## Amorphous Carbon 1591

Figure 8.20. Plot of the relation between the graphite particle size U and the Raman D-band to G-band intensity ratio /D/fe on a log-log scale. The straight line is a least-squares fit to the data points that provided the linear relationship Z.a=4.4 /G//D, where £.a is expressed in namometers (10A = 1 nm). [From D. S. Knight and W. B. White, J. Mater Sci. 4, 385 (1989).]

Figure 8.20. Plot of the relation between the graphite particle size U and the Raman D-band to G-band intensity ratio /D/fe on a log-log scale. The straight line is a least-squares fit to the data points that provided the linear relationship Z.a=4.4 /G//D, where £.a is expressed in namometers (10A = 1 nm). [From D. S. Knight and W. B. White, J. Mater Sci. 4, 385 (1989).]

### 8.2.4. Brillouin Spectroscopy

Brillouin scattering is a type of Raman scattering in which the difference frequency Ao) = Wphonon = (®inc ~ «»seal) corresponds to the acoustic branch of the phonon dispersion curves, with frequencies in the gigahertz (xlO'Hz) range, as was explained in Section 3.4.1. The negative and positive signs in the expression above for ®phonon correspond to Stokes and anti-Stokes lines, respectively.

Brillouin scattering has been used to study carbon films, and Fig. 8.21 compares the spectra of thick and thin films. The Ihick-film result (a) provides a bulk material response, namely, a strong central peak at zero frequency about 10 GHz wide, and a broad peak near 17 GHz attributed to longitudinal acoustic (LA) phonons. This latter frequency is consistent with the elastic moduli of carbon, which are measures of the stretching capability of solid carbon and its chemical bonds. The dotted line experimental spectrum of the 100 nm thick film at the top of Fig. 8.21b exhibits three peaks which come at positions close to the solid line theoretical spectrum

Figure 8.21. Brillouin spectra of (a) thick carbon fHm showing a Lorentzian fit to the data and (b) 100 nm thin carbon film. The upper experimentally measured spectrum of (b) is compared to the lower calculated spectrum, which does not take into account the scattering due to surface and structural irregularities that broaden the experimental Spectrum. [From P. Milani and C. E. Bottani, in Nalwa (2000), Vol. 2, Chapter 4, p. 262.]

BriNouin shift (GHz) (a)

Brillouin shift (GHz) (b)

plotted below the experimental one. The peaks of the experimental spectrum are much broader due to the roughness and structural irregularities of the surface. The theoretical fit provided values for the shear modulus p. 4.0 GPa, die bulk modulus a*3.7 GPa, and Poisson's ratio 1.0, consistent with the known value of die graphite elastic constant C44.

Acoustic phonons of nanoparticles exhibit Brillouin scattering that depends on particle size, and an example of this is the results shown in Fig. 8.22 for Ag nanoparticles. Figure 8.22a presents the spectra for particle diameters ¿=2.7, 4.1, and 5.2 nm, and Fig. 8.22b gives the dependence of the wavenumber (cm-1) on the reciprocal of the particle diameter, 1 /d. The latter figure also plots the theoretically expected results for torsional and spheroidal vibrational modes with angular momenta /= 1,2,3. A related work carried out with nucleated cordierite glass (Mg2Al4S5018) provided Brillouin scattering peaks with positions that were linear with the reciprocal of the diameter 1 /d (1 /£> on the figure) in the range of particle diameters from 15 to 40 nm, as shown in Fig. 8.23. Small-angle neutron scattering provided the nanoparticle diameters plotted in this figure.

Thus we have seen from Brillouin scattering data that acoustic modes shift to higher frequencies with reduced particle size, and we have seen from Raman data that optical modes shift to lower frequencies as the particle size is reduced.

12.1 cnr'. |
pc-Ag in Si02 |

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