Powder structure factors

Figure 8.5 SAXS patterns of oriented hexagonal morphologies in the system SI52QE516/D4 at 58°C (a) Azimuthally averaged scattering intensity, after background correction for (poA = 0.20 and 0.30 (with factor 10) (b) modeled powder structure factors (2" more likely).

Recently, the PDF method was extended to describe the local dynamics of disordered materials (Dmowski W, Vakhrushev SB, Jeong I-K, Hehlen M, Trouw F, Egami T (2006) Abstracts American conference on neutron scattering, St. Charles, IL, 18-22 June 2006, unpublished). The total PDF is obtained by the powder diffraction method so that S(Q) includes both elastic and inelastic intensities. To determine the dynamics we have to use an inelastic neutron scattering spectrometer and measure the dynamic structure factor, S(Q,a>), over a large Q and co space, and Fourier-transform along Q to obtain the dynamic PDF (DPDF). While the interpretation of the DPDF is a little... 

In order to obtain detailed structure, a knowledge of diffraction intensities is essential, the intensities being related to the structure factor. Computer-controlled single-crystal X-ray diffractometers with structure (software) packages have made structure elucidation a routine matter. The availability of synchrotron X-radiation of continuously variable wavelength has made X-ray diffraction a still more powerful structural tool for the study of solids. A technique of great utility to solid state chemists is the Rietveld treatment of powder X-ray diffraction profiles (Rietveld, 1969 Manohar, 1983). Automated structure packages for the determination of unknown structures by this method are now commercially available (see section 2.2.3). In Fig. 2.1, we show a typical set of profile data. 

In DLS computations, constant atomic distances are used and have been very useful. Therefore, it appears worthwhile to constrain the positional parameters of the framework with respect to the known distances for Si-O, Al-O, and 0-0 for normal structure factor least-squares computations. Constrained refinement essentially reduces the number and/or variability of the parameters and can be helpful for work with limited data sets (e.g., for powder diffraction). Constrained refinement has been discussed by Pawley (26, 27). 

Full-matrix least-squares refinement of the structure model was carried out with programs orfls (9). Since this program in the form we used refines structure factors Fhki rather than the intensities of the powder lines, it was necessary to decompose the intensity of each line having more than one component into contributions from the individual component reflections. This was done by assuming that the F°mz 2 for the several components of a powder line were in the same ratios as the corresponding FcftfcZ 2 obtained from a structure model—the original trial structure or the previous... 

By contrast to the /m3m refinement, the /43m refinement resulted in virtually no outstandingly bad disagreements between observed structure factors (obtained as defined above) and calculated ones. The powder line h2 + k2 + l2 = 34 is illustrative of the improvement of individual comparisons. 

In 1995, an elaborated method was developed for accurate structure analysis using X-ray powder diffraction data, that is, the MEM/Rietveld method [1,9]. The method enables us to construct the fine structural model up to charge density level, and is a self-consistent analysis with MEM charge density reconstruction of powder diffraction data. It also includes the Rietveld powder pattern fitting based on the model derived from the MEM charge density. To start the methods, it is necessary to have a primitive (or preliminary) structural model. The Rietveld method using this primitive structural model is called the pre-Rietveld analysis. It is well known that the MEM can provide useful information purely from observed structure factor data beyond a presumed crystal structure model used in the pre-Rietveld analysis. The flow chart of the method is shown in Fig. 2. 

Toraya s WPPD approach is quite similar to the Rietveld method it requires knowledge of the chemical composition of the individual phases (mass absorption coefficients of phases of the sample), and their unit cell parameters from indexing. The benefit of this method is that it does not require the structural model required by the Rietveld method. Furthermore, if the quality of the crystallographic structure is poor and contains disordered pharmaceutical or poorly refined solvent molecules, quantification by the WPPD approach will be unbiased by an inadequate structural model, in contrast to the Rietveld method. If an appropriate internal standard of known quantity is introduced to the sample, the method can be applied to determine the amorphous phase composition as well as the crystalline components.9 The Rietveld method uses structural-based parameters such as atomic coordinates and atomic site occupancies are required for the calculation of the structure factor, in addition to the parameters refined by the WPPD method of Toraya. The additional complexity of the Rietveld method affords a greater amount of information to be extracted from the data set, due to the increased number of refinable parameters. Furthermore, the method is commonly referred to as a standardless method, since the structural model serves the role of a standard crystalline phase. It is generally best to minimize the effect of preferred orientation through sample preparation. In certain instances models of its influence on the powder pattern can be used to improve the refinement.12... 

