## 3 INS Measurement--Outline of Some Results
### INS from Bulk Ice-Ih--Conventional Theory  
For illustration of the concepts and relations of the preceding subsection, we consider here a concrete experimental example: INS from H of H\({}_{2}\)O molecules of ice. Usually, INS from H-containing samples is done at very low temperatures, because this reduces the \"background\" and Debye-Waller-factor effects [8] on the measured \(S(K,E)\). The latter is the quantity of primary interest for quantifying material properties and testing theoretical predictions. Liquid and solid water (ice) have been studied extensively in the INS scientific literature. Figure 2 shows the experimentally observed \(S(K,E)\) of micro-crystalline ice Ih at \(T=20\) Kelvin and 1 bar pressure. Neutron's initial energy was \(E_{0}=750\) meV. The measurements were done with the TOFspectrometer MARI [16]. The most significant scattering contributions are caused by H, which has a large incoherent scattering cross-section [8]. The black parabola line represents the conventional recoil line of a free H being initially at rest  
\[E_{rec}^{free}=\frac{\hbar^{2}K^{2}}{2M_{H}}, \tag{8}\]  
(\(M_{H}\): atomic mass of H. Recall that H and neutron (\(n\)) have almost equal masses, \(m_{n}\approx M_{H}\)). Clearly, this is equivalent to Equation (3) above. The intensity peak around \(E=420\) meV and \(K=14\) A\({}^{-1}\) is due to the molecular stretching vibrational modes of H\({}_{2}\)O. Here one observes that the center of intensity peak lies on the free H-atom recoil line. In view of Equation (8) and relation \(m_{n}\approx M_{H}\), this experimental result shows that the observed intensity peak is well described with the common theoretical assumption that the collision is impulsive and essentially a two-body (i.e., neutron-H) scattering process., i.e., there is neither a significant free recoil of the H\({}_{2}\)O molecule, nor a \"dressing\" of H with environmental and/or intramolecular degrees of freedom. Thus the effective mass of H is here \(M_{H}^{eff}=M_{H}^{free}\approx 1.0\) a.m.u. The following physical insight provided by [PERSON] et al. [27] gives additional justification to the above physical picture. Namely, the very short (being in the pico- to femto-second time range) neutron-H scattering time, taken together with the significant spectral separation of the stretching modes from the other vibrational modes, effectuate a decoupling of the \"fast\" H-stretching motion from the other (slower) H-motions. We now consider the bending vibrational mode, i.e., the peak at energy transfer \(E\approx 200\) meV. Interestingly, this peak does not lie on the free H-recoil parabola, but its central \(K\)-value appears at lower \(E\)-values. (Equivalently, it lies at higher \(K\)-values, at its central \(E\)-value). i.e., this peak exhibits a negative \(E\)-shift (or, equivalently, a \"positive\" \(K\)-shift). However, also this result has a conventional explanation: This bending motion is hindered by adjacent H-bonded molecules, and thus the scattering H may be \"dressed\" by the aforementioned  
Figure 2: Experimental \(S(K,E)\) intensity map of ice Ih at 20 K and 1 bar, recorded on MARI [16]. \(E_{0}=750\) meV. The shown black line (parabola) is the conventional recoil trajectory of a free H. The strong intensity ribbon at \(E\approx 0\) is mainly due to the aluminum cell containing the water (ice). The intensity peak centered at \(E\approx 420\) meV and \(K\approx 14\) Å\({}^{-1}\) represents the vibrational stretching OH-vibrations. See the discussion in the text. (Reproduced from reference [17] with permission.) environmental degrees of freedom. Thus this effect leads to an increased effective mass of the struck H. Indeed this is here observed: An estimate derived from the peak-center position, as shown in Figure 2, yields for the effective mass of the struck H: \(M_{H}^{eff}\approx 1.3\) a.m.u. Thus we may conclude that, in contrast to the stretching modes, the bending vibration, having a lower frequency, is not fully decoupled from the other dynamical modes (i.e., librations, phonons, etc.)