Cubic crystal system

In crystallography, the cubic (or isometric) crystal 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.

A rock containing three crystals of pyrite (FeS₂). The crystal structure of pyrite is primitive cubic, and this is reflected in the cubic symmetry of its natural crystal facets.

A network model of a primitive cubic system

The primitive and cubic close-packed (also known as face-centered cubic) unit cells

There are three main varieties of these crystals:

• Primitive cubic (abbreviated cP and alternatively called simple cubic)
• Body-centered cubic (abbreviated cI or bcc)
• Face-centered cubic (abbreviated cF or fcc)

Note: the term fcc is often used in synonym for the cubic close-packed or ccp 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 are fcc but not close-packed. Each is subdivided into other variants listed below. Although the unit cells in these crystals are conventionally taken to be cubes, the primitive unit cells often are not.

Bravais lattices

Further information: Bravais lattice

The three Bravais latices in the cubic crystal system are:

Bravais lattice	Primitive cubic	Body-centered cubic	Face-centered cubic
Pearson symbol	cP	cI	cF
Unit cell

The primitive cubic lattice (cP) consists of one lattice 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 plus 1⁄2 × 6 from the faces).

The face-centered cubic lattice is closely related to the hexagonal 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 simple tetragonal Bravais 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

Further information: Crystallographic point group

The isometric crystal system class names, point groups (in Schönflies notation, Hermann–Mauguin notation, orbifold, and Coxeter notation), type, examples, international tables for crystallography space group number,^[2] and space groups are listed in the table below. There are a total 36 cubic space groups.

No.	Point group					Type	Example	Space groups
	Name[3]	Schön.	Intl	Orb.	Cox.			Primitive	Face-centered	Body-centered
195–197	Tetartoidal	T	23	332	[3,3]+	enantiomorphic	Ullmannite, Sodium chlorate	P23	F23	I23
198–199								P213		I213
200–204	Diploidal	Th	2/m3 (m3)	3*2	[3+,4]	centrosymmetric	Pyrite	Pm3, Pn3	Fm3, Fd3	I3
205–206								Pa3		Ia3
207–211	Gyroidal	O	432	432	[3,4]+	enantiomorphic	Petzite	P432, P4232	F432, F4132	I432
212–214								P4332, P4132		I4132
215–217	Hextetrahedral	Td	43m	*332	[3,3]		Sphalerite	P43m	F43m	I43m
218–220								P43n	F43c	I43d
221–230	Hexoctahedral	Oh	4/m32/m (m3m)	*432	[3,4]	centrosymmetric	Galena, Halite	Pm3m, Pn3n, Pm3n, Pn3m	Fm3m, Fm3c, Fd3m, Fd3c	Im3m, Ia3d

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

Single element structures

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 the orbital hybridization demands certain bond angles.) Accordingly, the primitive cubic structure, with especially low atomic packing factor, is rare in nature, but is found in polonium.^[4][5] The bcc and fcc, with their higher densities, are both quite common in nature. Examples of bcc include iron, chromium, tungsten, and niobium. Examples of fcc include aluminium, copper, gold and silver.

Another important cubic crystal structure is the diamond cubic structure, which can appear in carbon, silicon, germanium, and tin. Unlike fcc and bcc, this structure is not a lattice, since it contains multiple atoms in its primitive cell. Other cubic elemental structures include the A15 structure found in tungsten, and the extremely complicated structure of manganese.

Multi-element structures

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 the interstitial sites of the others.

Caesium chloride structure

See also: Category:Caesium chloride crystal structure

A caesium 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 the coordination 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 its cubic void.

In addition to caesium chloride itself, the structure also appears in certain other alkali 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).

The space group of the caesium chloride (CsCl) structure is called Pm3m (in Hermann–Mauguin notation), or "221" (in the International Tables for Crystallography). The Strukturbericht designation is "B2".^[8]

There are nearly a hundred rare earth intermetallic compounds that crystallize in the CsCl structure, including many binary compounds of rare earths with magnesium,^[9] and with elements in groups 11, 12,^[10][11] and 13. Other compounds showing caesium chloride like structure are CsBr, CsI, high-temperature RbCl, AlCo, AgZn, BeCu, MgCe, RuAl and SrTl.^{[citation needed]}

Rock-salt structure

See also: Category:Rock salt crystal structure

The rock-salt crystal structure. Each atom has six nearest neighbours, with octahedral geometry.

