# Cubic crystal system

(Redirected from Body-centered cubic)

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, and alternatively called cubic close-packed or ccp)

Each is subdivided into other variants listed below. Note that although the unit cell in these crystals is conventionally taken to be a cube, the primitive unit cell often is not.

## Bravais lattices

The three Bravais lattices 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. Each atom at a lattice point is then shared equally between eight adjacent cubes, and the unit cell therefore contains in total one atom (18 × 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 2 lattice points per unit cell (18 × 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 4 lattice points per unit cell (18 × 8 from the corners plus 12 × 6 from the faces). Each sphere in a cF lattice has coordination number 12. Coordination number is the number of nearest neighbours of a central atom in the structure.[1]

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) would result in a simple tetragonal Bravais lattice.

## Crystal classes

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.

# 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.

## Voids in the unit cell

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

Voids in the unit cell are also known as interstitial sites or holes. They are the empty space that exists between the packing of atoms (spheres) in the unit cells.[4]

The holes are easy to see if you try to pack circles together. No matter how close you get them or how you arrange them, you will have empty space in between. The same is true in a unit cell, no matter how the atoms are arranged, there will be interstitial sites present between the atoms. These sites or holes can be filled with other atoms.  The picture with packed circles is only a 2D representation.  In a crystal lattice, the atoms (spheres) would be packed in a 3D arrangement.  This results in different shaped interstitial sites depending on the arrangement of the atoms in the lattice.

A simple cubic unit cell, with stacks of atoms arranged as if at the eight corners of a cube would form a single cubic hole or void in the center.

A close packed unit cell, both face-centered cubic and hexagonal close packed, can form two different shaped holes.  Looking at the three green spheres in the hexagonal packing illustration, they form a triangle shaped hole.  If an atom is arranged on top of this triangular hole it forms a tetrahedral interstitial hole. If the three atoms in the layer above are rotated and their triangular hole sits on top of this one, it forms an octahedral interstitial hole.[4] In a close-packed structure there are 4 atoms per unit cell and it will have 4 octahedral voids (1:1 ratio) and 8 tetrahedral voids (1:2 ratio) per unit cell.[5] The tetrahedral void is smaller in size and could fit an atom with a radius 0.225 times the size of the atoms making up the lattice.  An octahedral void could fit an atom with a radius 0.441 times the size of the atoms making up the lattice.[5] An atom that fills this empty space could be larger than this ideal radius ratio, which would lead to a distorted lattice due to pushing out the surrounding atoms, but it cannot be smaller than this ratio.[5]

A body-centered cubic unit cell has six octahedral voids located at the center of each face of the unit cell, and twelve further ones located at the midpoint of each edge of the same cell, for a total of six net octahedral voids. Additionally, there are 24 tetrahedral voids located in a square spacing around each octahedral void, for a total of twelve net tetrahedral voids. These tetrahedral voids are not local maxima and are not technically voids, but they do occasionally appear in multi-atom unit cells.

One important characteristic of a crystalline structure is its atomic packing factor. This is calculated by assuming that all the atoms are identical spheres, with a radius large enough that each sphere abuts on the next. The atomic packing factor is the proportion of space filled by these spheres.

Assuming one atom per lattice point, in a primitive cubic lattice with cube side length a, the sphere radius would be a2 and the atomic packing factor turns out to be about 0.524 (which is quite low). This means that the volume of the unit cell actually occupied by atoms is only 52.4%. Similarly, in a bcc lattice, the atomic packing factor is 0.680, and in fcc it is 0.740. The fcc value is the highest theoretically possible value for any lattice, although there are other lattices which also achieve the same value, such as hexagonal close packed (hcp) and one version of tetrahedral bcc.

## Single element structures

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.[6][7] 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.

### Caesium chloride 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 the "caesium chloride" structure. This structure is often confused for a body-centered cubic structure, because the arrangement of atoms is the same. The true structure is shown in the graphic showing two individual primitive cubic structures that are superimposed within each other with the corner of one structure in the center of the cube of the other structure.  It helps to convince yourself that it is not body-centered cubic because there is no translational symmetry along the ½, ½, ½, plane, the chloride would be translated into a cesium, not another chloride.[8]

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 co - ordination number is eight. Altogether, the arrangement of atoms is the same as body-centered cubic, but with alternating types of atoms at the different lattice sites. 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.[9] 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 coordination number of each atom in the structure is 8: 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.

