About forbidden and weak reflections

Reflections forbidden under the single scattering approximation are not expected to remain extinct for crystal thicknesses for which multiple scattering becomes important, except for reflections of the Gjönnes-Moodie type. However, it has been observed that in many crystals with the incident beam along a zone axis, such as diamond-like crystals along the [110] zone, reflections forbidden under the single scattering approximation remain very weak up to large thicknesses. This is hard to explain in terms of many-beam dynamical scattering in Fourier space. The picture becomes clear if one describes the scattering in real space in terms of the channelling of the electrons along the atom columns parallel to the zone axis. In that case the exit wave of each atom column can be described by the S-state model, which is radially symmetric around the centre of the atom column. As a consequence the exit wave shows the same symmetry as the projected potential, so that the reflections forbidden under the single scattering approximation remain extinct. This condition only breaks down when the crystal thickness becomes so large that the S-state model becomes invalid, which is a function of the distance between neighbouring atom columns and/or the tilt from the exact zone axis. The sensitivity for small tilts is also in agreement with very old observations that have not been explained thus far.