On the phase problem in electron microscopy: the relationship between structure factors, exit waves, and HREM images

In electron microscopy, the word phase is used for different physical phenomena, including crystallographic structure-factor phases and the electron wave phases. This has resulted in great confusion, as to whether the phase information is present or lost when an image is recorded. The aim of this paper is to solve this phase confusion problem by studying the relationships between structure factors, exit waves, and high-resolution electron microscopy (HREM) images. Three approaches are taken. First phases at different stages of the imaging processes are compared analytically for a crystal that can be considered a weak phase object (WPO). Then these different phases are calculated by the multi-slice method based on dynamical diffraction theory, and their numerical values are compared. Finally, the validity of the theoretical description is checked by comparison with experimental data on a real crystal, Ti(2)S. It is demonstrated that it is possible to obtain accurate structure factor-phases directly from HREM images by crystallographic image processing. The two major methods for structure determination from HREM images-exit wave reconstruction and crystallographic image processing-are compared. It is shown that the information utilised by the two methods as well as the results are essentially the same.