Fourier Transform in X-ray Crystallography

A Fourier transform is performed when a monochomatic X-ray diffracts off a crystal. When the incidence angle is varied, the complete transform is produced.

The diffraction corresponding to a diffraction vector s and a single electron at position r multiplies the amplitude of the scattered wave by a phase factor e^(−2πirs). If ρ(r) is the electron density function in the crystal, the effect on s will sum to

F(s)=∫crystal ρ(r)e(−2πirs) dr.

Therefore, structure factor F(s) appears as the Fourier transform of the electron density function ρ(r).

How a monochromatic plane wave performs Fourier analysis on the electron density distribution:

Reference:
http://www.ams.org/samplings/feature-column/fc-2011-10

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