# Inelastic X-ray Scattering as a Probe of Terahertz Phonon Propagation in Nanoparticle Suspensions
## 3 What Can We Learn from an IXS Measurement on a Suspension? An extended treatment of the theoretical and practical aspects of the IXS technique goes beyond the scope of the present review, and the interested reader is deferred to the existing literature [16; 31]. Here, I recall some very general features we could expect from an IXS measurement on a simple system. As discussed in the introductory section, an inelastic scattering spectrometer can focus on dynamic events occurring over different time and length scales upon suitable \(Q\) and \(E\) variation. In principle, this strategy could enable the whole mapping of dynamic crossover between the hydrodynamic and the single-particle regimes. In practice, however, this mapping cannot be achieved using a single X-ray spectrometer, as the access to hydrodynamic scales would require using visible or UV light as a probe. Nonetheless, at the lowest Qs reachable by IXS, the dynamic response of a liquid is averaged over sufficiently long distances and time for a suitably generalized hydrodynamic description to hold validity. Evidence of the transition of the IXS spectrum from this generalized hydrodynamic regime to the single-particle one was provided, e.g., in an IXS work on lithium [32], whose main result is summarized in Figure 2. At the lowest Qs, the spectral shape consists of a relatively sharp triplet profile dominated by the generalized hydrodynamic modes. At low \(Q\), the inelastic shift of the two side peaks increases (linearly) with \(Q\), as suggested by the dashed line roughly connecting their maxima in the Stokes sides. As \(Q\) increases, the overallspectral shape evolves toward the Gaussian profile featuring the single-particle response. In this regime, the probed event essentially reduces to the free recoil of the single struck atom induced by the collision with the impinging probe particle, such as a photon, in the case of IXS. Similarly, probing the spectrum from a suspension at different \(Q\)s would enable a mapping of the dynamical processes occurring in the sample over different scales, as schematically illustrated in Figure 3.  
Suppose the measurement is performed at low \(Q\) values well below the IXS range. In this case, if the immersed NPs interact, one would expect inter-NP phonons to become a relevant part of the system's response over these large distances. These phonon modes consist of density fluctuations originating from displacements of the NPs from their equilibrium positions, which transmit from site to site due to the NPs' interactions. In principle, these modes overlap with collective excitations propagating through the hosting liquid and, more generally, with any other dynamic processes in such a liquid, such as relaxations and diffusions. At larger \(Q\) values, one should expect the emergence in the spectrum of phonons propagation modes inside the NPs (intra-NP phonons). Figure 2: The \(Q\)-transition of the IXS spectral shape of lithium (open circles) from the quasi-hydrodynamic regime (bottom) to the single-particle one (top). The spectra are collected at the indicated \(Q\) values and vertically offset for clarity. The dashed line is a guide to the eye that indicates the position of collective inelastic excitations in the spectrum. The spectral line shapes are from Ref. [33], while the whole figure is taken from Ref. [16]. The case of a dilute suspension is more straightforward than the one discussed above, as the long NP separations eliminate inter-NP phonons. Furthermore, the sparse character of the suspension likewise translates into a dynamic response similar to that of the pure hosting liquid. More interestingly, NP suspensions are simple prototypical systems ideally suited to investigate the manipulation of the acoustic response of a fluid via the inclusion of structural heterogeneity.