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Allotropy

Allotropy is the property exhibited by certain chemical elements to exist in two or more distinct structural forms, known as allotropes, within the same physical state, such as solid, liquid, or gas, where the forms differ primarily in the arrangement of atoms and resulting physical properties while sharing the same chemical composition.^[1]^[2] These allotropes arise due to variations in bonding and crystal structures, often influenced by environmental factors like temperature and pressure, leading to significant differences in characteristics such as hardness, conductivity, and density.^[3] A classic example of allotropy is observed in carbon, which manifests as diamond—a rigid, three-dimensional tetrahedral network of sp³-hybridized atoms making it the hardest known natural material and an electrical insulator—and graphite, featuring layered sp²-hybridized sheets with delocalized electrons that confer conductivity and lubricity, as seen in pencils and industrial applications.^[3]^[1] Other carbon allotropes include fullerenes like C₆₀ buckyballs, carbon nanotubes, and graphene, a single-layer honeycomb structure isolated in 2004, whose exceptional strength and electron mobility earned its discoverers the 2010 Nobel Prize in Physics.^[1] In metals, iron demonstrates allotropy through forms such as ferrite (body-centered cubic, stable at room temperature), austenite (face-centered cubic, above 910°C), and high-pressure ε-iron (hexagonal close-packed), which underpin processes like steel heat treatment for enhancing mechanical properties.^[1] The phenomenon of allotropy is distinct from polymorphism, which refers to multiple crystal structures in compounds rather than elements, and it plays a pivotal role in materials science by enabling tailored properties for engineering applications, such as improving steel's ductility or developing advanced nanomaterials like graphene for electronics and composites.^[2]^[1] Understanding allotropy also informs phase diagrams and thermodynamic stability, guiding synthesis methods that mimic natural conditions—e.g., subjecting carbon to extreme heat (2,500°F) and pressure (850,000 psi) to form diamonds.^[3] Elements like phosphorus (white, red, black) and sulfur (rhombic, monoclinic) further illustrate this versatility, highlighting allotropy's broad impact across chemistry and technology.^[1]

Fundamentals

Definition and Overview

Allotropy refers to the existence of a chemical element in two or more different structural modifications, known as allotropes, within the same physical state—such as solid, liquid, or gas—due to variations in atomic arrangement or bonding.^[4] These forms are chemically identical in composition but often exhibit distinct physical and chemical properties arising from their structural differences. Classic examples illustrate this phenomenon: carbon occurs as diamond, with a tetrahedral lattice of sp³-hybridized atoms, and graphite, featuring layered sp²-hybridized sheets; oxygen exists as dioxygen (O₂), a diatomic gas, and ozone (O₃), a triatomic gas; while phosphorus manifests as white phosphorus, a waxy solid with P₄ tetrahedral molecules, and red phosphorus, an amorphous polymeric solid. These cases highlight how allotropes can coexist under specific conditions, influencing the element's behavior in materials and reactions. Allotropy differs from polymorphism, which involves multiple crystal structures in chemical compounds rather than pure elements, and from isomerism, where compounds share the same molecular formula but differ in atomic connectivity or spatial arrangement.^[5] From a thermodynamic perspective, allotropes are distinct phases, each characterized by unique Gibbs free energy curves; the stable allotrope under given temperature and pressure is the one with the lowest free energy.^[6]

Types and Causes

Allotropes are classified into monotropic and enantiotropic types based on their thermodynamic stability relationships. In monotropic allotropy, one form is stable under all conditions, while the other is metastable with higher Gibbs free energy, leading to irreversible transitions; for example, graphite is the stable allotrope of carbon, with diamond being monotropic and thermodynamically unstable at ambient conditions. Enantiotropic allotropy involves two forms that are stable under different temperature or pressure conditions, with reversible transitions at a specific transition point; rhombic and monoclinic sulfur exemplify this, where rhombic sulfur is stable below approximately 95.5°C and monoclinic above it.^[7] Additionally, allotropes can be categorized as static or dynamic: static allotropy refers to fixed crystal structures differing in atomic arrangement, while dynamic allotropy involves reversible changes over a temperature range without a distinct transition point, often observed in elements like sulfur.^[8]^[9] The existence of allotropes arises primarily from differences in crystal lattice structures and bonding arrangements that minimize Gibbs free energy under specific conditions. For instance, diamond features a three-dimensional covalent network lattice with tetrahedral coordination, whereas graphite has a layered hexagonal structure with planar sheets; these structural variations lead to distinct stability and properties.^[10] Thermodynamically, the stable allotrope at a given temperature and pressure is the one with the lowest Gibbs free energy, G = H - TS, where phases coexist at boundaries defined by equal chemical potentials.^[11] Phase transitions between allotropes follow the Clapeyron equation, which relates the slope of the transition curve to enthalpy and volume changes:

\frac{dP}{dT} = \frac{\Delta H}{T \Delta V}

Here, \Delta H is the enthalpy of transition, T is temperature, and \Delta V is the volume change; positive \Delta V typically results in a positive slope, as seen in solid-solid transitions where denser phases stabilize under pressure.^[12] Temperature, pressure, and kinetic factors influence stability, with kinetics often hindering transitions to the global energy minimum, allowing metastable allotropes like diamond to persist.^[11] Bonding variations further explain allotropy, particularly through atomic orbital hybridization and intermolecular interactions. In carbon allotropes, diamond exhibits sp^3 hybridization, forming strong sigma bonds in a tetrahedral geometry with bond angles of 109.5°, while graphite uses sp^2 hybridization for trigonal planar sheets with 120° angles and delocalized pi electrons.^[10] Intermolecular forces, such as weak van der Waals (London dispersion) interactions, hold graphite layers together, enabling easy sliding and contrasting with the rigid covalent network in diamond. Computational approaches like density functional theory (DFT) have become essential for predicting new allotropes by calculating total energies, electronic structures, and stabilities of hypothetical lattices. DFT simulations screen vast structural databases, identifying low-energy configurations with mixed hybridizations or novel topologies, such as superhard or semiconducting carbon phases stable under extreme pressures; recent reviews highlight over 1,600 predicted 3D carbon allotropes, guiding experimental synthesis.^[13]

