiron,chromium,tungsten, andniobium. Examples offcc includealuminium,copper,gold andsilver.

Another important cubic crystal structure is thediamond cubic structure, which can appear incarbon,silicon,germanium, andtin. Unlike fcc and bcc, this structure is not a lattice, since it contains multiple atoms in itsprimitive cell. Other cubic elemental structures include theA15 structure found intungsten, and the extremely complicated structure ofmanganese.

Multi-element structures

[edit]

Compounds that consist of more than one element (e.g.binary compounds) often have crystal structures based on the cubic crystal system. 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.

Caesium chloride structure

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See also:Category:Caesium chloride crystal structure
Acaesium chloride unit cell. The two colors of spheres represent the two types of atoms.

One structure is the "interpenetrating primitive cubic" structure, also called a "caesium chloride" or B2 structure. This structure is often confused for a body-centered cubic structure because the arrangement of atoms is the same. However, the caesium chloride structure has a basis composed of two different atomic species. In a body-centered cubic structure, there would be translational symmetry along the [111] direction. In the caesium chloride structure, translation along the [111] direction results in a change of species. The structure can also be thought of as two separate simple cubic structures, one of each species, that are superimposed within each other. The corner of the chloride cube is the center of the caesium cube, and vice versa.[6]

This graphic shows the interlocking simple cubic lattices of cesium and chlorine. You can see them separately and as they are interlocked in what looks like a body-centered cubic arrangement

It works the same way for the NaCl structure described in the next section.  If you take out the Cl atoms, the leftover Na atoms still form an FCC structure, not a simple cubic structure.

In the unit cell of CsCl, each ion is at the center of a cube of ions of the opposite kind, so thecoordination number is eight. The central cation is coordinated to 8 anions on the corners of a cube as shown, and similarly, the central anion is coordinated to 8 cations on the corners of a cube. Alternately, one could view this lattice as a simple cubic structure with a secondary atom in itscubic void.

In addition to caesium chloride itself, the structure also appears in certain otheralkali halides when prepared at low temperatures or high pressures.[7] Generally, this structure is more likely to be formed from two elements whose ions are of roughly the same size (for example, ionic radius of Cs+ = 167 pm, and Cl− = 181 pm).

Thespace group of thecaesium chloride (CsCl) structure is called Pm3m (inHermann–Mauguin notation), or "221" (in the International Tables for Crystallography). TheStrukturbericht designation is "B2".[8]

There are nearly a hundredrare earthintermetallic compounds that crystallize in the CsCl structure, including manybinary compounds of rare earths withmagnesium,[9] and with elements in groups11,12,[10][11] and13. Other compounds showing caesium chloride like structure areCsBr,CsI, high-temperatureRbCl, AlCo, AgZn, BeCu, MgCe, RuAl and SrTl.[citation needed]

Rock-salt structure

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See also:Category:Rock salt crystal structure
The rock-salt crystal structure. Each atom has six nearest neighbours, withoctahedral geometry.

Thespace group of the rock-salt orhalite (sodium chloride) structure is denoted as Fm3m (inHermann–Mauguin notation), or "225" (in the International Tables for Crystallography). TheStrukturbericht designation is "B1".[12]

In the rock-salt structure, each of the two atom types forms a separate face-centered cubic lattice, with the two lattices interpenetrating so as to form a 3D checkerboard pattern. The rock-salt structure hasoctahedralcoordination: Each atom's nearest neighbors consist of six atoms of the opposite type, positioned like the six vertices of aregular octahedron. In sodium chloride there is a 1:1 ratio of sodium to chlorine atoms.  The structure can also be described as an FCC lattice of sodium with chlorine occupying eachoctahedral void or vice versa.[6]

Examples of compounds with this structure include sodium chloride itself, along with almost all other alkali halides, and "many divalent metal oxides, sulfides, selenides, and tellurides".[7] According to theradius ratio rule, this structure is more likely to be formed if the cation is somewhat smaller than the anion (a cation/anion radius ratio of 0.414 to 0.732).

The interatomic distance (distance between cation and anion, or half the unit cell lengtha) in some rock-salt-structure crystals are: 2.3 Å (2.3 × 10−10 m) for NaF,[13] 2.8 Å for NaCl,[14] and 3.2 Å for SnTe.[15] Most of thealkali metalhydrides andhalides have the rock salt structure, though a few have thecaesium chloride structure instead.

