X-ray Standing Waves in Crystals Distorted by a Constant Strain Gradient. A Theoretical Study

Abstract

X-ray Bragg-case diffraction in crystals distorted by a constant strain gradient is studied theoretically. The rocking curves and intensity distributions of the standing-wave fields are calculated for several data sets corresponding to real physical cases of diffraction in bent crystals (Si 111 reflection and Mo Kα 1 radiation, GaAs 004 reflection and Cu Kα 1 radiation, PbTe 002 reflection and Cr Kα 1 radiation). The obtained curves are interpreted using the quasi-classical approximation of the exact analytical solution of the problem. The oscillations on one flank of the rocking curves are shown to result from interference between the wave directly reflected at the crystal surface and the one issued from the wave field deflected back to the surface along a curved trajectory. Formulae for their distances are also derived. The sensitivity of the rocking curves and X-ray standing waves to the crystal curvature and absorption is discussed. In particular, the phase shift between the oscillations in the rocking curve and in the standing-wave intensity profile can be readily used to determine the positions of atoms in the bent crystal or at its surface.