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An example of the cubic crystals, halite

The cubic crystal system (or isometric) 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, called "simple cubic", "body-centered cubic" (bcc), and "face-centered cubic" (fcc), plus a number of 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.

Bravais lattices and point/space groups

The three Bravais lattices which form cubic crystal systems are

Simple cubic

Body-centered cubic

Face-centered cubic

The simple cubic system consists of one lattice point on each corner of the cube. Each atom at the lattice points is then shared equally between eight adjacent cubes, and the unit cell therefore contains in total one atom (1/8 * 8). The body-centered cubic system 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). Finally, the face-centered cubic has lattice points on the faces of the cube of which each unit cube gets exactly one half contribution, in addition to the corner lattice points, giving a total of 4 atoms per unit cell ((1/8 for each corner) * 8 corners + (1/2 for each face) * 6 faces).

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.

The point groups and space groups that fall under this crystal system are listed below, using the international notation.

Point group	#	Cubic space groups
	195-199	P23	F23	I23	P213	I213
	200-206	Pm	Pn	Fm	Fd	I	Pa	Ia
	207-214	P432	P4232	F432	F4132	I432	P4332	P4132	I4132
	215-220	P3m	F3m	I3m	P3n	F3c	I3d
	221-230	Pmm	Pnn	Pmn	Pnm	Fmm	Fmc	Fdm	Fdc
		Imm	Iad

There are 36 cubic space groups, of which 10 are hexoctahedral: Fd3c, Fd3m, Fm3c, Fm3m, Ia3d, Im3m, Pm3m, Pm3n, Pn3m, and Pn3n. Other terms for hexoctahedral are normal class, holohedral, ditesseral central class, galena type.

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 simple cubic lattice with cube side length a, the sphere size 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 that is theoretically possible for any lattice (see Close-packing), 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 hybridisation demands certain bond angles.) Accordingly, the simple-cubic structure, with especially-low atomic packing factor, is rare in nature, but is found in Polonium.^[1] The BCC and FCC, with their higher densities, are both quite common in nature. Examples of BCC include Fe-iron, Cr-chromium, W-tungsten, and Nb-niobium. Examples of FCC include lead (for example in lead(II) nitrate), Al- aluminium, Cu- copper, Au- gold and Ag- 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 simple cubic (caesium chloride) structure

A caesium chloride unit cell. The two colors of balls represent the two types of atoms.

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

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.^[2] More generally, this structure is more likely to be formed from two elements whose ions are of roughly the same size.

The space group of this structure is called "Pmm" (in Hermann-Mauguin notation), or "221" (in the International Tables for Crystallography). The Strukturbericht designation is "B2".^[3]

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

Rock-salt structure

Another structure is the "rock salt" or "sodium chloride" structure. In this, 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. (See picture here.)

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".^[2] 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).

Zincblende structure

A zincblende unit cell

Another common structure is the "zincblende" structure (also spelled "zinc blende"), named after the mineral zincblende. 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. Altogether, the arrangement of atoms is the same as diamond cubic structure, but with alternating types of atoms at the different lattice sites. (See picture here.)

Examples of compounds with this structure include zincblende itself, many compound semiconductors (such as gallium arsenide and cadmium telluride), and a wide array of other binary compounds.

The space group of this structure is called "F3m" (in Hermann-Mauguin notation), or "216" (in the International Tables for Crystallography)^[4][5]

See also

• Diamond cubic
• Reciprocal lattice
• Atomium: building which is a model of a bcc unit cell, with vertical body diagonal.
• Dislocations
• Crystal structure

References

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

1. ^ Greenwood, Norman N.; Earnshaw, A. (1997). Chemistry of the Elements, 2nd Edition, Oxford: Butterworth-Heinemann. ISBN 0-7506-3365-4. 
2. ^ ^{a b} Seitz, Modern Theory of Solids (1940), p.49
3. ^ [1]
4. ^ Quantum Theory of the Solid State, L. Kantorovich, page 32, Google Books link
5. ^ [2]

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• This page was last modified on 5 September 2008, at 16:59.

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