## Appendix A Experimental Context--Icoherent Neutron Scattering
### Details of Scattering Spectrometer, Calibration and \"What Is Measured\"  
An outline of a standard TOF scattering experiment is as follows; cf. Figure 11. A short pulse of neutrons produced by the spallation source reaches the monitor, (i.e., the clock that starts the TOF measurement). \(L_{0}\) is the distance between monitor and sample. A neutron of the pulse that scatters from the sample reaches the detector, which triggers the stop signal of the TOF measurement. \(L_{1}^{\theta}\) (or \(L_{1}\), for simplicity) is the sample-detector distance; the corresponding scattering angle is \(\theta\). Here we consider isotropic scattering only. For a measured TOF value \(t_{TOF}\) holds  
\[t_{TOF}=\frac{L_{0}}{v_{0}}+\frac{L_{1}^{\theta}}{v_{1}}+t_{0} \tag{11}\]  
\(v_{0}\) and \(v_{1}\) are the velocities of the neutron before and after scattering, respectively; \(t_{0}\) is a time-offset parameter caused by effects of the electronics. When \(v_{0}\) and thus also the initial kinetic energy \(E_{0}\) have experimentally predetermined fixed values, one speaks of a \"direct geometry TOF-spectrometer\". ARCS and MARI, which are mentioned in the present paper, belong to this class of spectrometers. The scattering intensity measured by the detector is a function of the neutron's energy transfer (or, energy loss)  
\[E = E_{0}-E_{1}=\frac{1}{2}mv_{0}^{2}-\frac{1}{2}mv_{1}^{2} \tag{12}\] \[\equiv \hbar\omega=\frac{(\hbar k_{0})^{2}}{2m}-\frac{(\hbar k_{1})^{2}}{ 2m}\]  
(\(m\): neutron mass) and the corresponding momentum transfer on the neutron  
\[\hbar\mathbf{K}_{n}=\hbar\mathbf{k}_{1}-\hbar\mathbf{k}_{0} \tag{13}\]  
(Bold letters represent vectors.) It holds  
\[|\mathbf{K}_{n}|=K_{n}=\sqrt{k_{0}^{2}+k_{1}^{2}-2k_{0}k_{1}\cos\theta} \tag{14}\]where \(k_{i}\) is the modulus of vector \(\mathbf{k}_{i}\). Since we consider here incoherent [7,8] scattering, the exchange of momentum and energy is between a neutron and (the nucleus of) a single atom [7,8]. Due to [PERSON]'s \"actio = reactio\", in strict two-body scattering the struck atom (nucleus) receives the momentum transfer  
\[\mathbf{K}_{A}=-\mathbf{K}_{n} \tag{10}\]  
and one defines  
\[K\equiv|\mathbf{K}_{A}|=|\mathbf{K}_{n}| \tag{11}\]  
Assuming that the sample is initially at rest, the neutron scatters from an atom with mass \(M\) and zero mean initial momentum, i.e., \(\langle P\rangle=0\). Here, assuming conservation of kinetic energy and momentum in a binary collision, it holds the simple kinematic relation  
\[\frac{k_{1}}{k_{0}}=\frac{\cos\theta+\sqrt{(M/m)^{2}-\sin^{2}\theta}}{M/m+1} \tag{12}\]  
Parenthetically, the shape analysis and theoretical interpretation of a measured intensity peak constitutes a large part of many neutron scattering investigations. The following facts should be pointed out. (a) The measured energy and momentum transfer values, which then are needed for the theoretical interpretation of the experimental data, are those of the neutron--and not those of the struck H, which are entering the dynamical structure factor \(S(K,E)\). (b) From the measured TOF value (10) follow the values of \(v_{1}\), Equation (10), of \(k_{1}=|\mathbf{k}_{1}|\), and consequently of energy transfer \(E=\hbar\omega\), see Equation (11). Note that the value of \(\theta\) plays no role here. The value of \(E\)--for each single elementary scattering process--is independent of any tacit theoretical beliefs or assumptions, because it follows from the actually measured TOF value immediately. (c) Momentum transfer \(\hbar\mathbf{K}\), Equation (12), is determined from both the measured TOF value and the direction given by \(\theta\) in which the detector is positioned. However, the direct experimental determination of \(\theta\) (which is given by the geometry of the spectrometer) is by no means an easy matter. Furthermore, one uses Equation (12) which connects the value of \(\theta\) and the atomic mass \(M\).