Abstract In electron scattering, diffuse scattering can be generated by atom vibrations, point vacancies and growth islands (or surface roughness). 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. In this paper, a formal dynamical theory is presented for calculating diffuse scattering with the inclusion of multiple diffuse scattering. By inclusion of a complex potential in dynamical calculation, a rigorous proof is given to show that the high-order diffuse scattering is 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 to numerical calculation. This conclusion establishes the basis for expanding the applications of the existing theories. The exact form of the optical potential is given using a general solution of the Green function, which can be computed numerically using existing dynamical diffraction theories. These conclusions are universal for both low- and high-energy electrons. For transmission electron diffraction, the final result is given in the Bloch wave representation for crystals containing distorted structure. The theory can also be applied to calculate electron images of diffusely scattered electrons in transmission and scanning transmission electron microscopy.
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