Branch points and the critical voltage effect in high-energy electron diffraction (Band theory models)

Band theory methods are used to study the form of the Bloch waves excited in a crystal by a beam of high-energy electrons. Because of absorption, the effective Hamiltonian is non-Hermitian and the branch points usually encountered in band theory at complex K can therefore occur at real K. Under systematic diffracting conditions, it is found that a branch point is either close to the centre of the Brillouin zone or close to the zone boundary at a particular incident beam energy E b which is close to the critical voltage. However, despite the divergence of the excitation amplitudes of the Bloch waves near a branch point, calculations have failed to display any diffraction effects which can be clearly ascribed to the branch point itself. Similarly, branch points can occur near a zone axis critical voltage under cross-grating diffraction conditions. To facilitate the calculations, a simple approximation is used to estimate the optical potential in the effective Hamiltonian.