Direct observation of elastic displacement modes by grazing-incidence X-ray diffraction

It is demonstrated that grazing-incidence X-ray diffraction is a direct tool for measuring the elastic displacement modes near the surface of a crystal. Due to the fact that X-ray diffraction is a Fourier transform of the electronic density, and thus, of the atomic positions, elastic displacement modes appear as additional spots in the reciprocal space. Their characteristics can be directly derived from the elastic constants of the material. Measuring the amplitude of the diffracted wave for these peaks allows direct determination of the force distribution at the surface, which is at the origin of the elastic displacements. Various examples of such determinations are given for self-organized surfaces and for vicinal surfaces.