Formation of an image by X-ray diffraction

We will now consider how to simulate this method of image formation in the X-ray diffraction experiment where we have to use a mathematical replacement for the objective lens. The studies by Porter are of great importance because they show how the Bragg reflections give the amplitude components of a Fourier series representing the electron density in the crystal (the electron-density map). In effect, Fourier analysis takes place in the diffraction experiment, so that the scattering of X rays by the electron density in the crystal produces Bragg reflections, each with a different amplitude F hkl) and relative phase Qhkl- 

The electron density in a crystal precisely fits the definition of a periodic function in which an exact repeat occurs at regularly fixed intervals in any direction (the crystal lattice translations). Therefore the electron density in a crystal with a periodicity d can be described by a Fourier synthesis in which each component cosine wave (which we will call an electron-density wave) has a periodicity (i.e., wavelength) d/n, and the amplitude of the rath-order Bragg reflection. 

An electron-density map is calculated by a Fourier synthesis, as shown in Equation 6.3. 

This electron density p xyz), at a point x,y,z in the unit cell volume V, is expressed in electrons per cubic A, and is highest near atomic centers. In the X-ray diffraction experiment the number of waves that must be 

FIGURE 6.9. Contributions to terms in a Fourier synthesis, (a) Individual terms from 000 to 10 00 h =10, k = 0, I = 0) are represented with positive areas shaded. These ten electron-density waves combine to give the electron density shown at the bottom of the diagram. This electron density is dependent on the phases (-f == 0°, - = 180 ) which are 000 + 100 200 300 -h 400 500 600 + 700 800 

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