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Body-centered cubic

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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.

There are three main varieties of these crystals:

  • Primitive cubic (abbreviated cP[1] and alternatively called simple cubic)
  • Body-centered cubic (abbreviated cI[1] or bcc),
  • Face-centered cubic (abbreviated cF[1] 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. This is related to the fact that in most cubic crystal systems, there is more than one atom per cubic unit cell.

Cubic space groups

The three Bravais lattices which form cubic crystal systems are:

Cubic Bravais lattices
Name Primitive cubic Body-centered cubic Face-centered cubic
Pearson symbol cP cI cF
Unit cell

The primitive cubic system (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).
The body-centered cubic system (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).
The face-centered cubic system (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).

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

Attempting to create a C-centered cubic crystal system (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, examples, Schönflies notation, Hermann-Mauguin notation, point groups, International Tables for Crystallography space group number,[2] orbifold, type, and space groups are listed in the table below. There are a total 36 cubic space groups.

# Point group Example Type Space groups
Class[3] Schönflies Intl Orbifold Coxeter
195-199 Tetartoidal T 23 332 [3,3]+ Ullmannite enantiomorphic P23 F23 I23 P213 I213  
200-206 Diploidal Th 2/m3
(m3)
3*2 [3+,4] Pyrite centrosymmetric Pm3 Pn3 Fm3 Fd3 I3 Pa3 Ia3  
207-214 Gyroidal O 432 432 [3,4]+ Petzite enantiomorphic P432 P4232 F432 F4132 I432 P4332 P4132 I4132
215-220 Hextetrahedral Td 43m *332 [3,3] Sphalerite P43m F43m I43m P43n F43c I43d  
221-230 Hexoctahedral Oh 4/m32/m
(m3m)
*432 [3,4] Galena centrosymmetric 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

A simple cubic unit cell has a single cubic void in the center.

A body-centered cubic unit cell has six octahedral voids located at the center of each face of the unit cell, for a total of three net octahedral voids. Additionally, there are 36 tetrahedral voids located in an octahedral spacing around each octahedral void, for a total of eighteen 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.

A face-centered cubic unit cell has eight tetrahedral voids located slightly towards the center from each corner of the unit cell, for a total of eight net tetrahedral voids. Additionally, there are twelve octahedral voids located at the center of edge of the unit cell as well as one octahedral hole in the very center, for a total of four net octahedral voids.

Atomic packing factors and examples

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 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). 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 and one version of tetrahedral bcc.

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] 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.

Multi-element compounds

Compounds that consist of more than one element (e.g. binary compounds) often have crystal structures based on a cubic crystal system. Some of the more common ones are listed here.

Interpenetrating primitive cubic (caesium chloride) structure


One structure is the "interpenetrating primitive cubic" structure, also called the "here.) Alternately, one could view this lattice as a simple cubic structure with a secondary atom in its cubic void.

Examples of compounds with this structure include caesium chloride itself, as well as certain other alkali halides when prepared at low temperatures or high pressures.[5] More 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 this structure is called Pm3m (in Hermann–Mauguin notation), or "221" (in the International Tables for Crystallography). The Strukturbericht designation is "B2".[6]


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.

Rock-salt structure

Another structure is the "rock salt" or "here.) Alternately, one could view this structure as a face-centered cubic structure with secondary atoms in its octahedral holes.

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".[5] More generally, this structure is more likely to be formed if the cation is slightly smaller than the anion (a cation/anion radius ratio of 0.414 to 0.732).

The space group of this structure is called Fm3m (in Hermann–Mauguin notation), or "225" (in the International Tables for Crystallography). The Strukturbericht designation is "B1".[7]

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,[8] 2.8 Å for NaCl,[9] and 3.2 Å for SnTe.[10]

Zincblende structure


Another common structure is the "zincblende" structure (also spelled "zinc blende"), named after the mineral here.)

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.

Additionally, if one were to take the two atoms to both be carbon, the zincblende structure is equivalent to the structure of diamond.

The space group of this structure is called F43m (in Hermann–Mauguin notation), or "#216" in the International Tables for Crystallography.[11][12] The Strukturbericht designation is "B3".[13]

See also

References

Further reading

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