Historical Development

Early Observations

Ancient civilizations, including those in India dating back to the 4th century BCE, were familiar with diamond as a hard, transparent gemstone, while graphite was utilized in Europe from the 16th century for marking and drawing due to its soft, black properties. These materials were regarded as distinct substances with vastly different characteristics, such as diamond's exceptional hardness and graphite's lubricity, without any recognition of their shared elemental composition.^[14] In 1772, French chemist Antoine Lavoisier conducted experiments burning diamonds in oxygen, producing carbon dioxide identical to that from charcoal combustion, thereby establishing diamond as a crystalline form of carbon and hinting at the possibility of multiple forms for the same element. Around the same time, graphite was confirmed to be impure carbon through similar combustion analyses by Lavoisier and others, marking one of the earliest empirical recognitions of what would later be understood as allotropy. In 1774, Joseph Priestley isolated oxygen gas (then called "dephlogisticated air") from heated mercuric oxide, laying the groundwork for identifying its variations; decades later, in 1839, Christian Friedrich Schönbein observed a pungent odor during water electrolysis and slow oxidation of phosphorus, leading to the discovery of ozone as a distinct, more reactive form of oxygen.^[15]^[16]^[17] The 18th century also saw initial isolations of reactive forms of other elements. In 1669, German alchemist Hennig Brand obtained white phosphorus by distilling urine residues, yielding a waxy, spontaneously igniting substance that glowed in the dark, though its elemental nature was not fully appreciated until later. Independently in 1770, Swedish chemist Carl Wilhelm Scheele produced white phosphorus from bone ash, but the existence of a more stable red form was not observed until 1845, when Austrian chemist Anton von Schrötter heated white phosphorus out of contact with air, producing a non-toxic, amorphous red variety with different solubility and reactivity. For sulfur, known since antiquity in its yellow, rhombic crystalline form, the monoclinic variant was first described in 1823 by German chemist Eilhard Mitscherlich, who noted its needle-like crystals forming upon cooling molten sulfur above 95.5°C, exhibiting a temporary stability and distinct optical properties compared to the rhombic phase.^[18]^[18] These early observations often led to confusion, with differing forms attributed to impurities, experimental artifacts, or even separate elements, as the prevailing phlogiston theory and lack of atomic theory hindered unified explanations. Only with advancing atomic hypotheses in the early 19th century, such as those proposed by John Dalton, did scientists begin to systematically link these variations to inherent structural differences within the same element, paving the way for formal recognition of allotropy.^[19]

Key Milestones and Terminology

The term "allotropy" was coined in 1841 by Swedish chemist Jöns Jakob Berzelius in a review of German physicist Moritz Frankenheim's work on the polymorphic forms of mercury(II) iodide and sulfur, where he described it as a third type of isomerism specific to elements, arising from differences in their internal atomic constitution without changes in composition.^[19] Berzelius derived the word from the Greek roots allos (other) and tropos (way or manner), emphasizing the varied manners in which elemental atoms could manifest distinct properties.^[19] Following the 1860 Karlsruhe Congress, where Stanislao Cannizzaro revived Avogadro's 1811 hypothesis on molecular volumes to establish accurate atomic weights, the concept of allotropy underwent significant refinement by linking crystal forms to atomic arrangements rather than vague compositional ambiguities. This advancement, building on Eilhard Mitscherlich's earlier 1819 law of isomorphism—which demonstrated that analogous compounds form similar crystals and aided atomic weight determinations by analogy—allowed chemists to better distinguish allotropic modifications as structural variants of pure elements. By 1888, Lothar Meyer had further integrated allotropy into atomic theory by attributing it to polyatomic molecular forms, aligning it with Avogadro's principles and solidifying its role in understanding elemental diversity.^[19] In the 20th century, experimental breakthroughs expanded allotropy through high-pressure techniques. American physicist Percy Williams Bridgman, starting in the 1910s, developed apparatus capable of generating pressures up to several thousand atmospheres, enabling the synthesis and discovery of new allotropic forms; for instance, in his pioneering high-pressure experiments starting in the early 1910s, he studied phase transitions in elements such as bismuth, identifying denser modifications like Bi II under pressures around 2.5 GPa.^[20] Bridgman's innovations, though unsuccessful in directly synthesizing diamond despite repeated attempts, laid the groundwork for later high-pressure allotropy studies. Bridgman's techniques inspired later successes; in December 1954, H. Tracy Hall at General Electric achieved the first verified synthesis of diamond crystals from graphite using a high-pressure belt apparatus, producing gem-quality diamonds by 1955 for industrial use.^[21] A landmark discovery occurred in 1985 when Harry Kroto, Robert Curl, and Richard Smalley identified buckminsterfullerene (C₆₀)—a spherical carbon cluster—as a new allotrope of carbon, produced via laser vaporization of graphite and confirmed through mass spectrometry. The terminology surrounding allotropy evolved notably in the early 20th century amid advances in X-ray crystallography. Initially overlapping with "polymorphism"—a broader term for multiple crystal forms—distinctions sharpened as X-ray diffraction revealed that allotropy specifically denoted elemental polymorphs driven by atomic bonding variations, while polymorphism applied to compounds. This formal separation gained traction in the 1910s–1920s crystallographic literature, with Wilhelm Ostwald in 1912 advocating the use of "polymorphism" universally, though the International Union of Pure and Applied Chemistry (IUPAC) retained "allotropy" for its elemental specificity to preserve historical precision.^[19]

