Fourier Transforms and Diffraction

The reader probably has it branded indelibly upon his/her mind by now that Fourier transform and diffraction pattern are mathematical and physical correlates. Hence the point will be belabored no more, and only one or the other will be used. The questions now become specific. How does a collection of atoms diffract X-rays, and how does a point lattice diffract X-rays ... 

Finally, we see from the Fourier transform equations, for the structure factor Fhu and the electron density p x, y, z), that any change in real space (e.g., the repositioning of an atom) affects the amplitude and phase of every reflection in diffraction space. Conversely, any change in the intensities or phases in reciprocal space (e.g., the inclusion of new reflections) affects all of the atomic positions and properties in real space. There is no point-to-point correspondence between real and reciprocal space. With the Fourier transform and diffraction phenomena, it is One for all, and all for one (Dumas, The Three Musketeers, 1844). 

Figure 6. TEM images of STAC-1 viewed down the a axis of a hexagonal unit cell (indicated by [M/]h) or the [110] direction of a cubic unit cell (indicated by [M/]c). The crystal is dominated by ABCABC close packing (indicated on (a)) with one stacking fault (marked by a horizontal line). A Fourier transform optical diffraction pattern with both Miller-Bravais indices to the hexagonal unit cell and Miller indices (in parentheses) to the cubic unit cell is inserted in (b). Simulated images based on a proposed model (right) are also inserted with specimen thickness of 30 nm, and lens focuses of—30 nm (a) and —10 nm (b).

Wrinch, D. (1946). Fourier Transforms and Structure Factors. American Society for X-ray and Electron Diffraction, Cambridge, Massachusetts. 

If the electron density is known correctly, then structure factors and their relative phases can be computed by Fourier transform techniques. The calculation of X-ray scattering factors from the computed orbital electron densities as a function of distance from the nucleus, shown in Figure 6.19, provides an example of this. In a crystal structure analysis it is possible, from the measured diffraction pattern (structure factors and their phases) to compute the Fourier transform and thereby obtain an image of the entire crystal structure. In practice, only the contents of one unit cell are computed because the reciprocal lattice is the Fourier transform of the direct lattice and vice versa, so that the two transforms can be multiplied (Figure 6.17). 

Despite the apparent simplicity with which a crystal structure can be restored by applying Fourier transformation to diffraction data (Eqs. 2.132 to 2.135), the fact that the structure amplitude is a complex quantity creates the so-called phase problem. In the simplest case (Eq. 2.133), both the absolute values of the structure amplitudes and their phases (Eq. 2.105) are needed to locate atoms in the unit cell. The former are relatively easily determined from powder (Eq. 2.65) or single crystal diffraction data but the latter are lost during the experiment. 

Figure 1.4 is a simple example. The object is the square shown on the mask in (a). If we look at a distance / behind the lens, then we see the diffraction pattern of the square in (b). A second example is shown in Figure 1.5. With the possible exception of the diffraction pattern of DNA, this is probably the most reproduced diffraction pattern in the history of X-ray crystallography (see Taylor and Lipson, 1964, for its origin). Here, the object in (a) is a duck (probably rubber), and in (b) we see the duck s Fourier transform, its diffraction pattern. Significantly, if we were to place the diffraction pattern in (b) in the place of the object in (a), then at distance / behind the lens we would now see the duck. In other words, (b) is the Fourier transform of (a), but (a) is also the Fourier transform of (b). The transform is symmetrical, and it tells us that either side of the Fourier transform contains all the information necessary to recreate the other side.

We will further see in later chapters that it is possible to combine the two kinds of transforms illustrated here, the continuous transform of a molecule with the periodic, discrete transform of a lattice. In so doing, we will create the Fourier transform, the diffraction pattern of a crystal composed of individual molecules (sets of atoms) repeated in three-dimensional space according to a precise and periodic point lattice. 

