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Binding energy per atomc

7.40 eV

[3.13]

Heat of formation (per g C atom)

10.16 kcal

[3.14]

Electron affinity

2.65±0.05 eV

[3.15]

Cohesive energy per C atom

1.4 eV/atom

[3.16]

Spin-orbit splitting of C(2p)

0.00022 eV

[3.17]

First ionization potential

7.58 eV

[3.18]

Second ionization potential

11.5 eV

[3.19]

Optical absorption edge''

1.65 eV

[3.13]

"This value was obtained from NMR measurements. The calculated geometric value for the diameter is 7.09 A (see text).

''This value for the outer diameter is found by assuming the thickness of the Qq shell to be 3.35 A. In the solid, the C^—C^ nearest-neighbor distance is 10.02 A (see Table 7.1).

cThe binding energy for Cm is believed to be —0.7 eV/C atom less than for graphite, though literature values for both are given as 7.4 eV/C atom. The reason for the apparent inconsistency is attributed to differences in calculational techniques.

^Literature values for the optical absorption edge (see §13.2.1) for the free C60 molecules in solution range between 1.55 and 2.3 eV.

the thickness of the 7r-electron cloud surrounding the carbon atoms on the C60 shell. The estimate of 3.35 A comes from the interplanar distance between graphite layers. The binding energy per carbon atom in graphite is 7.4 eV [3.20]. The corresponding energy has also been calculated by various authors for C60 [3.13,21], showing that the binding energy for C60 per carbon atom is smaller than that for graphite by 0.4-0.7 eV, although the absolute values of the binding energies of C60 and graphite are not as well established. The high binding energy for C60 accounts for the high stability of the C60 molecule.

By definition, a fullerene is a closed cage molecule containing only hexagonal and pentagonal faces. This requires that there be exactly 12 pentagonal faces and an arbitrary number h of hexagonal faces. This result follows from Euler's theorem for polyhedra f + v = e + 2 (3.1)

where /, v, and e are, respectively, the numbers of faces, vertices, and edges of the polyhedra. If we consider polyhedra formed by h hexagonal faces and p pentagonal faces, then f = p + h,

The three relations in Eq. (3.2) yield

from which we conclude that all fullerenes with only hexagonal and pentagonal faces must have 12 pentagonal faces, and the number of hexagonal faces is arbitrary. Since the addition of each hexagonal face adds two carbon atoms to the total number nc of carbon atoms in a fullerene, it follows that the number of hexagonal faces in a C„c fullerene can readily be determined. These arguments further show that the smallest possible fullerene is C20, which would form a regular dodecahedron with 12 pentagonal faces and no hexagonal faces.

It is, however, energetically unfavorable for two pentagons to be adjacent to each other, since this would lead to higher local curvature on the fullerene ball, and hence more strain. The resulting tendency for pentagons not to be adjacent to one another is called the isolated pentagon rule [3.22,23]. The smallest fullerene C„c to satisfy the isolated pentagon rule is C60 with nc = 60. For this reason, fullerenes with many fewer than 60 carbon atoms are less likely (see Fig. 1.4), and, in fact, no fullerenes with fewer than 60 carbon atoms are found in the soot commonly used to extract fullerenes. Thus far, it has been possible to stabilize small fullerenes only by saturating their dangling bonds with hydrogen, and C2oH2o is an example of such a molecule (see §2.17) [3.24]. While the smallest fullerene to satisfy the isolated pentagon rule is C60, the next largest fullerene to do so is C70, which is consistent with the absence of fullerenes between C60 and C70 in fullerene-containing soot. Furthermore, since the addition of a single hexagon adds two carbon atoms, all fullerenes must have an even number of carbon atoms, in agreement with the observed mass spectra for fullerenes as illustrated in Fig. 1.4 [3.25],

Although each carbon atom in C60 is equivalent to every other carbon atom, the three bonds emanating from each carbon atom are not equiva-

