# Inelastic X-ray Scattering as a Probe of Terahertz Phonon Propagation in Nanoparticle Suspensions
## 6 Using Nanoparticles to Damp High-Frequency Sound Waves in a Fluid  
As suggested by the above discussion, the most noticeable difference between the line shape of a diluted NP suspension and that of the pure solvent is that the inelastic modes of the hosting liquid are highly attenuated in the former profile. We achieved the most striking evidence of this effect when investigating the dynamic response of an Au-NP suspension in glycerol [39], whose IXS spectra are compared to the ones of pure glycerol in Figure 6. The pronounced inelastic shoulders in the pure solvent spectra are hardly visible in the suspension ones, which have inelastic wings barely emerging from the spectral background. At this stage, there is still a possibility that the comparison shown in Figure 6 is misled by the normalization to the elastic (\(E=0\)) maximum. One can argue that, since this maximum is more intense in the suspension spectrum, normalizing to its height may cause an apparent, but not particularly significant, attenuation of the inelastic shoulders. Figure 7 provides an answer to this legitimate doubt. Data in the plot were derived from Ref. [40] and suggest that, aside from the relative amplitude reduction, the dominant inelastic mode presents a significant increase of the relative damping, as defined by the ratio \(\Gamma/\Omega\) between acoustic damping and frequency. To understand this effect, it is helpful to recognize that, in the mesoscopic regime probed by IXS, the inequality \(Qd_{c}/2\pi\gg 1\) (with \(d_{c}\) being the colloid diameter) remains valid even for nm-sized colloids. At these scales, multiple scattering significantly impacts acoustic propagation, as acoustic waves are more prone to get reflected at the surface of the colloid [41]. These reflections prevent sound waves in the surrounding liquid from entering the colloid and vice versa. Due to multiple interface reflections, there is a considerable mutual dephasing between acoustic waves, leading to a significant enhancement in acoustic damping caused by interference. Figure 6: The IXS scattering profiles from a suspension of Au-NPs in glycerol and pure glycerol collected at the indicated \(Q\) values are reported after normalization to the respective maxima. The corresponding resolution profiles are also included for reference after similar normalization. The figure was reproduced from Ref. [22]. Another intriguing effect connected with the enhanced interface reflection is the appearance of an additional mode resulting from the propagation of interface density waves, customarily known as Stoneley waves [42]. An IXS work on glycerol documented the fingerprints of these waves in the terahertz spectrum of glycerol. Their primary signature is a low-energy branch with a nearly linear \(Q\) dependence, as shown in Figure 8.  
Figure 8: The plot displays the dispersion curves determined through the Bayesian analysis of the IXS spectra of an Au-NP suspension in glycerol. Red dots and black squares refer to the longitudinal acoustic mode and the interfacial mode, respectively (see the text). The data are from Ref. [43] and are compared with the IXS measurement on pure glycerol by [PERSON] and collaborators (Ref. [44]). The linear dispersion representing the high-frequency elastic response of glycerol is also reported as a solid line for comparison. Figure 7: The relative damping of the dominant inelastic mode in several suspensions of either charged or neutral SiO\({}_{2}\) NPs of various sizes (see legend) is reported as a function of \(Q\). The dashed line shows the critical damping condition. The data are from Ref. [40]. At this stage, a still unanswered question is whether a damping effect similar to the one observed in dilute liquid suspensions can also be observed in a more rigid and ordered system, such as a crystal with NPs embedded. A clear advantage in dealing with a non-amorphous solid is the sharpness of its phonon features, which makes any possible line shape modification induced by NPs more easily discernible.