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Fullerene



 
 
"C60" and "C-60" redirect here. For other uses, see C60 (disambiguation)
C60 (disambiguation)

C60, C.60 or C-60 could mean* C60, a carbon molecule, also known as Buckyballs* C60 , a band also known as Cobalt 60* C60 steam locomotive, a Japanese class of locomotives...
.


Fullerene are a family of carbon
Carbon

Carbon is a chemical element with chemical symbol C and atomic number 6. As a member of group 14 on the periodic table, it is nonmetallic and tetravalence?making four electrons available to form covalent bond chemical bonds....
 allotropes
Allotropy

Allotropy or allotropism is a behavior exhibited by certain chemical elements: these elements can exist in two or more different forms, known as allotropes of that element....
, molecule
Molecule

In chemistry, a molecule is defined as a sufficiently stable, electric charge neutral group of at least two atoms in a definite arrangement held together by very strong chemical bonds....
s composed entirely of carbon, in the form of a hollow sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
, ellipsoid
Ellipsoid

An ellipsoid is a type of Quadric that is a higher dimensional analogue of an ellipse. The equation of a standard axis-aligned ellipsoid body in an xyz-Cartesian coordinate system is...
, tube
Cylinder (geometry)

A cylinder is one of the most curvilinear basic geometric shapes: the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder....
, or plane. Spherical fullerenes are also called buckyballs, and cylindrical ones are called carbon nanotubes or buckytubes.






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Fullerene C540
"C60" and "C-60" redirect here. For other uses, see C60 (disambiguation)
C60 (disambiguation)

C60, C.60 or C-60 could mean* C60, a carbon molecule, also known as Buckyballs* C60 , a band also known as Cobalt 60* C60 steam locomotive, a Japanese class of locomotives...
.


Fullerene are a family of carbon
Carbon

Carbon is a chemical element with chemical symbol C and atomic number 6. As a member of group 14 on the periodic table, it is nonmetallic and tetravalence?making four electrons available to form covalent bond chemical bonds....
 allotropes
Allotropy

Allotropy or allotropism is a behavior exhibited by certain chemical elements: these elements can exist in two or more different forms, known as allotropes of that element....
, molecule
Molecule

In chemistry, a molecule is defined as a sufficiently stable, electric charge neutral group of at least two atoms in a definite arrangement held together by very strong chemical bonds....
s composed entirely of carbon, in the form of a hollow sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
, ellipsoid
Ellipsoid

An ellipsoid is a type of Quadric that is a higher dimensional analogue of an ellipse. The equation of a standard axis-aligned ellipsoid body in an xyz-Cartesian coordinate system is...
, tube
Cylinder (geometry)

A cylinder is one of the most curvilinear basic geometric shapes: the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder....
, or plane. Spherical fullerenes are also called buckyballs, and cylindrical ones are called carbon nanotubes or buckytubes. Graphene
Graphene

Graphene is a one-atom-thick planar sheet of sp2 bond carbon atoms that are densely packed in a honeycomb crystal lattice. It can be viewed as an chicken wire made of carbon atoms and their bonds....
 is an example of a planar fullerene sheet. Fullerenes are similar in structure to graphite
Graphite

The mineral graphite is one of the allotropes of carbon. It was named by Abraham Gottlob Werner in 1789 from the Greek language ??afe?? : "to draw/write", for its use in pencils, where it is commonly called lead, as distinguished from the actual metallic element lead....
, which is composed of stacked sheets of linked hexagonal rings, but may also contain pentagonal (or sometimes heptagonal) rings that would prevent a sheet from being planar.

The fullerene was discovered in 1985 by Robert Curl
Robert Curl

Robert Floyd Curl, Jr. the son of a Methodist Minister is an emeritus professor of chemistry at Rice University.He was awarded the Nobel Prize in Chemistry in 1996 for the discovery of fullerene ....
, Harold Kroto
Harold Kroto

Sir Harold Walter Kroto, Fellow of the Royal Society is an England chemistry and one of the three recipients to share the 1996 Nobel Prize in Chemistry....
 and Richard Smalley
Richard Smalley

Richard Errett Smalley was the Gene and Norman Hackerman Professor of Chemistry and a Professor of Physics and Astronomy at Rice University, in Houston, Texas....
 at the University of Sussex
University of Sussex

The University of Sussex is a British campus university situated next to the East Sussex village of Falmer, from Brighton. It was the first of the New Universities of Plate glass university....
 and Rice University
Rice University

William Marsh Rice University is a private university research university located in Houston, Texas, Texas, United States. The campus is located near the Houston Museum District and adjacent to the Texas Medical Center....
, who named it after Richard Buckminster Fuller, whose geodesic dome
Geodesic dome

A geodesic dome is a spherical or partial-spherical thin-shell structure based on a network of great circles lying on the surface of a sphere....
s it resembles.

Prediction and discovery

With mass spectroscopy, discrete peaks were observed corresponding to molecules with the exact mass of sixty or seventy or more carbon atoms. In 1985, Harold Kroto
Harold Kroto

Sir Harold Walter Kroto, Fellow of the Royal Society is an England chemistry and one of the three recipients to share the 1996 Nobel Prize in Chemistry....
 (then of the University of Sussex
University of Sussex

The University of Sussex is a British campus university situated next to the East Sussex village of Falmer, from Brighton. It was the first of the New Universities of Plate glass university....
), James R. Heath
James R. Heath

James R. Heath is an United States chemist and the Elizabeth W. Gilloon Professor of Chemistry at the California Institute of Technology....
, Sean O'Brien, Robert Curl
Robert Curl

Robert Floyd Curl, Jr. the son of a Methodist Minister is an emeritus professor of chemistry at Rice University.He was awarded the Nobel Prize in Chemistry in 1996 for the discovery of fullerene ....
 and Richard Smalley
Richard Smalley

Richard Errett Smalley was the Gene and Norman Hackerman Professor of Chemistry and a Professor of Physics and Astronomy at Rice University, in Houston, Texas....
, from Rice University
Rice University

William Marsh Rice University is a private university research university located in Houston, Texas, Texas, United States. The campus is located near the Houston Museum District and adjacent to the Texas Medical Center....
, discovered C60, and shortly thereafter came to discover the fullerenes. Kroto, Curl, and Smalley were awarded the 1996 Nobel Prize in Chemistry
Nobel Prize in Chemistry

