Incoherent structures

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 coherent structure factor of the Zimm model can be calculated [34] following the lines outlined in detail in connection with the Rouse model. As the incoherent structure factor (83), it is also a function of the scaling variable ( fz(Q)t)2/3 and has the form... 

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)... 

The self (incoherent) structure factor, which is given by the Fourier transformation of G% (r, f), then becomes the direct product of the structure factors corresponding to the two processes ... 

The incorporation of high amount of niobium results in the formation of an additional pore system and the achievement of an incoherent structure. A stirring with water (at RT, for 8 h) leads to the well ordered homogeneous material, confirmed by the adsorption/desorption isotherm and the pore size distribution (Figure 1 A). 

The elastic peak intensity is governed by Ao(Q), which is kipwn as he elastic incoherent structure fector (EISF). The variation of Ao(Q) with Q allows determination of the nature of the rotation (e.g., isotropic or uniaxial rotations). The dynamical behavior of protons is reflected by the quasi-elastic component, 5qei(Q, < >) This term can be expressed as a superposition of Lorentzians whose widths are related to the average time between jumps. 

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. 

Fig.3 Elastic ( = 0) and quasi-elastic ( = 1,2,3) incoherent structure factors for the rotational diffusion of methane (R = 1.1 A)...

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. ... 

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