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A rock containing three crystals of pyrite (FeS2). The crystal structure of pyrite is primitive cubic, and this is reflected in the cubic symmetry of its natural crystal facets.
A network model of a primitive cubic system
The primitive and cubic close-packed (also known as face-centered cubic) unit cells

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:

  • Primitive cubic (abbreviated cP and alternatively called simple cubic)
  • Body-centered cubic (abbreviated cI or bcc)
  • Face-centered cubic (abbreviated cF or fcc)

Note: the term fcc is often used in synonym for the cubic close-packed or ccp structure occurring in metals. However, fcc stands for a face-centered cubic Bravais lattice, which is not necessarily close-packed when a motif is set onto the lattice points. E.g. the diamond and the zincblende lattices are fcc but not close-packed. Each is subdivided into other variants listed below. Although the unit cells in these crystals are conventionally taken to be cubes, the primitive unit cells often are not.

Bravais lattices

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Further information: Bravais lattice

The three Bravais latices in the cubic crystal system are:

Bravais lattice Primitive
cubic 		Body-centered
cubic 		Face-centered
cubic
Pearson symbol cP cI cF
Unit cell

The primitive cubic lattice (cP) consists of one lattice point on each corner of the cube; this means each simple cubic unit cell has in total one lattice point. Each atom at a lattice point is then shared equally between eight adjacent cubes, and the unit cell therefore contains in total one atom (1⁄8 × 8).[1]

The body-centered cubic lattice (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 two lattice points per unit cell (1⁄8 × 8 + 1).[1]

The face-centered cubic lattice (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 four lattice points per unit cell (1⁄8 × 8 from the corners plus 1⁄2 × 6 from the faces).

primitive translations of FCC lattice

The face-centered cubic lattice is closely related to the hexagonal close packed (hcp) system, where two systems differ only in the relative placements of their hexagonal layers. The [111] plane of a face-centered cubic lattice is a hexagonal grid.

Attempting to create a base-centered cubic lattice (i.e., putting an extra lattice point in the center of each horizontal face) results in a simple tetragonal Bravais lattice.

Coordination number (CN) is the number of nearest neighbors of a central atom in the structure.[1] Each sphere in a cP lattice has coordination number 6, in a cI lattice 8, and in a cF lattice 12.

Atomic packing factor (APF) is the fraction of volume that is occupied by atoms. The cP lattice has an APF of about 0.524, the cI lattice an APF of about 0.680, and the cF lattice an APF of about 0.740.

Crystal classes

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Further information: Crystallographic point group

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,[2] and space groups are listed in the table below. There are a total 36 cubic space groups.

No. Point group Type Example Space groups
Name[3] Schön. Intl Orb. Cox. Primitive Face-centered Body-centered
195–197 Tetartoidal T 23 332 [3,3]+ enantiomorphic Ullmannite, Sodium chlorate P23 F23 I23
198–199 P213 I213
200–204 Diploidal Th 2/m3
(m3) 		3*2 [3+,4] centrosymmetric Pyrite Pm3, Pn3 Fm3, Fd3 I3
205–206 Pa3 Ia3
207–211 Gyroidal O 432 432 [3,4]+ enantiomorphic Petzite P432, P4232 F432, F4132 I432
212–214 P4332, P4132 I4132
215–217 Hextetrahedral Td 43m *332 [3,3] Sphalerite P43m F43m I43m
218–220 P43n F43c I43d
221–230 Hexoctahedral Oh 4/m32/m
(m3m) 		*432 [3,4] centrosymmetric Galena, Halite Pm3m, Pn3n, Pm3n, Pn3m Fm3m, Fm3c, Fd3m, Fd3c Im3m, Ia3d

Other terms for hexoctahedral are: normal class, holohedral, ditesseral central class, galena type.

Single element structures

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Visualisation of a diamond cubic unit cell: 1. Components of a unit cell, 2. One unit cell, 3. A lattice of 3 x 3 x 3 unit cells
See also: Periodic table (crystal structure)

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.[4][5] 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 structures

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Compounds that consist of more than one element (e.g. binary compounds) often have crystal structures based on the cubic crystal system. Some of the more common ones are listed here. These structures can be viewed as two or more interpenetrating sublattices where each sublattice occupies the interstitial sites of the others.

