# Allotropy

Allotropy or allotropism (from Ancient Greek ἄλλος (allos) 'other' and τρόπος (tropos) 'manner, form') is the property of some chemical elements to exist in two or more different forms, in the same physical state, known as allotropes of the elements. Allotropes are different structural modifications of an element: the atoms of the element are bonded together in different manners. For example, the allotropes of carbon include diamond (the carbon atoms are bonded together to form a cubic lattice of tetrahedra), graphite (the carbon atoms are bonded together in sheets of a hexagonal lattice), graphene (single sheets of graphite), and fullerenes (the carbon atoms are bonded together in spherical, tubular, or ellipsoidal formations).
The term allotropy is used for elements only, not for compounds. The more general term, used for any compound, is polymorphism, although its use is usually restricted to solid materials such as crystals. Allotropy refers only to different forms of an element within the same physical phase (the state of matter, such as a solid, liquid or gas). The differences between these states of matter would not alone constitute examples of allotropy. Allotropes of chemical elements are frequently referred to as polymorphs or as phases of the element.
For some elements, allotropes have different molecular formulae or different crystalline structures, as well as a difference in physical phase; for example, two allotropes of oxygen (dioxygen, O$_2$, and ozone, O$_3$) can both exist in the solid, liquid and gaseous states. Other elements do not maintain distinct allotropes in different physical phases; for example, phosphorus has numerous solid allotropes, which all revert to the same P$_4$ form when melted to the liquid state.

## History

The concept of allotropy was originally proposed in 1840 by the Swedish scientist Baron Jöns Jakob Berzelius (1779–1848). The term is derived from Greek άλλοτροπἱα (allotropia) 'variability, changeableness'. After the acceptance of Avogadro's hypothesis in 1860, it was understood that elements could exist as polyatomic molecules, and two allotropes of oxygen were recognized as O$_2$ and O$_3$. In the early 20th century, it was recognized that other cases such as carbon were due to differences in crystal structure.
By 1912, Ostwald noted that the allotropy of elements is just a special case of the phenomenon of polymorphism known for compounds, and proposed that the terms allotrope and allotropy be abandoned and replaced by polymorph and polymorphism. Although many other chemists have repeated this advice, IUPAC and most chemistry texts still favour the usage of allotrope and allotropy for elements only.

## Differences in properties of an element's allotropes

Allotropes are different structural forms of the same element and can exhibit quite different physical properties and chemical behaviours. The change between allotropic forms is triggered by the same forces that affect other structures, i.e., pressure, light, and temperature. Therefore, the stability of the particular allotropes depends on particular conditions. For instance, iron changes from a body-centered cubic structure (ferrite) to a face-centered cubic structure (austenite) above 906 °C, and tin undergoes a modification known as tin pest from a metallic form to a semimetallic form below 13.2 °C (55.8 °F). As an example of allotropes having different chemical behaviour, ozone (O$_3$) is a much stronger oxidizing agent than dioxygen (O$_2$).

## List of allotropes

Typically, elements capable of variable coordination number and/or oxidation states tend to exhibit greater numbers of allotropic forms. Another contributing factor is the ability of an element to catenate.
Examples of allotropes include:

### Non-metals

| Element | Allotropes |
| Carbon | Diamond – an extremely hard, transparent crystal, with the carbon atoms arranged in a tetrahedral lattice. A poor electrical conductor. An excellent thermal conductor. Lonsdaleite – also called hexagonal diamond. Graphene – is the basic structural element of other allotropes, nanotubes, charcoal, and fullerenes. Q-carbon – a ferromagnetic, tough, and brilliant crystal structure that is harder and brighter than diamonds. Graphite – a semimetallic, soft, black, flaky solid, a good electrical conductor. The C atoms are bonded in flat hexagonal lattices (graphene), which are then layered in sheets. Linear acetylenic carbon (carbyne) Amorphous carbon Fullerenes, including buckminsterfullerene, also known as "buckyballs", such as C$_{60}$. Carbon nanotubes – allotropes having a cylindrical nanostructure. Schwarzites Cyclocarbon Glassy carbon Superdense carbon allotropes – proposed allotropes |
| Nitrogen | Dinitrogen – by far the most common and stable form of nitrogen, found in the air. Hexazine Octaazacubane Tetranitrogen Trinitrogen Solid nitrogen |
| Phosphorus | White phosphorus – crystalline solid of tetraphosphorus (P$_4$) molecules Red phosphorus – amorphous polymeric solid Scarlet phosphorus Violet phosphorus with monoclinic crystalline structure Black phosphorus – semiconductor, analogous to graphite Diphosphorus – gaseous form composed of P$_2$ molecules, stable between 1200 °C and 2000 °C; created e.g. by dissociation of P$_4$ molecules of white phosphorus at around 827 °C |
| Oxygen | Dioxygen, O$_2$ – colorless (faint blue liquid and solid) Ozone, O$_3$ – blue Tetraoxygen, O$_4$ – metastable Octaoxygen, O$_8$ – red |
| Sulfur | Cyclo-Pentasulfur, Cyclo-S$_5$ Cyclo-Hexasulfur, Cyclo-S$_6$ Cyclo-Heptasulfur, Cyclo-S$_7$ Cyclo-Octasulfur, Cyclo-S$_8$ |
| Selenium | "Red selenium", cyclo-Se$_8$ Gray selenium, polymeric Se Black selenium, irregular polymeric rings up to 1000 atoms long Monoclinic selenium, dark red transparent crystals |
| Spin isomers of hydrogen | Orthohydrogen, H$_2$ with nuclear spins aligned parallel Parahydrogen, H$_2$ with nuclear spins aligned antiparallel These nuclear spin isomers have sometimes been described as allotropes, notably by the committee which awarded the 1932 Nobel prize to Werner Heisenberg for quantum mechanics and singled out the "allotropic forms of hydrogen" as its most notable application. |

