Incoherent dynamic structure factor

The self-correlation function leads directly to the mean square displacement of the diffusing segments Ar2n(t) = <(rn(t) — rn(0))2>. Inserting Eq. (20) into the expression for Sinc(Q,t) [Eq. (4b)] the incoherent dynamic structure factor is obtained... 

Now, the decay rate of the incoherent dynamic structure factor is proportional to fc(i+2v)/v xherefore, the -dependence of the decay rate for salt-free solutions is independent of whether the hydrodynamic interaction is present or not. 

The aqueous LiCl solution with the composition of LiC1.6.0H2O was prepared in the same way as described for the X-ray samples. In the aqueous solution the proton ( H) has a very large incoherent scattering cross section the observed differential scattering cross section can be approximated to the incoherent dynamic structure factor through... 

It is noteworthy that in the Rouse limit the diffusion constant is independent of the coil s size.) From the preceding, in the long-time limit, the characteristic time ti/2 of the incoherent dynamic structure factor exp[- Q B(0, t)] is linked to Q by the power law... 

Figure 8.5 Schematic representation of the incoherent dynamic structure factor Sj (q, co) for translational diffusion, at different q values.

Figure 8.19 Incoherent dynamic structure factor measured with poly(vinyl chloride) for q = 1.5A 1 at temperatures above and below the glass transition temperature.358 K. The broken curve is the instrumental resolution function measured. (From Colmenero euifr13)...

Fleischer [80] used PFGNMR to observe self-diffusion of 125 kDa polystyrene, M /M Ri 1.02, in toluene for concentrations 80-320 g/L or, equivalently 0.5ce < c < 2ce, estimating Ce from rheological data [81]. Fleischer s data agree well with eq. 15. Fleischer found that the incoherent dynamic structure factor 5, c(0 has only a single fast relaxation, even... 

Elementary excitations also include single particle diffusive excitations beside quantized vibrations (i.e., molecular vibrations and vibrations of the crystal as a whole associated with phonons/magnons). Consider the incoherent dynamic structure factor 5snc(Q,(o), which is the Fourier transform pair of the time-dependent self-correlation function, compare... 

The monomer mean-square displacement can be observed direcdy in computer simulations. It also enters in calculations for the coherent and incoherent dynamic structure factors. Most generally, we can consider a ftmaion ... 

The prerequisite for an experimental test of a molecular model by quasi-elastic neutron scattering is the calculation of the dynamic structure factors resulting from it. As outlined in Section 2 two different correlation functions may be determined by means of neutron scattering. In the case of coherent scattering, all partial waves emanating from different scattering centers are capable of interference the Fourier transform of the pair-correlation function is measured Eq. (4a). In contrast, incoherent scattering, where the interferences from partial waves of different scatterers are destructive, measures the self-correlation function [Eq. (4b)]. 

Comparing Eqs. (83), (84) and Eqs. (21), (22) it follows immediately that Rouse and Zimm relaxation result in completely different incoherent quasielastic scattering. These differences are revealed in the line shape of the dynamic structure factor or in the (3-parameter if Eq. (23) is applied, as well as in the structure and Q-dependence of the characteristic frequency. In the case of dominant hydrodynamic interaction, Q(Q) depends on the viscosity of the pure solvent, but on no molecular parameters and varies with the third power of Q, whereas with failing hydrodynamic interaction it is determined by the inverse of the friction per mean square segment length and varies with the fourth power of Q. 

The long-time behavior (Q(Q)t) > 1 of the coherent dynamic structure factors for both relaxations shows the same time dependence as the corresponding incoherent ones... 

Fig. 4.1 a Typical time evolution of a given correlation function in a glass-forming system for different temperatures (T >T2>...>T ), b Molecular dynamics simulation results [105] for the time decay of different correlation functions in polyisoprene at 363 K normalized dynamic structure factor at the first static structure factor maximum solid thick line)y intermediate incoherent scattering function of the hydrogens solid thin line), dipole-dipole correlation function dashed line) and second order orientational correlation function of three different C-H bonds measurable by NMR dashed-dotted lines)... 

After a correction similar to that for the PDF for sample independent scattering, absorption, multiple scattering, and incoherent scattering one obtains a dynamic structure factor ... 

Concerning the dynamic structure factor, we shall confine our attention to the incoherent case, where the self-correlation function B(0, ) only is required since we have q -4 1, it may be shown that the results are essentially valid for the coherent case as well (long-time limit) [86]. From,Eqn. (3.1.18) we get... 

Figure 8.17 Dynamic structure factor Ss (q, co) (predominantly incoherent) obtained, at 25 °C, with a sample of poly(methyl methacrylate) in which all the hydrogens, except the ones in the ester methyl group, were replaced by deuteriums. (From Gabrys et al.11)...

Inelastic neutron scattering is a technique that has been widely used both in the liquid and in the solid states to measure the stmcture and dynamics at small (that is, molecular) length scales. In an incoherent inelastic neutron-scattering experiment, the measured quantity is the self-dynamic structure factor Ss(Q, (o), which gives information, as in the liquid state, of the self-diSiision coefficient of the water molecules. Ss(Q, (o) is the Fourier transform of the intermediate self-scattering function Fg(Q, t), which is defined by... 

Dynamic structure factor S of (hexagonal) H2O ( H2O) ice (phase Ih), obtained from incoherent inelastic neutron scattering (Li 1996) vs. wave number v. Note the similarity of the above curve to results of conventional vibrational spectroscopic methods (IR, Raman)... 

Quasi-elastic Neutron Scattering. Coherent and incoherent inelastic neutron scattering are unique experimental techniques to characterize molecular motions on a time scale between 10 and 10 s. The continued development of high resolution inelastic scattering techniques in the past two decades (157-159) enables measurement of the dynamic structure factor S(Q, co) and the... 

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]. 

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. 

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