Domino phase-retrieval algorithm for structure determination using electron diffraction and high-resolution transmission electron microscopy patterns

Direct-method formalism to determine atomic structures using electron diffraction data is here aimed at a general solution of the phase-retrieval problem, consequently combining electron diffraction (ED) and high-resolution transmission electron microscopy (HRTEM) patterns in a `domino' fashion. While there are similarities to conventional (kinematical) direct methods, there remain major differences; in particular, owing to the dynamical effects in the data, the ED structure factors prove to be complex and then the positivity of the reconstructed electron density is no longer a valid constraint for `dynamical' direct methods. Besides, owing to the dynamical effects, heavy atoms no longer dominantly contribute to the HRTEM images. Thus, the `dynamical' direct-methods concept is based on the phase-retrieval algorithm utilizing both the dynamical ED and the HRTEM data. The fusion of the traditional direct-method technique, which is described here, allows realization of a full-phase restoration of complex structure factors. A numerical example, using the dynamical ED and HRTEM data for (Ga,In)₂SnO₅ ceramic, shows that the method is capable of yielding a unique phase-retrieval solution. The clear sense is that the domino transform algorithm proposed works well and represents a valuable method for phasing diffraction patterns in electron structural crystallography using an experiment that is readily performed when the ED and HRTEM data are collected.