Caesium chloride structure

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See also: Category:Caesium chloride crystal structure
A caesium chloride unit cell. The two colors of spheres represent the two types of atoms.

One structure is the "interpenetrating primitive cubic" structure, also called a "caesium chloride" or B2 structure. This structure is often confused for a body-centered cubic structure because the arrangement of atoms is the same. However, the caesium chloride structure has a basis composed of two different atomic species. In a body-centered cubic structure, there would be translational symmetry along the [111] direction. In the caesium chloride structure, translation along the [111] direction results in a change of species. The structure can also be thought of as two separate simple cubic structures, one of each species, that are superimposed within each other. The corner of the chloride cube is the center of the caesium cube, and vice versa.[6]

This graphic shows the interlocking simple cubic lattices of cesium and chlorine. You can see them separately and as they are interlocked in what looks like a body-centered cubic arrangement

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 coordination number is eight. 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. 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.[7] 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 space group of the caesium chloride (CsCl) structure is called Pm3m (in Hermann–Mauguin notation), or "221" (in the International Tables for Crystallography). The Strukturbericht designation is "B2".[8]

There are nearly a hundred rare earth intermetallic compounds that crystallize in the CsCl structure, including many binary compounds of rare earths with magnesium,[9] and with elements in groups 11, 12,[10][11] and 13. Other compounds showing caesium chloride like structure are CsBr, CsI, high-temperature RbCl, AlCo, AgZn, BeCu, MgCe, RuAl and SrTl.[citation needed]

Rock-salt structure

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See also: Category:Rock salt crystal structure
The rock-salt crystal structure. Each atom has six nearest neighbours, with octahedral geometry.

The space group of the rock-salt or halite (sodium chloride) structure is denoted as Fm3m (in Hermann–Mauguin notation), or "225" (in the International Tables for Crystallography). The Strukturbericht designation is "B1".[12]

In the rock-salt 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. The rock-salt structure has octahedral coordination: Each atom's nearest neighbors consist of six atoms of the opposite type, positioned like the six vertices of a regular octahedron. 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.[6]

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".[7] 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 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 × 10−10 m) for NaF,[13] 2.8 Å for NaCl,[14] and 3.2 Å for SnTe.[15] Most of the alkali metal hydrides and halides have the rock salt structure, though a few have the caesium chloride structure instead.