### Metalloids

| Element | Allotropes |
| Boron | Amorphous boron – brown powder – B$_{12}$ regular icosahedra α-rhombohedral boron β-rhombohedral boron γ-orthorhombic boron α-tetragonal boron β-tetragonal boron High-pressure superconducting phase |
| Silicon | Amorphous silicon α-silicon, a semiconductor, diamond cubic structure β-silicon - metallic, with the BCC similar to molybdenum and beta-tin (High Pressure Phase) Q-Silicon - a ferromagnetic (Similar to Q-Carbon) and highly conductive phase of silicon (similar to graphite) Silicene, buckled planar single layer Silicon, similar to Graphene |
| Germanium | Amorphous germanium α-germanium – semimetallic element or semiconductor, with the same structure as diamond (similar chemical properties with sulfur and silicon) β-germanium – metallic, with the same structure as beta-tin Germanene – Buckled planar Germanium, similar to graphene |
| Arsenic | Yellow arsenic – molecular non-metallic As$_4$, with the same structure as white phosphorus (Similar chemical properties with nitrogen and phosphorus) Gray arsenic, polymeric As (metallic, though heavily anisotropic) (similar to aluminum and antimony in chemical properties) Black arsenic – molecular and non-metallic, with the same structure as red phosphorus |
| Antimony | Blue-white antimony – stable form (metallic), with the same structure as gray arsenic (similar to arsenic in chemical properties) Black antimony (non-metallic and amorphous, only stable as a thin layer) |
| Tellurium | Amorphous tellurium – gray-black or brown powder Crystalline tellurium – hexagonal crystalline structure (metalloid) (similar chemical properties with selenium) |

