Elastic intensity

The amplitude of the elastic scattering, Ao(Q), is called the elastic incoherent structure factor (EISF) and is determined experimentally as the ratio of the elastic intensity to the total integrated intensity. The EISF provides information on the geometry of the motions, and the linewidths are related to the time scales (broader lines correspond to shorter times). The Q and ft) dependences of these spectral parameters are commonly fitted to dynamic models for which analytical expressions for Sf (Q, ft)) have been derived, affording diffusion constants, jump lengths, residence times, and so on that characterize the motion described by the models [62]. 

The intensity of a vibrational mode in HREELS on-specular is given by the ratio of the inelastic to elastic intensities... 

Fig. 4.30 Arrhenius plot of the characteristic frequencies (corresponding to the maximum of the dielectric loss) for PIB of the a- (filled triangle) and j0-relaxation (filled circle). The solid line represents a fit with an Arrhenius law. Dashed-dotted and dashed lines are the temperature laws shown in [135] for the a-relaxation and the secondary relaxation observed by NMR respectively. The squares correspond to the characteristic rates of the j0-process obtained from the quasi-elastic INI6 spectra and the thick solid line shows those deduced from the analysis of the elastic intensities (Reprinted with permission from [195]. Copyright 1998 American Chemical Society)...

For the quasi-elastic region ( 2 meV) that is of concern here, the vibrational motions affect only the elastic intensity through a Debye-Waller factor, exp(-Q (M )), where (u ) is the mean square hydrogen amplitude in the vibrational modes. The scattering law for rotations can be written as a sum of an elastic peak intensity and a quasi-elastic component ... 

Figure 26.10 Neutron spectra of a Nb(OH) 2 sample at 0.2 K (a) and 4.3 K (b). For both temperatures, the spectra are taken in the superconducting (OT) and normal-conducting (0.7 T) electronic state. The thick and thin solid lines represent the fit curves for the total and inelastic scattering intensity, respectively. The broken lines are for the elastic intensity (from Ref [112]).

In any case, the observed quasi-elastic intensity shows that the spatial extension of this localized motion is rather small, on the average. We think that in view of the multiplicity of sites available for hydrogen in these amorphous alloys, it is difficult to put forward a quantitative model. We can say that this localized motion of the hydrogen occurs certainly inside cages defined by the surrounding metal atoms. 

In chabazite, a zeolite with a 3-dimensional array of ellipsoidal cavities (0.37 x 0.42 nm), Stockmeyer followed the variation with q of the elastic intensity in the temperature range 40 K to 300 K and several loadings of H2 [59]. He found a continuous change in the spectrum, going from a solid-like behaviour at 25 K to an encapsulated gas-like motion at room temperature he interpreted the data with a model considering the motions of the orthohydrogen molecule in a potential well (depth U = 153 meV - size 1 = 0.08 nm) located inside a potential box of size L = 0.4 nm with infinite high barriers. 

Rotational motion or other localised motions give rise to an unbroadened elastic component and a broadened quasi-elastic component that may be comprised of sev l Lorentzian terms. The width of the broadened term is inversely related to the characteristic time of the motion. The ratio of elastic to elastic plus quasi-elastic intensity varies with Q and is determined by the geometry of the motion. This ratio... 

By emphasizing certain resonance features in the elastic intensity, one may use beams of light inert atomic particles to directly measure properties of the atom surface potential. From highly developed experiments it has been possible to infer very precise forms for the atom-surface interaction. Potentials inferred this way may have wider application than just atom-surface scattering since for many systems the solid-state nature of the target system may be unimportant. For such systems information such as the two-body potential established from gas-surface scattering may provide information about the two-body potential governing gas phase interactions. 

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