Alkali metal hydrides and halides with the rock salt structure
Hydrides Fluorides Chlorides Bromides Iodides
Lithium Lithium hydride Lithium fluoride[16] Lithium chloride Lithium bromide Lithium iodide
Sodium Sodium hydride Sodium fluoride[16] Sodium chloride Sodium bromide Sodium iodide
Potassium Potassium hydride Potassium fluoride[16] Potassium chloride Potassium bromide Potassium iodide
Rubidium Rubidium hydride Rubidium fluoride Rubidium chloride Rubidium bromide Rubidium iodide
Caesium Caesium hydride Caesium fluoride (CsCl structure)
Alkaline earth metal chalcogenides with the rock salt structure
Oxides Sulfides Selenides Tellurides Polonides
Magnesium Magnesium oxide Magnesium sulfide Magnesium selenide[17] Magnesium telluride[18] (NiAs structure)
Calcium Calcium oxide Calcium sulfide Calcium selenide[19] Calcium telluride Calcium polonide[20]
Strontium Strontium oxide Strontium sulfide Strontium selenide Strontium telluride Strontium polonide[20]
Barium Barium oxide Barium sulfide Barium selenide Barium telluride Barium polonide[20]
Rare-earth[21] and actinoid pnictides with the rock salt structure
Nitrides Phosphides Arsenides Antimonides Bismuthides
Scandium Scandium nitride Scandium phosphide Scandium arsenide[22] Scandium antimonide[23] Scandium bismuthide[24]
Yttrium Yttrium nitride Yttrium phosphide Yttrium arsenide[25] Yttrium antimonide Yttrium bismuthide[26]
Lanthanum Lanthanum nitride[27] Lanthanum phosphide[28] Lanthanum arsenide[25] Lanthanum antimonide Lanthanum bismuthide[29]
Cerium Cerium nitride[27] Cerium phosphide[28] Cerium arsenide[25] Cerium antimonide Cerium bismuthide[29]
Praseodymium Praseodymium nitride[27] Praseodymium phosphide[28] Praseodymium arsenide[25] Praseodymium antimonide[30] Praseodymium bismuthide[29]
Neodymium Neodymium nitride[27] Neodymium phosphide[28] Neodymium arsenide[25] Neodymium antimonide[30] Neodymium bismuthide[29]
Promethium ? ? ? ? ?
Samarium Samarium nitride[27] Samarium phosphide[28] Samarium arsenide[25] Samarium antimonide[30] Samarium bismuthide[29]
Europium Europium nitride[27] Europium phosphide (Na2O2 structure)[31] (unstable)[32]
Gadolinium Gadolinium nitride[27] Gadolinium phosphide Gadolinium arsenide[25] Gadolinium antimonide[30] Gadolinium bismuthide[29]
Terbium Terbium nitride[27] Terbium phosphide Terbium arsenide[25] Terbium antimonide[30] Terbium bismuthide[29]
Dysprosium Dysprosium nitride[27] Dysprosium phosphide Dysprosium arsenide Dysprosium antimonide Dysprosium bismuthide[29]
Holmium Holmium nitride[27] Holmium phosphide Holmium arsenide[25] Holmium antimonide Holmium bismuthide[29]
Erbium Erbium nitride[27] Erbium phosphide Erbium arsenide[25] Erbium antimonide Erbium bismuthide[29]
Thulium Thulium nitride[27] Thulium phosphide Thulium arsenide Thulium antimonide Thulium bismuthide[29]
Ytterbium Ytterbium nitride[27] Ytterbium phosphide Ytterbium arsenide[25] Ytterbium antimonide (unstable)[33][34]
Lutetium Lutetium nitride[27] Lutetium phosphide Lutetium arsenide Lutetium antimonide Lutetium bismuthide
Actinium ? ? ? ? ?
Thorium Thorium nitride[35] Thorium phosphide[35] Thorium arsenide[35] Thorium antimonide[35] (CsCl structure)
Protactinium ? ? ? ? ?
Uranium Uranium nitride[35] Uranium monophosphide[35] Uranium arsenide[35] Uranium antimonide[35] Uranium bismuthide[36]
Neptunium Neptunium nitride Neptunium phosphide Neptunium arsenide Neptunium antimonide Neptunium bismuthide[36]
Plutonium Plutonium nitride[35] Plutonium phosphide[35] Plutonium arsenide[35] Plutonium antimonide[35] Plutonium bismuthide[36]
Americium Americium nitride[36] Americium phosphide[36] Americium arsenide[36] Americium antimonide[36] Americium bismuthide[36]
Curium Curium nitride[37] Curium phosphide[37] Curium arsenide[37] Curium antimonide[37] Curium bismuthide[37]
Berkelium Berkelium nitride[37] Berkelium phosphide[37] Berkelium arsenide[37] ? Berkelium bismuthide[37]
Californium ? ? Californium arsenide[37] ? Californium bismuthide[37]
Rare-earth and actinoid chalcogenides with the rock salt structure
Oxides Sulfides Selenides Tellurides Polonides
Scandium (unstable)[38] Scandium monosulfide
Yttrium Yttrium monosulfide[39]
Lanthanum Lanthanum monosulfide[40]
Cerium Cerium monosulfide[40] Cerium monoselenide[41] Cerium monotelluride[41]
Praseodymium Praseodymium monosulfide[40] Praseodymium monoselenide[41] Praseodymium monotelluride[41]
Neodymium Neodymium monosulfide[40] Neodymium monoselenide[41] Neodymium monotelluride[41]
Promethium ? ? ? ?
Samarium Samarium monosulfide[40] Samarium monoselenide Samarium monotelluride Samarium monopolonide[42]
Europium Europium monoxide Europium monosulfide[40] Europium monoselenide[43] Europium monotelluride[43] Europium monopolonide[42]
Gadolinium (unstable)[38] Gadolinium monosulfide[40]
Terbium Terbium monosulfide[40] Terbium monopolonide[42]
Dysprosium Dysprosium monosulfide[40] Dysprosium monopolonide[42]
Holmium Holmium monosulfide[40] Holmium monopolonide[42]
Erbium Erbium monosulfide[40]
Thulium Thulium monosulfide[40] Thulium monopolonide[42]
Ytterbium Ytterbium monoxide Ytterbium monosulfide[40] Ytterbium monopolonide[42]
Lutetium (unstable)[38][44] Lutetium monosulfide[40] Lutetium monopolonide[42]
Actinium ? ? ? ?
Thorium Thorium monosulfide[35] Thorium monoselenide[35] (CsCl structure)[45]
Protactinium ? ? ? ?
Uranium Uranium monosulfide[35] Uranium monoselenide[35] Uranium monotelluride[35]
Neptunium Neptunium monosulfide Neptunium monoselenide Neptunium monotelluride
Plutonium Plutonium monosulfide[35] Plutonium monoselenide[35] Plutonium monotelluride[35]
Americium Americium monosulfide[36] Americium monoselenide[36] Americium monotelluride[36]
Curium Curium monosulfide[37] Curium monoselenide[37] Curium monotelluride[37]
Transition metal carbides and nitrides with the rock salt structure
Carbides Nitrides
Titanium Titanium carbide Titanium nitride
Zirconium Zirconium carbide Zirconium nitride
Hafnium Hafnium carbide Hafnium nitride[46]
Vanadium Vanadium carbide Vanadium nitride
Niobium Niobium carbide Niobium nitride
Tantalum Tantalum carbide (CoSn structure)
Chromium (unstable)[47] Chromium nitride