The Nobel Prize in Chemistry is awarded annually by the Royal Swedish Academy of Sciences to scientists in the various fields of chemistry. It is one of the five Nobel Prizes established by the will of Alfred Nobel in 1895, awarded for outstanding contributions in chemistry, Nobel Prize in Physics, Nobel Prize in Literature, Nobel Peace Pri...
 for their roles in the discovery of this class of compounds. C60 and other fullerenes were later noticed occurring outside the laboratory (e.g., in normal candle
Candle

A candle is a source of light, and sometimes a source of heat, consisting of a solid block of fuel and an embedded candle wick.Today, most candles are made from paraffin....
 soot
Soot

Soot is a general term that refers to impure carbon particles resulting from the incomplete combustion of a hydrocarbon. It is more properly restricted to the product of the gas-phase combustion process but is commonly extended to include the residual pyrolyzed fuel particles such as cenospheres, charred wood, petroleum coke, etc....
). By 1991, it was relatively easy to produce gram-sized samples of fullerene powder using the techniques of Donald Huffman and Wolfgang Krätschmer. Fullerene purification remains a challenge to chemists and to a large extent determines fullerene prices. So-called endohedral fullerenes
Endohedral fullerenes

Endohedral fullerenes are fullerenes that have additional atoms, ions, or clusters enclosed within their inner spheres. The first lanthanum C60 complex was synthesed in 1985 called La@C60....
 have ions or small molecules incorporated inside the cage atoms. Fullerene is an unusual reactant in many organic reaction
Organic reaction

Organic reactions are chemical reactions involving organic compounds. The basic organic chemistry reaction types are addition reactions, elimination reactions, substitution reactions, pericyclic reactions, rearrangement reactions and organic redox reaction....
s such as the Bingel reaction
Bingel reaction

The Bingel reaction in fullerene chemistry is a fullerene cyclopropane to a methanofullerene first discovered by C. Bingel in 1993 with the bromo derivative of diethyl malonate in the presence of a base such as sodium hydride or DBU ....
 discovered in 1993. The first nanotubes were obtained in 1991.

Minute quantities of the fullerenes, in the form of C60, C70, C76, and C84 molecules, are produced in nature, hidden in soot
Soot

Soot is a general term that refers to impure carbon particles resulting from the incomplete combustion of a hydrocarbon. It is more properly restricted to the product of the gas-phase combustion process but is commonly extended to include the residual pyrolyzed fuel particles such as cenospheres, charred wood, petroleum coke, etc....
 and formed by lightning discharges in the atmosphere. Recently, Buckminsterfullerenes were found in a family of minerals known as Shungites in Karelia
Karelia

Karelia , the land of the Karelians, is an area in Northern Europe of historical significance for Finland, Russia, and Sweden. It is currently divided between the Russian Republic of Karelia, the Russian Leningrad Oblast, and Finland ....
, Russia.

The existence of C60 was predicted by Eiji Osawa of Toyohashi University of Technology
Toyohashi University of Technology

Toyohashi University of Technology , often abbreviated to TUT, is a national university institute of technology located in Toyohashi, Aichi, Japan....
 in a Japanese magazine in 1970. He noticed that the structure of a corannulene
Corannulene

Corannulene is a polycyclic aromatic hydrocarbon with chemical formula Carbon20Hydrogen10. The molecule consists of a cyclopentane ring ring fusion with 5 benzene rings....
 molecule was a subset of a soccer-ball shape, and he made the hypothesis that a full ball shape could also exist. His idea was reported in Japanese magazines, but did not reach Europe or America.

Naming

Buckminsterfullerene (C60) was named after Richard Buckminster Fuller, a noted architectural modeler who popularized the geodesic dome
Geodesic dome

A geodesic dome is a spherical or partial-spherical thin-shell structure based on a network of great circles lying on the surface of a sphere....
. Since buckminsterfullerenes have a similar shape to that sort of dome, the name was thought to be appropriate. As the discovery of the fullerene family came
after buckminsterfullerene, the shortened name 'fullerene' was used to refer to the family of fullerenes.

For illustrations of geodesic dome structures, see Montreal Biosphere
Montreal Biosphère

The Biosph?re of Environment Canada is a museum in Montreal dedicated to water and the Natural environment. It is located at Parc Jean-Drapeau, on Saint Helen's Island in the building of the United States pavilion for the 1967 World Exhibition Expo 67....
, Eden Project
Eden Project

The Eden Project is a visitor attraction in the United Kingdom, including the world's largest greenhouse.The project is located in a reclaimed Kaolinite clay pit, located from the town of St Blazey and from the larger town of St Austell, Cornwall, England....
, Missouri Botanical Garden
Missouri Botanical Garden

The Missouri Botanical Garden is a botanical garden located in St. Louis, Missouri, and is also known informally as "Shaw's Garden" .Founded in 1859, the Missouri Botanical Garden is one of the oldest botanical institutions in the United States and a National Historic Landmark....
, Science World at Telus World of Science
Science World at TELUS World of Science

TELUS World of Science, Vancouver is a Science museum run by a non-profit organization in Vancouver, British Columbia, Canada. It is located at the end of False Creek, and features many permanent interactive exhibits and displays, as well as areas with varying topics throughout the years....
, Mitchell Park Horticultural Conservatory
Mitchell Park Horticultural Conservatory

Mitchell Park Horticultural Conservatory is a conservatory located at Mitchell Park in Milwaukee, Wisconsin, Wisconsin, U.S.. It is owned and operated by the Milwaukee County Park System, and replaced the original Milwaukee Conservatory which stood from 1898 to 1955....
, Gold Dome
Gold Dome

The Gold Dome is a geodesic dome in Oklahoma City, United States, considered to be a major landmark. Located at the intersection of North West 23rd Street and North Classen Boulevard, it sits in the heart of the city's Asia District....
, Tacoma Dome
Tacoma Dome

The Tacoma Dome is an list of indoor arenas located in Tacoma, Washington, USA, approximately 30 miles South of Seattle....
, Reunion Tower
Reunion Tower