Property Variations

Physical Differences

Allotropes of the same element can exhibit striking differences in physical properties due to variations in atomic arrangement, leading to diverse applications in materials science. For instance, the carbon allotropes graphite and diamond differ significantly in density, with graphite at 2.26 g/cm³ and diamond at 3.51 g/cm³, reflecting graphite's layered structure versus diamond's compact tetrahedral lattice.^[22]^[23] These density variations influence mechanical behavior, as seen in hardness: diamond achieves a Mohs scale value of 10, making it the hardest known natural material, while graphite scores only 1–2, enabling its use as a lubricant.^[24] Electrical and thermal conductivity also diverge markedly; graphite conducts electricity anisotropically due to delocalized π electrons in its planar layers, whereas diamond is an electrical insulator but excels in thermal conductivity, up to 2200 W/m·K along certain directions, owing to its rigid sp³-bonded framework.^[25]^[26] Optical properties further highlight allotropic distinctions, often tied to electronic structure and light interaction. Diamond's high refractive index of 2.42 contributes to its exceptional brilliance and dispersion, responsible for the "fire" in gem-quality stones, while graphite appears opaque and dull.^[27] Color variations are evident in phosphorus allotropes, where white phosphorus displays a waxy, translucent white-to-yellow hue, contrasting with the amorphous red phosphorus's reddish-brown appearance, both arising from different molecular packings.^[18] Thermal stability among allotropes is governed by phase transition temperatures, affecting practical usability. In tin, gray tin (α-Sn) is stable below 13.2°C, transitioning to the metallic white tin (β-Sn) above this point, a change known as "tin pest" that can cause structural degradation in cold environments.^[28] Such transitions underscore the sensitivity of allotropes to temperature, with implications for material design in varying conditions. External factors like pressure profoundly alter physical properties by favoring denser phases. For carbon, compression of glassy carbon above 40 GPa yields amorphous diamond, a superhard allotrope with fully sp³-bonded structure and diamond-like density, demonstrating enhanced mechanical strength under extreme conditions.^[29]

Chemical and Structural Differences

Allotropy manifests in distinct atomic arrangements and bonding configurations that profoundly influence the chemical behavior of elements. In carbon, for instance, diamond adopts a cubic crystal system with the space group Fd-3m, where each carbon atom is tetrahedrally coordinated to four others, forming a three-dimensional network of sp³ hybridized bonds with a uniform bond length of approximately 1.54 Å.^[30] In contrast, graphite features a hexagonal crystal system with layers of sp² hybridized carbon atoms arranged in a planar honeycomb lattice, where each atom has a coordination number of three within the plane and bond lengths of about 1.42 Å, connected by weaker van der Waals forces between layers.^[31] These structural differences dictate the overall rigidity and interlayer mobility, directly impacting chemical accessibility. Chemical reactivity among allotropes varies markedly due to these structural motifs. White phosphorus consists of discrete P₄ tetrahedral molecules with strained P-P bonds, rendering it highly reactive and prone to spontaneous ignition in air, necessitating storage under water to prevent oxidation.^[32] Red phosphorus, however, forms an amorphous polymeric network of linked P₄ units with more stable covalent bonds, exhibiting lower reactivity and stability in air without ignition.^[32] Such variations arise because the molecular isolation in white phosphorus facilitates easier bond cleavage and oxidation to phosphorus oxides like P₄O₁₀, whereas the extended structure in red phosphorus hinders initial attack, influencing the effective oxidation pathways despite the elemental oxidation state remaining zero in the pure forms.^[33] Stability differences further highlight allotropic distinctions, with some forms being thermodynamically favored while others persist due to kinetic barriers. Graphite is the thermodynamically stable allotrope of carbon at standard conditions, whereas diamond is metastable, maintained by high activation energies that prevent spontaneous conversion despite graphite's lower free energy.^[34] Transitions between allotropes can be catalyzed; for example, iron, a Group VIII metal, facilitates the conversion of graphite to diamond under high-pressure, high-temperature conditions by dissolving carbon and lowering the energy barrier for nucleation, enabling sp² to sp³ bond rearrangement.^[35] Spectroscopic techniques exploit these structural variances for allotrope identification. Raman spectroscopy reveals diamond through a sharp peak at 1332 cm⁻¹, corresponding to the triply degenerate vibration of its sp³ C-C bonds in the cubic lattice.^[36] Graphite, conversely, displays characteristic G-band at 1582 cm⁻¹ from in-plane sp² vibrations and a D-band around 1350 cm⁻¹ indicating disorder, allowing differentiation from diamond even in mixed samples.^[36] Infrared spectroscopy complements this by probing bond symmetries, though Raman's sensitivity to carbon's vibrational modes makes it particularly effective for non-destructive analysis.^[37]