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".[10]

There are nearly a hundred rare earth intermetallic compounds that crystalize in the CsCl structure, including many binary compounds of rare earths with magnesium,[11] and with elements in groups 11, 12,[12][13] 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

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

The space group of the rock-salt (NaCl) structure is called Fm3m (in Hermann–Mauguin notation), or "225" (in the International Tables for Crystallography). The Strukturbericht designation is "B1".[14]

In the rock-salt or sodium chloride (halite) 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. Earlier it was stated that there is one octahedral hole per atom in a cubic close packed structure.  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.[8]

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".[9] 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 coordination number of each atom in this structure is 6: each cation is coordinated to 6 anions at the vertices of an octahedron, and similarly, each anion is coordinated to 6 cations at the vertices of an octahedron.

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,[15] 2.8 Å for NaCl,[16] and 3.2 Å for SnTe.[17] 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[18] Lithium chloride Lithium bromide Lithium iodide
Sodium Sodium hydride Sodium fluoride[18] Sodium chloride Sodium bromide Sodium iodide
Potassium Potassium hydride Potassium fluoride[18] 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[19] Magnesium telluride[20] (NiAs structure)
Calcium Calcium oxide Calcium sulfide Calcium selenide[21] Calcium telluride Calcium polonide[22]
Strontium Strontium oxide Strontium sulfide Strontium selenide Strontium telluride Strontium polonide[22]
Barium Barium oxide Barium sulfide Barium selenide Barium telluride Barium polonide[22]
Rare-earth[23] and actinoid pnictides with the rock salt structure
Nitrides Phosphides Arsenides Antimonides Bismuthides
Scandium Scandium nitride Scandium phosphide Scandium arsenide[24] Scandium antimonide[25] Scandium bismuthide[26]
Yttrium Yttrium nitride Yttrium phosphide Yttrium arsenide[27] Yttrium antimonide Yttrium bismuthide[28]
Lanthanum Lanthanum nitride[29] Lanthanum phosphide[30] Lanthanum arsenide[27] Lanthanum antimonide Lanthanum bismuthide[31]
Cerium Cerium nitride[29] Cerium phosphide[30] Cerium arsenide[27] Cerium antimonide Cerium bismuthide[31]
Praseodymium Praseodymium nitride[29] Praseodymium phosphide[30] Praseodymium arsenide[27] Praseodymium antimonide[32] Praseodymium bismuthide[31]
Neodymium Neodymium nitride[29] Neodymium phosphide[30] Neodymium arsenide[27] Neodymium antimonide[32] Neodymium bismuthide[31]
Promethium ? ? ? ? ?
Samarium Samarium nitride[29] Samarium phosphide[30] Samarium arsenide[27] Samarium antimonide[32] Samarium bismuthide[31]
Europium Europium nitride[29] Europium phosphide (Na2O2 structure)[33] (unstable)[34]
Terbium Terbium nitride[29] Terbium phosphide Terbium arsenide[27] Terbium antimonide[32] Terbium bismuthide[31]
Dysprosium Dysprosium nitride[29] Dysprosium phosphide Dysprosium arsenide Dysprosium antimonide Dysprosium bismuthide[31]
Holmium Holmium nitride[29] Holmium phosphide Holmium arsenide[27] Holmium antimonide Holmium bismuthide[31]
Erbium Erbium nitride[29] Erbium phosphide Erbium arsenide[27] Erbium antimonide Erbium bismuthide[31]
Thulium Thulium nitride[29] Thulium phosphide Thulium arsenide Thulium antimonide Thulium bismuthide[31]
Ytterbium Ytterbium nitride[29] Ytterbium phosphide Ytterbium arsenide[27] Ytterbium antimonide (unstable)[35][36]
Lutetium Lutetium nitride[29] Lutetium phosphide Lutetium arsenide Lutetium antimonide Lutetium bismuthide
Actinium ? ? ? ? ?
Thorium Thorium nitride[37] Thorium phosphide[37] Thorium arsenide[37] Thorium antimonide[37] (CsCl structure)
Protactinium ? ? ? ? ?
