Cubic crystal system: Difference between revisions
Content deleted Content added
add a link to japanese version |
←Blanked the page |
||
| Line 1: | Line 1: | ||
[[Image:ImgSalt.jpg|thumb|right|200px|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 metallic crystals. |
|||
==Bravais lattices and point/space groups== |
|||
The three [[Bravais lattice]]s which form the cubic crystal system are |
|||
<gallery> |
|||
Image:Lattic_simple_cubic.svg|Simple cubic <math>a=2 \cdot R</math> |
|||
Image:Lattice_body_centered_cubic.svg|Body-centered cubic <math>\sqrt{3}a=4 \cdot R</math> |
|||
Image:Lattice_face_centered_cubic.svg|Face-centered cubic <math>\sqrt{2}a=4 \cdot R</math> |
|||
</gallery> |
|||
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 contribution 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 would result in a simple tetragonal Bravais lattice. |
|||
There are 8 lattice points on a simple cubic for each corner of the shape. There are 9 lattice points for a body centered because of the extra point in the center of the unit. There are 14 lattice points on a face centered cubic. |
|||
The [[crystallographic point group|point groups]] and [[space groups]] that fall under this crystal system are listed below, using the international notation. |
|||
{| align=center cellpadding=2 |
|||
|----- align=center |
|||
| bgcolor=#c0ffff | Point group |
|||
| bgcolor=#a0ff80 | # |
|||
| bgcolor=#ffffc0 colspan=8| Cubic space groups |
|||
|----- valign=top |
|||
| bgcolor=#c0ffff align=center valign=center| <math>23\,\!</math> |
|||
| bgcolor=#a0ff80 align=center valign=center| <small>195-199</small> |
|||
| P23 || F23 || I23 || P2<sub>1</sub>3 || I2<sub>1</sub>3 || colspan=3| |
|||
|----- valign=top bgcolor=#f4f4f4 |
|||
| bgcolor=#c0ffff align=center valign=center| <math>m\bar3\,\!</math> |
|||
| bgcolor=#a0ff80 align=center valign=center| <small>200-206</small> |
|||
| 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> || |
|||
|----- valign=top |
|||
| bgcolor=#c0ffff align=center valign=center| <math>432\,\!</math> |
|||
| bgcolor=#a0ff80 align=center valign=center| <small>207-214</small> |
|||
| P432 || P4<sub>2</sub>32 || F432 || F4<sub>1</sub>32 || I432 || P4<sub>3</sub>32 || P4<sub>1</sub>32 || I4<sub>1</sub>32 |
|||
|----- valign=top bgcolor=#f4f4f4 |
|||
| bgcolor=#c0ffff align=center valign=center| <math>\bar4 3m\,\!</math> |
|||
| bgcolor=#a0ff80 align=center valign=center| <small>215-220</small> |
|||
| 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 || colspan=2| |
|||
|----- valign=top |
|||
| bgcolor=#c0ffff align=center valign=center rowspan=2| <math>m\bar3 m\,\!</math> |
|||
| bgcolor=#a0ff80 align=center valign=center rowspan=2| <small>221-230</small> |
|||
| 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 |
|||
|----- valign=top |
|||
| 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. |
|||
==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. |
|||
==Single-element lattices== |
|||
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, but is found in [[Polonium]] <ref>{{Greenwood&Earnshaw}}</ref>. 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 Fe-[[iron]], Cr-[[chromium]], W-[[tungsten]], and Nb-[[niobium]]. |
|||
Finally, the face centered cubic crystals have a density of approximately 0.740<!-- pi/sqrt(18) -->, 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]]), Al- [[aluminium]], Cu- [[copper]], Au- [[gold]] and Ag- [[silver]]. |
|||
==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''' structure, 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. This structure actually has a simple cubic lattice with a two atom basis, the atom positions being atom A at (0,0,0) and atom B at(0.5,0.5.0.5) |
|||
However, if the cation is slightly smaller than the anion (a cation/anion radius ratio of 0.414 to 0.732), the crystal forms a different structure, '''interpenetrating FCC'''. When drawn separately, both atoms are arranged in an FCC structure. This structure has an FCC lattice, with a two atom basis, the atom positions being atom A at (0,0,0) and atom B at (0.5,0.5,0.5). [[Sodium chloride]] is a common example of this type of structure |
|||
==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 |
|||
<references/> |
|||
[[Category:Crystallography]] |
|||
[[de:Kubisches Kristallsystem#Kubisch-raumzentriertes_Gitter]] |
|||
[[et:Kuubiline süngoonia]] |
|||
[[eo:Kuba kristalsistemo]] |
|||
[[ja:立方晶]] |
|||
[[ko:입방정계]] |
|||
[[lv:Kubiskā singonija]] |
|||
[[nl:Kubisch (kristallografie)]] |
|||
[[pl:Układ regularny]] |
|||
[[ru:Кубическая сингония]] |
|||
[[sk:Kubická sústava]] |
|||
[[sr:Тесерална кристална система]] |
|||
[[vi:Hệ tinh thể lập phương]] |
|||
[[uk:Гранецентрована кубічна ґратка]] |
|||
[[zh:等轴晶系]] |
|||