Schemes to determine the crystal potential under dynamical conditions using voltage variation

Abstract

Charge densities and crystal structures can be determined routinely from X-ray diffraction as X-ray scattering is relatively weak and single scattering can be assumed. The strong dynamical diffraction of high-energy electrons has prevented electron diffraction from being used in the same way. Dynamical diffraction describes both the propagation of the Bragg diffracted wave in the crystal and the scattering by the crystal potential. The balance between these two processes changes as a function of voltage due to relativistic effects. The difference in diffracted intensities recorded at two voltages is shown to be directly proportional to the crystal potential. This is confirmed by calculations using first-order perturbation theory which show negligible differences compared to exact calculation. It should therefore be possible to use differences in intensity measured as a function of voltage to determine the crystal potential directly. If the full complex wave function is available, then there is a particularly simple procedure to recover the potential, even under dynamical conditions.