Dynamical Electron Diffraction: An Analytic Solution for Five Beams

In the kinematical limit, an electron beam transiting a crystal diffracts once and then leaves. However, in thicker crystals an electron beam diffracts multiple times in what is called dynamical diffraction. Beams undergoing multiple diffraction interact with each other via the pendell?sung effect, producing intricate interference patterns inside the convergent beam electron diffraction (CBED) disks. In the case of two or three beams, the diffraction patterns can be solved analytically in terms of coupled harmonic oscillators. Here we show that, using a more sophisticated matrix analysis, the five beam case can also be solved under certain conditions. Applying this result to a four dimensional scanning transmission electron microscopy (4DSTEM) data hypercube, we determine a sample's thickness and extinction distance. The five-beam solution provides a possible path for high precision interrogation of crystalline samples.