The space group of the rock-salt or halite (sodium chloride) structure is denoted as Fm3m (in Hermann–Mauguin notation), or "225" (in the International Tables for Crystallography). The Strukturbericht 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 has octahedral coordination: Each atom's nearest neighbors consist of six atoms of the opposite type, positioned like the six vertices of a regular 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 each octahedral 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 the radius 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 length a) 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 the alkali metal hydrides and halides have the rock salt structure, though a few have the caesium chloride structure instead.

Alkali metal hydrides and halides with the rock salt structure

	Hydrides	Fluorides	Chlorides	Bromides	Iodides
Lithium	Lithium hydride	Lithium fluoride[16]	Lithium chloride	Lithium bromide	Lithium iodide
Sodium	Sodium hydride	Sodium fluoride[16]	Sodium chloride	Sodium bromide	Sodium iodide
Potassium	Potassium hydride	Potassium fluoride[16]	Potassium chloride	Potassium bromide	Potassium iodide
Rubidium	Rubidium hydride	Rubidium fluoride	Rubidium chloride	Rubidium bromide	Rubidium iodide
Caesium	Caesium hydride	Caesium fluoride	(CsCl structure)

Alkaline earth metal chalcogenides with the rock salt structure

	Oxides	Sulfides	Selenides	Tellurides	Polonides
Magnesium	Magnesium oxide	Magnesium sulfide	Magnesium selenide[17]	Magnesium telluride[18]	(NiAs structure)
Calcium	Calcium oxide	Calcium sulfide	Calcium selenide[19]	Calcium telluride	Calcium polonide[20]
Strontium	Strontium oxide	Strontium sulfide	Strontium selenide	Strontium telluride	Strontium polonide[20]
Barium	Barium oxide	Barium sulfide	Barium selenide	Barium telluride	Barium polonide[20]

Rare-earth^[21] and actinoid pnictides with the rock salt structure

	Nitrides	Phosphides	Arsenides	Antimonides	Bismuthides
Scandium	Scandium nitride	Scandium phosphide	Scandium arsenide[22]	Scandium antimonide[23]	Scandium bismuthide[24]
Yttrium	Yttrium nitride	Yttrium phosphide	Yttrium arsenide[25]	Yttrium antimonide	Yttrium bismuthide[26]
Lanthanum	Lanthanum nitride[27]	Lanthanum phosphide[28]	Lanthanum arsenide[25]	Lanthanum antimonide	Lanthanum bismuthide[29]
Cerium	Cerium nitride[27]	Cerium phosphide[28]	Cerium arsenide[25]	Cerium antimonide	Cerium bismuthide[29]
Praseodymium	Praseodymium nitride[27]	Praseodymium phosphide[28]	Praseodymium arsenide[25]	Praseodymium antimonide[30]	Praseodymium bismuthide[29]
Neodymium	Neodymium nitride[27]	Neodymium phosphide[28]	Neodymium arsenide[25]	Neodymium antimonide[30]	Neodymium bismuthide[29]
Promethium	?	?	?	?	?
Samarium	Samarium nitride[27]	Samarium phosphide[28]	Samarium arsenide[25]	Samarium antimonide[30]	Samarium bismuthide[29]
Europium	Europium nitride[27]	Europium phosphide	(Na2O2 structure)[31]	(unstable)[32]
Gadolinium	Gadolinium nitride[27]	Gadolinium phosphide	Gadolinium arsenide[25]	Gadolinium antimonide[30]	Gadolinium bismuthide[29]
Terbium	Terbium nitride[27]	Terbium phosphide	Terbium arsenide[25]	Terbium antimonide[30]	Terbium bismuthide[29]
Dysprosium	Dysprosium nitride[27]	Dysprosium phosphide	Dysprosium arsenide	Dysprosium antimonide	Dysprosium bismuthide[29]
Holmium	Holmium nitride[27]	Holmium phosphide	Holmium arsenide[25]	Holmium antimonide	Holmium bismuthide[29]
Erbium	Erbium nitride[27]	Erbium phosphide	Erbium arsenide[25]	Erbium antimonide	Erbium bismuthide[29]
Thulium	Thulium nitride[27]	Thulium phosphide	Thulium arsenide	Thulium antimonide	Thulium bismuthide[29]
Ytterbium	Ytterbium nitride[27]	Ytterbium phosphide	Ytterbium arsenide[25]	Ytterbium antimonide	(unstable)[33][34]
Lutetium	Lutetium nitride[27]	Lutetium phosphide	Lutetium arsenide	Lutetium antimonide	Lutetium bismuthide
Actinium	?	?	?	?	?
Thorium	Thorium nitride[35]	Thorium phosphide[35]	Thorium arsenide[35]	Thorium antimonide[35]	(CsCl structure)
Protactinium	?	?	?	?	?
Uranium	Uranium nitride[35]	Uranium monophosphide[35]	Uranium arsenide[35]	Uranium antimonide[35]	Uranium bismuthide[36]
Neptunium	Neptunium nitride	Neptunium phosphide	Neptunium arsenide	Neptunium antimonide	Neptunium bismuthide[36]
Plutonium	Plutonium nitride[35]	Plutonium phosphide[35]	Plutonium arsenide[35]	Plutonium antimonide[35]	Plutonium bismuthide[36]
Americium	Americium nitride[36]	Americium phosphide[36]	Americium arsenide[36]	Americium antimonide[36]	Americium bismuthide[36]
Curium	Curium nitride[37]	Curium phosphide[37]	Curium arsenide[37]	Curium antimonide[37]	Curium bismuthide[37]
Berkelium	Berkelium nitride[37]	Berkelium phosphide[37]	Berkelium arsenide[37]	?	Berkelium bismuthide[37]
Californium	?	?	Californium arsenide[37]	?	Californium bismuthide[37]

