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, 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 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 (​1⁄8 × 8).[1]

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 (​1⁄8 × 8 + 1).[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 (​1⁄8 × 8 from the corners plus ​1⁄2 × 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 system 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 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, 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

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, 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.

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

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 ​a⁄2 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 (hcp) 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][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.

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.

Caesium chloride structure
A caesium chloride unit cell. The two colors of spheres represent the two types of atoms.

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".[6]

One structure is the "interpenetrating primitive cubic" structure, also called the "caesium chloride" structure. Each of the two atom types forms a separate primitive cubic lattice, with an atom of one type at the center of each cube of the other type. 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 (see picture here). 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 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.

Other compounds showing caesium chloride like structure are CsBr, CsI, high-temp RbCl, AlCo, AgZn, BeCu, MgCe, RuAl and SrTl.

Rock-salt structure
The rock-salt crystal structure. Each atom has six nearest neighbors, 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".[8]

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. 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".[7] More generally, 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,[9] 2.8 Å for NaCl,[10] and 3.2 Å for SnTe.[11]

Other compounds showing rock salt like structure are LiF[12], LiCl, LiBr, LiI, NaF[12], NaBr, NaI, KF[12], KCl, KBr, KI, RbF, RbCl, RbBr, RbI, CsF, MgO, PbS, AgF, AgCl, AgBr and ScN.[13]

The fluorite structure and anti-fluorite structures (AB2) are also Fm3m structures but have a 1:2 ratio of ions. They are designated Wyckoff positions 4a and 8c whereas the rock-salt structure positions are 4a and 4b.[14][15]

Zincblende structure
A zincblende unit cell

The space group of the Zincblende structure is called F43m (in Hermann–Mauguin notation), or 216.[16][17] The Strukturbericht designation is "B3".[18]

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. 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.

Other compounds showing zinc blende-like structure are α-AgI, β-BN, diamond, CuBr, β-CdS, BP and BAs.

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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See also

References
  1. P. M. de Wolff, N. V. Belov, E. F. Bertaut, M. J. Buerger, J. D. H. Donnay, W. Fischer, Th. Hahn, V. A. Koptsik, A. L. Mackay, H. Wondratschek, A. J. C. Wilson and S. C. Abrahams (1985). "Nomenclature for crystal families, Bravais-lattice types and arithmetic classes. Report of the International Union of Crystallography Ad-Hoc Committee on the Nomenclature of Symmetry". Acta Crystallographica Section A. 41: 278. doi:10.1107/S0108767385000587.CS1 maint: multiple names: authors list (link)
  2. Prince, E., ed. (2006). International Tables for Crystallography. International Union of Crystallography. doi:10.1107/97809553602060000001. ISBN 978-1-4020-4969-9.
  3. Crystallography and Minerals Arranged by Crystal Form, Webmineral
  4. Greenwood, Norman N.; Earnshaw, Alan (1997). Chemistry of the Elements (2nd ed.). Butterworth-Heinemann. ISBN 978-0-08-037941-8.
  5. The original discovery was in J. Chem. Phys. 14, 569 (1946).
  6. The CsCl (B2) Structure Archived 2008-09-15 at the Wayback Machine
  7. Seitz, Modern Theory of Solids (1940), p.49
  8. The NaCl (B1) Structure Archived 2008-10-19 at the Wayback Machine
  9. Sundquist, J. J.; Lin, C. C. (1981). "Electronic structure of the F centre in a sodium fluoride crystal". Journal of Physics C: Solid State Physics. 14 (32): 4797–4805. Bibcode:1981JPhC...14.4797S. doi:10.1088/0022-3719/14/32/016.
  10. Abrahams, S. C.; Bernstein, J. L. (1965). "Accuracy of an automatic diffractometer. Measurement of the sodium chloride structure factors". Acta Crystallogr. 18 (5): 926–932. doi:10.1107/S0365110X65002244.
  11. Kao, W.; Peretti, E. (1970). "The ternary subsystem Sn4As3-SnAs-SnTe". Journal of the Less Common Metals. 22: 39–50. doi:10.1016/0022-5088(70)90174-8.
  12. J. Aigueperse, P. Mollard, D. Devilliers, M. Chemla, R. Faron, R. Romano, J. P. Cuer, "Fluorine Compounds, Inorganic" (section 4) in Ullmann’s Encyclopedia of Industrial Chemistry, Wiley-VCH, Weinheim, 2005. doi:10.1002/14356007.a11_307.
  13. Gu, Zheng; Edgar, J H; Pomeroy, J; Kuball, M; Coffey, D W (August 2004). "Crystal Growth and Properties of Scandium Nitride". Journal of Materials Science: Materials in Electronics. 15 (8): 555–559. doi:10.1023/B:JMSE.0000032591.54107.2c.
  14. "Fluorite". aflow.org. Retrieved 2020-05-22.
  15. "Rock Salt". aflow.org. Retrieved 2020-05-22.
  16. L. Kantorovich (2004). Quantum Theory of the Solid State. Springer. p. 32. ISBN 1-4020-2153-4.
  17. Birkbeck College, University of London
  18. The Zincblende (B3) Structure Archived October 19, 2008, at the Wayback Machine

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