Abstract Dynamical theories are developed to calculate the diffraction intensities of double-inelastically scattered high-energy electrons (i.e. the electrons that have been inelastically scattered twice as the result of exciting two distinct crystal states) in crystals within the framework of quantum mechanics. These theories are needed to quantify the data of thermal diffusely scattered electron energy loss spectroscopy. The Bloch wave and Green's function approaches proposed here take into account the full dynamical diffraction effects of the electrons before and after each inelastic event. The Bloch wave theory gives a full three-dimensional description of double-inelastic scattering in crystals including higher-order Laue zone reflections, but it may not be convenient to calculate the intensity of high-angle thermal diffuse scattering of electrons because of the difficulty of including all of the possible multiphonon excitations. Green's function theory, as an alternative approach, is most suitable for integrating the contributions made by all possible multiphonon excitations. Details are given based on the first principle considerations of the inelastic processes. The diffraction intensity derived based on the Green function approach has been given in explicit analytical forms that are adequate for numerical calculations.
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