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).[37]

Fluorite structure

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Main article: Fluorite structure
See also: Category:Fluorite crystal structure

Much like the rock salt structure, the fluorite structure (AB2) is also an Fm3m 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.[48][49]

Zincblende structure

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See also: Category:Zincblende crystal structure
A zincblende unit cell

The space group of the Zincblende structure is called F43m (in Hermann–Mauguin notation), or 216.[50][51] The Strukturbericht designation is "B3".[52]

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.[6] 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. The structure can also be described as an FCC lattice of zinc with sulfur atoms occupying half of the tetrahedral voids or vice versa.[6]

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.[citation needed] 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.[53][54]

Copper halides with the zincblende structure
Fluorides Chlorides Bromides Iodides
Copper Copper(I) fluoride Copper(I) chloride Copper(I) bromide Copper(I) iodide
Beryllium and Group 12 chalcogenides with the zincblende structure
Sulfides Selenides Tellurides Polonides
Beryllium Beryllium sulfide Beryllium selenide Beryllium telluride Beryllium polonide[55][56]
Zinc Zinc sulfide Zinc selenide Zinc telluride Zinc polonide
Cadmium Cadmium sulfide Cadmium selenide Cadmium telluride Cadmium polonide
Mercury Mercury sulfide Mercury selenide Mercury telluride –

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.

Group 13 pnictogenides with the zincblende structure
Nitrides Phosphides Arsenides Antimonides
Boron Boron nitride* Boron phosphide Boron arsenide Boron antimonide
Aluminium Aluminium nitride* Aluminium phosphide Aluminium arsenide Aluminium antimonide
Gallium Gallium nitride* Gallium phosphide Gallium arsenide Gallium antimonide
Indium Indium nitride* Indium phosphide Indium arsenide Indium antimonide

This group is also known as the III-V family of compounds.

The structure of the Heusler compounds with formula X2YZ (e. g., Co2MnSi).

Heusler structure

[edit]
Main article: Heusler compound

The Heusler structure, based on the structure of Cu2MnAl, is a common structure for ternary compounds involving transition metals. It has the space group Fm3m (No. 225), and the Strukturbericht designation is L21. Together with the closely related half-Heusler and inverse-Huesler compounds, there are hundreds of examples.

