Electron Statistical Dynamical Diffuse Scattering in Crystals Containing Short-Range-Order Point Defects

Abstract

A formal statistical dynamical theory is developed to calculate diffuse scattering produced by short-range order (SRO) in a distorted crystal structure with consideration of atomic thermal vibrations. Diffuse scattering not only produces fine details in diffraction patterns but also introduces a non-local imaginary potential function that reduces the intensities of the Bragg reflected beams. The distribution of the diffusely scattered electrons and the Fourier coefficients of the absorption potential are directly related to a dynamic form factor S(Q,Q'), which has been calculated with consideration of SRO in the distorted lattices. The statistical structure average on imperfections is performed analytically and the final result is correlated to Cowley's short-range-order parameters. The theory is formulated in the Bloch-wave scheme (Bethe theory) for the convenience of numerical calculation in transmission electron diffraction. A rigorous theoretical proof is given to show that the inclusion of a complex potential in the dynamical 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 simply expands the capability of conventional single diffuse scattering theories. Therefore, the complex potential has a much richer meaning than the conventional interpretation of absorption effect.

Thermal diffuse scattering in sub-angstrom quantitative electron microscopy—phenomenon, effects and approaches

“…A detailed application of Eq. (18) to the diffraction of diffuselly scattered electrons arisen from TDS and SRO has been given elsewhere (Wang, 1996a).…”

Phonon Scattering in Quantitative High-Resolution Electron Microscopy – effects, problems and approaches

“…In electron scattering, most of the existing dynamical theories have been developed under the first order diffuse scattering approximation, thus, they are restricted to cases where the lattice distortion is small. A formal dynamical theory is presented for calculating diffuse scattering with the inclusion of multiple diffuse scattering [3,4,5]. By inclusion of a complex potential in dynamical calculation, a rigorous proof is given to show that the high order diffuse scattering are fully recovered in the calculations using the equation derived under the distorted wave Born approximation, and more importantly, the statistical time and structure averages over the distorted crystal lattices are evaluated analytically prior numerical calculation.…”

“…Based on a rigorous quantum mechanical phonon excitation theory, we have proved that an identical result would be obtained using the frozen lattice model and the formal phonon excitation model if, 1) the incoherence between different orders of thermal diffuse scattering is considered in the frozen lattice model calculation, and 2) the specimen thickness and the mean-freepath length for phonon excitation both are smaller than the distance traveled by the electron within the life-time of the phonon (~ 5 µm for 100 kV electrons) [6]. Condition 2) is usually absolutely satisfied and condition 1) can be precisely accounted for in the calculation with the introduction of the mixed dynamic form factor S(Q,Q') [4]. The conclusion holds for each and all of the orders of diffuse scattering, thus, the quantum mechanical basis of the frozen lattice model is established, confirming the validity, reliability and accuracy of using the frozen lattice model in quantitative dynamical electron diffraction and imaging calculations.…”