Diffuse scattering probability

The set of pictures [Fig. (3)] show some characteristic patterns which have been obtained. Picture (3a) corresponds to unswollen filled rubber and is characterized by weak diffuse scattering, probably due to important depolarizing reflections at the particle (n = 1.486) and rubber (n = 1.519) interface. 

In the diffuse mismatch model, the scattering destroys the correlation between the wave vector of the impinging phonon and that of the diffused one. In other words, the scattering probability is the same independent of which of the two materials the phonon comes from. This probability is proportional to the phonon state density in the material (Fermi golden rule). 

Figure 15 The probability of diffuse scattering plotted versus E, the sum of surface oscillator energy (kTs), incident molecular rotational energy (kTmu n). and translational energy (l/2mv ). Note that the data with varying rotational energy are taken with a translational energy of 0.13 eV. From Glatzer et al. [124].

Fig. 15. (a) Intramolecular hydrogen bonds in urea crystal with displacement ellipsoids at 50% probability, (b) Static deformation density obtained from the multipolar analysis of the experimental data corrected for the thermal diffuse scattering. Theoretical deformation density obtained using (c) the Hartree-Fock method (d) the DFT method by generalized gradient approximation (contours at 0.0675 eA-3) (reproduced with permission from Zavodnik et al. [69]). 

Note. B and C are probably 10/ diffuse scattering associated with a nestled structure akin to that of C,4AsFe (30). 

The residual difference after a successful DDM refinement or/and decomposition can be considered as a scattering component of the powder pattern free of Bragg diffraction. The separation of this component would facilitate the analysis of the amorphous fraction of the sample, the radial distribution function of the non-crystalline scatterers, the thermal diffuse scattering properties and other non-Bragg features of powder patterns. The background-independent profile treatment can be especially desirable in quantitative phase analysis when amorphous admixtures must be accounted for. Further extensions of DDM may involve Bayesian probability theory, which has been utilized efficiently in background estimation procedures and Rietveld refinement in the presence of impurities.DDM will also be useful at the initial steps of powder diffraction structure determination when the structure model is absent and the background line cannot be determined correctly. The direct space search methods of structure solution, in particular, may efficiently utilize DDM. 

The increase in resistivity at narrow fine widths has been attributed to surface scattering and grain-boundary scattering. The Fuchs and Sondheimer (FS) model attributes the resistivity increase in thin and narrow fines to diffuse scattering of electrons at the exterior surfaces with a probability of 1 — p, where p is the specular scattering coefficient. The length scales in the FS model are the thickness and line width of the conductor and the mean free path A. The simplified expression for resistivity as a function of thickness (T) and linewidth (W) of the conductor is given by ... 

Let us now briefly mention the important applications of WAXS in polymer physics. We have already presented a whole range of applications in Chapters 6, 7 and 9. First of all, WAXS provides direct evidence of the physical structure. The polymer can be considered as semicrystalline provided that sharp Bragg reflections are observed. A diffuse scattering pattern indicates that only short-range order is present. The polymer is probably fully amorphous. However, it is possible that it is liquid-crystalline with a nematic mesomorphism (see Chapter 6). WAXS is the standard method for the assessment of crystallinity. Details of the different X-ray methods used to determine crystallinity are presented in Chapter 7. X-ray diffraction is the major tool used in crystallographic work. The assessment of the unit cell of polymers with unknown crystal structure, and... 

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