Structure factors from powder diffraction data

5 Structure factors from powder diffraction data 

In addition to the determination of individual observed structure factors, the full pattern deconvolution carries several supplementary functions  

The two related full pattern decomposition methods in common use today were suggested by Pawley and by Le Bail et al Pawley s approach is based on Eq. 2.48 and full pattern decomposition in the case of dual wavelength data, when Kai/Ka2 doublets are present, is performed by solving the following system of equations using a least squares minimization  

The notations used in Eq. 6.6 are identical to Eq. 2.48. Individual integrated intensities are treated as free least squares parameters. Peak shape function parameters are represented as described in section 2.9, and Bragg peak positions, which affect the values of Xk, are established by the unit cell dimensions, see section 2.8. The background, where 1 n 

When peak shape functions and their parameters, including Bragg reflection positions, are known precisely and the background is modeled by a polynomial function with j coefficients, the solution of Eq. 6.6 is trivial because all equations are linear with respect to the unknowns (Bj, see Eq. 4.1, and / ). It facilitates the use of a linear least squares algorithm described in section 5.13.1. In practice, it is nearly always necessary to refine both peak shape and lattice parameters in addition to Bj and h to achieve a better precision of the resultant integrated intensities. Thus, a non-linear least squares minimization technique (see next section) is usually employed during full pattern decomposition using Eq. 6.6. 

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We note that a figure-of-merit with fixed weighting parameter has been employed previously [85] in structure solution from powder diffraction data, although the figure-of-merit used in that paper differs in several respects from G(T) defined here. The figure-of-merit used in Ref. [85] did not consider normalized energy and normalized R-factor functions, and the R-factor was based on the use of integrated peak intensities rather than a whole-profile fit to the powder diffraction pattern. 

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. 

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-... 

When a powder is examined, many diffracted beams overlap, (see Section 6.11), so that the procedure of structure determination is more difficult. In particular this makes space group determination less straightforward. Nevertheless, powder diffraction data is now used routinely to determine the structures of new materials. An important technique used to solve structures from powder diffraction data is that of Rietveld refinement. In this method, the exact shape of each diffraction line, called the profile, is calculated and matched with the experimental data. Difficulties arise not only because of overlapping reflections, but also because instrumental factors add significantly to the profile of a diffracted beam. Nevertheless, Rietveld refinement of powder diffraction patterns is routinely used to determine the structures of materials that cannot readily be prepared in a form suitable for single crystal X-ray study. 

Among the most recent advances of the powder method is the determination of crystal structures from powder diffraction data. It is an application for which the resolution of the pattern is of prime importance. A series of successive stages are involved in the analysis, including the determination of cell dimensions and identification of the space group from systematic reflection absences, the extraction of structure factor moduli I hkl y the solution to the phase problem to elaborate a structure model and, finally, the refinement of the atomic coordinates with the Rietveld method. 

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. 

As follows from Eqs. 2.94 and 2.95, the relationships between By and Uij are identical to that given in Eq. 2.92 and both are measured in A. The P,y parameters in Eq. 2.93 are dimensionless but may be easily converted into By or Uy. Very high quality powder diffraction data are needed to obtain dependable anisotropic displacement parameters and even then, they may be reliable only for those atoms that have large scattering factors (see next section). On the other hand, the refinement of anisotropic displacement parameters is essential for those crystal structures, where strongly scattering atoms are distinctly anisotropic. 

Solving the crystal structure using either heavy atom or direct techniques does not always work in a straightforward fashion even when the well-resolved and highly accurate diffraction data from a single crystal are available. The complicating factor in powder diffraction is borne by the intrinsic overlap of multiple Bragg peaks. The latter may become especially severe when the unit cell volume and complexity of the structure increase. 

As briefly mentioned in the previous chapter, the determination of a crystal structure may be considered complete only when multiple pattern variables and crystallographic parameters of a model have been fully refined against the observed powder diffraction data. Obviously, the refined model should remain reasonable from both physical and chemical standpoints. The refinement technique, most commonly employed today, is based on the idea suggested in the middle 1960 s by Rietveld. The essence of Rietveld s approach is that experimental powder diffraction data are utilized without extraction of the individual integrated intensities or the individual structure factors, and all structural and instrumental parameters are refined by fitting a calculated profile to the observed data. 

Early data analysis attempted to extract values of the individual structure factors from peak envelopes and then apply standard single crystal methods to obtain structural information. This approach was severely limited because the relatively broad peaks in a powder pattern resulted in substantial reflection overlap and the number of usable structure factors that could be obtained in this way was very small. Consequently, only very simple crystal structures could be examined by this method. For example, the neutron diffraction study of defects in CaF2-YF3 fluorite solid solutions used 20 reflection intensities to determine values for eight structural parameters. To overcome this limitation, H. M. Rietveld realized that a neutron powder diffraction pattern is a smooth curve that consists of Gaussian peaks on top of a smooth background... 

The experimental diffraction data were analyzed by a combined technique involving Rietveld analysis, the maximum entropy method (MEM), and MEM-based pattern fitting (MPF) [10-15]. Rietveld analysis, which is used to refine the crystal structure from the powder diffraction data by a least squares method, was carried out using the RIETAN-2000 program [27], which yields structure factors and their errors after structural refinement. It is known that MEM can be used to obtain a nuclear density distribution map based on neutron structure factors and their errors [5, 6, 8, 10-15, 26-29] any type of complicated nuclear density distribution is allowed so long as it satisfies the symmetry requirements. MEM calculations were carried out using the PRIMA program [29]. To reduce the bias imposed by the simple structural model in the Rietveld refinement, an iterative procedure known as the REMEDY cycle [29] was applied after MEM analysis (Fig. 6.3). In this procedure, structure factors... 

The crystal structures of four chlorinated derivatives of di-benzo-p-dioxin have been determined by x-ray diffraction from diffractometer data (MoKa radiation). The compounds, their formulae, cell dimensions, space groups, the number of molecules per unit cell, the crystallographic B.-factors, and the number of observed reflections are given. The dioxin crystal structures were performed to provide absolute standards for assignment of isomeric structures and have been of considerable practical use in combination with x-ray powder diffraction analysis. 

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. 

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