On the dynamical theory of different types of electron microscopic transmission fringe patterns

The amplitudes of the waves transmitted and diffracted by a series of superposed parallel plate-shaped crystals are calculated. The two-beam case is considered, i.e. in each crystal only one family of lattice planes is near to the exact Bragg orientation for reflection. Each crystal is defined by its extinction distance, its absorption length, the phase-angle of the Fourier coëfficiënt of the potential, and the reciprocal lattice vector determining the set of reflecting lattice planes. These quantities are generally assumed to be different for each crystal. The formulae remain valid if one or more of the crystals are replaced by vacuum layers. The expressions for the case of a sequence of parallel stacking faults can be obtained as particular results. The formulae can also be applied if the superposed crystals are not plate-shaped and are stacked in an arbitrary way, provided the column approximation is assumed. A short, but not exhaustive, review is given of the different types of fringe patterns which may occur.