Cubic crystal system
Learn more about Cubic crystal system
The cubic 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 metallic crystals.
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[edit] Bravais lattices and point/space groups
The three Bravais lattices that form the cubic crystal system are:
The simple cubic system consists of one lattice point on each corner of the cube. Each lattice point is then shared equally between eight adjacent cubes, and the unit cell therefore contains in total one lattice point (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 in total 2 lattice points per cell ((1/8)*8 + 1). Finally, the face centered cubic lattice has lattice points on the faces of the cube, giving a total of 4 lattice points (1/8 * 8 + 1/2*6).
The point groups and space groups that fall under this crystal system are listed below, using the international notation.
Point group | # | Cubic space groups | |||||||
23 | 195-199 | P23 | F23 | I23 | P2_{1}3 | I2_{1}3 | |||
m<math>\bar3</math> | 200-206 | Pm<math>\bar3</math> | Pn<math>\bar3</math> | Fm<math>\bar3</math> | Fd<math>\bar3</math> | I<math>\bar3</math> | Pa<math>\bar3</math> | Ia<math>\bar3</math> | |
432 | 207-214 | P432 | P4_{2}32 | F432 | F4_{1}32 | I432 | P4_{3}32 | P4_{1}32 | I4_{1}32 |
<math>\bar4</math>3m | 215-220 | P<math>\bar4</math>3m | F<math>\bar4</math>3m | I<math>\bar4</math>3m | P<math>\bar4</math>3n | F<math>\bar4</math>3c | I<math>\bar4</math>3d | ||
m<math>\bar3</math>m | 221-230 | Pm<math>\bar3</math>m | Pn<math>\bar3</math>n | Pm<math>\bar3</math>n | Pn<math>\bar3</math>m | Fm<math>\bar3</math>m | Fm<math>\bar3</math>c | Fd<math>\bar3</math>m | Fd<math>\bar3</math>c |
Im<math>\bar3</math>m | Ia<math>\bar3</math>d |
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.
[edit] Atomic packing factors and examples
The cubic crystal system is one of the most common crystal systems found in elemental metals, and naturally occurring crystals and minerals. One very useful way to analyse a crystal is to consider the atomic packing factor. In this approach, the amount of space which is filled by the atoms is calculated under the assumption that they are spherical.
[edit] Single-element compounds
Assuming one atom per lattice point, the atomic packing factor of the simple cubic system is only 0.524. Due to its low density, this is a high energy structure and is rare in nature. Similarly, the body centered structure has a density of 0.680. The higher density makes this a low energy structure which is fairly common in nature. Examples include iron, chromium, and tungsten.
Finally, the face centered cubic crystals have a density of 0.741, a ratio that it shares with several other systems, including hexagonal close packed and one version of tetrahedral BCC. This is the most tightly packed crystal possible with spherical atoms. Due to its low energy, FCC is extremely common, examples include lead (for example in lead(II) nitrate), aluminum, copper, and gold.
[edit] Multi-element compounds
When the compound is formed of two elements whose ions are of roughly the same size, they have what is called the interpenetrating simple cubic, where two atoms of a different type have individual simple cubic crystals. However, the unit cell consists of the atom of one being in the middle of the 8 vertices, structurally resembling body centered cubic. The most common example is caesium chloride CsCl.
However, if the cation is slightly smaller than the anion (a cation/anion radius ratio of 0.414 to 0.732), it forms a different structure, interpenetrating FCC. When drawn separately, both atoms are arranged in an FCC structure. The unit cell for this is shown to the left.
[edit] See also
- Reciprocal lattice
- Atomium - building which is a model of a bcc unit cell, with vertical body diagonal.
[edit] References
- Hurlbut, Cornelius S.; Klein, Cornelis, 1985, Manual of Mineralogy, 20th ed., Wiley, ISBN 0-471-80580-7de:Kubisches Kristallsystem
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