" n, is the number of isomers obeying the isolated pentagon rule for given C„c fullerenes.

fcThe symmetries of the 5 isomers of C78 are: C78i(D,), C782(C2„), C783(C2J, Cn4(D]t,), CyS5(Dih).

cThe symmetries of the 7 isomers of C80 are: C80l(Du), Cm2(D2), Cm3(C2v), C804(£>3))

dThe symmetries of the 9 isomers of C82 are: C821(C2), C822(C,), C823(C2), C824(Cs), C825(C2), C826(Cs), C827(C3„), C828(C3„), C829(C2„).

eThe symmetries of the 19 isomers of C66 and the 35 isomers of C88 are listed in Table 4 of Reference [3.60],

" n, is the number of isomers obeying the isolated pentagon rule for given C„c fullerenes.

fcThe symmetries of the 5 isomers of C78 are: C78i(D,), C782(C2„), C783(C2J, Cn4(D]t,), CyS5(Dih).

cThe symmetries of the 7 isomers of C80 are: C80l(Du), Cm2(D2), Cm3(C2v), C804(£>3))

dThe symmetries of the 9 isomers of C82 are: C821(C2), C822(C,), C823(C2), C824(Cs), C825(C2), C826(Cs), C827(C3„), C828(C3„), C829(C2„).

eThe symmetries of the 19 isomers of C66 and the 35 isomers of C88 are listed in Table 4 of Reference [3.60], not having inversion symmetry. Liquid chromatography can be used to separate various isomers of C„c according to their shapes, although this can be a more delicate distinction than separation according to mass. Experiments such as NMR and electron paramagnetic resonance (EPR) can in some cases identify the symmetry and shape of isomers of a given C„c. Such studies have been made on C78 and C82 [3.56,57,63], although no definitive agreement has yet been reached about the relative abundances of the various isomers, which may be sensitive to preparation methods.

It is now possible to obtain sufficient quantities of some of the higher fullerenes for characterization experiments using NMR techniques and various properties measurements as well, including EPR spectroscopy (§16.2), STM, and surface science studies (§17.9.5). Measurements of the electron affinity EA have already been reported for C70 (2.72 ± 0.05 eV), C76 (2.88 ±0.05 eV), C78 (3.1 ±0.07 eV), and C84 (3.05 ±0.08 eV), using Knud-

3.2. Structure of C70 and Higher Fullerenes a _ b

D6h Td D2(leapfrog)

Fig. 3.7. Perspective views of several isomers of Cw: (a) structure with lowest energy D2, (b) D2d, (c) C2, (d) D6Ii, (e) Td, and (f) leapfrog D2 isomer of C84 [3.21].

sen cell mass spectrometry [3.45], showing a general increase in EA with increasing nc. The NMR technique, for example, is sensitive to the number and characteristics of the distinct carbon sites on the fullerene surface. The number of distinct NMR sites, the cohesive energy of the various isomers and their symmetry types, and the HOMO-LUMO band gap have been calculated for fullerenes in the range C60 to C94 [3.21]. Experimental NMR studies have already been initiated [3.56,64] to study the isomers of C84. To facilitate the analysis of such NMR spectra, calculations of the molecular electronic density of states have been carried out for a variety of isomers of C84 [3.35], Density of states calculations for C76 have also been carried out [3.35] and have been used to compare with measured photoemission spectra [3.65]. The synthesis and isolation of these highermass fullerenes is now an active research field, and it is expected that as the availability of larger amounts of these materials becomes widespread, more extensive properties measurements of these higher-mass fullerenes will be carried out (see §17.9.5).

Whereas small-mass C„c fullerenes (nc < 100) are likely to be singlewalled, it is not known presently whether high-mass fullerenes (nc > 200)

Fig. 3.8. (a) Formation of a pentagonal defect in a honeycomb lattice removes the area designated by the shaded outlined triangular wedge, (b) The vector (n, m) which connects two pentagonal defects fully specifies the icosahedral fullerene. The diagram is constructed for (n, m) = (2,1) and shows the related basis vectors of the unit cell.

Fig. 3.8. (a) Formation of a pentagonal defect in a honeycomb lattice removes the area designated by the shaded outlined triangular wedge, (b) The vector (n, m) which connects two pentagonal defects fully specifies the icosahedral fullerene. The diagram is constructed for (n, m) = (2,1) and shows the related basis vectors of the unit cell.

are single-walled or perhaps may be multiwalled and described as a fullerene onion (see §19.10).

