Bloch wave HREM

An analysis of the effects of beam convergence is carried out for high resolution electron microscope imaging, including the changes in diffraction across the illumination aperture by a Bloch wave approach. It is shown that the effect of a small tilt u is to introduce a perturbation u·{down triangle, open}_ρ{variant} into the projected 2D Schrödinger equation for the Bloch waves. By using perturbation theory (essentially a k·p analysis), it is shown that the dominant effect is a modification of the non-linear convergence envelope term between different Bloch waves. The effects should be larger for defects than for single crystals, and important for thicknesses of more than ∼ 50 nm, thicknesses similar to those at which non-linear effects are important. The general trends with structure, elemental composition and incident electron voltage are briefly described.