Neutron scattering diffusion model

Specification of. S SkCG, CO) requires models for the diffusive motions. Neutron scattering experiments on lipid bilayers and other disordered, condensed phase systems are often interpreted in terms of diffusive motions that give rise to an elastic line with a Q-dependent amplitude and a series of Lorentzian quasielastic lines with Q-dependent amplitudes and widths, i.e.. 

Two physically reasonable but quite different models have been used to describe the internal motions of lipid molecules observed by neutron scattering. In the first the protons are assumed to undergo diffusion in a sphere [63]. The radius of the sphere is allowed to be different for different protons. Although the results do not seem to be sensitive to the details of the variation in the sphere radii, it is necessary to have a range of sphere volumes, with the largest volume for methylene groups near the ends of the hydrocarbon chains in the middle of the bilayer and the smallest for the methylenes at the tops of the chains, closest to the bilayer surface. This is consistent with the behavior of the carbon-deuterium order parameters,. S cd, measured by deuterium NMR ... 

Lateral diffusion of phospholipids in model membranes at ambient pressure has been studied over the years by a variety of techniques including fluorescence recovery after photobleaching (FRAP), spin-label ESR, pulse field gradient NMR (PFG-NMR), quasielastic neutron scattering (QENS), excimer fluorescence and others.In general, the values reported for the lateral diffusion coefficient (D) range from 10 to 10 cm /s in the... 

X-ray and diffuse neutron scattering and diffraction studies of PMN have been interpreted in terms of the spherical layer model of Vakhrushev et al. [25,26]. The Pb atom is not situated at the (000) position as it should be for an ideal perovskite lattice, but is distributed over a sphere of radius R around this position. 

Physically this description corresponds to putting an atom (mass M) in an external time-dependent harmonic potential (frequency co0). The potential relaxes exponentially in time (time constant l/x0) so that eventually the atom experiences only a frictional force. Compared with other models2 which have been proposed for neutron scattering calculation, the present model treats oscillatory and diffusive motions of an atom in terms of a single equation. Both types of motion are governed by the shape of the potential and the manner in which it decays. The model yields the same velocity auto-correlation function v /(r) as that obtained by Berne, Boon, and Rice2 using the memory function approach. 

As already described, the upper three portions of Figure 2 summarize the differences in the way the constraints are applied in the constrained-junction theory, constrained-chain theory, and the diffused-constraints theory, respectively [4], Additional comparisons between theory and experiment for a variety of elastomeric properties should be very helpful [20], Also, neutron-scattering measurements conducted on series of networks having different values of the junction functionality , which is the number of chains emanating from a junction (cross-link), would be extremely useful in suggesting how to position the constraints along a chain in refining such models, since should have a pronounced effect on the... 

Example. As a model for two-stage diffusion take i = 1,2 and F as in (7.1). Then 7i,2 = 2 and 72,1 =7i- F°r computing the cross-section for neutron scattering one needs to know the probability density Gs(r, t) that a molecule that, at t = 0, was at r = 0 will, at time t, be at r. The differential cross-section is its double Fourier transform GS(A , co). It is convenient to apply the Fourier transformation in space right away to (7.4) so that both operators Ff reduce to factors,... 

Spectral lineshapes were first expressed in terms of autocorrelation functions by Foley39 and Anderson.40 Van Kranendonk gave an extensive review of this and attempted to compute the dipolar correlation function for vibration-rotation spectra in the semi-classical approximation.2 The general formalism in its present form is due to Kubo.11 Van Hove related the cross section for thermal neutron scattering to a density autocorrelation function.18 Singwi et al.41 have applied this kind of formalism to the shape of Mossbauer lines, and recently Gordon15 has rederived the formula for the infrared bandshapes and has constructed a physical model for rotational diffusion. There also exists an extensive literature in magnetic resonance where time-correlation functions have been used for more than two decades.8... 

Incoherent quasielastic neutron scattering measured as a function of hydration for powders of deuterated phycocyanin has been used to probe water motions (Middendorf et al., 1984). The simplest model accounting for the data was jump diffusion of water molecules between localized-sorption sites and the development of clusters of surface water at higher hydration (half-coverage of the surface, 0.15 h). This model is consistent with the picture developed from sorption thermodynamics. 

Except for the fullerenes, carbon nanotubes, nanohoms, and schwarzites, porous carbons are usually disordered materials, and cannot at present be completely characterized experimentally. Methods such as X-ray and neutron scattering and high-resolution transmission electron microscopy (HRTEM) give partial structural information, but are not yet able to provide a complete description of the atomic structure. Nevertheless, atomistic models of carbons are needed in order to interpret experimental characterization data (adsorption isotherms, heats of adsorption, etc.). They are also a necessary ingredient of any theory or molecular simulation for the prediction of the behavior of adsorbed phases within carbons - including diffusion, adsorption, heat effects, phase transitions, and chemical reactivity. 

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