### Metals

Among the metallic elements that occur in nature in significant quantities (56 up to U, without Tc and Pm), almost half (27) are allotropic at ambient pressure: Li, Be, Na, Ca, Ti, Mn, Fe, Co, Sr, Y, Zr, Sn, La, Ce, Pr, Nd, Sm, Gd, Tb, Dy, Yb, Hf, Tl, Th, Pa and U. Some phase transitions between allotropic forms of technologically relevant metals are those of Ti at 882 °C, Fe at 912 °C and 1,394 °C, Co at 422 °C, Zr at 863 °C, Sn at 13 °C and U at 668 °C and 776 °C.
| Element | Phase name(s) | Space group | Pearson symbol | Structure type | Description |
| Lithium | α-Li | R3m | hR9 | α-Sm | Forms below 70 K. |
| Lithium | β-Li | Im3m | cI2 | W | Stable at room temperature and pressure. |
| Lithium | | Fm3m | cF4 | Cu | Forms above 7GPa |
| Lithium | | R3m | hR1 | α-Hg | An intermediate phase formed ~40GPa. |
| Lithium | | I43d | cI16 | | Forms above 40GPa. |
| Lithium | | | oC88 | | Forms between 60 and 70 GPa. |
| Lithium | | | oC40 | | Forms between 70 and 95 GPa. |
| Lithium | | | oC24 | | Forms above 95 GPa. |
| Beryllium | α-Be | P6$_3$/mmc | hP2 | Mg | Stable at room temperature and pressure. |
| Beryllium | β-Be | Im3m | cI2 | W | Forms above 1255 °C. |
| Sodium | α-Na | R3m | hR9 | α-Sm | Forms below 20 K. |
| Sodium | β-Na | Im3m | cI2 | W | Stable at room temperature and pressure. |
| Sodium | | Fm3m | cF4 | Cu | Forms at room temperature above 65 GPa. |
| Sodium | | I43d | cI16 | | Forms at room temperature, 108GPa. |
| Sodium | | Pnma | oP8 | MnP | Forms at room temperature, 119GPa. |
| Sodium | | | tI19* | | A host-guest structure that forms above between 125 and 180 GPa. |
| Sodium | | | hP4 | | Forms above 180 GPa. |
| Magnesium | | P6$_3$/mmc | hP2 | Mg | Stable at room temperature and pressure. |
| Magnesium | | Im3m | cI2 | W | Forms above 50 GPa. |
| Aluminium | α-Al | Fm3m | cF4 | Cu | Stable at room temperature and pressure. |
| Aluminium | β-Al | P6$_3$/mmc | hP2 | Mg | Forms above 20.5 GPa. |
| Potassium | | Im3m | cI2 | W | Stable at room temperature and pressure. |
| Potassium | | Fm3m | cF4 | Cu | Forms above 11.7 GPa. |
| Potassium | | I4/mcm | tI19* | | A host-guest structure that forms at about 20 GPa. |
| Potassium | | P6$_3$/mmc | hP4 | NiAs | Forms above 25 GPa. |
| Potassium | | Pnma | oP8 | MnP | Forms above 58GPa. |
| Potassium | | I4$_1$/amd | tI4 | | Forms above 112 GPa. |
| Potassium | | Cmca | oC16 | | Formas above 112 GPa. |
| Iron | α-Fe, ferrite | Im3m | cI2 | Body-centered cubic | Stable at room temperature and pressure. Ferromagnetic at T<770 °C, paramagnetic from T=770–912 °C. |
| Iron | γ-iron, austenite | Fm3m | cF4 | Face-centered cubic | Stable from 912 to 1,394 °C. |
| Iron | δ-iron | Im3m | cI2 | Body-centered cubic | Stable from 1,394 – 1,538 °C, same structure as α-Fe. |
| Iron | ε-iron, Hexaferrum | P6$_3$/mmc | hP2 | Hexagonal close-packed | Stable at high pressures. |
| Cobalt | α-Cobalt | | | hexagonal-close packed | Forms below 450 °C. |
| Cobalt | β-Cobalt | | | face centered cubic | Forms above 450 °C. |
| Cobalt | ε-Cobalt | P4$_1$32 | | primitive cubic | Forms from thermal decomposition of [Co$_2$CO$_8$]. Nanoallotrope. |
| Rubidium | α-Rb | Im3m | cI2 | W | Stable at room temperature and pressure. |
| Rubidium | | | cF4 | | Forms above 7 GPa. |
| Rubidium | | | oC52 | | Forms above 13 GPa. |
| Rubidium | | | tI19* | | Forms above 17 GPa. |
| Rubidium | | | tI4 | | Forms above 20 GPa. |
| Rubidium | | | oC16 | | Forms above 48 GPa. |
| Tin | α-tin, gray tin, tin pest | Fd3m | cF8 | d-C | Stable below 13.2 °C. |
| Tin | β-tin, white tin | I4$_1$/amd | tI4 | β-Sn | Stable at room temperature and pressure. |
| Tin | γ-tin, rhombic tin | I4/mmm | tI2 | In | Forms above 10 GPa. |
| Tin | γ'-Sn | Immm | oI2 | MoPt$_2$ | Forms above 30 GPa. |
| Tin | σ-Sn, γ"-Sn | Im3m | cI2 | W | Forms above 41 GPa. Forms at very high pressure. |
| Tin | δ-Sn | P6$_3$/mmc | hP2 | Mg | Forms above 157 GPa. |
| Tin | Stanene | | | |
| Polonium | α-Polonium | | | simple cubic | |
| Polonium | β-Polonium | | | rhombohedral | |
Most stable structure under standard conditions.   Structures stable below room temperature.   Structures stable above room temperature.   Structures stable above atmospheric pressure.

#### Lanthanides and actinides

* Cerium, samarium, dysprosium and ytterbium have three allotropes.
* Praseodymium, neodymium, gadolinium and terbium have two allotropes.
* Plutonium has six distinct solid allotropes under "normal" pressures. Their densities vary within a ratio of some 4:3, which vastly complicates all kinds of work with the metal (particularly casting, machining, and storage). A seventh plutonium allotrope exists at very high pressures. The transuranium metals Np, Am, and Cm are also allotropic.
* Promethium, americium, berkelium and californium have three allotropes each.

## Nanoallotropes

In 2017, the concept of nanoallotropy was proposed. Nanoallotropes, or allotropes of nanomaterials, are nanoporous materials that have the same chemical composition (e.g., Au), but differ in their architecture at the nanoscale (that is, on a scale 10 to 100 times the dimensions of individual atoms). Such nanoallotropes may help create ultra-small electronic devices and find other industrial applications. The different nanoscale architectures translate into different properties, as was demonstrated for surface-enhanced Raman scattering performed on several different nanoallotropes of gold. A two-step method for generating nanoallotropes was also created.