__FORCETOC__ Reunion Tower is a 560 foot...
, and Spaceship Earth (Epcot).

Variations

Since the discovery of fullerenes in 1985, structural variations on fullerenes have evolved well beyond the individual clusters themselves. Examples include:
  • buckyball clusters: smallest member is (unsaturated version of dodecahedrane
    Dodecahedrane

    Dodecahedrane is a chemical compound first organic synthesis by Leo Paquette of Ohio State University in 1982, primarily for the "aesthetically pleasing symmetry of the dodecahedral framework"....
    ) and the most common is ;
  • nanotubes: hollow tubes of very small dimensions, having single or multiple walls; potential applications in electronics industry;
  • megatubes: larger in diameter than nanotubes and prepared with walls of different thickness; potentially used for the transport of a variety of molecules of different sizes;
  • polymers: chain, two-dimensional and three-dimensional polymers are formed under high pressure high temperature conditions
  • nano"onions": spherical particles based on multiple carbon layers surrounding a buckyball core; proposed for lubricants;
  • linked "ball-and-chain" dimers: two buckyballs linked by a carbon chain;
  • fullerene rings


The "Buckyball"

C60a

Buckminsterfullerene

Buckminsterfullerene (IUPAC name (C60-Ih)[5,6]fullerene) is the smallest fullerene molecule in which no two pentagons share an edge (which can be destabilizing; see pentalene
Pentalene

Pentalene is a polycyclic hydrocarbon composed of two fused cyclopentadiene rings. It has chemical formula C8H6. It is antiaromaticity, because it has H?ckel's rule pi electrons....
). It is also the most common in terms of natural occurrence, as it can often be found in soot
Soot

Soot is a general term that refers to impure carbon particles resulting from the incomplete combustion of a hydrocarbon. It is more properly restricted to the product of the gas-phase combustion process but is commonly extended to include the residual pyrolyzed fuel particles such as cenospheres, charred wood, petroleum coke, etc....
.

The structure of C60 is a truncated (T = 3) icosahedron
Truncated icosahedron

The truncated icosahedron is an Archimedean solid. It comprises 12 regular pentagon faces, 20 regular hexagon faces, 60 vertices and 90 edges....
, which resembles a soccer ball
Football (ball)

A football is a ball used to play one of the various sports known as football.In the distant past, crude balls such as inflated pigs' bladders were used, but balls are now designed by teams of engineers to exacting specifications....
 of the type made of twenty hexagons and twelve pentagons, with a carbon atom at the vertices of each polygon and a bond along each polygon edge.

The van der Waals diameter of a C60 molecule is about 1 nanometer (nm). The nucleus to nucleus diameter of a C60 molecule is about 0.7 nm.

The C60 molecule has two bond lengths. The 6:6 ring bonds (between two hexagons) can be considered "double bonds" and are shorter than the 6:5 bonds (between a hexagon and a pentagon). Its average bond length is 1.4 angstroms.

Boron buckyball

A new type of buckyball utilizing boron
Boron

Boron is a chemical element with atomic number 5 and the chemical symbol B. Boron is a trivalent metalloid element which occurs abundantly in the evaporite ores borax and ulexite....
 atoms instead of the usual carbon has been predicted and described by researchers at Rice University. The B-80 structure is predicted to be more stable than the C-60 buckyball. One reason for this given by the researchers is that the B-80 is actually more like the original geodesic dome structure popularized by Buckminster Fuller which utilizes triangles rather than hexagons. However, this work has been subject to much criticism by quantum chemists as it was concluded that the predicted Ih symmetric structure was vibrationally unstable and the resulting cage undergoes a spontaneous symmetry breaking yielding a puckered cage with rare Th symmetry (symmetry of the volleyball
Volleyball (ball)

A volleyball is a ball used to play indoor volleyball, beach volleyball, or other less common volleyball variations. Volleyballs are round and traditionally consist of eighteen nearly rectangular panels of synthetic or genuine leather, arranged in six identical sections of three panels each, wrapped around a bladder....
) number of six rings of carbon in this molecule is 20 and number of five member rings is 12

Variations of buckyballs

Another fairly common buckminsterfullerene is C70, but fullerenes with 72, 76, 84 and even up to 100 carbon atoms are commonly obtained.

In mathematical
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
 terms, the structure of a fullerene is a trivalent
Valence (chemistry)

In chemistry, valence, also known as valency or valency number, is a measure of the number of chemical bonds formed by the atoms of a given chemical element....
 convex polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 with pentagonal and hexagonal faces. In graph theory
Graph theory

In mathematics and computer science, graph theory is the study of graph : mathematical structures used to model pairwise relations between objects from a certain collection....
, the term fullerene refers to any 3-regular
Regular graph

In graph theory, a regular graph is a graph where each vertex has the same number of neighbors, i.e. every vertex has the same Degree or valency....
, planar graph
Planar graph

In graph theory, a planar graph is a graph which can be graph embedding in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints....
 with all faces of size 5 or 6 (including the external face). It follows from Euler's polyhedron formula
Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....
, |V|-|E|+|F| = 2, (where |V|, |E|, |F| indicate the number of vertices, edges, and faces), that there are exactly 12 pentagons in a fullerene and |V|/2-10 hexagons.

20-fullerene
(dodecahedral graph)
26-fullerene graph 60-fullerene
(truncated icosahedral graph)
70-fullerene graph


The smallest fullerene is the dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
--the unique C20. There are no fullerenes with 22 vertices. The number of fullerenes C2n grows with increasing n = 12,13,14..., roughly in proportion to n9. For instance, there are 1812 non-isomorphic fullerenes C60. Note that only one form of C60, the buckminsterfullerene alias truncated icosahedron
Truncated icosahedron

The truncated icosahedron is an Archimedean solid. It comprises 12 regular pentagon faces, 20 regular hexagon faces, 60 vertices and 90 edges....
, has no pair of adjacent pentagons (the smallest such fullerene). To further illustrate the growth, there are 214,127,713 non-isomorphic fullerenes C200, 15,655,672 of which have no adjacent pentagons.

Trimetasphere carbon nanomaterials were discovered by researchers at Virginia Tech and licensed exclusively to Luna Innovations
Luna Innovations

Luna Innovations is a company that develops and manufactures products for the healthcare, telecommunications, energy and defense markets. It is headquartered in Roanoke, Virginia....
. This class of novel molecules comprises 80 carbon atoms (C80) forming a sphere which encloses a complex of three metal atoms and one nitrogen atom. These fullerenes encapsulate metals which puts them in the subset referred to as metallofullerenes. Trimetaspheres have the potential for use in diagnostics (as safe imaging agents), therapeutics and in organic solar cells.

