**3. Inelastic neutron scattering**

As mentioned, the general aim of an INS measurement is to measure the spectrum of density fluctuations *S Q*ð Þ , *E* , which conveys insights on positions and movements of the atoms in a sample. Oversimplifying, depending on the spectrometer we use to determine it, we can have access to different aspects of *S Q*ð Þ , *E* , either relating to collective movements of atoms (e.g., acoustic waves, structural relaxation processes), single-particle ones (e.g., translational diffusion, rotations, librations … ), or both. To measure *S Q*ð Þ , *E* of a given system one can use two different types of neutron spectrometers: triple axis spectrometers (TAS) and time-of-flight (TOF) ones.

Here, we assume to execute measurements with a TOF spectrometer [13] where *S Q*ð Þ , *E* surfaces are sampled, ideally, for each *Q* and *E* values simultaneously [14]. The rate of neutrons scattered at the different scattering angles 2Θ (see Appendix A) and impinging on the sensitive area of the detector after a time (of flight) *t* defines the timedependent intensity function *I*ð Þ 2Θ, *t* . The latter is converted into an intensity *I Q*ð Þ , *E* , which is a function of the momentum, *Q*, and the energy, *E* ¼ ℏ*ω*, exchanged between sample and probe particles, with ℏ and *ω* being the reduced Planck constant and the exchanged frequency, respectively. In Appendix A, a sketch of the BRISP spectrometer and few hints about the principles of the TOF technique are shortly recalled.

Aside from instrumental effects such as energy resolution and signal background, *I Q*ð Þ , *E* is proportional to *S Q*ð Þ , *E* , which is the physical variable, INS (and IXS)

investigators usually seek for. To sample and gather this intensity function with adequate counting statistics, providing us the needed information, a certain acquisition time is required, which depends on the characteristics of the instruments (incident neutron flux, detector efficiency, resolution … ) and on the scattering properties of the sample. These are embodied in its double differential cross section *d*<sup>2</sup> *σ=d*Ω*dE* [12, 14] defined as the number of neutrons deviated in a second into the small solid angle ΔΩ subtended by a detector along the 2Θ direction, with final energy included in the interval between *E* and *E* þ Δ*E* [14]. More explicitly:

$$
\dot{N} = \frac{d^2 \sigma}{d\Omega dE} J\Delta\Omega dE.\tag{1}
$$

where *J* represents the incident flux of neutrons.

To summarize, the *image* of the *S Q*ð Þ , *E* is being built up as the measurement runs, and the larger the acquisition time, the more precise is the *S Q*ð Þ , *E* rendering. To avoid data loss of the entire measurement in case of instrument failure, the measurement is usually split into different sub-runs which, for the sake of simplicity we will hereafter assume to have the same acquisition time.
