The `Frozen-Lattice' Approach for Incoherent Phonon Excitation in Electron Scattering. How Accurate Is It?

The `frozen-lattice' model is a semi-classical approach for calculating electron diffuse scattering in crystals that has arisen from thermal vibration of crystal atoms. This quasi-elastic scattering approach is, however, unproven since its equivalence with the incoherent phonon-excitation model is not yet established. As quantitative electron microscopy is becoming a realistic method, it is necessary to examine the accuracy of the model. In this paper, based on a rigorous quantum-mechanical phonon excitation theory, it is proved that an identical result would be obtained using the frozen-lattice model and the formal phonon-excitation model if (i) the incoherence between different orders of thermal diffuse scattering is considered in the frozen-lattice-model calculation and (ii) the specimen thickness and the mean-free-path length for phonon excitation are both smaller than the distance travelled by the electron within the lifetime of the phonon (τ0v, which is 5 µm for 100 kV electrons). Condition (ii) is usually absolutely satisfied and condition (i) can be precisely accounted for in the calculation with the introduction of the mixed dynamic form factor S(Q, Q′). The conclusion holds for each and all orders of diffuse scattering, thus, the quantum-mechanical basis of the frozen-lattice model is established, confirming the validity, reliability and accuracy of using this model in quantitative dynamical electron diffraction and imaging calculations. It has also been shown that the frozen-lattice model is suitable for low-energy electrons.