Fourier transforms diffraction patterns

Fig. 5. Illustration of the convolution theorem applied to a crystal structure and its diffraction pattern. (A) is a lattice and (B) is the motif or repeating unit on the lattice. The full crystal (C) is a convolution of (A) and (B). The diffraction pattern (F) of the crystal (C) is the product of the diffraction patterns (Fourier transforms) (D) and (E) from (A) and (A), respectively. For details, see text. (Based on Squire, 1981.)...

As I stated in Chapter 2, computation of the Fourier transform is the lens-simulating operation that a computer performs to produce an image of molecules in the crystal. The Fourier transform describes precisely the mathematical relationship between an object and its diffraction pattern. The transform allows us to convert a Fourier-series description of the reflections to a Fourier-series description of the electron density. A reflection can be described by a structure-factor equation, containing one term for each atom (or each volume element) in the unit cell. In turn, the electron density is described by a Fourier series in which each term is a structure factor. The crystallographer uses the Fourier transform to convert the structure factors to p(.x,y,z), the desired electron density equation. 

When applied to the XRD patterns of Fig. 4.5, average diameters of 4.2 and 2.5 nm are found for the catalysts with 2.4 and 1.1 wt% Pd, respectively. X-ray line broadening provides a quick but not always reliable estimate of the particle size. Better procedures to determine particle sizes from X-ray diffraction are based on line-profile analysis with Fourier transform methods. 

Unlike the wave function, the electron density can be experimentally determined via X-ray diffraction because X-rays are scattered by electrons. A diffraction experiment yields an angular pattern of scattered X-ray beam intensities from which structure factors can be obtained after careful data processing. The structure factors F(H), where H are indices denoting a particular scattering direction, are the Fourier transform of the unit cell electron density. Therefore we can obtain p(r) experimentally via ... 

These are two important special cases. The power and simplicity of diffraction pattern analysis (crystallography) for the analysis of regular structure is a result of Eq. (2.27) and Eq. (2.25). No information is lost if infinite abstract lattices are subjected to Fourier transformation. 

A prime example of a Refolding model is that of the insulin protofilament (Jimenez et al., 2002). Insulin is a polypeptide hormone composed of two peptide chains of mainly o -helical secondary structure (Fig. 3A Adams et al., 1969). Its chains (21- and 30-amino acids long) are held together by 3 disulfide bonds, 2 interchain and 1 intrachain (Sanger, 1959). These bonds remain intact in the insulin amyloid fibrils of patients with injection amyloidosis (Dische et al., 1988). Fourier transform infrared (FTIR) and circular dichroic (CD) spectroscopy indicate that a conversion to jS-structure accompanies insulin fibril formation (Bouchard et al., 2000). The fibrils also give a cross-jS diffraction pattern (Burke and Rougvie, 1972). 

Figure 3.4 Steps in X-ray structure determination. X-ray scattering by the crystal gives rise to a diffraction pattern. From the diffraction pattern, the molecular structure can be determined using Fourier transformation mathematical calculations. Source For diffraction photograph, Nicholls Ft. Double hehx photo not taken by Franklin, BioMedNet News and Comments, 2003. http //news.bmn.com/news/story day=030425 story=l caption name [accessed April 28, 2003].)...

Figure 2. Fonnation of ring and oblique texture patterns, a - several randomly rotated artificial crystallites and its Fourier transform (inset) b - reciprocal space with rings and zero tilt Ewald sphere c - 60° tilt of the Ewald sphere (reflection centers lie on the ellipse) d - the diffraction pattern as it is seen on the image plane.

The number of satellite peaks will depend on the shape of the interface between the units. It is convenient to think of the diffraction pattern in the kinematic approximation as the Fourier transform of the structure. If the layers in the units were graded so that the overall structure factor variation were sinusoidal, this would have ordy one Fourier component and thus only one pair of satellites. If the interface is abrapt, this is equivalent to the Fourier transform of a square wave, which consists of an infinite number of odd harmonics the corresponding diffraction pattern is also an infinite number of odd satellites. The intensities of the satellites therefore contain information about the interface sharpness and grading. 

For obvious reasons, Fourier transformations are widely used to solve problems in X-ray crystallography [129]. With innumerable replications of a molecule in a crystal, all being oriented the same way, approximate periodic boundary conditions are given. Periodic functions become discrete when Fourier transformed. In fact, the diffraction pattern of an X-ray shot on a crystal amounts to the Fourier transform of the square of the absolute values of the real space function [130]. The measurements of intensities and different reflection angles from the crystal relate to the Fourier transform of the electron densities in the crystal. 

The most spectacular success of the theory in its quasistatic limit is to show how to film atomic motions during a physicochemical process. As is widely known, photographing atomic positions in a liquid can be achieved in static problems by Fourier sine transforming the X-ray diffraction pattern [22]. The situation is particularly simple in atomic liquids, where the well-known Zernicke-Prins formula provides g(r) directly. Can this procedure be transfered to the quasistatic case The answer is yes, although some precautions are necessary. The theoretical recipe is as follows (1) Build the quantity F q)q AS q,x), where F q) = is the sharpening factor ... 

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