Allotropes by Element Category

Non-metals

Non-metal elements exhibit a rich variety of allotropes due to their ability to form diverse covalent networks and molecular structures, leading to stark differences in physical and chemical properties. Carbon, a quintessential non-metal, displays multiple allotropes that highlight this diversity. Diamond consists of a three-dimensional tetrahedral network of sp³-hybridized carbon atoms, resulting in exceptional hardness (Mohs scale 10) and thermal conductivity, making it an ideal abrasive and heat sink material. In contrast, graphite features planar layers of sp²-hybridized carbon atoms arranged in hexagonal rings, held together by weak van der Waals forces, which imparts lubricity, electrical conductivity along the planes, and softness (Mohs scale 1-2).^[38] Fullerenes, such as buckminsterfullerene (C_{60}), form closed-cage structures with 60 carbon atoms in a truncated icosahedral geometry, exhibiting high symmetry and stability due to delocalized π-electrons, with properties including solubility in organic solvents and potential for superconductivity when doped.^[39] Carbon nanotubes, cylindrical extensions of graphene sheets, can be single-walled or multi-walled with helical arrangements of hexagonal carbon rings, diameters ranging from 1 to tens of nanometers, and exceptional tensile strength (up to 100 GPa) alongside electrical conductivity that varies from metallic to semiconducting based on chirality.^[40] Oxygen, another key non-metal, exists primarily as dioxygen (O_2), a diatomic molecule with a double bond that is paramagnetic and colorless in the gas phase but takes on a pale blue hue in the liquid state due to its electronic transitions. Ozone (O_3), a triatomic bent molecule, is a diamagnetic gas with a pungent odor and pale blue color, highly reactive as an oxidant, and less stable than O_2 owing to its higher energy structure.^[41] Under high pressure (8-96 GPa), oxygen forms the ε-phase, a red polymeric solid composed of O_8 molecular units in a rhombohedral lattice, exhibiting a loss of magnetism, intense infrared absorption, and a darker red color with increasing pressure due to band gap narrowing.^[42] Sulfur demonstrates temperature-dependent allotropes centered on S_8 rings. Rhombic sulfur (α-sulfur), the most stable form below 95.5°C, crystallizes in an orthorhombic lattice with puckered crown-shaped S_8 molecules, appearing as yellow crystals that are insoluble in water but soluble in carbon disulfide.^[43] Monoclinic sulfur (β-sulfur), stable between 95.5°C and 119°C, features a similar S_8 ring but in a needle-like monoclinic crystal packing, with slightly lower density and solubility properties akin to the rhombic form.^[43] Plastic sulfur, an amorphous polymer formed by rapid quenching of molten sulfur, consists of long entangled chains that confer viscoelastic behavior, allowing it to stretch like rubber when warm before reverting to a brittle solid upon cooling.^[44] Phosphorus allotropes vary dramatically in reactivity and structure. White phosphorus comprises discrete P_4 tetrahedral molecules, a waxy white solid that is highly toxic, glows in the dark via chemiluminescence, and ignites spontaneously in air at approximately 30°C due to its strained bonds. Red phosphorus is an amorphous network of linked phosphorus atoms, less reactive and non-toxic compared to the white form, with an ignition temperature around 240°C, making it suitable for safety matches. Black phosphorus adopts a layered orthorhombic structure analogous to graphite, with puckered sheets of phosphorus atoms forming a semiconductor with a direct band gap of about 0.3 eV, exhibiting high carrier mobility and anisotropic properties. Other non-metals like selenium show polymorphic forms. Gray selenium, the stable hexagonal allotrope, features helical chains of selenium atoms in a metallic-gray crystalline lattice, behaving as a photoconductor with electrical resistivity decreasing under light exposure.^[45] Red selenium is a monoclinic crystalline form with discrete Se_n rings (n=6-8), appearing brick-red and amorphous in powder, while vitreous selenium is a black, glassy amorphous solid formed by rapid cooling, all less stable than the gray form and convertible to it upon heating.^[45]

Metalloids

Metalloids exhibit allotropy that bridges non-metallic covalent bonding and metallic conductivity, often resulting in semiconductor behaviors critical for electronics and optoelectronics. These elements, including silicon, germanium, boron, arsenic, and antimony, display diverse structures ranging from crystalline frameworks to amorphous and molecular forms, influencing their electrical, optical, and thermal properties.^[46] Silicon, a cornerstone of semiconductor technology, primarily exists in a crystalline diamond cubic structure at ambient conditions, where each silicon atom bonds tetrahedrally to four others in a face-centered cubic lattice with space group Fd̅3m and lattice parameter a = 5.44 Å, yielding an indirect bandgap of approximately 1.1 eV that enables efficient charge carrier transport in devices like transistors.^[47] Amorphous silicon, lacking long-range order, features a continuous random network of Si-Si bonds with a higher optical bandgap around 1.7-1.8 eV, though it suffers from lower electron mobility (about 1 cm²/V·s) due to defect states, making it suitable for thin-film solar cells and displays where flexibility is prioritized over performance.^[48] Silicon clathrates represent open-framework allotropes, such as type I (Si₄₆) and type II (Si₁₃₆) structures, composed of polyhedral cages hosting guest atoms like sodium; these exhibit direct or quasi-direct bandgaps (1.4-2.4 eV), enhancing light absorption for photovoltaic applications compared to the indirect bandgap of diamond silicon.^[49] In 2023, Q-silicon was discovered through laser melting and quenching of amorphous silicon, forming a high-density phase (60% denser than diamond silicon) with sp³ bonding and random packing of diamond tetrahedra subunits, which introduces ferromagnetism (Curie temperature >400 K) absent in standard silicon, potentially revolutionizing spintronic devices.^[50] Germanium shares structural similarities with silicon but displays greater phase diversity under pressure, starting with the stable diamond cubic allotrope (space group Fd̅3m), a semiconductor with a narrower indirect bandgap of 0.66 eV that supports high-speed electronics due to superior hole mobility (1900 cm²/V·s).^[51] The white metallic allotrope adopts a body-centered tetragonal β-tin structure under high pressure (above ~10 GPa), transitioning from semiconducting to metallic behavior with increased conductivity, though it is metastable at ambient conditions and reverts upon decompression.^[52] Layered germanium forms, such as puckered hexagonal sheets analogous to silicene, exhibit tunable bandgaps (0.5-1.5 eV) depending on stacking, enabling potential use in 2D semiconductors, while high-pressure hexagonal phases (e.g., simple hexagonal at ~100 GPa) further diversify its structural motifs with metallic properties.^[51]^[53] Boron allotropes are characterized by complex icosahedral clusters (B₁₂ units) that dictate their semiconductor properties, with amorphous boron consisting of a disordered network of these clusters linked by three-center two-electron bonds, resulting in a wide bandgap (~2 eV) and high hardness suitable for abrasives.^[54] The α-rhombohedral form features a primitive lattice of isolated B₁₂ icosahedra with weak inter-icosahedral bonding, yielding semiconducting behavior with anisotropic conductivity and a bandgap of 1.5-2 eV, while the β-tetragonal allotrope incorporates linked icosahedra in chains, enhancing stability and electrical resistivity up to 10¹⁰ Ω·cm.^[46] These icosahedral-based structures underscore boron's structural diversity, where cluster packing variations lead to varying degrees of covalent-metallic hybridization, influencing applications in neutron detectors and high-temperature ceramics.^[54] Arsenic demonstrates pronounced allotropy tied to its pnictogen nature, with the gray allotrope forming stable layered sheets in a rhombohedral lattice (space group R̅3m), behaving as a semimetal with overlapping valence and conduction bands (small Fermi surface) that confer anisotropic semiconductor-like transport properties for thermoelectric devices.^[55] Yellow arsenic comprises discrete tetrahedral As₄ molecules, akin to white phosphorus, with a molecular bandgap exceeding 2.5 eV and high reactivity due to weak van der Waals interactions, limiting it to low-temperature studies but highlighting covalent molecular diversity.^[56] The black polymeric allotrope, synthesized under high pressure (>0.7 GPa), adopts a puckered layered structure similar to black phosphorus, featuring a direct bandgap (~0.3 eV) and higher stability below 300°C, though it irreversibly converts to gray arsenic upon heating or decompression, illustrating pressure-induced structural transitions in metalloids.^[55] Antimony's allotropes reflect its semimetallic character, dominated by the stable metallic rhombohedral form (space group R̅3m) with puckered layers and a small bandgap (~0.15 eV), enabling moderate electrical conductivity (comparable to graphite) and applications in alloys for enhanced thermoelectric efficiency through Seebeck coefficients up to 50 μV/K.^[57] The explosive form, an ill-defined amorphous variant produced by electrolysis in acidic media, exists in a strained, highly disordered state with expanded interatomic distances, detonating upon shock due to rapid recombination of polymer chains, though it is not a thermodynamically distinct phase.^[58] This form's instability contrasts with the metallic allotrope's robustness, underscoring antimony's potential in energy conversion where structural sensitivity tunes thermal and electrical responses.^[57]