Carbon nanotubes

Kohlenstoffnanoroehre Animation
Nanotubes are cylindrical fullerenes. These tubes of carbon are usually only a few nanometres wide, but they can range from less than a micrometer to several millimeters in length. They often have closed ends, but can be open-ended as well. There are also cases in which the tube reduces in diameter before closing off. Their unique molecular structure results in extraordinary macroscopic properties, including high tensile strength, high electrical conductivity, high ductility, high resistance to heat, and relative chemical inactivity (as it is cylindrical and "planar"—that is, it has no "exposed" atoms that can be easily displaced). One proposed use of carbon nanotubes is in paper batteries
Paper battery

A paper battery is a Battery engineered to use a paper-thin sheet of cellulose infused with aligned carbon nanotubes. The nanotubes act as electrodes; allowing the storage devices to conduct electricity....
, developed in 2007 by researchers at Rensselaer Polytechnic Institute
Rensselaer Polytechnic Institute

Rensselaer Polytechnic Institute, or RPI, is a Private university research university located in Troy, New York, New York, United States. RPI was founded in 1824 by Stephen Van Rensselaer III for the "application of science to the common purposes of life", and is the oldest technological university in the English-speaking world....
. Another proposed use in the field of space technologies and science fiction is to produce high-tensile carbon cables required by a space elevator
Space elevator

A space elevator is a proposed structure designed to transport material from a Astronomical object's surface into space. Many variants have been proposed, all of which involve traveling along a fixed structure instead of using rocket powered space launch....
.

Carbon nanobuds

Nanobuds have been obtained by adding Buckminsterfullerenes to carbon nanotubes.

Properties

For the past decade, the chemical and physical properties of fullerenes have been a hot topic in the field of research and development, and are likely to continue to be for a long time. Popular Science
Popular science

Popular science, sometimes called literature of science, is interpretation of science intended for a general audience. While science journalism focuses on recent scientific developments, popular science is broad-ranging, often written by scientists as well as journalists, and is presented in many formats, which can include books, televi...
 has published articles about the possible uses of fullerenes in armor. In April 2003, fullerenes were under study for potential medicinal use
Nanomedicine

Nanomedicine is the medicine application of nanotechnology. The approaches to nanomedicine range from the medical use of nanomaterials, to Nanoelectronics biosensors, and even possible future applications of molecular nanotechnology....
: binding specific antibiotic
Antibiotic

In common usage, an antibiotic is a substance or compound that kills or inhibits the growth of bacteria. Antibiotics belong to the group of antimicrobial compounds used to treat infections caused by microorganisms, including fungus and protozoa....
s to the structure to target resistant bacteria and even target certain cancer
Cancer

Cancer is a class of diseases in which a group of cell display uncontrolled growth , invasion , and sometimes metastasis . These three malignant properties of cancers differentiate them from benign tumors, which are self-limited, do not invade or metastasize....
 cells such as melanoma
Melanoma

Melanoma is a malignant tumor of melanocytes which are found predominantly in skin but also in the bowel and the eye . It is one of the rarer types of skin cancer but causes the majority of skin cancer related deaths....
. The October 2005 issue of Chemistry and Biology contains an article describing the use of fullerenes as light-activated antimicrobial
Antimicrobial

An antimicrobial is a substance that kills or inhibits the growth of microbes such as bacteria, fungi, protozoals or viruses. Antimicrobial drugs either kill microbes or prevent the growth of microbes ....
 agents.

In the field of nanotechnology
Nanotechnology

Nanotechnology, shortened to "Nanotech", is the study of the control of matter on an atomic and molecular scale. Generally nanotechnology deals with structures of the size 100 nanometers or smaller, and involves developing materials or devices within that size....
, heat resistance
Thermal conductivity

In physics, thermal conductivity, , is the List of materials properties of a material that indicates its ability to conduct heat. It appears primarily in Heat conduction#Fourier's law for heat conduction....
 and superconductivity
Superconductivity

Superconductivity is a phenomenon occurring in certain materials generally at very low temperatures, characterized by exactly zero electrical resistance and the exclusion of the interior magnetic field ....
 are some of the more heavily studied properties.

A common method used to produce fullerenes is to send a large current between two nearby graphite
Graphite

The mineral graphite is one of the allotropes of carbon. It was named by Abraham Gottlob Werner in 1789 from the Greek language ??afe?? : "to draw/write", for its use in pencils, where it is commonly called lead, as distinguished from the actual metallic element lead....
 electrodes in an inert
Inert

In English, to be inert is to be in a state of doing little or nothing....
 atmosphere. The resulting carbon
Carbon

Carbon is a chemical element with chemical symbol C and atomic number 6. As a member of group 14 on the periodic table, it is nonmetallic and tetravalence?making four electrons available to form covalent bond chemical bonds....
 plasma
Plasma (physics)

In physics and chemistry, plasma is a partially ionized gas, in which a certain proportion of electrons are free rather than being bound to an atom or molecule....
 arc between the electrodes cools into sooty residue from which many fullerenes can be isolated.

There are many calculations that have been done using ab-initio Quantum Methods applied to fullerenes. By DFT
Density functional theory

Density functional theory is a quantum mechanics theory used in physics and chemistry to investigate the electronic structure of Many-body problem, in particular atoms, molecules, and the condensed phases....
 and TD-DFT
Time-dependent density functional theory

Time-dependent density functional theory is a quantum mechanical theory used in physics and chemistry to investigate the properties and Molecular dynamics of many-body systems in the presence of time-dependent potentials, such as electric or magnetic fields....
 methods one can obtain IR, Raman
Raman spectroscopy

Raman spectroscopy is a Spectroscopy technique used in condensed matter physics and chemistry to study vibrational, rotational, and other low-frequency modes in a system....
 and UV
Ultraviolet-visible spectroscopy

Ultraviolet-visible spectroscopy or ultraviolet-visible spectrophotometry involves the spectroscopy of photons in the UV-visible region....
 spectra. Results of such calculations can be compared with experimental results.

