This work presents a detailed characterization, using high-resolution x-ray diffraction, of multilayered Si1-xGex heterostructures grown on (001), (011), and (111) Si substrates by reduced pressure chemical vapor deposition. Reciprocal space mapping has been used to determine both the strain and Ge concentration depth profiles within each layer of the heterostructures after initially determining the crystallographic tilt of all the layers. Both symmetric and asymmetric reciprocal space maps were measured on each sample, and the evaluation was performed simultaneously for the whole data set. The ratio of misfit to threading dislocation densities has been estimated for each individual layer based on an analysis of diffuse x-ray scattering from the defects.

The dynamical theory for X-ray diffraction from the bilayer crystal structure with different lateral periods of the crystal unit cells (lateral mismatch) is considered in the present paper. The amplitudes of the principal diffraction waves and all harmonics conditioned by the lateral mismatch are calculated.The formation of the Bragg peaks is analysed taking into account the sphericity of the incident beam wave front set. The connection between the parameters of the coherent diffraction potential in the partially relaxed (epitaxial) crystals and the microscopic characteristics of the dislocations is also discussed.

A covariant matrix method based on many-beam dynamical x-ray diffraction is reported for simulation of the wide x-ray diffraction profiles from multilayered crystalline samples both in coplanar and noncoplanar geometries. Because of many-wave scattering, the approximations of the second-order on x-ray polarizability both for dispersion equations and boundary conditions are taken into account. The influence of various factors on calculation of the x-ray profiles containing multiple Bragg reflections is investigated.

In the presence of texture, the concept of X-ray elastic constants as well as Sin2ψ law is inapplicable and the X-ray stress factors (XSF) connecting average strain and stress have to be used [1-2]. The SO(3) vector parameterization with smart composition law [3-4] proved to be a powerful tool for handling transformations between reference systems used in XSF calculation. Decomposition of the 4-th rank elastic constant tensor on the SO(3) irreducible representations (IR) allows one to highlight the symmetry properties and to separate isotropic and anisotropic parts. Joint use of the vector parameterization and IR decomposition enables to obtain transparent analytical expressions for XSF in case of textures described by preferred spherical/fiber components.

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