Iron monosilicide structure

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See also: Category:Iron monosilicide structure type
Diagram of the iron monosilicide structure.

The space group of the iron monosilicide structure is P213 (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.

Transition metal silicides and germanides with the FeSi structure
Silicides Germanides
Manganese Manganese monosilicide Manganese germanide
Iron Iron monosilicide Iron germanide
Cobalt Cobalt monosilicide Cobalt germanide
Chromium Chromium(IV) silicide Chromium(IV) germanide

Weaire–Phelan structure

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Weaire–Phelan structure

A Weaire–Phelan structure has Pm3n (223) symmetry.

It has three 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 a "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.

See also

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References

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Further reading

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Cubic crystal system

View on Grokipedia
from Grokipedia
The cubic crystal system, also known as the isometric system, is one of the seven fundamental crystal systems in , defined by a with three equal edge lengths (a = b = c) and all interaxial angles equal to 90° (α = β = γ = 90°), forming a symmetrical cube-shaped lattice. This system exhibits the highest degree of among all crystal systems, featuring 4-fold rotational axes along the <100> directions, 3-fold axes along <111>, and 2-fold axes along <110>, which contributes to its isotropic optical and elastic properties in certain materials. The cubic system encompasses three distinct Bravais lattices: the primitive (simple) cubic lattice, where lattice points occupy only the corners of the unit cell; the body-centered cubic (BCC) lattice, with an additional point at the cube's center; and the face-centered cubic (FCC) lattice, featuring points at the centers of each face in addition to the corners. These lattices form the basis for numerous important crystal structures, including the simple FCC arrangement in metals like (a ≈ 3.615 ), the structure in (a ≈ 3.567 ), as well as the BCC structure in some alloys and the rock-salt (NaCl) structure (a ≈ 5.642 ) in ionic compounds. Due to its high symmetry and prevalence in elemental and compound solids, the cubic system is central to , influencing properties such as , , and electrical conductivity in applications from semiconductors to structural metals.

Fundamental Properties

Definition and Lattice Parameters

The cubic crystal system is one of the seven crystal systems in , defined by a with three equal lattice parameters and orthogonal axes./07%3A_Molecular_and_Solid_State_Structure/7.01%3A_Crystal_Structure) Specifically, it features equal edge lengths (a=b=ca = b = c) and all interaxial angles at right angles (α=β=γ=90∘\alpha = \beta = \gamma = 90^\circ), making it the most symmetric of the systems. This geometric configuration distinguishes the cubic system from less symmetric ones, such as tetragonal or orthorhombic, where axes or angles differ./07%3A_Molecular_and_Solid_State_Structure/7.01%3A_Crystal_Structure) The volume of the cubic unit cell is calculated simply as V=a3V = a^3, where aa is the representing the edge length. This formula arises directly from the cubic geometry, providing a straightforward measure of the repeating unit's size in the crystal lattice. The high of the cubic system imparts significant to single crystals, meaning many physical properties, such as and electrical conductivity, are independent of direction. This directional uniformity contrasts with lower-symmetry crystals, where properties vary along different axes, and arises from the equivalent treatment of all three perpendicular directions in the lattice. The geometric foundations of the cubic system were formalized in early through the work of Auguste Bravais, who in 1850 identified 14 distinct lattice types across all crystal systems, emphasizing the primacy of cubic in describing atomic arrangements. In cubic crystals, the conventional often serves as a practical description with full lattice , but it may encompass a larger than the minimal primitive cell, which contains exactly one lattice point and the smallest repeating . The primitive cell is a fraction of the conventional one depending on the specific arrangement, yet both maintain the defining a=b=ca = b = c and 90∘90^\circ angles.