3.3. The Projection Method for Specifying Fullerenes

A useful construction for specifying the geometry of fullerene molecules is based on considering a curvature-producing pentagon as a defect in a planar network of hexagons. Shown in Fig. 3.8(a) is one such pentagonal defect. If a 60° wedge is introduced at each defect of this type, as indicated in Fig. 3.8(a), a five-sided regular polygon is formed. If the triangular wedge is cut out and the hexagons along the cut are joined together to each other, then a curved surface is produced. Two such pentagonal defects are shown in Fig. 3.8(b). The vector d„m between the two pentagonal defects is d„m = na1 + ma2, (3.6)

and the corresponding length is dnm = «o ("2 + ™2 + nmf'2 (3.7)

in which a! and a2 are basis vectors of the honeycomb lattice and (n, m) are two integers which specify dnm. The set of integers (n, m) further specify the fullerene. Using this concept, a planar projection of a fullerene on a honeycomb lattice can be made, as described below.

For the case of an icosahedral fullerene, only one vector d„m needs be used to specify the fullerene, and the fullerene can be constructed from 20 equilateral triangles of length dmn on a side, as shown in Fig. 3.9 [3.66]. Ten of these equilateral triangles form the belt section and five identical equilateral triangles form each of the two cap sections of the icosahedral fullerene. In this projection, the locations of the 12 pentagons needed to provide the curvature for the closed polyhedral structures are indicated by

Fig. 3.9. Projection construction for an icosahedral fullerene. The size of each triangle is appropriate for C140 with (n, m) = (2,1), as defined in Fig. 3.8(b). The 12 pentagonal defects are indicated by 1-12. The C140 fullerene is obtained by superimposing pentagons with the same number to form a closed cage molecule. The parallelogram ABCD represents the "belt" region.

the numbers 1,..., 12. By superimposing the numbered hexagons in Fig. 3.9 (which become pentagonal defects in the fullerene) with the same numbers wherever they appear, closed cage fullerenes are easily constructed. For illustrative purposes, the projections for the icosahedral C60, C80, and Ci40 molecules are shown in Fig. 3.10 [3.66].

If the lattice vector between two nearest-neighbor pentagonal defects is nax + ma2, where aj and a2 are basis vectors of the honeycomb lattice, a0 is the lattice constant of the 2D lattice (a0 = 2.46 A), and (n, m) are both integers, then a regular truncated icosahedron contains exactly 20 equilateral triangles. If the fullerene has an icosahedral shape, then the distance between nearest-neighbor pentagons is constant and is given by dnm, as shown in Fig. 3.8(b). Using dnm as the length of a side of an equilateral triangle, then the area of this equilateral triangle Anm is

where a0 is the lattice constant of the honeycomb lattice as indicated by vectors a, and a2 in Fig. 3.8(b).

Every regular truncated icosahedron can be constructed from 20 equilateral triangles, each having an area Anm given by Eq. (3.8). The number of carbon atoms per triangle is (ti2 + nm + m2), so that the number of carbon atoms in the fullerene nc is given by Eq. (3.4). Figure 3.9 shows the generic format for constructing icosahedral fullerenes.

In Fig. 3.10 we show examples of projections of specific fullerenes, C60, C80, and Ci40 corresponding to (n, m) — (1,1), (2,0), and (2,1), respectively. The connection between Figs. 3.9 and 3.10 can be easily understood by comparing the two projections for C140 and drawing 10 equilateral triangles in

the belt area and 5 equilateral triangles in the cap regions of the projection in Fig. 3.10(c) [3.66]. Some rotations by 72° of the equilateral triangles in the cap regions are necessary to bring Fig. 3.9 into an equivalent form that looks identical to the projection shown in Fig. 3.10(c).

Although shown here only for icosahedral fullerenes, the projection method can also be used to construct polyhedra that do not have an icosahedral shape. The projection method is also valuable for specifying the structure of the end caps of carbon tubules (see §19.2.3). For the nanotubes, the tube wall is represented by many rows of hexagons, which are inserted parallel to AB and CD in Fig. 3.9, and the pentagons

'7' at points 'A and 'B' are joined, as are pentagons '1' at points 'C' and 'D' to form a cylinder [3.66].

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