X-ray diffraction of spatially bounded beams in lateral periodic structures

Lateral periodic structures (LPS), including multilayer and crystal diffraction gratings, systems modulated by an external influence (for example, by an ultrasonics), are widely used in nano- and optoelectronics, as well as in X-ray optics. Usually, X-ray scattering on LPS is described by means of a model of an incident plane wave, spatially unbounded in lateral direction. However, when one compares experimental and simulated data, such model is not entirely appropriate, since in any experiment X- ray beams are always spatially bounded. On the other hand, within the framework of the plane wave model, it is impossible to perform a numerical calculation of the scattering intensity distribution in vicinity of a reciprocal lattice node, since an angular dimension of diffraction orders is described by the Dirac delta function. Within the framework of the theory of elasticity, numerical calculations of atomic displacement fields for lithium niobate and silicon crystals are performed. In the first case, periodic elastic deformations are caused by an action of a surface acoustic wave with a modulation period of 4 µm. In the second case, deformations arisedue to intermolecular interactions caused by elastic force at the interface between medium of different chemical composition - wolfram stripes with a width of 0.5 µm periodically located on a silicon surface. Using the found solutions, numerical simulations of reciprocal space maps for crystals with periodically distributed elastic strains were performed.It was established, that for crystals modulated by an accoustic wave, diffraction orders consist of the main reflection vertical band and a pair of inclined bands, induced by spatially bounded X-ray beams. In the case of diffraction in crystals with the surface grating, one observes additional satellites along inclined bands caused by the spatial modulation of incident X-ray wave. Therefore, a general problem of the dynamical theory of X-ray diffraction in lateral periodic structures for a case of spatially bounded incident and reflected beams is considered. Effects, due to elastic deformations of a crystal lattice on an angular distribution of the scattered intensity, are investigated. Results of X-ray diffraction on a crystal modulated by a surface acoustic wave and on a crystal with a metal surface grating are presented.