Structure factor elastic incoherent

The elastic incoherent structure factor (EISF), Aq(Q), is defined as [17]... 

Figure 6 Apparent elastic incoherent structure factor A q(Q) for ( ) denatured and ( ) native phosphoglycerate kinase. The solid line represents the fit of a theoretical model in which a fraction of the hydrogens of the protein execute only vihrational motion (this fraction is given by the dotted line) and the rest undergo diffusion in a sphere. For more details see Ref. 25.

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]. 

Figure 10 Elastic incoherent structure factors for lipid H atoms obtained from an MD simulation of a fully hydrated DPPC bilayer, and quasielastic neutron scattering experiments on DPPC bilayers at two hydration levels for (a) motion in the plane of the bilayer and (b) motion m the direction of the bilayer normal.

The elastic contribution is also called elastic incoherent structure factor (EISF). It may be interpreted as the Fourier transformed of the asymptotic distribution of the hopping atom for infinite times. In an analogous way to the relaxation functions (Eq. 4.6 and Eq. 4.7), the complete scattering function is obtained by averaging Eq. 4.22 with the barrier distribution function g E) obtained, e.g. by dielectric spectroscopy (Eq. 4.5)... 

Second, the Q-dependence of the measured elastic incoherent structure factor (EISF) appears to be in excellent agreement with the predictions of the model of localized atomic motion over a hexagon (Eq. (26.13)) with the distance between the nearest-neighbor sites equal to the experimental value. As an example of these results, Eig. 26.5 shows the behavior of the EISE for TaV2Hj j as a function of Q at several temperatures. The solid curves represent the fits of the six-site model to the data. In these fits the distance between the nearest-neighbor sites has been fixed to its value resulting from the structure, = 0.99 A, so that the... 

Figure 26.5 The elastic incoherent structure factor for TaVjH,, as a function of Q at T = 105, 200, 250 and 300 K [76], The solid lines represent the fits ofthe six-site model with the fixed r, = 0.99 A to the data.

This elastic scattering term is known as the elastic incoherent structure factor. It decreases from unity at (2 = 0 to 0 at large Q. As the area of S" (Q, (o) in the 00 direction is unity, there is an additional quasi-elastic component that increases from 0 at (2 = 0 to unity at large Q. The form of the quasi-elastic component depends on the nature of the localised diffusion. In the simplest case, where the jumping is between two trapping sites, the quasi-elastic term is a Lorentzian with a (2-independent width which is just 1/t where x is the mean residence time on either site. Two specific models will be noted here (a) random jumping round a ring of sites, the Barnes model [38] and (b)... 

Figure 8.7 Elastic incoherent structure factor Ao(q) calculated according to Equation (8.53) for the jump rotational motion between two sites.

Figure 8.18 Elastic incoherent structure factor A (q) obtained with poly (methyl methacrylate) at 150 K (open circles) and 290 K (solid circles). The theoretical prediction based on a model of rotation among three symmetric sites is given by the solid curve, whereas the broken curve was obtained by modifying the theoretical curve for the amount of contamination by coherent scattering in the experimental results. (From Gabrys et al.n)...

The elastic (f = 0) and quasi-elastic incoherent structure factors for the isotropic rotation of methane are shown in Fig. 3. It appears from this figure that only the first three terms of the summation in expression 23 have to be considered in the Q range, which is usually covered by QENS instruments. The self-diffusivity will be obtained by first fitting the QENS spectra with expression 23 and then from the broadening of As with Q. 

Souaille. M. Guillaume, F. Smith. J.C. Molecular dynamics simulation of n-nonadecane in urea inclusion 67. compound. FT. Rotational distribution and elastic incoherent structure factor. J. Chem. Phys. 1996. 105. pp. 1516. 

But their elastic incoherent structure factor (EISF) curve is very favourable to a proton motion resulting, over 250 K, from two coordinated rotations, as for the very acidic water in clays. ... 

Superimposed on the term S containing the time dependence autocorrelation, there is a purely elastic component. From equation (12) one can see that its intensity is the spatial Fourier transform of the trajectory of the atom performing the motion m, weighted by a probability of occupation. It has the dimension of a structure factor and is called the Elastic Incoherent Structure Factor (EISF) of the corresponding motion. The EISF of the total scattering function is the direct product of the partial EISF. At each q, the value Aq of the EISF is the ratio of the elastic component intensity le versus the overall scattering intensity. Keeping in mind that the sum rule is valid on the elastic and quasielastic part of the spectrum, Ie(q) + Iqe(q) is constant for any q, and one can write ... 

The figure shows that the proportion of elastic scattering decreases as Q increases from zero. In fact the ratio of elastic to total (elastic plus quasi-elastic) scattering intensity provides a very useful quantity which is known as the elastic incoherent structure factor (EISF). It is often experimentally measurable and can be calculated easily from a theoretical model of the molecular motion... 

A)(Q) = elastic incoherent structure factor (EISF) B = rotational constant Dchem = chemical diffusion coefficient D, = tracer diffusion coefficient Jq x) = zeroth-order spherical Bessel function L(< ) = Lorentzian line shape = mass of cfth atom n(p) = atomic momentum distribution Q = momentum transfer 5 " (Q, to) = coherent scattering function tt>) = incoherent scattering function ... 

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