Fourier transforms between crystal and diffraction space

5 Fourier transforms (between crystal and diffraction space) 

FIGURE 6.12. Mathematical expressions for Fourier transforms. These show that one can use a Fourier transform to convert structure factors (with phases) to electron density and electron density to structure factors with phases. 

FIGURE 6.13. Graphical examples of the Fourier transforms of (a) a cosine and (b) a sine function. Note that the Fourier transform contains information on phase, but that this information is lost when intensities (which involve the square of the displacement) are measured. The designation real and imaginary derives from the presence of i = in the Fourier transform Equation 6.14.1. 

FIGURE 6.15. The Fourier transforms (FTs) of the 200, 300, and 500 electron-density waves. Shown on the left is one unit cell and an electron-density wave, and, on the right, its Fourier transform (Bragg reflection). 

The Fourier transform thus provides the bridge between Bragg reflections and the electron-density map. The Bragg reflection, order n, contributes to the electron-density map (Equation 6.3) an electron-density wave with a periodicity d/n. For example, the Bragg reflection 300 represents an electron-density wave that repeats three times in the a direction of the unit cell and has an amplitude F(300). If only the 300 Bragg reflection is observed, then the electron density repeats three times 

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Fourier Transforms (Between Crystal and Diffraction Space)... 

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