Rare-earth and actinoid chalcogenides with the rock salt structure

	Oxides	Sulfides	Selenides	Tellurides	Polonides
Scandium	(unstable)[38]	Scandium monosulfide
Yttrium		Yttrium monosulfide[39]
Lanthanum		Lanthanum monosulfide[40]
Cerium		Cerium monosulfide[40]	Cerium monoselenide[41]	Cerium monotelluride[41]
Praseodymium		Praseodymium monosulfide[40]	Praseodymium monoselenide[41]	Praseodymium monotelluride[41]
Neodymium		Neodymium monosulfide[40]	Neodymium monoselenide[41]	Neodymium monotelluride[41]
Promethium		?	?	?	?
Samarium		Samarium monosulfide[40]	Samarium monoselenide	Samarium monotelluride	Samarium monopolonide[42]
Europium	Europium monoxide	Europium monosulfide[40]	Europium monoselenide[43]	Europium monotelluride[43]	Europium monopolonide[42]
Gadolinium	(unstable)[38]	Gadolinium monosulfide[40]
Terbium		Terbium monosulfide[40]			Terbium monopolonide[42]
Dysprosium		Dysprosium monosulfide[40]			Dysprosium monopolonide[42]
Holmium		Holmium monosulfide[40]			Holmium monopolonide[42]
Erbium		Erbium monosulfide[40]
Thulium		Thulium monosulfide[40]			Thulium monopolonide[42]
Ytterbium	Ytterbium monoxide	Ytterbium monosulfide[40]			Ytterbium monopolonide[42]
Lutetium	(unstable)[38][44]	Lutetium monosulfide[40]			Lutetium monopolonide[42]
Actinium		?	?	?	?
Thorium		Thorium monosulfide[35]	Thorium monoselenide[35]	(CsCl structure)[45]
Protactinium		?	?	?	?
Uranium		Uranium monosulfide[35]	Uranium monoselenide[35]	Uranium monotelluride[35]
Neptunium		Neptunium monosulfide	Neptunium monoselenide	Neptunium monotelluride
Plutonium		Plutonium monosulfide[35]	Plutonium monoselenide[35]	Plutonium monotelluride[35]
Americium		Americium monosulfide[36]	Americium monoselenide[36]	Americium monotelluride[36]
Curium		Curium monosulfide[37]	Curium monoselenide[37]	Curium monotelluride[37]

