Fourier transform images, electron densities

FIGURE 5.5 Electron-density mapping corresponding to the Fourier transforms (A) for denatured (extruded at 100 °C) and native WPI, an (B) inverse reciprocal spacing of electron-density images of native and denatured WPI (Onwulata et al., 2006). 

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. 

First I will discuss Fourier series and the Fourier transform in general terms. I will emphasize the form of these equations and the information they contain, in the hope of helping you to interpret the equations — that is, to translate the equations into words and visual images. Then I will present the specific types of Fourier series that represent structure factors and electron density and show how the Fourier transform inter con verts them. 

When we describe structure factors and electron density as Fourier series, we find that they are intimately related. The electron density is the Fourier transform of the structure factors, which means that we can convert the crystallographic data into an image of the unit cell and its contents. One necessary piece of information is, however, missing for each structure factor. We can measure only the intensity Ihkl of each reflection, not the complete structure factor Fhkl. What is the relationship between them It can be shown that the amplitude of structure factor Fhkl is proportional to the square root of... 

In order to understand the procedures needed to obtain an image of a crystal on an atomic scale, it is best to use mathematical tools, such as the Fourier analysis, the Fourier synthesis and the Fourier transform. These greatly simplify our understanding of how to analyze a crystal diffraction pattern. In this Chapter we will describe the use of a Fourier synthesis to obtain an electron-density map, and the significance of a Fourier analysis in terms of the Bragg reflections measured in the diffraction experiment. In addition we will describe what a Fourier transform is, and its role as the bridge between the atomic arrangement in a crystal and its diffraction pattern. 

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). 

We have seen that the diffracted waves Fhki, from a particular family of planes hkl, when Bragg s law is satisfied, depends only on the perpendicular distances of all of the atoms from those hkl planes, which are h xj for all atoms j. Therefore each Fhki carries information regarding atomic positions with respect to a particular family hkl, and the collection of Fhki for all families of planes hkl constitutes the diffraction pattern, or Fourier transform of the crystal. If we calculate the Fourier transform of the diffraction pattern (each of whose components Fhki contain information about the spatial distribution of the atoms), we should see an image of the atomic structure (spatial distribution of electron density in the crystal). What, then, is the mathematical expression that we must use to sum and transform the diffraction pattern (reciprocal space) back into the electron density in the crystal (real space) ... 

The electron density equation very simple structures such as NaCl can be solved by comparison of the relative intensities of the diffraction spots. For more complicated structures, the power of Fourier transform methods was soon appreciated [27]. In order to produce an image of the structure, the diffracted rays must be combined. In the light microscope this is achieved by the focussing power of the objective lens (Fig. 3b). For X-rays the refractive index of almost all substances is close to 1 and it is not possible to construct a lens. The diffracted rays must be combined mathematically. This is achieved with the electron density equation. 

---