Uranium Uranium nitride[37] Uranium monophosphide[37] Uranium arsenide[37] Uranium antimonide[37] Uranium bismuthide[38]
Neptunium Neptunium nitride Neptunium phosphide Neptunium arsenide Neptunium antimonide Neptunium bismuthide[38]
Plutonium Plutonium nitride[37] Plutonium phosphide[37] Plutonium arsenide[37] Plutonium antimonide[37] Plutonium bismuthide[38]
Americium Americium nitride[38] Americium phosphide[38] Americium arsenide[38] Americium antimonide[38] Americium bismuthide[38]
Curium Curium nitride[39] Curium phosphide[39] Curium arsenide[39] Curium antimonide[39] Curium bismuthide[39]
Berkelium Berkelium nitride[39] Berkelium phosphide[39] Berkelium arsenide[39] ? Berkelium bismuthide[39]
Californium ? ? Californium arsenide[39] ? Californium bismuthide[39]
Rare-earth and actinoid chalcogenides with the rock salt structure
Oxides Sulfides Selenides Tellurides Polonides
Scandium (unstable)[40] Scandium monosulfide
Yttrium Yttrium monosulfide[41]
Lanthanum Lanthanum monosulfide[42]
Cerium Cerium monosulfide[42] Cerium monoselenide[43] Cerium monotelluride[43]
Praseodymium Praseodymium monosulfide[42] Praseodymium monoselenide[43] Praseodymium monotelluride[43]
Neodymium Neodymium monosulfide[42] Neodymium monoselenide[43] Neodymium monotelluride[43]
Promethium ? ? ? ?
Samarium Samarium monosulfide[42] Samarium monoselenide Samarium monotelluride Samarium monopolonide[44]
Europium Europium monoxide Europium monosulfide[42] Europium monoselenide[45] Europium monotelluride[45] Europium monopolonide[44]
Terbium Terbium monosulfide[42] Terbium monopolonide[44]
Dysprosium Dysprosium monosulfide[42] Dysprosium monopolonide[44]
Holmium Holmium monosulfide[42] Holmium monopolonide[44]
Erbium Erbium monosulfide[42]
Thulium Thulium monosulfide[42] Thulium monopolonide[44]
Ytterbium Ytterbium monoxide Ytterbium monosulfide[42] Ytterbium monopolonide[44]
Lutetium (unstable)[40][46] Lutetium monosulfide[42] Lutetium monopolonide[44]
Actinium ? ? ? ?
Thorium Thorium monosulfide[37] Thorium monoselenide[37] (CsCl structure)[47]
Protactinium ? ? ? ?
Uranium Uranium monosulfide[37] Uranium monoselenide[37] Uranium monotelluride[37]
Neptunium Neptunium monosulfide Neptunium monoselenide Neptunium monotelluride
Plutonium Plutonium monosulfide[37] Plutonium monoselenide[37] Plutonium monotelluride[37]
Americium Americium monosulfide[38] Americium monoselenide[38] Americium monotelluride[38]
Curium Curium monosulfide[39] Curium monoselenide[39] Curium monotelluride[39]
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[48]
Niobium Niobium carbide Niobium nitride
Tantalum Tantalum carbide (CoSn structure)
Chromium (unstable)[49] 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).[39] Other compounds showing rock salt like structure are TiB,[50] ZrB,[51] PbS, PbSe, PbTe, SnTe, AgF, AgCl, and AgBr.

### Fluorite 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.[52][53]

### Zincblende structure

A zincblende unit cell

The space group of the Zincblende structure is called F43m (in Hermann–Mauguin notation), or 216.[54][55] The Strukturbericht designation is "B3".[56]

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.[8] 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.

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.[57][58]

Copper halides with the zincblende structure
Fluorides Chorlorides 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[59][60]
Zinc Zinc sulfide Zinc selenide Zinc telluride Zinc 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.

### Heusler structure

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

The Huesler 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

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

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

It has 3 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 the "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.

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