There are other factors affecting the intensity of the peaks on a x-ray diffraction profile of a powdered sample. We have analyzed the structure factor, the polarization factor, and the broadening of the lines because of the dimensions of the crystallites. Now, we will analyze the multiplicity factor, the Lorentz factor, the absorption factor, the temperature factor, and the texture factor [21,22,24,26],... 

The factors that are included when calculating the intensity of a powder diffraction peak in a Bragg-Brentano geometry for a pure sample, composed of three-dimensional crystallites with a parallelepiped form, are the structure factor Fhkl 2=l/ TS )l2, the multiplicity factor, mm, the Lorentz polarization factor, LP(0), the absorption factor, A, the temperature factor, D(0), and the particle-size broadening factor, Bp(0). Then, the line intensity of a powder x-ray diffraction pattern is given by [20-22,24-26]... 

In the direct-space approach [26-28] for solving crystal structures from powder diffraction data, trial crystal structures are generated in direct space, independently of the experimental powder diffraction data. The powder diffraction pattern for the trial structure is calculated automatically using Eq. (1) in Sect. 2.2 [the structure factor amplitudes F(h) obtained using this equation are used to determine the relative intensities 1(h) of the diffraction maxima in the powder diffraction pattern]. The suitability of each trial structure is then assessed by direct comparison between the experimental powder diffraction pattern and the powder diffraction pattern calculated for the trial structure. The comparison between the experimental and calculated powder diffraction patterns is quanti-... 

According to Drits (private communication, 1979) structure-factor calculations for the hM reflections of tochilinite I show that the reflections with k = 5n and k = 6n ought to be among the strongest reflections because of the subperiodidties of metal atoms in the two component structures. On the X-ray powder patterns and rotation photographs... 

Such a refinement program was very useful in these cases but is in general of limited application, for it is only with very simple structures (both nuclear and magnetic) that a sufficient number of nuclear intensities can be accurately resolved at 4.2K to provide a basis for refinement and the determination of the scale factor. A more general refinement procedure has been recently introduced (69) which fits the measured profile of the powder diffraction pattern rather than individual intensities or structure factors. With data of high resolution obtained over a wide... 

In any of the reciprocal space methods, which are based exclusively on the use of the observed structure factors, the powder diffraction pattern must be deconvoluted and the integrated intensities of all, or as many as possible, individual Bragg reflections determined with a maximum precision. Only then, Patterson or direct phase angle determination techniques may be employed to create a partial or compete structural model. Theoretical background supporting these two methods was reviewed in section 2.14. 

Currently, Patterson and direct methods are the most frequently employed classical structure solution approaches. The direct phase determination methods are especially successful in solving structures from single crystal data, but their use in powder diffraction increases progressively as the quality of powder data improves, better deconvolution techniques are developed and more precise individual structure factors become available. 

Structure factors from powder diffraction data... 

There is a variety of freely available software, which enables one to deconvolute a powder diffraction pattern and determine either or all individual intensities, lattice and peak shape function parameters, and observed structure factors of all possible Bragg reflections. Freeware codes include EXPO, FullProf, GSAS, LHPM-Rietica, and others. In addition to free programs, nearly all manufacturers of commercial powder diffractometers offer software for sale either as a package with the sale of the equipment or as stand-alone products. ... 

By solving crystal structures of different classes of materials," we will illustrate only a few of the possible approaches to the ab initio structure solution from powder diffraction data. Whenever possible the structure factors obtained from full pattern decompositions should be used until the coordinates of all atoms are established. In some cases it may be necessary to re-determine individual structure factors based on the nearly completed structural model, especially when locations of lightly scattering atoms are of concern after all strongly scattering species have been correctly positioned in the unit cell. This re-determination may be routinely performed during Rietveld refinement and will be briefly discussed in Chapter 7. 

Table 6.4. The list of Bragg reflections with their corresponding observed structure factors squared determined from Le Bail s full pattern decomposition of the powder diffraction...

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