Thermal diffuse scatter

Hall, C.R. and Hirsch, P.B. (1965) Effect of thermal diffuse scattering on propagation of high energy electron through crystals, Proc. R. Soc. London A, 286, 158-177. 

To model diffraction intensities, detector effects and the background intensity from thermal diffuse scattering must be included. A general expression for the theoretical intensity considering all of these factors is... 

Early studies, which did not include many high-order reflections, revealed systematic differences between spherical-atom X-ray- and neutron-temperature factors (Coppens 1968). Though the spherical-atom approximation of the X-ray treatment is an important contributor to such discrepancies, differences in data-collection temperature (for studies at nonambient temperatures) and systematic errors due to other effects cannot be ignored. For instance, thermal diffuse scattering (TDS) is different for neutrons and X-rays. As the effect of TDS on the Bragg intensities can be mimicked by adjustment of the thermal parameters, systematic differences may occur. Furthermore, since neutron samples must be... 

The X-N technique is sensitive to systematic errors in either data set. As discussed in chapter 4, thermal parameters from X-ray and neutron diffraction frequently differ by more than can be accounted for by inadequacies in the X-ray scattering model. In particular, in room-temperature studies of molecular crystals, differences in thermal diffuse scattering can lead to artificial discrepancies between the X-ray and neutron temperature parameters. Since the neutron parameters tend to be systematically lower, lack of correction for the effect leads to sharper atoms being subtracted, and therefore to larger holes at the atoms, but increases in peak height elsewhere in the X-N deformation maps (Scheringer et al. 1978). 

It is encouraging that the correction for thermal diffuse scattering (TDS), applied to the 123 K data set of Dam et al. (1983), has very little influence on the deformation density, though it significantly affects the thermal parameters, as may be expected. 

Another reason why high angle reflections are better measured at low temperature is the decreased thermal diffuse scattering (see Sect. 2.2) which allows a more accurate integration of those intensities. 

Thermal diffuse scattering (TDS) is ascribed to low-frequency lattice vibrations. The atoms in a perfect crystal are not fixed to their sites and oscillate about their positions. The Bragg intensities are reduced by the Debye-Waller factor, proportional to... 

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]). 

If one looks between diffraction peaks at high resolution, one finds "streaks" due to lattice phonons, which sharpen gradually at low temperatures this is called thermal diffuse scattering. 

Below Tp, the static ( locked ) CDW of the conduction electrons at 2 kp (or 4 kp), couple with the other atomic or molecular electrons in the lattice, cause a slight lattice distortion, and gives rise to extra X-ray reflections. Above Tp, these CDW are mobile, with no phase locking between excitations on nearby chains one sees X-ray diffuse reflections, similar to thermal diffuse scattering, which sharpen as T is lowered. Below Tp the static distortion produces new, usually weak, reflections between reciprocal-lattice layer lines. When the band filling is a rational fraction (1/4, 1/2, 2/3,1, etc.) then these reflections overlap with certain Bragg reflections of the background lattice, and are more difficult to detect. 

Preliminary photographs of a crystal may be taken in order to check for a cracked or twinned crystal, or for thermal diffuse scattering or superlattice formation. Some of the instruments for doing this are those that were historically used for data collection. It is now debated whether it is necessary to take such preliminary photographs, because the more sophisticated data-collecting devices, together with a high-speed computer, can provide much of the same information. The reader, however, may encounter these other methods, which are briefly described here. More details can be obtained from the listed references. 

Thermal diffuse scattering Diffuse scattering results from a departure from a regular periodic character of a crystal lattice. It is evident as diffuse spots or blurs around normal diffraction spots. If it is a temperature-dependent effect, it is called thermal diffuse scattering. 

Additional ripples of diffuse scattering observed at 17 and 22 A-1 for KHCOs are rather weak and broad. As they survive in the deuterated analogue they are not related to quantum statistics. They were tentatively attributed to thermal diffuse scattering [Wilson 2002], These properties are further documented in Ref. [Fillaux 2003 (a)]. 

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. 

Gjpnnes, J. Disorder and defect scattering, thermal diffuse scattering, amorphous materials. In Electron Diffraction Techniques, Cowley, J.M., Ed. Oxford University Press New York, 1993 Vol. 2, 223-259. 

Uncertainties of the conventional parameters of H-atoms have been addressed since the early applications of X-ray charge density method. Support from ND measurements appears to be essential, because the neutron scattering power is a nuclear property (it is independent of the electronic structure and the scattering angle). The accuracy of nuclear parameters obtained from ND data thus depends mainly on the extent to which dynamic effects (most markedly thermal diffuse scattering) and extinction are correctable. Problems associated with different experimental conditions and different systematic errors affecting the ND and XRD measurements have to be addressed whenever a joint interpretation of these data is attempted. This has become apparent in studies which aimed either to refine XRD and ND data simultaneously [59] (commonly referred to as the X+N method), or to impose ND-derived parameters directly into the fit of XRD data (X—N method) [16]. In order to avoid these problems, usually only the ND parameters of the H-atoms are used and fixed in the XRD refinement (X-(X+N) method). 

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