Powder diffraction data resolution

Information content in a powder diffraction pattern is reduced as compared to that in single crystal diffraction, due to the collapse of the three dimensional reciprocal space into a one dimensional space where the only independent variable is the scattering angle. The poorer the resolution of the diffraction method, the less the information content in the pattern (Altomare et al. 1995 David 1999). As a consequence, structure of less complex phases can be determined from power diffraction alone (fewer atoms in the asymmetric unit of the unit cell). However, refinement of the structure is not limited so seriously with resolution issues, so powder diffraction data are used in Rietveld refinement more frequently than in structure determination. Electron powder diffraction patterns can be processed and refined using public domain computer programs. The first successful applications of electron diffraction in this field were demonstrated on fairly simple structures. 

A third problem with simulation of high resolution diffraction data is that there is no unique instrament function. In the analysis of powder diffraction data, the instalment function can be defined, giving a characteristic shape to all diffraction peaks. Deconvolution of these peaks is therefore possible and fitting techniques such as that of Rietveld can be used to fit overlapping diffraction peaks. No such procedure is possible in high resolution diffraction as the shape of the rocking curve profile depends dramatically on specimen thickness and perfection. Unless you know the answer first, you cannot know the peak shape. 

Patterson methods have also been successfiilly used for structure solution from powder diffraction data. By taking advantage of the Patterson function Fi, usefiil information about the crystal structure can be deduced. Compared to Direct methods, Patterson techniques are more suitable for powder diffraction data with lower resolution, and peak overlap causing significant difficulties. The Patterson function can be calculated by using the equation... 

Overall, powder diffractometers equipped with point detectors offer the best resolution of the resulting powder diffraction data. While the instrumental resolution increases with the increasing goniometer radius, the intensity of the diffracted beam unfortunately decreases because the incident beam produced by an analytical x-ray tube is always divergent. Therefore, typical goniometer radii vary between 150 and -300 mm. 

You are using Fe Ka radiation to collect powder diffraction data employing powder diffractometer C (see problem 4). After several quick scans, you established that the receiving slit with the aperture of 0.03° results in both acceptable resolution and intensity. Bragg peaks appear to have a full width at half maximum between 0.4 and 0.5° 20. What is the largest allowable step during data collection and why ... 

Unfortunately, a straightforward Rietveld refinement of model A fails because it is far from reality, in addition to the low resolution of powder diffraction data. Therefore, the initial model A must be improved before attempting the Rietveld refinement. The improvement was achieved using the following two approaches to geometry optimization ... 

As established in the previous chapter (section 6.17), the anhydrous iron phosphate has a relatively simple crystal structure, however, poor crystallinity of the powder results in broad peaks where full widths at half maximum vary from 0.2 to 0.5° when Mo Ka radiation is employed. Therefore, the resolution of the diffraction data is quite low, as was illustrated in Figure 6.33. There are only 6 atoms in the asymmetric unit but Rietveld refinement of the model is complicated by the inadequate quality of the diffraction data. The model, derived from a suspected analogy with the hydrated FeP04 2H20, cannot be completed based solely on the powder diffraction data due to problems with the experiment. Thus, Rietveld refinement considered in this section starts from the model improved by... 

High time-resolution is afforded with energy dispersive diffraction (see above). While systematic errors are problematic, reliable structural refinements are possible for a limited class of experiment. The software and method to enable structure refinement using the Rietveld method and ED powder diffraction data are now well-established (Chen and Weidner 1997, Larson and von Dreele 1986). In a recent study of the partitioning of iron between the olivine and ringwoodite polymorphs of (Mg,Fe)2Si04, energy dispersive data were sufficiently accurate to allow derivation of unit cell volumes... 

Complementary structural techniques have recently helped to clarify the structure of bimessite and todorokite and to limit possible structural models for vemadite. High-resolution transmission electron microscopy (HRTEM) revealed the tunnel structure of todorokite in marine manganese deposits (30). Post and Bish (31) used these results to carry out a Rietveld refinement on powder diffraction data of todorokite, which confirmed the basic (3 X 3) tunnel structure of this mineral. 

The above description is actually a simplified version of reality since a high-resolution analysis of the spectral lines of Cu Koc shows that both the oci and 0C2 peaks are distinctly asymmetric. An understanding of the origin of this asymmetry is important in implementing the so-called fundamental parameters approach to the profile fitting of powder diffraction data peaks, described in Chapters 5, 6, 9 and 13, in which the detailed spectrum of the incident X-rays must be known. A combination of five Lorentzian functions is commonly used to model the peak shape of Cu radiation, though detailed investigations to characterize the X-ray spectrum continue. ... 

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