# 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
In a recent IXS work [45], we addressed this issue by investigating the phonon spectrum of a frozen aqueous suspension of Au-NPs. We found that, even at a low concentration (about 0.1% in volume), embedded NPs substantially impact the phonon spectrum of the hosting medium, at least at some \(Q\) values. In fact, after looking at Figures 9 and 10, one readily notices that the embedded nanoparticles significantly attenuate the dominant phonon peak of ice (at about 8 meV). This conclusion stems from comparing the non-normalized line shapes, suggesting that the inclusion of NPs causes an attenuation of the phonon mode at about 8 meV, which is more pronounced than that experienced by the elastic peak of the same spectrum. This trend is unlikely to be a unique consequence of the more significant absorption of embedded NPs, as any absorption increase would have affected all spectral modes, elastic or inelastic, in equal measure. Furthermore, the line shapes in the left plot of Figure 9 exhibit a side feature in the 9.5-12 meV range (encircled by the ellipsis), whose intensity is nearly the same in the ice and suspension spectra despite the different absorption of the two samples. However, in ice, it is shifted at a slightly larger energy transfer. In summary, the comparison in Figures 9 and 10 evidences distinctive features not trivially amenable to the X-ray absorption increase caused by embedded NPs or their additional, intra-NP phonon modes. Therefore, we can conclude that the impact of embedded NPs on the phonon spectrum of a solid is a genuine effect, which, however, seems to affect the various spectral modes to a different extent in a way not very well understood at present. Figure 9: **(Left panel)** Comparison between the IXS spectrum measured at Q = 19 nm\({}^{-1}\) of the ice sample and the frozen suspension. The spectral profiles are displayed in an expanded view, and the dashed ellipses encircle the regions where the difference between the phonon modes of the two samples is most evident. **(Right panel)** The same comparison is reported at full scale. The Bayesian Inference at Work: Two Examples  
Let us briefly discuss how Bayesian inference can help with line shape analysis and interpretation. To provide a practical example, the typical outcome of the Bayesian analysis of the _IXS_ spectra from a nanoparticle suspension is summarized in Figure 11, which refers to the IXS results discussed in Ref. [22]. In Panel A, I compare the _IXS_ spectra from an aqueous suspension of gold NPs, the corresponding best fitting model line shapes, and their spectral components, all already displayed in Figure 4.  
Figure 11: **(Panel A)** displays the same spectral line shapes already shown in Figure 4. **(Panel B)** shows the posterior distribution of the number of inelastic components drawn in the same IXS study for three \(Q\) values. **(Panel C)**, finally, illustrates the posterior of the inelastic shift of the lower frequency phonon mode for the \(Q=3.5\,\mathrm{nm}^{-1}\) spectrum. Figure 10: The same as Figure 9, but for \(Q=21\,\mathrm{nm}^{-1}\). Panel B shows that the relative probability of the most plausible model option (\(k=2\)) decreases upon an increase in \(Q\), while the probability of the alternative model option (\(k=1\)) correspondingly increases. Most importantly, the sharp and unimodal posterior distributions of the inelastic shifts (e.g., that in panel C) support the physical significance of the two spectral modes detected and the robustness of the related line shape model. A second noticeable example of the outcome of a Bayesian analysis can be found in Figure 12, which refers to our INS work on liquid silver. There, we used Bayesian inference to settle a controversy about the presence of a second spectral excitation of a transverse character. The plot displays the posteriors \(P(\Omega_{l}|y)_{k}\) of the \(l\)-th mode's inelastic shift drawn for a \(k\) excitation model option and for the IXS spectrum at \(Q=16\) nm\({}^{-1}\).