C60 and Nanotubes: Carbon's Third Structure

A third way to arrange carbon

Diamond and graphite were, for two centuries, the only known pure forms of carbon — one a rigid sp3 lattice, the other stacked sp2 sheets. In 1985 Kroto, Curl and Smalley fired a laser at graphite in a helium jet trying to reproduce conditions in a carbon star’s atmosphere, and the mass spectrometer kept returning an oddly dominant peak at mass 720: exactly 60 carbon atoms. The only structure that fit — symmetric, closed, no dangling bonds — was a hollow cage shaped like a football, which they named buckminsterfullerene after Buckminster Fuller’s geodesic domes. The 1996 Nobel Prize in Chemistry followed.

C60 is built entirely from sp2-hybridised carbon, the same bonding as graphite: each atom bonds to three neighbours in a plane, leaving one delocalised p-orbital electron per atom free to join a shared π system. The difference from flat graphene is topology — to close a sheet of hexagons into a sphere you must introduce curvature, and Euler’s polyhedron formula says exactly how much: any closed convex structure built purely from hexagons and pentagons needs exactly 12 pentagons, regardless of size. C60 is the smallest fullerene in which every pentagon is completely surrounded by hexagons and no two pentagons touch — the isolated pentagon rule — which is precisely what makes it the most stable and most abundant.

Counting the cage: Euler's formula

V − E + F = 2                     (Euler's formula for a closed polyhedron)
C60: V = 60 atoms, E = 90 bonds, F = 32 faces
     F = 12 pentagons + 20 hexagons
     60 − 90 + 32 = 2  ✓

Every fullerene Cn with n ≥ 20 obeys the same 12-pentagon rule; only the hexagon count grows, which is why larger cages like C70 (ellipsoidal, with an extra belt of hexagons) and C84 exist as stable, isolable molecules while C20 — all pentagons, no hexagons — is a real but highly strained dodecahedral cage.

Rolling the sheet into a tube

Take a single sheet of graphene and roll it into a seamless cylinder and you get a single-walled carbon nanotube (SWCNT). How you roll it is captured by a chiral vector (n, m) connecting two lattice points that become superimposed at the seam. Three families follow directly from that pair of integers: armchair tubes (n = m), zigzag tubes (m = 0), and everything in between is chiral — the tube wall traces a spiral rather than straight rings around the circumference, and mirror-image (n,m) and (m,n) tubes are genuinely non-superimposable, like left and right hands.

d = a · sqrt(n² + nm + m²) / π         (tube diameter, a ≈ 0.246 nm graphene lattice constant)
metallic if (n − m) mod 3 = 0, otherwise semiconducting

Why chirality decides metal or semiconductor

This one arithmetic condition on (n, m) is the single most consequential fact in nanotube electronics: roughly one third of all possible tubes are metallic and conduct like a wire with essentially no scattering over micron lengths, while the other two-thirds are semiconducting, with a bandgap that shrinks as the diameter grows — roughly inversely proportional to d. A (10,10) armchair tube is always metallic; a (10,0) zigzag tube is semiconducting. This is why bulk nanotube synthesis, which produces a statistical mix of chiralities, has been such a stubborn obstacle to nanotube transistors: you need to sort tubes by chirality after the fact, since growth methods do not yet reliably select just one.

What actually holds it together

Both cage and tube are held rigid by strong in-plane sp2 σ bonds (carbon-carbon bond energy around 346 kJ/mol) plus a delocalised π system spread over the whole surface, which is why fullerenes and nanotubes are chemically fairly inert and mechanically extremely stiff along the bonded directions — nanotubes have measured Young’s moduli approaching a terapascal, among the highest of any material. Curvature in C60 does strain the ideal 120-degree sp2 bond angle down toward roughly 108 degrees at the pentagon vertices, which is exactly why the isolated-pentagon C60 cage is a local energy minimum rather than flat graphene simply wrapping into a ball at random.

Frequently asked questions

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