X-ray diffraction of multilayers and superlattices

Recursion formulae for calculating the reflected amplitude ratio of multilayers and superlattices have been derived from the Takagi-Taupin differential equations, which describe the dynamical diffraction of X-rays in deformed crystals. Calculated rocking curves of complicated layered structures, such as non-ideal superlattices on perfect crystals, are shown to be in good agreement with observed diffraction profiles. The kinematical theory can save computing time only in the case of an ideal superlattice, for which a geometric series can be used, but the reflectivity must be below 10% so that multiple reflections can be neglected. For a perfect crystal of arbitrary thickness the absorption at the center of the dynamical reflection is found to be proportional to the square root of the reflectivity. Sputter-deposited periodic multilayers of tungsten and carbon can be considered as an artificial crystal, for which dynamical X-ray diffraction calculations give results very similar to those of a macroscopic optical description in terms of the complex index of refraction and Fresnel reflection coefficients.