Metals

Metallic elements exhibit allotropy through phase transitions driven by temperature, pressure, or composition, resulting in distinct crystal structures that influence mechanical properties such as ductility and strength. These transformations are particularly significant in metals used for structural applications, where the stability of phases determines alloy performance. For instance, many transition metals display body-centered cubic (BCC), face-centered cubic (FCC), or hexagonal close-packed (HCP) lattices, with transitions occurring at specific critical temperatures. Iron demonstrates classic allotropy with three primary phases at atmospheric pressure: alpha-iron (ferrite), which adopts a BCC structure and is ferromagnetic up to the Curie temperature of 770°C; gamma-iron (austenite), an FCC structure stable between approximately 912°C and 1394°C; and delta-iron, another BCC form existing from 1394°C to the melting point at 1538°C. In steels, rapid cooling from the austenite phase produces martensite, a supersaturated BCC variant distorted to body-centered tetragonal due to trapped carbon, enhancing hardness but reducing ductility. These phase changes underpin heat treatment processes in steel production.^[59]^[60] Tin exists in two allotropes at ambient conditions: white tin (beta-tin), a metallic tetragonal structure stable above 13.2°C with density of 7.31 g/cm³, and gray tin (alpha-tin), a non-metallic diamond cubic structure that forms below this temperature, leading to the brittle "tin pest" degradation where objects crumble due to the 27% volume expansion. This transformation is autocatalytic and historically affected tin alloys in cold environments, such as organ pipes in cathedrals.^[61] Titanium undergoes an allotropic transformation at 882°C, shifting from the low-temperature alpha phase with HCP structure to the high-temperature beta phase with BCC structure, enabling the design of alpha, beta, and alpha-beta alloys with tailored properties. Under high pressure, titanium can form the omega phase, a hexagonal structure that appears in pure titanium or alloys subjected to gigapascal stresses, altering electronic properties and hardness.^[62]^[63] Among lanthanides and actinides, plutonium exhibits remarkable complexity with seven allotropic forms under varying conditions, including the low-temperature alpha phase (monoclinic) and the ductile delta phase (FCC), stabilized at room temperature by small additions of elements like gallium for nuclear applications. These phases arise from the delocalized 5f electrons, leading to anomalous properties like negative thermal expansion in delta-plutonium, which impacts handling, machining, and safety in nuclear fuel cycles due to phase instability.^[64]^[65]^[66] Uranium similarly shows three allotropes: alpha-uranium (orthorhombic, stable up to 668°C), beta-uranium (tetragonal, 668–776°C), and gamma-uranium (BCC, above 776°C to melting at 1132°C), with the alpha phase's anisotropy causing dimensional instability in nuclear reactor components. These transformations affect uranium's reactivity and mechanical integrity, critical for fuel fabrication and waste storage.^[67]^[68]^[69] Other metals like sodium reveal allotropy under high pressure, transitioning from ambient BCC through FCC, complex structures such as cI16 (at ~100 GPa), oP8, and tI19, with these phases linked to an anomalous melting curve minimum around 120 GPa due to electronic restructuring. Such high-pressure forms provide insights into planetary interiors but are not stable at ambient conditions.^[70]^[70]

Advanced Topics

Nanoallotropes

Nanoallotropes refer to structural variants of elements at the nanoscale, typically below 100 nm, that exhibit distinct properties from their bulk counterparts due to quantum confinement, high surface-to-volume ratios, and altered bonding configurations. These forms often display unique electronic, optical, mechanical, or catalytic behaviors not observed in macroscopic allotropes. The concept of nanoallotropy was formally proposed in 2017 to describe nanoporous materials composed of the same elemental nanoparticles but arranged in different architectures, analogous to traditional allotropy but confined to nanoscale dimensions.^[71] A prominent example is found in carbon nanoallotropes, which extend beyond bulk forms like diamond and graphite into low-dimensional structures. Graphene, a single layer of sp²-hybridized carbon atoms in a 2D honeycomb lattice, possesses exceptional mechanical strength, with a tensile strength approximately 200 times that of steel and a Young's modulus of about 1 TPa. Carbon nanotubes, representing 1D rolled-up graphene sheets, exhibit high electrical conductivity and tensile strength up to 100 GPa, depending on chirality. Fullerenes, such as C₆₀ in 0D buckyball form, feature curved sp² networks with unique electronic properties enabling superconductivity in doped variants. These carbon nanoallotropes demonstrate how dimensionality reduction alters charge transport and reactivity compared to bulk carbon. For other elements, silicon nanoallotropes like nanowires contrast sharply with bulk silicon's brittleness. Silicon nanowires, with diameters of 20–100 nm, show enhanced ductility and can undergo reversible plastic deformation under bending, achieving strains up to 10% without fracture, due to surface-mediated dislocation activity. In metals, gold nanoparticles illustrate size-dependent phases; for instance, arrays of 10–15 nm gold particles can form non-close-packed structures such as tetrahedral clusters (vac₁Au₅) or layered quartets and septets (vac₁Au₁₁), which enhance surface-enhanced Raman scattering (SERS) signals by factors of 10³–10⁴ compared to close-packed bulk films, owing to tunable nanoscale gaps.^[71] Synthesis of nanoallotropes commonly employs methods like chemical vapor deposition (CVD) for graphene and silicon nanowires, where precursors decompose on substrates at 800–1000°C to grow aligned structures. Arc discharge, used for carbon nanotubes and fullerenes, involves vaporizing graphite electrodes in inert gas to produce 1–10 nm diameter tubes or clusters. For metallic nanoallotropes, post-assembly etching of binary superlattices—such as selectively removing iron oxide from gold-iron oxide arrays with HCl—yields stable porous architectures.^[71] However, nanoallotropes face stability challenges, including aggregation driven by high surface energies (up to 1–10 J/m²) and phase instability under ambient conditions, often requiring passivation layers or low-temperature processing to prevent coalescence into bulk-like forms.