Alkali metalhydrides andhalides with the rock salt structure
HydridesFluoridesChloridesBromidesIodides
LithiumLithium hydrideLithium fluoride[16]Lithium chlorideLithium bromideLithium iodide
SodiumSodium hydrideSodium fluoride[16]Sodium chlorideSodium bromideSodium iodide
PotassiumPotassium hydridePotassium fluoride[16]Potassium chloridePotassium bromidePotassium iodide
RubidiumRubidium hydrideRubidium fluorideRubidium chlorideRubidium bromideRubidium iodide
CaesiumCaesium hydrideCaesium fluoride(CsCl structure)
Alkaline earth metalchalcogenides with the rock salt structure
OxidesSulfidesSelenidesTelluridesPolonides
MagnesiumMagnesium oxideMagnesium sulfideMagnesium selenide[17]Magnesium telluride[18](NiAs structure)
CalciumCalcium oxideCalcium sulfideCalcium selenide[19]Calcium tellurideCalcium polonide[20]
StrontiumStrontium oxideStrontium sulfideStrontium selenideStrontium tellurideStrontium polonide[20]
BariumBarium oxideBarium sulfideBarium selenideBarium tellurideBarium polonide[20]
Rare-earth[21] andactinoidpnictides with the rock salt structure
NitridesPhosphidesArsenidesAntimonidesBismuthides
ScandiumScandium nitrideScandium phosphideScandium arsenide[22]Scandium antimonide[23]Scandium bismuthide[24]
YttriumYttrium nitrideYttrium phosphideYttrium arsenide[25]Yttrium antimonideYttrium bismuthide[26]
LanthanumLanthanum nitride[27]Lanthanum phosphide[28]Lanthanum arsenide[25]Lanthanum antimonideLanthanum bismuthide[29]
CeriumCerium nitride[27]Cerium phosphide[28]Cerium arsenide[25]Cerium antimonideCerium bismuthide[29]
PraseodymiumPraseodymium nitride[27]Praseodymium phosphide[28]Praseodymium arsenide[25]Praseodymium antimonide[30]Praseodymium bismuthide[29]
NeodymiumNeodymium nitride[27]Neodymium phosphide[28]Neodymium arsenide[25]Neodymium antimonide[30]Neodymium bismuthide[29]
Promethium?????
SamariumSamarium nitride[27]Samarium phosphide[28]Samarium arsenide[25]Samarium antimonide[30]Samarium bismuthide[29]
EuropiumEuropium nitride[27]Europium phosphide(Na2O2 structure)[31](unstable)[32]
GadoliniumGadolinium nitride[27]Gadolinium phosphideGadolinium arsenide[25]Gadolinium antimonide[30]Gadolinium bismuthide[29]
TerbiumTerbium nitride[27]Terbium phosphideTerbium arsenide[25]Terbium antimonide[30]Terbium bismuthide[29]
DysprosiumDysprosium nitride[27]Dysprosium phosphideDysprosium arsenideDysprosium antimonideDysprosium bismuthide[29]
HolmiumHolmium nitride[27]Holmium phosphideHolmium arsenide[25]Holmium antimonideHolmium bismuthide[29]
ErbiumErbium nitride[27]Erbium phosphideErbium arsenide[25]Erbium antimonideErbium bismuthide[29]
ThuliumThulium nitride[27]Thulium phosphideThulium arsenideThulium antimonideThulium bismuthide[29]
YtterbiumYtterbium nitride[27]Ytterbium phosphideYtterbium arsenide[25]Ytterbium antimonide(unstable)[33][34]
LutetiumLutetium nitride[27]Lutetium phosphideLutetium arsenideLutetium antimonideLutetium bismuthide
Actinium?????
