Cubic crystal system
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Template:Confusing In crystallography, the cubic crystal system (or isometric crystal system) is the most symmetric of the 7 crystal systems. The system is composed of the three Bravais lattices whose symmetry group is that of a cube.
The three Bravais lattices that form the cubic crystal system are:
| simple cubic (sc) | body-centered cubic (bcc) | face-centered cubic (fcc) |
| Image:Cubic crystal shape.png | Image:Cubic-body-centered.png | Image:Cubic-face-centered.png |
The cubes drawn are the conventional unit cells. For a cube whose vertices include 000 and 200, bcc has additional lattice point 111, while fcc has 110, 101, and 011. For bcc the primitive cells have a volume of 1/2 of the cube, e.g. the parallelepiped 000 200 020 220 111 311 131 331 with primitive translation vectors 200, 020, and 111, with determinant 4. For fcc the primitive cells have a volume of 1/4 of the cube, e.g. the parallelepiped 110 220 020 130 101 211 011 121 with primitive translation vectors 110, -1 1 0, and 0 -1 1, with determinant 2.
As can be seen by turning the base plane 45°, bcc and fcc only differ by a vertical scaling: in both cases the lattice points in the middle layer are above the centers of the squares of the base layer. Both scales are "special", allowing a cubic symmetry: for bcc the middle layer has a height of 1/2 of the grid size of the square grid of each layer, while for fcc the middle layer has a height of 1/2 √2 of that grid size. For other scalings both are the same, body-centered tetragonal.
Perpendicular to each body diagonal, fcc has hexagonal layers, with three positionings, which are cyclically changed. Two opposite vertices of the cube have two layers in between. See also: close-packing
The point groups that fall under this crystal system are listed below, followed by their representations in international notation and Schoenflies notation, and mineral examples.
| name | international | Schoenflies | example | |
| hexoctahedral | <math>\frac4m\overline{3}\frac2m</math> | or <math>m3m</math> | Oh | galena |
| gyroidal | 432 | O | no known minerals | |
| diploidal | <math>\frac2m \overline{3}</math> | or <math>m3</math> | Th | pyrite |
| tetrahedral | <math>\overline{4}3m</math> | Td | sphalerite | |
| tetartohedral | 23 | T | ||
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.
Halite structure
Image:Sodium chloride crystal.png Sodium chloride forms fcc crystals. In these, the larger chloride ions are arranged in a cubic close-packing, while the smaller sodium ions fill the octahedral gaps between them. Each ion is surrounded by six of the other kind. This same basic structure is found in many other minerals, and is known as the halite structure.
The sodium ions indicated in blue show that the fcc lattice can also be represented by mid-edge and centered lattice points, without lattice points at the vertices of the cube.
See also
- Reciprocal lattice
- Atomium - building which is a model of a bcc unit cell, with vertical body diagonal.
References
- Hurlbut, Cornelius S.; Klein, Cornelis, 1985, Manual of Mineralogy, 20th ed., Wiley, ISBN 0471805807de:Kubisches Kristallsystem
et:Kuubiline süngoonia nl:Kubisch ru:Кубическая сингония zh:等轴晶系