Brillouin zones and Kikuchi lines for crystals under electron channeling conditions

The geometry of the Kikuchi lines in high- and low-energy electron diffraction patterns is defined in terms of intersections of the Brillouin zone boundaries with a sphere of reflections. Full treatment of the cases of two-dimensional and one-dimensional real lattices reveals previously unknown boundaries in the form of parabolic surfaces (2D) and paraboloids of revolution (1D). These boundaries are applied to K lines which arise from electron channeling. Correlation is made with the previous explanations of parabolas and circles in the reflection high-energy electron diffraction (RHEED) and transmission high-energy electron diffraction (THEED) patterns and K2 and K1 lines in low-energy electron diffraction (LEED) patterns. Experimental convergent-beam RHEED patterns of a reconstructed MgO (111) surface are presented in which parabolas due to the surface periodicity are shown for the first time.