ThoriumThorium nitride[35]Thorium phosphide[35]Thorium arsenide[35]Thorium antimonide[35](CsCl structure)
Protactinium?????
UraniumUranium nitride[35]Uranium monophosphide[35]Uranium arsenide[35]Uranium antimonide[35]Uranium bismuthide[36]
NeptuniumNeptunium nitrideNeptunium phosphideNeptunium arsenideNeptunium antimonideNeptunium bismuthide[36]
PlutoniumPlutonium nitride[35]Plutonium phosphide[35]Plutonium arsenide[35]Plutonium antimonide[35]Plutonium bismuthide[36]
AmericiumAmericium nitride[36]Americium phosphide[36]Americium arsenide[36]Americium antimonide[36]Americium bismuthide[36]
CuriumCurium nitride[37]Curium phosphide[37]Curium arsenide[37]Curium antimonide[37]Curium bismuthide[37]
BerkeliumBerkelium nitride[37]Berkelium phosphide[37]Berkelium arsenide[37]?Berkelium bismuthide[37]
Californium??Californium arsenide[37]?Californium bismuthide[37]
Rare-earth andactinoidchalcogenides with the rock salt structure
OxidesSulfidesSelenidesTelluridesPolonides
Scandium(unstable)[38]Scandium monosulfide
YttriumYttrium monosulfide[39]
LanthanumLanthanum monosulfide[40]
CeriumCerium monosulfide[40]Cerium monoselenide[41]Cerium monotelluride[41]
PraseodymiumPraseodymium monosulfide[40]Praseodymium monoselenide[41]Praseodymium monotelluride[41]
NeodymiumNeodymium monosulfide[40]Neodymium monoselenide[41]Neodymium monotelluride[41]
Promethium????
SamariumSamarium monosulfide[40]Samarium monoselenideSamarium monotellurideSamarium monopolonide[42]
EuropiumEuropium monoxideEuropium monosulfide[40]Europium monoselenide[43]Europium monotelluride[43]Europium monopolonide[42]
Gadolinium(unstable)[38]Gadolinium monosulfide[40]
TerbiumTerbium monosulfide[40]Terbium monopolonide[42]
DysprosiumDysprosium monosulfide[40]Dysprosium monopolonide[42]
HolmiumHolmium monosulfide[40]Holmium monopolonide[42]
ErbiumErbium monosulfide[40]
ThuliumThulium monosulfide[40]Thulium monopolonide[42]
YtterbiumYtterbium monoxideYtterbium monosulfide[40]Ytterbium monopolonide[42]
Lutetium(unstable)[38][44]Lutetium monosulfide[40]Lutetium monopolonide[42]
Actinium????
ThoriumThorium monosulfide[35]Thorium monoselenide[35](CsCl structure)[45]
Protactinium????
UraniumUranium monosulfide[35]Uranium monoselenide[35]Uranium monotelluride[35]
NeptuniumNeptunium monosulfideNeptunium monoselenideNeptunium monotelluride
PlutoniumPlutonium monosulfide[35]Plutonium monoselenide[35]Plutonium monotelluride[35]
AmericiumAmericium monosulfide[36]Americium monoselenide[36]Americium monotelluride[36]
CuriumCurium monosulfide[37]Curium monoselenide[37]Curium monotelluride[37]
Transition metalcarbides andnitrides with the rock salt structure
CarbidesNitrides
TitaniumTitanium carbideTitanium nitride
ZirconiumZirconium carbideZirconium nitride
HafniumHafnium carbideHafnium nitride[46]
VanadiumVanadium carbideVanadium nitride
NiobiumNiobium carbideNiobium nitride
TantalumTantalum carbide(CoSn structure)
Chromium(unstable)[47]Chromium nitride