Aromaticity

Researchers have been able to increase the reactivity of fullerenes by attaching active groups to their surfaces. The Buckminsterfullerene does not exhibit "superaromaticity
Aromaticity

Aromaticity is a chemical property in which a conjugated system ring of unsaturated bonds, lone pairs, or empty orbitals exhibit a stabilization stronger than would be expected by the stabilization of conjugation alone....
:" that is, the electrons in the hexagonal rings do not delocalize
Delocalized electron

In chemistry delocalized electrons are electrons in a molecule that are not associated with a single atom or to a covalent bond. Delocalized electrons are contained within an Molecular_orbital that extends over several adjacent atoms....
 over the whole molecule.

A spherical fullerene of
n carbon atoms has n pi-bonding electrons. These should try to delocalize over the whole molecule. The quantum mechanics of such an arrangement should be like one shell only of the well-known quantum mechanical structure of a single atom, with a stable filled shell for n = 2, 8, 18, 32, 50, 72, 98, 128, etc.; i.e. twice a perfect square
Perfect square

Perfect square may refer to:...
; but this series does not include 60. As a result, C60 in water tends to pick up two more electrons and become an anion. The nC60 described below may be the result of C60's trying to form a loose metallic bonding.

Chemistry

Fullerenes are stable, but not totally unreactive. The sp2-hybridized carbon atoms, which are at their energy minimum in planar graphite, must be bent to form the closed sphere or tube, which produces angle strain
Angle strain

The presence of angle strain in a molecule indicates that in a specific chemical conformation there exist bond angles that deviate from the ideal bond angles required to achieve maximum bond strength....
. The characteristic reaction of fullerenes is electrophilic addition
Electrophilic addition

In organic chemistry, an electrophilic addition reaction is an addition reaction where, in a chemical compound, a pi bond is removed by the creation of two new covalent bonds....
 at 6,6-double bonds, which reduces angle strain by changing sp2-hybridized carbons into sp3-hybridized ones. The change in hybridized orbital
Atomic orbital

An atomic orbital is a mathematical function that describes the wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus....
s causes the bond angles to decrease from about 120 degrees in the sp2 orbitals to about 109.5 degrees in the sp3 orbitals. This decrease in bond angles allows for the bonds to bend less when closing the sphere or tube, and thus, the molecule becomes more stable.

Other atoms can be trapped inside fullerenes to form inclusion compound
Inclusion compound

In host-guest chemistry an inclusion compound is a complex in which one chemical compound forms a cavity in which molecules of a second "guest" compound are located....
s known as endohedral fullerenes
Endohedral fullerenes

Endohedral fullerenes are fullerenes that have additional atoms, ions, or clusters enclosed within their inner spheres. The first lanthanum C60 complex was synthesed in 1985 called La@C60....
. An unusual example is the egg shaped fullerene Tb3N@C84, which violates the isolated pentagon rule. Recent evidence for a meteor impact at the end of the Permian
Permian

The PermianThe term "Permian" was introduced into geology in 1841 by Sir Roderick Murchison, president of the Geological Society of London, who identified typical strata in extensive Russian explorations undertaken with Edouard de Verneuil; Murchison asserted in 1841 that he named his "Permian system" after the ancient kingdom...
 period was found by analyzing noble gas
Noble gas

|}The noble gases are a group of chemical elements with very similar properties: under standard conditions, they are all odorless, colorless, monatomic gases, with a very low chemical reactivity....
es so preserved. Metallofullerene
Metallofullerene

Metallofullerenes are spherical molecular structures made of carbon and metal. Their foundations are based in the fullerene family and hold a lot of potential in the advancement of materials design....
-based inoculates using the rhonditic
Rhondite

Rhondite is a nano-scale helical carbon-based structure that may be used in the production of steels and alloys to increase cohesion, strength, and uniformity....
 steel process are beginning production as one of the first commercially-viable uses of buckyballs.

Solubility

Fullerenes are sparingly soluble in many solvent
Solvent

A solvent is a liquid or gas that dissolves a solid, liquid, or gaseous solute, resulting in a solution.The most common solvent in everyday life is water....
s. Common solvents for the fullerenes include aromatics, such as toluene
Toluene

Toluene, also known as methylbenzene or phenylmethane, is a clear, Water -insoluble liquid with the typical smell of paint thinners, redolent of the sweet smell of the related compound benzene....
, and others like carbon disulfide
Carbon disulfide

Carbon disulfide is a colorless, volatile liquid with the chemical formula CS2. The compound is used frequently as a building block in organic chemistry as well as an industrial and chemical non-polar solvent....
. Solutions of pure Buckminsterfullerene have a deep purple color. Solutions of C70 are a reddish brown. The higher fullerenes C76 to C84 have a variety of colors. C76 has two optical forms, while other higher fullerenes have several structural isomers. Fullerenes are the only known allotrope of carbon that can be dissolved in common solvents at room temperature.

Some fullerene structures are not soluble because they have a small band gap
Band gap

In solid state physics and related applied fields, a band gap, also called an energy gap or bandgap, is an energy range in a solid where no electron states exist....
 between the ground and excited state
Excited state

Excitation is an elevation in energy level above an arbitrary baseline energy state. In physics there is a specific technical definition for energy level which is often associated with an atom being excited to an excited state....
s. These include the small fullerenes C28, C36 and C50. The C72 structure is also in this class, but the endohedral version with a trapped lanthanide
Lanthanide

According to the IUPAC terminology, the lanthanoid series comprises the fifteen chemical elements with atomic numbers 57 through 71, from lanthanum to lutetium....
-group atom is soluble due to the interaction of the metal atom and the electronic states of the fullerene. Researchers had originally been puzzled by C72 being absent in fullerene plasma-generated soot extract, but found in endohedral samples. Small band gap fullerenes are highly reactive and bind to other fullerenes or to soot particles.

