Structure factors, neutron scattering

The Q and ft) dependence of neutron scattering structure factors contains infonnation on the geometry, amplitudes, and time scales of all the motions in which the scatterers participate that are resolved by the instrument. Motions that are slow relative to the time scale of the measurement give rise to a 8-function elastic peak at ft) = 0, whereas diffusive motions lead to quasielastic broadening of the central peak and vibrational motions attenuate the intensity of the spectrum. It is useful to express the structure factors in a form that permits the contributions from vibrational and diffusive motions to be isolated. Assuming that vibrational and diffusive motions are decoupled, we can write the measured structure factor as... 

Figure 9 Fit of an incoherent neutron scattering structure factor, S(Q, O)), computed for iipid H atom motion in the piane of the biiayer in a simuiation of a DPPC biiayer, by the sum of an eiastic iine, a naiTow Lorentzian with width T , and a broad Lorentzian with width T2, convoiuted with a Gaussian resoiution function with AE = 0.050 meV.

The coherent tunneling case is experimentally dealt with in spectroscopic studies. For example, the neutron-scattering structure factor determining the spectral line shape is... 

C. Landron, A. K. Soper, T. Jenkins, G. N. Greaves, L. Heima, and J. P. Coutures, Measuring neutron scattering structure factor for liquid alumina and analysing the radial distribution function by empirical potential structure refinement, J. Non-Ciyst Solids 293, 453-457 (2001). 

Fig. 3 Neutron and X-ray scattering structure factors of [C4mim][PF6] obtained from CG simulation (IL-MSl) and experiment at 200 K. The inset shows the positions of the two major peaks of the X-ray scattering structure factors of [C4mim][PF6] for the two mapping schemes (IL-MSl and IL-MS2) at different temperatures. Neutron and X-ray scattering data are from [23]...

The correlation fiinction G(/) quantifies the density fluctuations in a fluid. Characteristically, density fluctuations scatter light (or any radiation, like neutrons, with which they can couple). Then, if a radiation of wavelength X is incident on the fluid, the intensity of radiation scattered through an angle 0 is proportional to the structure factor... 

We first examine the reiationship between particie dynamics and the scattering of radiation in the case where both the energy and momentum transferred between the sampie and the incident radiation are measured. Linear response theory aiiows dynamic structure factors to be written in terms of equiiibrium flucmations of the sampie. For neutron scattering from a system of identicai particies, this is [i,5,6]... 

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 data taken is normally presented as the total structure factor, F(Q). This is related to the neutron scattering lengths hi, the concentrations C , and the partial structure factor Sy(Q) for each pair of atoms i and j in the sample, by Equation 4.1-1 ... 

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)]. 

How can one hope to extract the contributions of the different normal modes from the relaxation behavior of the dynamic structure factor The capability of neutron scattering to directly observe molecular motions on their natural time and length scale enables the determination of the mode contributions to the relaxation of S(Q, t). Different relaxation modes influence the scattering function in different Q-ranges. Since the dynamic structure factor is not simply broken down into a sum or product of more contributions, the Q-dependence is not easy to represent. In order to make the effects more transparent, we consider the maximum possible contribution of a given mode p to the relaxation of the dynamic structure factor. This maximum contribution is reached when the correlator in Eq. (32) has fallen to zero. For simplicity, we retain all the other relaxation modes = 1 for s p. 

Performing neutron scattering not on perdeuterated samples but on a single deuterated chain in a protonated matrix (or vice versa both ways provide the same contrast) gives the single-chain structure factor,... 

Figure 5.4 The structure factor S(Q) determined using the method of Ashcroft and Lekener9 compared with experimental data gathered by Ottewill11 (points) using neutron scattering r is the distance between particle centres...

Neutron Scattering and the Single Chain Structure Factor. . . . . 209... 

Neutron Scattering and the Single Chain Structure Factor... 

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