Elastic scattering of high-energy electrons by dopant atoms within a crystal in transmission electron microscopy

A Bloch-wave model of dopant-atom scattering is developed using perturbation theory for parallel illumination in a transmission electron microscope. Dopant-atom scattering causes a change in the Bloch-wave excitations, with transitions from one Bloch state to another being governed by the amplitudes of the Bloch states at the dopant-atom position. The scattering mechanisms therefore depend on whether the dopant atom is substitutional or interstitial as well as the orientation of the crystal. The model is used to calculate the electron wavefunction for substitutional and interstitial Mo atoms in [111]- and [001]-oriented body-centred cubic Fe and the results overall are consistent with multislice simulations. However, subtle differences are also observed. For example, in the Bloch-wave model the phase change of the incident electrons (with respect to the perfect crystal) due to dopant-atom scattering varies with the crystallographic orientation while in multislice theory the phase change is constant. This is likely to be due to the assumptions made in the Bloch-wave model, such as neglect of elastic diffuse scattering. Apart from providing a fundamental understanding of dopant-atom scattering, the model can also potentially be extended to analyse dopant-atom imaging in a scanning transmission electron microscope.