Solvents that are able to dissolve buckminsterfullerene (C60) are listed below in order from highest solubility. The value in parentheses is the approximate saturated concentration.
  1. 1-chloronaphthalene
    1-Chloronaphthalene

    1-Chloronaphthalene is an aromatic compound. It is a colorless, oily liquid which may be used to determine the refractive index of crystals by immersion....
     (51 mg/mL)
  2. 1-methylnaphthalene
    1-Methylnaphthalene

    1-Methylnaphthalene is an aromatic hydrocarbon. It has a cetane number of zero, and was previously used as the lower reference for cetane number....
     (33 mg/mL)
  3. 1,2-dichlorobenzene
    1,2-Dichlorobenzene

    1,2-Dichlorobenzene, or ortho-dichlorobenzene, is an organic compound with the formula C6H4Cl2. This colourless liquid is poorly soluble in water but miscible with most organic solvents....
     (24 mg/mL)
  4. 1,2,4-trimethylbenzene
    1,2,4-Trimethylbenzene

    1,2,4-Trimethylbenzene is a colorless liquid with chemical formula C9H12. It is flammable aromatic hydrocarbon with a strong odor....
     (18 mg/mL)
  5. tetrahydronaphthalene (16 mg/mL)
  6. carbon disulfide
    Carbon disulfide

    Carbon disulfide is a colorless, volatile liquid with the chemical formula CS2. The compound is used frequently as a building block in organic chemistry as well as an industrial and chemical non-polar solvent....
     (8 mg/mL)
  7. 1,2,3-tribromopropane (8 mg/mL)
  8. bromoform
    Bromoform

    Bromoform is a pale yellowish liquid with a sweet odor similar to chloroform, a halomethane or haloform. Its refractive index is 1.595 . Small amounts are formed naturally by plants in the ocean....
     (5 mg/mL)
  9. toluene
    Toluene

    Toluene, also known as methylbenzene or phenylmethane, is a clear, Water -insoluble liquid with the typical smell of paint thinners, redolent of the sweet smell of the related compound benzene....
     (3 mg/ml)
  10. benzene
    Benzene

    Benzene, or benzol, is an organic compound chemical compound and a known carcinogen with the molecular formula Carbon6Hydrogen6....
     (1.5 mg/ml)
  11. cyclohexane
    Cyclohexane

    Cyclohexane is a cycloalkane with the molecular formula Carbon6Hydrogen12. Cyclohexane is used as a nonpolar solvent for the chemical industry, and also as a raw material for the industrial production of adipic acid and caprolactam, both of which are intermediates used in the production of nylon....
     (1.2 mg/ml)
  12. carbon tetrachloride
    Carbon tetrachloride

    Carbon tetrachloride, also known by many other names is the organic compound with the chemical formula CCl4. It is a reagent in organic synthesis chemistry and was formerly widely used in fire extinguishers, as a precursor to refrigerations, and a cleaning agent....
     (0.4 mg/ml)
  13. chloroform
    Chloroform

    Chloroform, also known as trichloromethane and methyl trichloride, is a chemical compound with chemical formula CarbonHydrogenChlorine3....
     (0.25 mg/ml)
  14. n-hexane
    Hexane

    Hexane is an alkane hydrocarbon with the chemical formula CH34CH3 or C6H14. The "hex" prefix refers to its six carbons, while the "ane" ending indicates that its carbons are connected by single bonds....
     (0.046 mg/ml)
  15. tetrahydrofuran
    Tetrahydrofuran

    Tetrahydrofuran is a colorless, water-miscible organic liquid with low-viscosity at "room" temperature and pressure . It is a Heterocyclic compound compound with a chemical formula C4H8O, and is the fully Hydrogenation analog of the aromatic organic compound furan....
     (0.006 mg/ml)
  16. acetonitrile
    Acetonitrile

    Acetonitrile is the chemical compound with chemical formula CH3CN. This colourless liquid is the simplest organic nitrile and is widely used as a solvent....
     (0.004 mg/ml)
  17. methanol
    Methanol

    Methanol, also known as methyl alcohol, carbinol, wood alcohol, wood naphtha or wood spirits, is a chemical compound with chemical formula carbonhydrogen3oxygenhydrogen ....
     (0.00004 mg/ml)
  18. water
    Water

    Water is a common chemical substance that is essential for the survival of all known forms of life. In typical usage, water refers only to its liquid form or States of matter, but the substance also has a solid state, ice, and a gaseous state, water vapor or steam....
     (1.3x10-11 mg/mL)


Solubility of C60 in some solvents shows unusual behaviour due to existence of solvate phases (analogues of crystallohydrates). For example, solubility of C60 in benzene
Benzene

Benzene, or benzol, is an organic compound chemical compound and a known carcinogen with the molecular formula Carbon6Hydrogen6....
 solution shows maximum at about 313 K. Crystallization from benzene solution at temperatures below maximum results in formation of triclinic solid solvate with four benzene molecules C60•4C6H6 which is rather unstable in air. Out of solution, this structure decomposes into usual fcc C60 in few minutes' time. At temperatures above solubility maximum the solvate is not stable even when immersed in saturated solution and melts with formation of fcc C60. Crystallization at temperatures above the solubility maximum results in formation of pure fcc C60. Large millimetre size crystals of C60 and C70 can be grown from solution both for solvates and for pure fullerenes.

Quantum mechanics

In 1999, researchers from the University of Vienna
University of Vienna

The University of Vienna is a public university located in Vienna, Austria. Having opened in 1365, it is one of the oldest universities in Europe....
 demonstrated that wave-particle duality applied to molecules such as fullerene. One of the co-authors of this research, Julian Voss-Andreae
Julian Voss-Andreae

Julian Voss-Andreae is a German sculptor living and working in the U.S.Voss-Andreae started out as a painter and later studied experimental physics at Free University of Berlin, University of Edinburgh and University of Vienna....
, has since created several sculptures symbolizing wave-particle duality in fullerenes
(see Fullerenes in popular culture
Fullerenes in popular culture

The following is a list of references to fullerenes in popular culture....
 for more detail).

Science writer Marcus Chown stated on the CBC radio show
Quirks and Quarks
Quirks and Quarks

Quirks & Quarks is a Canadian weekly science news program heard over CBC Radio One of the Canadian Broadcasting Corporation .Airing since October 8, 1975, Quirks & Quarks is consistently rated among the most popular CBC programs, attracting over 500,000 listeners each Saturday from 12:00 to 13:00....
in May 2006 that scientists are trying to make buckyballs exhibit the quantum behavior of existing in two places at once (quantum superposition
Quantum superposition

Quantum superposition is the fundamental law of quantum mechanics. It defines the allowed state space of a quantum mechanical system.In Probability theory, every possible event has a non-negative real number between zero and one associated to it, the probability, which gives the chance that it happens....
).

