Scattering theory 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]... 

In this section we will discuss perturbation methods suitable for high-energy electron diffraction. For simplicity, in this section we will be concerned with only periodic structures and a transmission diffraction geometry. In the context of electron diffraction theory, the perturbation method has been extensively used and developed. Applications have been made to take into account the effects of weak beams [44, 45] inelastic scattering [46] higher-order Laue zone diffraction [47] crystal structure determination [48] and crystal structure factors refinement [38, 49]. A formal mathematical expression for the first order partial derivatives of the scattering matrix has been derived by Speer et al. [50], and a formal second order perturbation theory has been developed by Peng [22,34],... 

The incorporation of non-Gaussian effects in the Rouse theory can only be accomplished in an approximate way. For instance, the optimized Rouse-Zimm local dynamics approach has been applied by Guenza et al. [55] for linear and star chains. They were able to obtain correlation times and results related to dynamic light scattering experiments as the dynamic structure factor and its first cumulant [88]. A similar approach has also been applied by Ganazzoli et al. [87] for viscosity calculations. They obtained the generalized ZK results for ratio g already discussed. 

The method of strueture analysis developed by the Soviet group was based on the kinematieal approximation that ED intensity is directly related (proportional) to the square of structure factor amplitudes. The same method had also been applied by Cowley in Melbourne for solving a few structures. In 1957 Cowley and Moodie introdueed the -beam dynamical diffraction theory to the seattering of eleetrons by atoms and crystals. This theory provided the basis of multi-sliee ealeulations whieh enabled the simulation of dynamieal intensities of eleetron diffraetion patterns, and later electron microscope images. The theory showed that if dynamical scattering is signifieant, intensities of eleetron diffraetion are usually not related to strueture faetors in a simple way. Sinee that day, the fear of dynamical effects has hampered efforts to analyze struetures by eleetron diffraction. 

We began the discussion of kinematical theory in Chapter 1, showing how the scattering from atoms is added up with regard to the phase to form the scattering from a unit cell, the structure factor. We repeat this important eqrration here. The structure factor for the hkl reflection is... 

For both polymer systems the static structure factors were investigated using small angle neutron scattering and the results interpreted in terms of RPA theory. Figure 6.6 displays the temperature-dependent static structure factor obtained from a PE-PEE melt (sample IV). 

Whereas the values of h, k, l in theory should span from — oo to + oo, they are limited in practice to those finite values which are accessible to the diffraction experiment for a given radiation. The structure factor Fhkl is the resultant of N waves scattered in the direction of the reflection hkl by the N atoms in the unit cell, as expressed by equations 2 or 3 ... 

Here we present a different prescription to calculate the dynamic structure factor or the intermediate scattering function in the supercooled regime. This is a quantitative approach based on the basic result of the mode coupling theory. The effect of the mode coupling term in the intermediate scattering function is written in a simpler way by the following expression ... 

The 8 part in (2.53) is responsible for elastic scattering, whereas the second term, which is proportional to the Fourier transform of C(f), leads to the gain and loss spectral lines. When the system undergoes undamped oscillations with frequency A0, this leads to two delta peaks in the structure factor, placed at spectral line. The spectral theory clearly requires knowing an object different from (o-2(/)), the correlation function [Dattaggupta et al., 1989]. 

To allow for these more general circumstances, wc rewrite Eq. (16-22) in terms of the structure factor. At the same time, we may convert from 1/t to resistivity. Transport theory leads to an expression for the resistivity in terms of the scattering rate, p = ml Ne z), with N again the electron density, Z/Qq. Wc may combine this with Eq. (16-22) to give the resistivity as... 

Figure 19 Static structure factor (minus the trivial self-scattering term) versus wave vector. Circles are X-ray diffraction results for polyethylene at 1 atm and 430 K. Curves are predictions of RISM theory for a RIS model with hard site diameters of 3.70 A (dashed line) and 3.90 A (solid line). The solid square is experimental S(0) - 1. An expanded view of the first peak is shown in the inset. (From Ref. 123.)...

In the work of Haymet and Oxtoby the direct correlation function is approximated through statistical mechanical perturbation theory about its value for a uniform liquid of density Pq. This approach relies on the earlier work of Ramakrishnan and Youssouff who showed that such a jjerturbation approach gave rather accurate results for the equilibrium phase diagram for atomic liquids. One advantage of this approach is that the only external input to the functional 2 is the direct correlation of the liquid, which can be related to the structure factor, a quantity measurable by x-ray or neutron scattering... 

In addition to the repulsive part of the potential given by Eq. (4), a short-range attraction between the macroions may also be present. This attraction is due to the van der Waals forces [17,18], and can be modelled in different ways. The OCF model can be solved for the macroion-macroion pair-distribution function and thermodynamic properties using various statistical-mechanical theories. One of the most popular is the mean spherical approximation (MSA) [40], The OCF model can be applied to the analysis of small-angle scattering data, where the results are obtained in terms of the macroion-macroion structure factor [35], The same approach can also be applied to thermodynamic properties Kalyuzhnyi and coworkers [41] analyzed Donnan pressure measurements for various globular proteins using a modification of this model which permits the protein molecules to form dimers (see Sec. 7). 

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