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. There are three main varieties of these crystals: 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: 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 ( × 8). 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 ( × 8 + 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 ( × 8 from the corners plus  × 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. 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  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, and space groups are listed in the table below. There are a total 36 cubic space groups. Other terms for hexoctahedral are: normal class, holohedral, ditesseral central class, galena type. Voids in the unit cell. Voids in the unit cell are also known as interstitial sites or holes. They are the  empty space that exists between the packing of atoms (spheres) in the unit cells. The holes are easy to see if you try to pack circles together. No matter how close you get them or how you arrange them, you will have empty space in between. The same is true in a unit cell, no matter how the atoms are arranged, there will be interstitial sites present between the atoms. These sites or holes can be filled with other atoms.  The picture with packed circles is only a 2D representation.  In a crystal lattice, the atoms (spheres) would be packed  in a 3D arrangement.  This results in different shaped interstitial sites depending on the arrangement of the atoms in the lattice. A simple cubic unit cell, with stacks of atoms arranged as if at the eight corners of a cube would form a single cubic hole or void in the center. A close packed unit cell, both face-centered cubic and hexagonal close packed, can form two different shaped holes.  Looking at the three green spheres in the hexagonal packing illustration, they form a triangle shaped hole.  If an atom is arranged on top of this triangular hole it forms  a tetrahedral interstitial hole. If the three atoms in the layer above are rotated and their triangular hole sits on top of this one, it forms an octahedral interstitial hole. In a close-packed structure there are 4 atoms per unit cell and it will have 4 octahedral voids (1:1 ratio) and 8 tetrahedral voids (1:2 ratio) per unit cell. The tetrahedral void is smaller in size and could fit an atom with a radius 0.225 times the size of the atoms making up the lattice.  An octahedral void could fit an atom with a radius 0.441 times the size  of the atoms making up the lattice. An atom that fills this empty space could be larger than this ideal radius ratio, which would lead to a distorted lattice due to pushing out the surrounding atoms, but it cannot be smaller than this ratio. 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. 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 and the atomic packing factor turns out to be about 0.524 (which is quite low). This means that the volume of the unit cell actually occupied by atoms is only 52.4%. 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". Single element structures. 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. 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. Another important cubic crystal structure is the diamond cubic structure, which can appear in carbon, silicon, germanium, and tin . Unlike fcc and bcc, this structure is not a lattice, since it contains multiple atoms in its primitive cell. Other cubic elemental structures include the A15 structure found in Tungsten, and the extremely complicated structure of manganese. 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. One structure is the "interpenetrating primitive cubic" structure, also called the "caesium chloride" structure. This structure is often confused for a body-centered cubic structure, because the  arrangement of atoms is the same. The true structure is shown in the graphic showing two individual primitive cubic structures that are superimposed within each other with the corner of one structure in the center of the cube of the other structure.  It helps to convince yourself that it is not body-centered cubic because there is no translational symmetry along the ½, ½, ½, plane, the chloride would be translated into a cesium, not another chloride. It works the same way for the NaCl structure described in the next section.  If you take out the Cl atoms, the leftover  Na atoms still form an FCC structure, not a simple cubic structure. 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. 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. 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. The space group of the caesium chloride (CsCl) structure is called Pmm (in Hermann–Mauguin notation), or "221  (in the International Tables for Crystallography). The Strukturbericht designation is B2". There are nearly a hundred rare earth intermetallic compounds that crystalize in the CsCl structure, including many binary compounds of rare earths with magnesium, and with elements in groups 11, 12, and 13. Other compounds showing caesium chloride like structure are CsBr, CsI, high-temperature RbCl, AlCo, AgZn, BeCu, MgCe, RuAl and SrTl. Rock-salt structure. The space group of the rock-salt (NaCl) structure is called Fmm (in Hermann–Mauguin notation), or "225" (in the International Tables for Crystallography). The Strukturbericht designation is "B1". 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. Earlier it was stated that there is one octahedral hole per atom in a cubic close packed structure.  In sodium chloride there is a 1:1 ratio of sodium to chlorine atoms.  The structure can also be described as an FCC lattice of sodium with chlorine occupying each octahedral void or vice versa. 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". According to the radius ratio rule, 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 m) for NaF, 2.8 Å for NaCl, and 3.2 Å for SnTe. Most of the alkali metal hydrides and halides have the rock salt structure, though a few have the caesium chloride structure instead. Many transition metal monoxides also have the rock salt structure (TiO, VO, CrO, MnO, FeO, CoO, NiO, CdO). The early actinoid monocarbides also have this structure (ThC, PaC, UC, NpC, PuC). Other compounds showing rock salt like structure are TiB, ZrB, PbS, PbSe, PbTe, SnTe, AgF, AgCl, and AgBr. Fluorite structure. Much like the rock salt structure, the fluorite  structure (AB) is also an Fmm structure but has 1:2 ratio of ions. The anti-fluorite structure is nearly identical, except the positions of the anions and cations are switched in the structure. They are designated Wyckoff positions 4a and 8c whereas the rock-salt structure positions are 4a and 4b. Zincblende structure. The space group of the Zincblende structure is called F3m (in Hermann–Mauguin notation), or 216. The Strukturbericht designation is "B3". 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. In zinc sulfide the ratio of zinc to sulfur is 1:1. 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. The boron group pnictogenides usually have a zincblende structure, though the nitrides are more common in the wurtzite structure, and their zincblende forms are less well known polymorphs. This group is also known as the II-VI family of compounds, most of which can be made in both the zincblende (cubic) or wurtzite (hexagonal) form. This group is also known as the III-V family of compounds. Heusler structure. The Huesler structure, based on the  structure of CuMnAl, is a common structure for ternary compounds involving transition metals. It has the space group Fmm (No. 225), and the Strukturbericht designation is L2. Together with the closely related half-Heusler and inverse-Huesler compounds, there are hundreds of examples. Iron monosilicide structure. The space group of the iron monosilicide structure is P23 (No. 198), and the Strukturbericht designation is B20. This is a chiral structure, and is sometimes associated with helimagnetic properties. There are four atoms of each element for a total of eight atoms in the unit cell. Examples occur among the transition metal silicides  and germanides, as well as a few other compounds such as gallium palladide. Weaire–Phelan structure. The Weaire–Phelan structure has Pmn (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.