Safety and toxicity


When considering toxicological data, care must be taken to distinguish as necessary between what are normally referred to as fullerenes: (C60, C70,...); fullerene derivatives: C60 or other fullerenes with covalently bonded chemical groups; fullerene complexes (e.g., water-solubilized with surfactants, such as C60-PVP; host-guest complexes, such as with cyclodextrin; ), where the fullerene is physically bound to another molecule; C60 nanoparticles, which are extended solid-phase aggregates of C60 crystallites; and nanotubes, which are generally much larger (in terms of molecular weight and size) compounds, and are different in shape to the spheroidal fullerenes C60 and C70, as well as having different chemical and physical properties.

The above different compounds span the range from insoluble materials in either hydrophilic or lipophilic media, to hydrophilic, lipophilic, or even amphiphilic compounds, and with other varying physical and chemical properties. Therefore any broad generalization extrapolating for example results from C60 to nanotubes or vice versa is not possible, though technically all are fullerenes, as the term is defined as a close-caged all-carbon molecule. Any extrapolation of results from one compound to other compounds must take into account considerations based on a Quantitative Structural Analysis Relationship Study (QSARS), which mostly depends on how close the compounds under consideration are in physical and chemical properties.

In 1996 and 1997, Moussa
et al. studied the in vivo toxicity of C60 after intra-peritoneal administration of large doses. No evidence of toxicity was found and the mice tolerated a dose of 5 000 mg/kg of body weight (BW). Mori et al. (2006) could not find toxicity in rodents for C60 and C70 mixtures after oral administration of a dose of 2 000 mg/kg BW and did not observed evidence of genotoxic or mutagenic potential in vitro. Other studies could not establish the toxicity of fullerenes: on the contrary, the work of Gharbi et al. (2005) suggested that aqueous C60 suspensions failing to produce acute or subacute toxicity in rodents could also protect their livers in a dose-dependent manner against free-radical damage.

A comprehensive and recent review on fullerene toxicity is given by Kolosnjaj
et al. (2007a,b, c). These authors review the works on fullerene toxicity beginning in the early 1990s to present, and conclude that very little evidence gathered since the discovery of fullerenes indicate that C60 is toxic.

With reference to nanotubes, a recent study of Poland
et al. (2008) on carbon nanotubes introduced into the abdominal cavity of mice led the authors to suggest comparisons to "asbestos-like pathogenicity". It should be noted that this was not an inhalation study, though there have been several performed in the past, therefore it is premature to conclude that nanotubes should be considered to have a toxicological profile similar to asbestos. Conversely, and perhaps illustrative of how the various classes of compounds which fall under the general term fullerene cover a wide range of properties, Sayes, et al., found that in vivo inhalation of C60(OH)24 and nano-C60 in rats gave no effect, whereas in comparison quartz particles produced an inflammatory response under the same conditions (Nano Letters, 2007, Vol. 7, No. 8, 2399-2406). As stated above, nanotubes are quite different in chemical and physical properties to C60, i.e., molecular weight, shape, size, physical properties (such as solubility) all are very different, so from a toxicological standpoint, different results for C60 and nanotubes are not suggestive of any discrepancy in the findings.

Superconductivity


After the synthesis of macroscopic amounts of Fullerenes, their physical properties could be investigated. Very soon Haddon et al., found that intercalation of alkali-metal atoms in solid C60 leads to metallic behavior. In 1991, It was revealed that potassium-doped fullerenes becomes superconducting at 18K. This was the highest transition temperature for a molecular superconductor. Since then, Superconductivity has been reported in fullerene doped with various metals as well as potassium. It is shown that the superconducting transition temperature in alkaline-metal-doped fullerene increase in the unit cell volume V. As Cesium is the largest alkali ion, Cesium doped fullerene is an important material in this Family. Recently, superconductivity at 38K is reported in bulk Cs3C60.

The increase of transition temperature with the unit-cell volume had been believed to be an evidence for the BCS mechanism of C60 solid superconductivity. Because inter C60 separation can be related to an increase in the density of states on the Fermi level, N(eF). Therefore, there have been many efforts to increasing the interfullerene separation. In particular, intercalating neutral molecules into the A3 C60 lattice to increase the interfullerene spacing while the valence of C60 is kept unchanged. However, this ammoniation technique has revealed a new aspect of fullerene intercalation compound: the Mott-Hubbard transition and the correlation between the orientation/orbital order of C60 molecules and the magnetic structure.

The C60 molecules compose a solid of weakly bound molecules. The Fullerites are therefore molecular solids, in which the molecular properties still survive. The discrete levels of a free C60 molecule are only weakly broadened in the solid, which leads to a set of essentially nonoverlapping bands with a narrow width of about 0.5 eV. For an undoped C60 solid, the 5-fold hu band is HOMO level, and the 3-fold t1u band is LUMO level(empty), and this system is a band insulator. But, when The C60 solid is doped with metal atoms, this metal atoms give electrons to the t1u band or the upper 3-fold t1g band. So, this electron partial occupation of band leads to sometimes metallic behavior. However, A4C60 is an insulator, although the t1u band is only partially filled and should be a metal according to band theory. this unpredicted behavior can be probably explained by the Jahn-Teller effect, where spontaneous deformations of high-symmetry molecules induce the splitting of degenerate levels to gain the electronic energy. The Jahn-Teller type electron-phonon interaction is strong enough in C60 solids to destroy the band picture for particular valence states.

A narrow band or strongly correlated electronic system and degenerated ground states are important points to understand superconductivity in fullerene solids. When the interelectron repulsion U is greater than bandwidth, insulating localized electron ground state is produced in the simple Mott-Hubbard model. This explain well about the absence of superconductivity at ambient pressure in Cesium doped C60 solids. electron-correlation driven localization of the t1u electrons exceeds the critical value, this leads to the Mott-insulator. The application of high pressure decreases the interfullerene spacing, therefore Cesium doped C60 solids turns to metallic and superconducting. Actually, a fully developed theory of C60 solids superconductivity is still lacking. But, it has accepted widely that strong electronic correlations and Jahn-Teller electron-phonon coupling. produce local electron pairing show high transition temperature close to the insulator-metal transition. .

