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order spectrum is a peak close to 2(1360 cm"1) = 2730 cm"1 with a shoulder at 2698 cm"1, where the lineshape reflects the density of two-phonon states in 3D graphite. Another important benchmark spectrum shown in Fig. 19.56 is that for a typical glassy carbon specimen in Fig. 19.56(a) [19.149] showing a significantly broadened feature at 1600 cm"1 and a broad disorder-induced band with a maximum near 1359 cm"1, both arising from a high density of phonon states in their respective spectral ranges. The strongest second-order feature for this glassy carbon sample is located at 2973 cm-1, somewhat upshifted from a combination mode expected in graphite at (1359 + 1600) cm"1 = 2959 cm"1, where the Raman intensity near 1600 cm"1 is associated with a midzone maximum of the uppermost optical phonon branch which has a T-point frequency of 1582 cm"1 [19.149],

The Raman spectrum [Fig. 19.56(b)] of the "as-synthesized" carbon nanosoot, prepared by laser pyrolysis of a mixture of benzene, ethylene, and iron carbonyl [19.153], is very similar to that of glassy carbon [Fig. 19.56(a)] and has peaks in the first-order Raman spectrum at 1359 and 1600 cm-1 and a broad second-order feature near 2950 cm-1. In addition, the laser pyrolysis-derived carbon nanosoot has weak features in the second-order spectrum at 2711 and 3200 cm-1, similar to HOPG, but appearing much closer to twice the frequency of the first-order lines, namely 2(1359 cnr1) = 2718 cm"1 and 2(1600 cm"1) = 3200 cnr1, indicative of a much weaker 3D phonon dispersion in the carbon nanosoot than found in HOPG. Figure 19.56(c) shows the Raman spectrum of the laser pyrolysis-derived carbon black, heat treated to 2820°C, indicating enhanced crystallinity, consistent with a decrease in the intensity of the disorder-induced band at 1360 cm"1 [19.149],

The sharp features at 1566 and 1592 cm1 (see Fig. 19.57) can be explained by the frequency-independent Raman-active modes near 1590 cm"1 in Figs. 19.52 and 19.54 which appear as a doublet for the pertinent range of tubule diameters. Figure 19.56 further suggests a broad feature near 1370 cm"1 and another weaker and broader feature in the 740-840 cm"1 range. Because of the reduced dimensionality of the carbon nanotubes, the second-order spectral features are expected to be located at harmonics of the first-order features, consistent with the observations in Fig. 19.57.

Raman spectra have also been reported by Hiura et al. [19.150] for the material in the core of their carbon arc cathode deposit, which contained predominantly carbon-coated bundles of nested nanotubes and likely had a different distribution of diameters from the material used for the spectrum in Fig. 19.56(e). Hiura et al. identified two Raman lines in their spectrum with these nanotubes: at 1574 (FWHM = 23 cm"1) and at 2687 cm"1. The linewidth of their first-order peak at 1574 cm"1 is more than twice as broad as either of the first-order lines in Fig. 19.56(c) and lies between the two sharp first-order lines at 1566 and 1592 cm"1 for Co-catalyzed carbons, consistent with a distribution of tube diameters in the tubule bundles. In addition, Hiura reported a wide peak at ~1346 cnr1, close to the disorder-induced line at 1348 cm"1 in glassy carbon. While the second-order feature of Hiura et al. at 2687 cm"1 is slightly broader than, and upshifted by 6 cm-1 from, the second-order feature in Figs. 19.56(e) and 19.57, the second-order tube-related features in both spectra [19.149,150,154] are significantly downshifted from the corresponding features for other sp2 carbons, consistent with the very weak dispersion expected for the phonon branches of carbon nanotubes. A weak feature at 2456 cm"1 was also reported by Hiura et al. [19.150] in their second-order spectrum for the carbon nanotubes, which they also identified in the second-order spectra for HOPG, glassy carbon, and the outer shell material in the carbon arc deposit, corresponding to the large density of phonon states near 860 and 1590 cm"1.

Raman spectra were also reported by Chandrabhas et al. [19.151] for the central core deposit from a dc carbon arc discharge, which was characterized by x-ray and TEM measurements, showing tubules with diameters ranging from 15 to 50 nm and inner diameters down to 2 nm. For this tubule diameter distribution, the observed first-order Raman spectra would be expected to be close to that of graphite, and the observations showed first-order features at 49 cm""1, a small feature at 58 cm-1, 470 cm"1, 700 cm"1, 1353 cm"1, and 1583 cm"1, and second-order features at 2455 cm ', 2709 cm"1, and 3250 cm"1, which were interpreted in terms of disordered graphite, consistent with the x-ray and TEM data, which also indicated the presence of structural defects. The dominant features in the spectra of Chandrabhas et al. at 1583 cm"1 and 2709 cm"1 and a weaker feature at 1353 cm"1 are basically in agreement with the observations of Hiura et al. [19.150], although all peaks were upshifted relative to the frequencies observed by Hiura et al. These differences in behavior may be attributed to different distributions in the diameter and chirality of the nanotubes measured by the two groups [19.150,151]. Most of the remaining features in the Raman spectrum reported by Chandrabhas et al. [19.151] can be identified with graphitic modes or local maxima in the density of states. The features reported by Chandrabhas et al. [19.151] are rather different from those reported by Holden et al. [19.149], presumably because of differences in the diameter distribution of the nanotube samples. Further progress awaits the preparation, characterization, and spectral measurements on nanotubes with diameters less than ~ 4 nm.

19.8. Elastic Properties

The elastic properties of fullerene tubules have been discussed both theoretically and experimentally. Direct observations, mostly using highresolution TEM, have shown that small diameter carbon nanotubes are remarkably flexible. As shown in Fig. 19.58, even relatively large diameter (~10 nm) carbon nanotubes grown from the vapor phase can bend, twist, and kink without fracturing [19.17,155]. The basic mechanical properties of the nanotubes are very different from those of conventional PAN-based and vapor-grown carbon fibers, which are much more fragile and are easily broken when bent or twisted. It is of interest to note that when bent or twisted, the nanotubes appear to flatten in cross section, especially single wall nanotubes with diameters greater than 2.5 nm [19.155,156].

Fig. 19.58. High-resolution TEM images of bent and twisted carbon nanotubes. The length scales for these images are indicated [19.17].

Theoretical studies have focused on several issues, including the effect of curvature, tubule diameter, and chirality on the electronic structure, and the calculation of the bending modes for tubules. Regarding the dependence of the strain energy on the tubule diameter d„ Mintmire and White [19.102] have shown from continuum elasticity theory that the strain energy per carbon atom a/N is given by a Ea3jrfc

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