Dynamics of thermal diffuse scattering in high-energy electron diffraction and imaging: Theory and experiments

Abstract

A dynamical theory is proposed to calculate the diffraction patterns of electrons which have undergone thermal diffuse scattering (TDS). The lattice dynamics and electron diffraction dynamics are comprehensively combined in this approach. Full dynamical simulations are presented for Mo(001). Under the single-inelastic-scattering approximation and in the near-zone-axis case, the sharpness of the TDS streaks is determined by the phonon dispersion relationships of the acoustic branches, the optical branches contribute only a diffuse background, and dynamical scattering effects can change the intensity distribution of TDS electrons but have almost no effect on the sharpness of TDS streaks. The TDS streaks are defined by the q_x-q_y curves which satisfy δ_i(q)=O, where δ_i(q) is the phonon dispersion relationship determined by the two-dimensional atomic vibrations in the (hkl) plane perpendicular to the incident beam direction B = [hkl]. The directions of TDS streaks predicted according to these procedures are consistent with those predicted according to the q·[r(1)-r(l₁) = 0 rule given by Wang and Bentley, where the summation of l₁, is restricted to the first-nearest neighbours of the lth atom that are located in the same atomic plane as the lth atom perpendicular to the incidence beam direction.

Atomic resolution images can be formed by TDS electrons in transmission electron microscopy. The image theory is equivalent to the incoherent imaging theory if the spherical aberration coefficient C₈ for the objective lens is small. The image resolution may be higher than that formed by pure elastically scattered electrons, but the ‘inclined-incidence effect’ (i.e. the correction of momentum transfer q to the electron wave-vector K) in phonon scattering may distort the image. The phase coupling of vibrating atoms does not affect calculations for images but does affect diffraction patterns. Thus the image simulation can be performed based on the Einstein model as long as the correct vibration amplitude for each atom is used.

Equivalent results are obtained for the TDS electrons based on either the quantum inelastic scattering theory or the ‘frozen’-lattice model of the semiclassical approach if the temperature is not much higher than room temperature.