The Phase Problem of X-Ray Crystallography from the Viewpoint of Signal Recovery

The diffraction of X-rays by crystals was discovered in 1912 by Laue who proposed a theory of the phenomenon based on its analogy with optical diffraction by gratings. It is most easily understood from the standpoint of Fourier analysis, in which language it was reformulated shortly afterwards. The electron density in a crystal is periodic, and may therefore be conceived as a superposition of plane waves whose wave vectors belong to a "reciprocal lattice" dual to the crystal lattice. The contribution of each wave, or system of fringes, is described by a complex Fourier coefficient, whose amplitude gives the strength of this system of fringes, and whose phase determines the position of the fringes relative to some fixed origin. If all the amplitudes and phases of these waves are known, it is possible to obtain a picture of the electron distribution in the crystal by superposing them; that is, by a simple Fourier synthesis.