Recent and Emerging Forms

In 2023, Q-silicon emerged as a novel silicon allotrope synthesized through nanosecond laser melting of ion-implanted amorphous silicon followed by rapid quenching from an undercooled state. This process yields a structure composed of randomly packed tetrahedra in amorphous variants (Q1, Q2, Q3) and a crystalline phase with subunit cells of higher atomic density—60% greater than that of diamond cubic silicon—arranged along <110> directions with alternating voids. Q-silicon demonstrates robust ferromagnetism at room temperature, with a blocking temperature exceeding 400 K and a Curie temperature above 600 K, alongside enhanced hardness and metallic conductivity derived from delocalized electrons during the molten phase transition.^[72] Recent investigations into tin's high-pressure phases, detailed in a 2022 review combining X-ray diffraction, density functional theory calculations, and Gibbs energy modeling, have confirmed multiple allotropes beyond the familiar α-Sn (diamond cubic) and β-Sn (body-centered tetragonal).^[73] These include γ-Sn (body-centered tetragonal, stable above ~10 GPa), γ'-Sn (body-centered cubic, above 45 GPa), σ-Sn (a complex phase at intermediate pressures), and δ-Sn (body-centered cubic form at over 120 GPa, potentially transitioning to hexagonal close-packed). Such phases exhibit metallic behavior and hold promise as high-temperature superconductors, with critical temperatures potentially reaching 10-20 K under compression.^[73] High-pressure studies have also uncovered advanced allotropes of black phosphorus, including a body-centered cubic phase identified as a cI16 superlattice derived from simple cubic black phosphorus, stable above 260 GPa. This form, explored through experimental synthesis and theoretical modeling, contrasts with the orthorhombic structure of ambient black phosphorus and enables tunable electronic properties for extreme-condition applications.^[74] Ongoing theoretical research continues to predict low-dimensional metallic phases of hydrogen, such as 2D configurations, which could exhibit superconductivity and serve as lightweight energy carriers if stabilized at accessible pressures. Computational advancements in 2024, including Bayesian machine learning interatomic potentials, have accelerated the discovery of stable elemental allotropes like those of boron (α-B, β-B, γ-B), by evaluating thermodynamic stability and phonon spectra to prioritize synthesizable candidates over exhaustive searches.^[75] These emerging forms fill critical gaps in quantum materials, such as spintronic devices from ferromagnetic Q-silicon and high-pressure superconductors from tin and phosphorus phases, expanding allotropy's role in next-generation technologies.