Symmetry Elements and Operations

The cubic crystal system exhibits the highest degree of among the seven crystal systems, defined by a set of axes and reflection planes that operate on the lattice while leaving it unchanged. The core operations include three 4-fold axes aligned with the normals to the faces of the unit cell (along the x, y, and z directions), four 3-fold axes directed along the body diagonals opposite vertices, and six 2-fold axes passing through the midpoints of opposite edges. Complementing these are nine mirror planes: three parallel to the principal faces and six diagonal planes oriented at 45 degrees to the faces, which reflect the lattice across these surfaces. Most cubic crystal classes incorporate an inversion center at the origin, which maps every point (x, y, z) to (-x, -y, -z), ensuring centrosymmetric arrangements; however, the pyritohedral class lacks this element, resulting in chiral structures without . In the holosymmetric class, denoted O_h, these elements combine to yield a total of 48 distinct , encompassing rotations, reflections, inversions, and roto-inversions. The 48-fold symmetry in the holosymmetric case arises from the pure rotational subgroup O, which has 24 elements: the identity operation (1), nine 4-fold rotations (three axes each contributing 90°, 180°, and 270°), eight 3-fold rotations (four axes each contributing 120° and 240°), and six 2-fold rotations (180° about six axes). Doubling this through inclusion of the inversion center and associated improper rotations accounts for the full set. This high enforces equivalence among the x, y, and z directions, rendering certain physical properties isotropic or highly constrained; for instance, the in cubic crystals reduces to just three independent components due to the identical response along all principal axes.

Bravais Lattices

Primitive Cubic Lattice

The primitive cubic lattice, also referred to as the simple cubic lattice, is the most basic within the cubic crystal system, featuring lattice points exclusively at the eight corners of a cubic . Each corner point is shared equally among eight adjacent s, yielding one net lattice point per primitive . This arrangement results in a of 6, where each atom bonds to six nearest neighbors positioned along the cube's edges at a equal to the lattice aa. To quantify the efficiency of atomic packing in this structure, the is derived for a model of touching along the edges. The rr is a/2a/2, so the volume of one spherical atom is 43πr3=43π(a2)3=πa36\frac{4}{3}\pi r^3 = \frac{4}{3}\pi \left(\frac{a}{2}\right)^3 = \frac{\pi a^3}{6}. The unit cell volume is a3a^3, leading to: APF=πa36a3=π6≈0.52\text{APF} = \frac{\frac{\pi a^3}{6}}{a^3} = \frac{\pi}{6} \approx 0.52 This value indicates that only about 52% of the unit cell volume is occupied by atoms, significantly lower than in other cubic lattices. The sparse geometry can be visualized as a three-dimensional grid of cubes, each with atoms solely at the vertices, creating open spaces along the faces and body diagonals. The primitive cubic lattice is exceedingly rare among elemental structures due to its low packing efficiency, which offers minimal energetic stability for most metals under ambient conditions. α-Polonium (α-Po) stands as the sole elemental example of this lattice type, where atoms occupy the corner positions of the simple cubic unit cell at room temperature. This unusual configuration in α-Po persists owing to relativistic effects that favor the structure despite its inefficiency, though β-polonium adopts a different rhombohedral form at higher temperatures.

Body-Centered Cubic Lattice

The body-centered cubic (BCC) lattice is characterized by lattice points located at each of the eight corners of a cubic , with an additional lattice point at the center of the cube./06%3A_Metals_and_Alloys-_Structure_Bonding_Electronic_and_Magnetic_Properties/6.04%3A_Crystal_Structures_of_Metals) This arrangement results in two atoms per conventional , as the corner atoms contribute 1/8 each (totaling 1) and the central atom contributes 1. The BCC can be viewed as a primitive cubic lattice with additional centering at the vector (a2,a2,a2)\left( \frac{a}{2}, \frac{a}{2}, \frac{a}{2} \right), where aa is the ./Physical_Properties_of_Matter/States_of_Matter/Properties_of_Solids/Crystal_Lattice/Closest_Pack_Structures) In the BCC lattice, each atom has a of 8, with nearest neighbors being the eight corner atoms surrounding the central atom (or vice versa)./12%3A_Solids/12.02%3A_The_Arrangement_of_Atoms_in_Crystalline_Solids) The to these nearest neighbors is 32a\frac{\sqrt{3}}{2} a
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