An optical potential approach to incoherent multiple thermal diffuse scattering in quantitative HRTEM

The theory for the absorption potential (or optical potential) in electron scattering was first proposed by Yoshioka in 1957 based on an approximation that the Greenʹs function is replaced by its form in free-space. This approximation has dramatically simplified the calculation, but the function of the optical potential has been partially lost. In this paper, a rigorous theoretical proof is given based on the quantum inelastic excitation theory to show that the inclusion of the optical potential in the dynamic calculation automatically recovers the contributions made by the high-order diffuse scattering although the calculation is done using the equation derived for single diffuse scattering. This conclusion generalizes the existing first-order diffuse scattering theories to cases in which the incoherent multiple diffuse scattering are important. It is suggested that multiple thermal diffuse scattering (or phonon excitations) is likely to be the dominant source for affecting image contrast in quantitative electron microscopy because of the channeling effect and the localized scattering nature of phonon excitation. Surprisingly, phonon scattering is anticipated to contribute fine atomic-scale contrast in the image, which is believed to be more pronounced than the “white noise” effect of conventional understanding. The optical potential is no longer a simple potential function, rather it is a non-local function strongly dependent on the dynamic diffraction in the crystal because of the involvement of Greenʹs function. These characteristics can be properly taken into the computation using a proposed Bloch wave–multislice approach, in which the non-local effect is resolved in the Bloch wave matrix diagonalization and the dynamical effect is taken care of using Greenʹs function calculated by the multislice theory under the small angle (or high-energy) scattering approximation. In the ground-state approximation, by which we mean that the crystal is in its ground state before each and every inelastic excitation, an introduction of the mixed dynamic form factor and the density matrix automatically produces the incoherence between different order and different phonon excitation processes, resolving a big problem encountered by many other theories. In the conventional dynamical calculation for elastic wave, the inclusion of the Debye–Waller factor and the optical potential accounts only for the effects of the phonon excitation on the elastic wave rather than the contribution made by the diffusely scattered electrons to the image/diffraction pattern.