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Allotropes can be defined as different types of compounds made out of the same single element but in different chemical formulas and different arrangements. ...
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[PDF] Lecture 15: 11.02.05 Phase changes and phase diagrams of single
Feb 11, 2005 · The free energy curves of each phase are unique- even though both phases are composed of the same atoms in single-component systems, they have ...
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Modelling temperature-dependent properties of polymorphic organic ...
At any given pressure, each polymorph pair is either enantiotropic or monotropic. In monotropic pairs, one polymorph has a larger free energy than the other ...
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The Origin of the Term Allotrope
Jun 6, 2006 · The term allotrope or allotropic was introduced by. Berzelius in 1841 ... of the same element, such as monoclinic and rhombic sulfur or ...Missing: enantiotropic | Show results with:enantiotropic
[9]
The following communications to the Discussion were read by the ...
and sulphur is clear. Dynamic allotropy may wcur over a range of temperature, and the term. '' allotropy ” here is used in a wide sense. Let us confine ...
[10]
Polymorphism - Allotropy - Read Chemistry
Jun 18, 2023 · – This form of allotropy, known as dynamic allotropy, resembles enantiotropy in that it is reversible but there is no fixed transition point. – ...
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Mixed sp2–sp3 Nanocarbon Materials: A Status Quo Review - PMC
The hybridization of the carbon atom determines the most prominent types of arrangement—either sp2 giving rise to a strictly planar configuration or sp3 ...
[12]
[PDF] 7. Phase Transitions and Phase Equilibria - CalTech GPS
At a given T, P; a thermodynamic system will seek to minimize the Gibbs energy: G = E + PV - TS. As one increases pressure, this will clearly favor phases of ...
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Density Function Theory Predicted Carbon Allotropes: Recent ...
Jun 22, 2023 · The most recent data on experimentally undiscovered carbon allotropes, predicted by the Density Function Theory (DFT), are reviewed.
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A Brief History of Lab-Grown Diamonds - International Gem Society
Jul 10, 2025 · People have been trying to create diamonds since their true nature as pure carbon was discovered by Antoine Lavoisier in 1772. (Shigley, 2024) ...
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NOVA | Transcripts | The Diamond Deception - PBS
Feb 1, 2000 · All that is left is carbon dioxide gas - proving that diamonds are nothing but pure carbon. The only pure carbon substance known in Lavoisier's ...
[17]
Synthesis of Diamonds and Their Identification - GeoScienceWorld
Jul 1, 2022 · Since 1797, when Tennant demonstrated that diamond consists solely of elemental carbon by comparing the volume of carbon dioxide formed by ...
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History of the study of atmospheric ozone - NASA
The history of ozone begins with the clear apprehension, in the year 1840, by Schönbein, that in the odor given off in the electrolysis of water, and ...
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Phosphorus - Element information, properties and uses
Phosphorus was first made by Hennig Brandt at Hamburg in 1669 when he evaporated urine and heated the residue until it was red hot, whereupon phosphorus vapour ...Missing: Scheele | Show results with:Scheele
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None
### Summary of Early Historical Observations of Allotropy (Pre-19th Century and Early 19th Century)
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[PDF] General survey of certain results in the field of high-pressure physics
In this lecture I shall attempt to present a general survey of those parts of the field of high-pressure physics with which I have had direct contact, ...Missing: synthesis allotropes 1910s
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[PDF] Graphite - USGS Publications Warehouse
This gives graphite a density of. 2.09 to 2.26 grams per cubic centimeter (g/cm 3), perfect basal. (one-plane) cleavage, softness (0.5 to 1 on the Mohs scale), ...
[23]
Diamond - HyperPhysics
Diamond is a crystalline form of carbon which has a density of has a density of 3.51 gm/cm 3 compared to 2.26 gm/cm 3 in the graphite form.
[24]
Diamond Graphite | Miles Mineral Museum
Graphite is very soft and has a hardness of 1 to 2 on this scale. Diamonds are the hardest known natural substance and have a hardness of 10.
[25]
The Atomic Difference Between Diamonds and Graphite
Feb 18, 2014 · Diamond is one of the hardest materials known, is transparent to light, and does not conduct electricity at all. Graphite is soft, gray, and can ...Missing: electrical conductivity
[26]
Thermal Conductivity of Diamond Composites - PMC - NIH
It was shown that the sample density reaches a maximum of ~3.4 g/cm3 after 10–12 s of sintering, after which it no longer depends on sintering time. The density ...
[27]
Index of Refraction of Diamond | Physics Van | Illinois
Dec 22, 2015 · Although diamond has a very high index of refraction, 2.42, there are several other substances with higher values, for example Rutile (TiO2) ...
[28]
Tin - Element information, properties and uses | Periodic Table
For pure tin the transition occurs at 13.2 oC but the transition temperature is lower, or does not occur at all, if there are enough impurities present, for ...
[29]
Amorphous Diamond: A High-Pressure Superhard Carbon Allotrope
Dec 8, 2011 · Compressing glassy carbon above 40 GPa, we have observed a new carbon allotrope with a fully sp{sup 3}-bonded amorphous structure and ...<|separator|>
[30]
mp-66: C (Cubic, Fd-3m, 227) - Materials Project
C is diamond structured and crystallizes in the cubic Fd̅3m space group. C is bonded to four equivalent C atoms to form corner-sharing CC₄ tetrahedra.
[31]
Giant covalent structures: diamond and graphite - Book chapter
In diamond there is only one characteristic bond length, which is 1.55 Å. In graphite there are two relevant interatomic distances; namely 1.42 Å within the ...
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Chemistry of Phosphorus (Z=15)
Jun 30, 2023 · White phosphorus has to be stored in water to prevent natural combustion, but red phosphorus is stable in air. Figure 2: The four common ...Discovery of Phosphorus · Allotropes of Phosphorus · Chemistry of Phosphorus
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White phosphorus - American Chemical Society
Sep 7, 2020 · White phosphorus is one of three allotropes of the element phosphorus. The other two are red, an amorphous polymer, and black, a graphitelike polymer.Missing: Scheele | Show results with:Scheele
[34]
4.6: Day 32- Gibbs Free Energy and Work, Kinetic Metastability
Aug 3, 2023 · Exercise 3: Diamond and Graphite ... Diamond is kinetically metastable. Thermodynamics says it should change to graphite, but the change is so ...
[35]
Catalytic Effects in the Transformation of Graphite to Diamond
The transformation graphite to diamond is catalyzed by the Group VIII metals, manganese and chro-. mIUm. The fact that these metals are good solvents for ...
[36]
Raman Spectroscopy of Carbon — More Information Than You ...
Before the modern age of Raman spectroscopy, Raman was used to identify diamond, graphite, and disorder in graphitic–carbon materials, including carbon- ...
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[PDF] Raman spectroscopy: a simple, non-destructive way to characterise ...
The identification of pure diamond by. Raman spectroscopy is trivial, so men bear that in mind if your fiancée is a spec- troscopist and you are thinking of ...