Manytransition metal monoxides also have the rock salt structure (TiO,VO,CrO,MnO,FeO,CoO,NiO,CdO). The early actinoid monocarbides also have this structure (ThC,PaC,UC,NpC,PuC).[37]

Fluorite structure

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Main article:Fluorite structure
See also:Category:Fluorite crystal structure

Much like the rock salt structure, thefluorite structure (AB2) is also an Fm3m structure but has 1:2 ratio of ions. The anti-fluorite structure is nearly identical, except the positions of the anions and cations are switched in the structure. They are designatedWyckoff positions 4a and 8c whereas the rock-salt structure positions are 4a and 4b.[48][49]

Zincblende structure

[edit]
See also:Category:Zincblende crystal structure
A zincblende unit cell

Thespace group of the Zincblende structure is called F43m (inHermann–Mauguin notation), or 216.[50][51] The Strukturbericht designation is "B3".[52]

The Zincblende structure (also written "zinc blende") is named after the mineral zincblende (sphalerite), one form ofzinc sulfide (β-ZnS). As in the rock-salt structure, the two atom types form two interpenetrating face-centered cubic lattices. However, it differs from rock-salt structure in how the two lattices are positioned relative to one another. The zincblende structure hastetrahedralcoordination: Each atom's nearest neighbors consist of four atoms of the opposite type, positioned like the four vertices of aregular tetrahedron. In zinc sulfide the ratio of zinc to sulfur is 1:1.[6] Altogether, the arrangement of atoms in zincblende structure is the same asdiamond cubic structure, but with alternating types of atoms at the different lattice sites. The structure can also be described as an FCC lattice of zinc with sulfur atoms occupying half of thetetrahedral voids or vice versa.[6]

Examples of compounds with this structure include zincblende itself,lead(II) nitrate, many compound semiconductors (such asgallium arsenide andcadmium telluride), and a wide array of other binary compounds.[citation needed] Theboron grouppnictogenides usually have a zincblende structure, though thenitrides are more common in thewurtzite structure, and their zincblende forms are less well knownpolymorphs.[53][54]

Copperhalides with the zincblende structure
FluoridesChloridesBromidesIodides
CopperCopper(I) fluorideCopper(I) chlorideCopper(I) bromideCopper(I) iodide
Beryllium andGroup 12chalcogenides with the zincblende structure
SulfidesSelenidesTelluridesPolonides
BerylliumBeryllium sulfideBeryllium selenideBeryllium tellurideBeryllium polonide[55][56]
ZincZinc sulfideZinc selenideZinc tellurideZinc polonide
CadmiumCadmium sulfideCadmium selenideCadmium tellurideCadmium polonide
MercuryMercury sulfideMercury selenideMercury telluride–

This group is also known as theII-VI family of compounds, most of which can be made in both the zincblende (cubic) orwurtzite (hexagonal) form.

Group 13pnictogenides with the zincblende structure
NitridesPhosphidesArsenidesAntimonides
BoronBoron nitride*Boron phosphideBoron arsenideBoron antimonide
AluminiumAluminium nitride*Aluminium phosphideAluminium arsenideAluminium antimonide
GalliumGallium nitride*Gallium phosphideGallium arsenideGallium antimonide
IndiumIndium nitride*Indium phosphideIndium arsenideIndium antimonide

This group is also known as theIII-V family of compounds.

The structure of theHeusler compounds with formula X2YZ (e. g., Co2MnSi).

Heusler structure

[edit]
Main article:Heusler compound

The Heusler structure, based on the structure of Cu2MnAl, is a common structure forternary compounds involvingtransition metals. It has the space group Fm3m (No. 225), and theStrukturbericht designation is L21. Together with the closely related half-Heusler and inverse-Huesler compounds, there are hundreds of examples.

Iron monosilicide structure

[edit]
See also:Category:Iron monosilicide structure type
Diagram of theiron monosilicide structure.

The space group of the iron monosilicide structure is P213 (No. 198), and theStrukturbericht designation is B20. This is achiral structure, and is sometimes associated withhelimagnetic properties. There are four atoms of each element for a total of eight atoms in the unit cell.

Examples occur among the transition metal silicides and germanides, as well as a few other compounds such asgallium palladide.

Transition metal silicides and germanides with the FeSi structure
SilicidesGermanides
ManganeseManganese monosilicideManganese germanide
IronIron monosilicideIron germanide
CobaltCobalt monosilicideCobalt germanide
ChromiumChromium(IV) silicideChromium(IV) germanide

Weaire–Phelan structure

[edit]
Weaire–Phelan structure

AWeaire–Phelan structure has Pm3n (223) symmetry.

It has three orientations of stackedtetradecahedrons withpyritohedral cells in the gaps. It is found as acrystal structure inchemistry where it is usually known as a "type Iclathrate structure".Gas hydrates formed by methane, propane, and carbon dioxide at low temperatures have a structure in which water molecules lie at the nodes of the Weaire–Phelan structure and arehydrogen bonded together, and the larger gas molecules are trapped in the polyhedral cages.

See also

[edit]

References

[edit]
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Further reading

[edit]
  • Hurlbut, Cornelius S.; Klein, Cornelis, 1985,Manual of Mineralogy, 20th ed., Wiley,ISBN 0-471-80580-7

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Four 2D systems
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