Reflection high-energy electron diffraction is a Bragg case, in which the beams to be detected are those reflected from the bulk crystal surface. Owing to the strong interaction between the incident electron and the crystal atoms, multiple (or dynamical) scattering is very strong. Although the positions of RHEED beams can generally be predicted by kinematical scattering theory, quantitative analysis of RHEED patterns relies on dynamical calculations. In this chapter, we will first illustrate the quantum mechanical approach for electron diffraction. Then we will outline a few commonly used dynamical theories accompanied by some calculated results.

Based on the first-principles approach, we will consider the fundamental equation that governs the scattering of high-energy electrons in crystals. Before we show the mathematical description, it is important to consider the nature of the events that we are studying. The average distance between successive electrons that strike the crystal in a transmission electron microscope is about 0.2 mm (for 100 keV electrons) if the electron flux is on the order of 1012 s−1 This distance is much larger than the thickness (typically less than 0.5 μm) of the specimen, thus the interaction between any successive incident electrons is extremely weak. Therefore, the interaction between the incident beam and the crystal can be treated one electron at a time. In other words, electron diffraction theory is basically a single-electron scattering theory.

Strictly speaking, scattering of high-energy electrons obeys the Dirac equation. The Dirac equation contains not only the relativistic effects but also electron spin.