Examples of fullerenes in popular culture

Examples of fullerenes in popular culture
Popular culture

Popular culture is the totality of Distinction memes, ideas, Perspective s and Attitude s that are deemed preferred per an informal consensus within the mainstream of a given culture....
 are numerous. Fullerenes appeared in fiction well before scientists took serious interest in them. In
New Scientist
New Scientist

New Scientist is a liberal weekly international science magazine and website covering recent developments in science and technology for a general English language-speaking audience....
there used to be a weekly column called "Daedalus" written by David Jones, which contained humorous descriptions of unlikely technologies. In 1966 the columnist included a description of C60 and other forms of graphite. This was meant as pure entertainment. Fullerenes are part of the plot in a science fiction novel named Decipher
Decipher (novel)

Decipher is a speculative fiction novel by Stel Pavlou , published in 2001 in England by Simon and Schuster and 2002 in the United States by St....
written by Stel Pavlou
Stel Pavlou

Stelios Grant Pavlou is a United Kingdom author and screenwriter....
 and published in 2001. Bucky Balls were also mentioned in A&E's 2008 remake of
The Andromeda Strain
The Andromeda Strain

The Andromeda Strain , by Michael Crichton, is a techno-thriller novel documenting the efforts of a team of scientists investigating a deadly extraterrestrial life microorganism that rapidly and fatally clots human blood....
, which is based on the Michael Crichton
Michael Crichton

John Michael Crichton, Doctor of Medicine , was an United States author, film producer, film director, and physician, best known for his work in the science fiction, medical fiction, and techno-thriller genres....
 novel of the same name; John Ringo
John Ringo

John Ringo is an American science fiction and military fiction author who writes full time. He has had several The New York Times New York Times Best Seller list....
 uses fullerene as a carrier for antimatter
Antimatter

In particle physics, antimatter is the extension of the concept of the antiparticle to matter, where antimatter is composed of antiparticles in the same way that normal matter is composed of particles....
 in his legacy of the aldenata series other novels to make mention of them include Kim Stanley Robinson
Kim Stanley Robinson

Kim Stanley Robinson is an United States science fiction writer, probably best known for his award-winning Mars trilogy.His work delves into ecological and sociological themes regularly, and many of his novels appear to be the direct result of his own scientific fascinations, such as the 15 years of research and lifelong fascination with M...
's Mars Trilogy
Mars trilogy

The Mars trilogy is a series of award-winning science fiction novels by Kim Stanley Robinson, chronicling the settlement and Terraforming of Mars through the intensely personal and detailed viewpoints of a wide variety of characters spanning almost two centuries....
;
Iron by Poul Anderson
Poul Anderson

Poul William Anderson was an American science fiction author who wrote during a Golden Age of Science Fiction of the genre. Anderson also authored several works of fantasy....
 and Neal Stephenson
Neal Stephenson

Neal Town Stephenson is an American writer, known for his speculative fiction works, which have been variously categorized science fiction, historical fiction, maximalism, cyberpunk, and postcyberpunk....
's
The Diamond Age
The Diamond Age

The Diamond Age or, A Young Lady's Illustrated Primer is a postcyberpunk novel by Neal Stephenson. It is a bildungsroman focused on a young girl named Nell, and set in a world in which nanotechnology affects all aspects of life....
. Buckyballs are referenced in similarity with a large spherical meteorite with the potential to unleash a virus that could kill all organic life on Earth In Clive Cussler's novel The Sacred Stone. Buckyball is the state molecule of Texas
Texas

Texas is a U.S. state located in the South Central United States, nicknamed the Lone Star State. Texas is the second largest U.S. state in both area and population, spanning , and with a growing population of 24.3 million residents....
.

Fullerite (solid state)


C60 Fulleren Kristallin
Fullerites are the solid-state manifestation of fullerene
Fullerene

Fullerene are a family of carbon Allotropy, molecules composed entirely of carbon, in the form of a hollow sphere, ellipsoid, cylinder , or plane....
s and related compounds and materials.

Polymerized single-walled nanotubes
Carbon nanotube

Carbon nanotubes are allotropes of carbon with a nanostructure that can have a length-to-diameter ratio of up to 28,000,000:1, which is significantly larger than any other material....
 (P-SWNT) are a class of fullerites and are comparable to diamond
Diamond

In mineralogy, diamond is the Allotropes of carbon where the carbon atoms are arranged in an isometric-hexoctahedral crystal lattice. After graphite, diamond is the second most stable form of carbon....
 in terms of hardness. However, due to the way that nanotube
Nanotube

A nanotube is a nanometer-scale tube-like structure. It may refer to:* Carbon nanotube* Inorganic nanotube* DNA nanotechnology#DNA nanotubes...
s intertwine, P-SWNTs do not have the corresponding crystal lattice that makes it possible to cut diamonds neatly. This same structure results in a less brittle material, as any impact that the structure sustains is spread out throughout the material. Because nanotubes are still very expensive to produce in useful quantities, uses for a material lighter and stronger than steel
Steel

Steel is an alloy consisting mostly of iron, with a carbon content between 0.2% and 2.14% by weight , depending on grade. Carbon is the most cost-effective alloying material for iron, but various other alloying elements are used such as manganese, chromium, vanadium, and tungsten....
 will have to wait until nanotube production becomes more economically viable.

Ultrahard fullerite, Buckyball

Ultrahard fullerite (C60) is a form of carbon synthesized under high pressure high temperature conditions. It is believed that fullerene molecules are three-dimensionally polymerized in this material.

See also

  • Buckypaper
    Buckypaper

    Buckypaper is a thin sheet made from an aggregate of carbon nanotubes. The nanotubes are approximately 50,000 times thinner than a human hair. Originally, it was fabricated as a way to handle carbon nanotubes, but in 2008 is being studied and developed into applications by several research groups, showing promise as a building material for ae...
  • Truncated rhombic triacontahedron
    Truncated rhombic triacontahedron

    The truncated rhombic triacontahedron is a convex set polyhedron constructed from the rhombic triacontahedron by truncating the twelve vertices where five faces meet at their acute corners....


Further reading


External links

  • at www.cbc.ca
  • , University of Vienna
  • Fullerene-based architectures for quantum computing in and in at the