Holographic methods in X-ray crystallography. II. Detailed theory and connection to other methods of crystallography

Equations are derived for finding an unknown part of the electron density in the unit cell of a crystal when part of the structure of its contents is already known. These equations are based on the similarity of the X-ray diffraction pattern to a hologram. The X-ray field scattered by the known part of the structure is identified as the holographic reference beam. It interferes with the waves scattered from the unknown part of the structure. The interference pattern contains phase information that can be exploited to recover fully the unknown part of the structure. This paper discusses mathematical properties of the resulting equations and some methods for their solution. A strong similarity to inverse problems of image processing is pointed out and connections to other known methods of X-ray crystallography are established. In paper III [Maalouf, Hoch, Stem, Szöke & Szöke (1993). Acta Cryst. A49, 866-871], some modest numerical simulations are presented.