Unique diffraction data

The reader should recall that the fitting of a structure to diffraction data is not unique. We have shown that both the constructed modified random network model of Polk, as well as the network simulated by allowing Gaussian distributions of atom-atom distances can fit the observed structure functions for low density H20(as), and the latter, with modification to include small OOO... 

As a stmcture becomes more complex and the number of unique atoms increases, phases derived by direct methods become less reliable, especially when the electron diffraction data deviate from the kinematical approximation because of dynamic effects. HREM combined with crystallographic image processing provides a unique method for determining such stmctures. HREM images from a number of projections along different zone axes may be combined into a 3D potential map. 

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. 

X-ray fiber diffraction can be used to visualize highly hydrated polymer specimens at atomic resolution. An essential part of such an analysis is the inclusion of reliable stereochemical information to supplement the diffraction data. Structure determination involves modelling and refinement of putative structures, and adjudication amongst the optimized models. This technique has been successfully applied to a number of polysaccharides. The precision of resulting structures is often sufficient to identify the critical interactions within and between molecules, that are responsible for the unique properties of these materials. 

Fig. 9a and b. Difference Fourier maps calculated from Laue diffraction data showing maltoheptose bound in phosphorylase b. The Laue map shown in a is calculated with a subset of 9029 unique data at 2.5 A resolution. A positive contour at half maximal peak height is shown, b is an enlargement of a and shows 4 of the 7 sugar units, the 3 central units have the highest occupancies. Side chain movements produce the two extra lobes of density. (Figures courtesy of J. Hajdu)... 

Let us now see how these conditions for systematic absences are used. Suppose we have established from the X-ray diffraction data that a crystal is monoclinic. See Table 11.7 for the monoclinic space groups. We can next see if the unit cell is primitive or centered. If we choose the unique axis to be c, we look for absences indicative of A centering (hkl, k + 2 n) or B centering... 

Machinery now exists to permit, in many cases, very detailed analyses of fibrous structures using the under-appreciated X-ray diffraction data supplied by the polymers themselves. Some of this machinery can be adapted to tackle the problem of providing unique solutions statistical tests can be applied to (least-squares) optimised versions of competing models. However, additional or alternative tests of the creditability of different models should not be ignored. 

X-ray powder diffraction files (PDFs) for explosive materials are available as subsets of PDF libraries, such as Xpowder and that of the International Centre for Diffraction Data (ICDD). Commercial X-ray diffractometers can yield lattice spacings, d, to a precision of 0.0001 A. At this accuracy, it is not difficult to distinguish uniquely between many thousands of chemical compounds. Hence, XRD is a standard analytical technique for precisely identifying crystalline materials. 

A data set with 2°<20 140° CuKa was collected for a barrelshaped crystal on an Enraf Nonius CAD4 diffractometer with a 3.5° variable-speed 6-20 scan to accommodate the broad diffractions. From 196 unique diffractions above background (2o) out of 2170 measured diffractions, the structure was solved independently by the Multan direct method and a search of hypothetical structures. The resulting framework (Figure 2) is 81 predicted by J.V. Smith(4). Refinement to R 16% yielded the framework geometry, but the T-0 distances did not correspond satisfactorily to Al, P alternation, and the TPAOH was not located. Nevertheless the framework topology appeared correct, and the calculated powder pattern (Figure lb) was satisfactory. 

Crystals of Xe(OSeFs)2 are rhombohedral, space group R3m. At 23.5 C the hexagonal axes are fl - 6 = 8.588 (3) and c - 11.918 (3) A Z = 3, da cd = 3.345 g cm"3, and V = 761.23 AL The molecule lies on a threefold axis, and there is orientational disorder of the oxygen and fluorine positions. X-ray diffraction data obtained with an automatic diffractometer were analyzed on the basis of a molecular model with some constraints based on chemical considerations to reduce the number of independent parameters of the poorly resolved oxygen and fluorine atoms. For 122 unique reflections with > a(F ) and with anisotropic thermal parameters, R = 0.064. Bond distances are Xe-0 2.12 (5), Se-0 1.53 (5), and Se-F = 1.70 (2) A (uncorrected) and Se-F = 1.77 A (corrected for thermal motion). 

Another approach to the analysis of a WAXS pattern is called Debye function analysis (DFA) and has been applied by several groups (Reinhard et al. 1997, 1998 Gnutzmann and Vogel 1990). The main difficulty in any diffraction experiment is that a unique structural model cannot usually be extracted from the data. This is obvious with powder diffraction data where for a complex structure there are far more structural... 

When the space group symmetry is unknown, i.e. when reflection conditions are analyzed from diffraction data, the answer may not be unique. For example, the combination of systematic absences listed above also... 

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