Phase determination via Sayre-type equations with anomalous-scattering data

The necessary background for the analysis of complex-valued electron-density maps is established. Various systems of structure-factor equations of convolutional type akin to Sayre's squaring method equations are tested for agreement on the real and imaginary parts of the electron density as well as approximations thereof. A system of convolutional structure-factor equations holding in a complex-valued electron density generated by two atom types is developed. The scope of application of these equations is determined and it is shown that the equations provide a method of extrapolating high-resolution phases from a low-resolution base phase set without introducing further model bias. Additional applications to phase refinement are explored.