Fig. 3.3. Use of heptagons to yield a concave surface [3.28].

lent, two being electron-poor single bonds (on the pentagonal edges) and one being an electron-rich double bond (joining two hexagons). The alternating bond structure found around the hexagons of C60 (see Fig. 3.1) is called the Kekule structure and stabilizes the C60 structure, consistent with x-ray and neutron diffraction evidence discussed above. We note that the icosahedral Ih symmetry is preserved for unequal values for a5 and a6. The single bonds that define the pentagonal faces are increased from the average bond length of 1.44 A to 1.46 A, and the double bonds are decreased to 1.40 A [3.5,10,26,27].

With regard to curvature for fullerene-related molecules, we note that pentagons are needed to produce closed (convex) surfaces, and hexagons by themselves lead to a planar surface. To produce a concave surface that spreads outward, heptagons need to be introduced. There are numerous references in the literature [3.28—30] to the use of heptagons to yield a concave surface (see Fig. 3.3). As mentioned in §19.2.3, heptagons are commonly observed in elbows and bends in carbon nanotubes.

Since the 60 carbon atoms in C60 form a cage with a very small value for the nearest-neighbor carbon-carbon distance ac_c, it is expected that the C60 molecule is almost incompressible [3.31]. Of course, in the solid state the large van der Waals spacings between C60 molecules result in a solid that is quite compressible and "soft" [3.32] (see §7.3).

Fig. 3.4. (a) The icosahedral C60 molecule, (b) The rugby ball-shaped C,0 molecule (Dsh symmetry), (c) The C80 isomer, which is an extended rugby ball (Dsi symmetry), (d) The Cgo isomer, which is an icosahedron (Ih symmetry).

3.2. Structure of C70 and Higher Fullerenes

In the arc synthesis of C60 (see §5.1.2), larger molecular weight fullerenes C„c (nc > 60) are also formed. By far the most abundant higher molecular weight fullerene present in the mass spectra for fullerenes (see Fig. 1.4) is the rugby ball-shaped C70 [Fig. 3.4(b)]. The high relative abundance of C70 is connected to its stability, which in part relates to its high binding energy, which is 7.42 eV/C atom or ~0.02 eV/C atom greater than that of C60 [3.13]. Although higher in binding energy, C70 is probably less abundant than C60 for reasons of kinetics rather than energetics. Even if the potential well for C70 may be deeper, the barriers that must be overcome to reach C70 upon formation are greater than for C60, so that the synthesis rate is less favorable.

As the fullerene mass increases, the next fullerene beyond C7o to satisfy the isolated pentagon rule is C76. For masses higher than C70, the abundance decreases dramatically. However, significant quantities of C76, C7g, C82, and C84 have also been isolated and the structure and properties of these higher-mass fullerenes have now been studied in some detail [3.21,33-35], Bands of carbon clusters as large as C240 and C330 and clusters in the range C600-C700 have been observed in mass spectra [3.36-40], but these larger clusters, in most cases, have not yet been proved to be fullerenes, nor have they been isolated or studied in detail.

The rugby ball shape of C70 [3.33] can be envisioned either by adding a ring of 10 carbon atoms or, equivalently, adding a belt of 5 hexagons around the equatorial plane of the C60 molecule which is normal to one of the fivefold axes [see Fig. 3.4(a)], and suitably rotating the two hemispheres of C60 by 36° so that they fit continuously onto the belt hexagons. (See §4.4.1 for a discussion of the symmetry of the C70 molecule.) In contrast to C60, which has only one unique carbon site, the C70 molecule has five inequivalent sites, shown in Fig. 3.5 as numbered circles, and whereas C60 has two unique bond lengths, C70 has eight distinct bond lengths, labelled in Fig. 3.5 by boxed numbers [3.13]. The longer bond lengths tend to pertain

Table 3.2

Physical constants for the C70 molecule.

Table 3.2

Physical constants for the C70 molecule.

Quantity

Value

Reference

Average C-C distance" (Â)

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