Transition metal carbides and nitrides with the rock salt structure

	Carbides	Nitrides
Titanium	Titanium carbide	Titanium nitride
Zirconium	Zirconium carbide	Zirconium nitride
Hafnium	Hafnium carbide	Hafnium nitride[46]
Vanadium	Vanadium carbide	Vanadium nitride
Niobium	Niobium carbide	Niobium nitride
Tantalum	Tantalum carbide	(CoSn structure)
Chromium	(unstable)[47]	Chromium nitride

Many transition 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

Main article: Fluorite structure

See also: Category:Fluorite crystal structure

Much like the rock salt structure, the fluorite 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 designated Wyckoff positions 4a and 8c whereas the rock-salt structure positions are 4a and 4b.^[48][49]

Zincblende structure

See also: Category:Zincblende crystal structure

A zincblende unit cell

The space group of the Zincblende structure is called F43m (in Hermann–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 of zinc 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 has tetrahedral coordination: Each atom's nearest neighbors consist of four atoms of the opposite type, positioned like the four vertices of a regular 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 as diamond 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 the tetrahedral voids or vice versa.^[6]

Examples of compounds with this structure include zincblende itself, lead(II) nitrate, many compound semiconductors (such as gallium arsenide and cadmium telluride), and a wide array of other binary compounds.^{[citation needed]} The boron group pnictogenides usually have a zincblende structure, though the nitrides are more common in the wurtzite structure, and their zincblende forms are less well known polymorphs.^[53][54]

Copper halides with the zincblende structure

	Fluorides	Chlorides	Bromides	Iodides
Copper	Copper(I) fluoride	Copper(I) chloride	Copper(I) bromide	Copper(I) iodide

Beryllium and Group 12 chalcogenides with the zincblende structure

	Sulfides	Selenides	Tellurides	Polonides
Beryllium	Beryllium sulfide	Beryllium selenide	Beryllium telluride	Beryllium polonide[55][56]
Zinc	Zinc sulfide	Zinc selenide	Zinc telluride	Zinc polonide
Cadmium	Cadmium sulfide	Cadmium selenide	Cadmium telluride	Cadmium polonide
Mercury	Mercury sulfide	Mercury selenide	Mercury telluride	–

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

Group 13 pnictogenides with the zincblende structure

	Nitrides	Phosphides	Arsenides	Antimonides
Boron	Boron nitride*	Boron phosphide	Boron arsenide	Boron antimonide
Aluminium	Aluminium nitride*	Aluminium phosphide	Aluminium arsenide	Aluminium antimonide
Gallium	Gallium nitride*	Gallium phosphide	Gallium arsenide	Gallium antimonide
Indium	Indium nitride*	Indium phosphide	Indium arsenide	Indium antimonide

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

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

Heusler structure

Main article: Heusler compound

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

Iron monosilicide structure

See also: Category:Iron monosilicide structure type

Diagram of the iron monosilicide structure.

The space group of the iron monosilicide structure is P2₁3 (No. 198), and the Strukturbericht designation is B20. This is a chiral structure, and is sometimes associated with helimagnetic 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 as gallium palladide.

Transition metal silicides and germanides with the FeSi structure

	Silicides	Germanides
Manganese	Manganese monosilicide	Manganese germanide
Iron	Iron monosilicide	Iron germanide
Cobalt	Cobalt monosilicide	Cobalt germanide
Chromium	Chromium(IV) silicide	Chromium(IV) germanide

Weaire–Phelan structure

Weaire–Phelan structure

A Weaire–Phelan structure has Pm3n (223) symmetry.

It has three orientations of stacked tetradecahedrons with pyritohedral cells in the gaps. It is found as a crystal structure in chemistry where it is usually known as a "type I clathrate 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 are hydrogen bonded together, and the larger gas molecules are trapped in the polyhedral cages.

See also

• Atomium: building which is a model of a bcc unit cell, with vertical body diagonal.
• Close-packing
• Dislocations
• Reciprocal lattice

References

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Further reading

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

External links

• JMol simulations by Graz University:
  • Simple cubic
  • BCC
  • FCC
  • HCP
• Making crystal structure with Molview