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C60: Buckminsterfullerene - Nature
### Abstract
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Helical microtubules of graphitic carbon - Nature
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The Diverse Modes of Oxygen Reactivity in Life & Chemistry - PMC
Ozone (O3). 3.1. General. Ozone, the triatomic allotrope of oxygen, is less stable than triplet oxygen and also much more reactive toward alkenes and alkynes.<|separator|>
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O 8 Cluster Structure of the Epsilon Phase of Solid Oxygen
Aug 25, 2006 · The red color of the ε phase becomes darker with increasing pressure. The near-ir reflectivity increased at around 100 GPa suggesting ...Missing: paper | Show results with:paper
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Structure and General Properties of the Nonmetals – Chemistry
Both rhombic sulfur and monoclinic sulfur contain S8 molecules in which atoms form eight-membered, puckered rings that resemble crowns, as illustrated in [link] ...
[44]
[PDF] Plastic Sulfur Sulfur Rings Hexasulfur Heptasulfur, Nonasulfur ... - MIT
If molten liquid sulfur is quenched from 350 ◦C rapidly to 20 ◦C by spraying into cold water, the resulting solid sulfur is known as “plastic sulfur”.
[45]
Biological Activity of Selenium and Its Impact on Human Health - PMC
Jan 30, 2023 · Gray selenium is the most stable and dense form with a chiral hexagonal crystal lattice consisting of a helical polymeric chain and behaves as ...
[46]
22.2 The Elements of Group 14
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[47]
13.4B: Structures of the Elements - Chemistry LibreTexts
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[48]
mp-149: Si (Cubic, Fd-3m, 227) - Materials Project
Si is diamond structured and crystallizes in the cubic Fd̅3m space group. Si is bonded to four equivalent Si atoms to form corner-sharing SiSi₄ tetrahedra.
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Electronic and Optical Properties of Amorphous Silicon
Dec 30, 2016 · This chapter deals with the electronic and optical properties of amorphous silicon, particularly hydrogenated amorphous silicon (a-Si:H), ...
[50]
Advancements in low-density crystalline silicon allotropes
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[PDF] Synthesis and novel properties of Q-silicon J. Narayan, S.S. Sahoo ...
Q-silicon has a 60% higher atomic density than crystalline silicon, formed by laser melting and quenching, and can exhibit robust ferromagnetism.Missing: clathrates | Show results with:clathrates
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Accurate electronic and optical properties of hexagonal germanium ...
Mar 11, 2019 · Ge, one of the most important and widely used semiconductors, crystallizes like Si in the cubic diamond structure (space group F d 3 ¯ m ) with ...Missing: white layered
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(PDF) Germanium at high pressures - ResearchGate
Aug 6, 2025 · Abstract. X-ray diffraction experiments have been carried out on germanium in a diamond anvil cell to 26 GPa. The pressure at the transition to ...Missing: gray layered
[54]
Reversible dc-Ge to (⁠ β-Sn)-Ge transformation under high shear
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[55]
Spectroscopic properties of crystalline elemental boron and the ...
The allotropes of solid elemental boron include six crystalline phases (α-rhombohedral, β-rhombohedral, α-tetragonal, β-tetragonal, γ-orthorhombic, and α-Ga ...
[56]
[PDF] Thermodynamic Origins of Structural Metastability in Two ... - arXiv
Black arsenic (b-As) can only survive below 0.7 GPa, and then irreversibly transforms to gray arsenic. It sublimates at 300 ℃.
[57]
Stabilization of Tetrahedral P-4 and As-4 Molecules as Guests in ...
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Antimony - an overview | ScienceDirect Topics
The other allotropes of antimony are thermodynamically unstable and at elevated temperatures they rapidly convert to its most stable rhombohedral structure. ...
[59]
[PDF] Mineral Commodity Profiles: Antimony
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[60]
[PDF] Thermodynamic and Experimental Investigation of the Mn-Ni-B ...
mainly those with high oxygen affinity for obtaining high-performance PM alloys, based on the need to fulfill several requirements [20, 21]. The master ...
[61]
[PDF] materials studies for - magnetic fusion energy
(b.c.c.) or body-centered-tetragonal (b.c.t.) martensite in steels. ... before the fcc-bcc phase transition, the magnetic state undergoes a change, which ...
[62]
Tin - an overview | ScienceDirect Topics
At 1 bar pressure, two allotropic crystal forms exist for tin: the α-form (face-centered diamond-cubic form) or gray tin that is stable below − 13.2 °C and ...Missing: non- | Show results with:non-
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New mechanism reveals the secrets of a Ti alloy transformation
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[64]
Omega Phase Formation in Ti–3wt.%Nb Alloy Induced by High ... - NIH
Apr 27, 2021 · A study of the fundamentals of α → ω phase transformations induced by high-pressure torsion (HPT) in Ti–Nb-based alloys is presented in the current work.
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[PDF] Photoelectron Spectroscopy of α- and δ-Plutonium
Among the metals in this group, plutonium appears to be the most exotic. It displays six allotropic phases, and its 5f electrons are at the transition point.Missing: seven implications<|separator|>
[66]
[PDF] Plutonium Futures - Los Alamos National Laboratory
Feb 7, 2019 · By knowing that the δ-Pu fcc phase can be retained by alloying with a few percent of Al, Ga, or Am. This phase, used in applications due to its ...
[67]
Emergent magnetic moments produced by self-damage in plutonium
Among the interesting properties of plutonium is a complex phase diagram, which at ambient pressure exhibits six distinct solid-state phases below the melting ...Missing: seven | Show results with:seven
[68]
[PDF] Elemental Solubility Tendency for the Phases of Uranium by ...
Traditional alloy theory models have been applied relative to the three solid phases of uranium: alpha (orthorhombic), beta (tetragonal), and mainly gamma (body ...
[69]
Uranium - Los Alamos National Laboratory
Uranium metal exhibits in three crystallographic modifications: alpha --> (688°C) --> beta --> (776°C) --> gamma. Uranium is pyrophoric when finely divided.
[70]
Xenon mobility in γ-uranium and uranium–molybdenum alloys
Jan 10, 2022 · Pure uranium has three allotropes at atmospheric pressure: α (base-centered orthorhombic), β (tetragonal), and γ (body-centered cubic). At ...<|separator|>
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Electronic origin of the anomalous melting of sodium under pressure
Oct 7, 2025 · For example, sodium exhibits a series of structural phase transitions: body-centered cubic (BCC) → face-centered cubic (FCC) → cI16 → oP8 → tI19 ...Missing: allotropes | Show results with:allotropes
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Tunable porous nanoallotropes prepared by post-assembly etching ...
Oct 27, 2017 · We show that non–close-packed nanoparticle arrays can be fabricated through the selective removal of one of two components comprising binary nanoparticle ...
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Synthesis and novel properties of Q-silicon - Taylor & Francis Online
Jun 21, 2023 · These allotropes of silicon may have different electrical, optical, and mechanical properties, as demonstrated by robust ferromagnetism in ...
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High-Pressure and High-Temperature Chemistry of Phosphorus and ...
The thermodynamically stable allotrope of phosphorus at ambient conditions ... body-centered cubic structure (bcc), later identified as a superlattice ...