FIGURE 1.9 The basic X-ray diffraction experiment is shown here schematically. X rays, produced by the impact of high-velocity electrons on a target of some pure metal, such as copper, are collimated so that a parallel beam is directed on a crystal. The electrons surrounding the nuclei of the atoms in the crystal scatter the X rays, which subsequently combine (interfere) with one another to produce the diffraction pattern on the film, or electronic detector face. Each atom in the crystal serves as a center for scattering of the waves, which then form the diffraction pattern. The magnitudes and phases of the waves contributed by each atom to the interference pattern (the diffraction pattern) is strictly a function of each atom s atomic number and its position x, y, z relative to all other atoms. Because atomic positions x, y, z determine the properties of the diffraction pattern, or Fourier transform, the diffraction pattern, conversely, must contain information specific to the relative atomic positions. The objective of an X-ray diffraction analysis is to extract that information and determine the relative atomic positions. 

In summary then, a crystal can be conceived of as an electron density wave in three-dimensional space, which can be resolved into a spectrum of components. The spectral components of the crystal correspond to families of planes having integral, Miller indexes, and these can, as we will see, give rise to diffracted rays. The atoms in the unit cell don t really lie on the planes, but we can adjust for that when we calculate the intensity and phase with which each family of planes scatter X rays. The diffracted ray from a single family of planes (which produces a single diffraction spot on a detector) is the Fourier transform of that family of planes. The set of all diffracted rays scattered by all of the possible families of planes having integral Miller indexes is the Fourier transform of the crystal. Thus the diffraction pattern of a crystal is its Fourier transform, and it is composed of the individual Fourier transforms of each of the families of planes that sample the unit cells. 

The concept of a repeated distribution is important because it can be shown (we will forego a painful formal proof here) that the Fourier transform (or diffraction pattern) of the convolution of two spatial functions is the product of their respective Fourier transforms. This was demonstrated physically using optical diffraction in Figure 1.8 of Chapter 1. In principle, this means that if we can formulate an expression for the Fourier transform of a single unit cell, and if we can do the same for a lattice, then if we multiply them together, we will have a mathematical statement for how a crystal diffracts waves, its Fourier transform. 

Now let us ask what is needed to numerically evaluate this expression for the Fourier transform. The diffraction vector s = k — ko, as well as X, are experimental variables that are chosen, and the Zj are known for each atom as well. The only remaining variables are the xj, and these can be generated from the atomic coordinates xj, yj, zj. Thus all we really need to compute the resultant waves making up the diffraction pattern, for any array of scattering points, are their relative positions in space. 

G.18 H. Lipson and C. A. Taylor. Fourier Transforms and X-Ray Diffraction (London George Bell, 1958). On the interpretation of x-ray scattering by Fourier transforms. 

X-ray photoelectron spectroscopy Mossbauer spectroscopy X-ray singlecrystal diffraction XPS (ESCA) Inner-shell electron transitions. Excitation of nuclear spin by y rays. Fourier transform of diffraction data reveals location of electron density. Oxidation state of metal. Oxidation and spin state. Antiferromagnetic coupling (Fe only). Precise three-dimensional structure, bond distances and angles for small molecules. Lower resolution and precision for proteins. 

As will be shown below, the diffraction pattern on the screen is a Fourier transform of the intensity distribution of the light passing the electrode after interacting with the chromophore. Therefore, the relationship between chromophore generation and diffracted intensity is embodied in the Fourier transform, and all that can be learned about events at the electrode can be learned by analyzing the diffraction pattern using Fourier transform techniques. 

The functions and F(J fornn a reciprocal pair of Fourier transforms, and Equations 83 and 85 are considered as the main expressions in solving the chief task of diffraction techniques to determine the object s structure from the observed (measured) diffraction pattern F(s). In the operational notation... 

In most cases, structural analysis of polymer crystals is carried out using uniaxially oriented samples (fibers or films). The basic procedures include (1) determination of the fiber period (2) indexing (hkl) diffractions and determining the unit cell parameters (5) determination of the space group symmetry (4) structural analysis and (5) Fourier transforms and syntheses and Patterson functions. The first three aspects of the procedure are discussed here, and the last two aspects are left for further references. 

BraceweU R (1999) The Fourier Transform and Its Applications, Me Graw-Hill, New York, 3rd edn. Guinier A and Fbumet G (1955) Small-Angle Scattering of X-Rays, Chapman and Hall, London, UK. Hosemann R and Bagchi S N (1962) Direct Analysis of Diffraction by Matter, North-Holland, Amsterdam, Netherland. 

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