Phases of Fourier coefficients directly from crystal diffraction data

It is shown that the application of Schwarz's and Cauchy's Inequalities to the formulas for calculating F_hkl from, respectively, the density of scattering matter and the atomic positions in a crystal leads to relations between the magnitudes of some F_hkl's and the signs or phases of others. These relations are in the form of inequalities, which vary with the symmetry of the crystal under consideration. A table of the simplest inequalities applicable to crystals possessing each of the simple symmetry elements is included. Examples of the inequalities arising from the presence of combinations of symmetry elements are presented.