X-ray diffraction in a crystal with lattice strains is studied theoretically using two-dimensional recurrence relations in Laue geometry. Based on these relations, an algorithm for calculating the coherent scattering intensity near a reciprocal-lattice node is developed. Simulation of reciprocal-space mapping was performed for a model of a silicon crystal with Si(Al) thermomigration channels. The change in reciprocal-space maps is shown depending on the strain magnitude in the channel and scanning of the X-ray beam along the input surface of the crystal.

In the kinematical approximation, new analytical solutions are obtained that describe the diffraction of a restricted X-ray beam from a thin crystal. Calculation of the angular distribution of reflected X-ray beams within the framework of the developed approach significantly reduces the computational cost compared with numerical methods. For a thin silicon crystal, X-ray reciprocal-space mapping was simulated using analytical solutions, as well as calculated using numerical methods based on 2D recurrence relations and the Takagi-Taupin equations.

Numerical modeling of kinematical and dynamical X-ray diffraction in a bent crystal was performed on the basis of two approaches to integrating the Takagi–Taupin equations, and using two-dimensional recurrence relations. Within the framework of kinematical diffraction, a new equation is obtained that describes the distribution of diffracted intensity inside a bent crystal. The time taken for numerical calculations based on this equation is significantly reduced in comparison with the use of algorithms of the dynamical diffraction theory. The simulation shows for the first time that, for strongly bent crystals, the maximum value of the diffraction intensity is formed inside the deformed structure and not on its surface. In the case of strong bending of the crystal structure, the deviation of the X-ray beam from the Bragg angle does not change the diffraction pattern but shifts it along the lateral direction. The results of calculations of diffraction in a strongly bent crystal based on the equations of dynamical and kinematical diffraction coincide, while the computations for weakly bent crystals differ. The possibility of estimating the primary extinction length of a bent crystal as a function of the bending radius is shown. In the case of kinematical diffraction in bent crystalline microsystems, a new method has been developed to calculate X-ray reciprocal-space mapping.

In the kinematical approximation, a method for rapid numerical calculation of X-ray diffraction from thin crystalline microsystems has been developed. The speed of calculating of reciprocal space maps using this approach is three to four orders of magnitude higher than calculations based on the Takagi–Taupin equations or two-dimensional recurrence relations. Within the framework of the obtained solutions, numerical simulation of X-ray reciprocal space mapping was performed for three models of crystal chips of microsystems.

A dynamical theory is developed of X-ray diffraction on a crystal with surface relief for the case of high-resolution triple-crystal X-ray diffractometry. Crystals with trapezoidal, sinusoidal and parabolic bar profile models are investigated in detail. Numerical simulations of the X-ray diffraction problem for concrete experimental conditions are performed. A simple new method to resolve the crystal relief reconstruction problem is proposed.

Using two-dimensional recurrence relations, a description of dynamical X-ray diffraction in crystals is presented. It is shown that this approach makes it possible to calculate Xray fields inside the crystal and reciprocal space maps.

Si(111) wafers patterned with an array of vertical 120 µm-wide Al-doped (1 × 10 19 cm −3 ) p-channels extending through the whole wafer were studied with the X-ray double- and triple-crystal diffraction technique in Bragg geometry with copper radiation. Reciprocal space maps (RSMs) of diffraction intensity far from the channels and near them were measured, and their non-trivial shape was observed. The obtained experimental RSMs demonstrate high sensitivity to the structural distortions of the crystal in the subsurface layer owing to the influence of the surface on the elastic strain field in the channel. These features result from the small difference of the ionic radii of Si and Al, leading to the absence of misfit dislocations on the borders of the channel. Simulations of RSMs using the Takagi–Taupin dynamical diffraction theory taking into account the influence of the surface on the elastic strain field in the channel and the effect of the instrumental function were carried out. Finally, numerical RSM calculations showed that the proposed model of the surface effect on the elastic strain field in a semi-infinite crystal with a vertical Si(Al) channel can be used to retrieve the information on the concentration of aluminium in the thermomigrated Si(Al) channel from the diffraction data obtained in the Bragg geometry.

The dynamical diffraction of spatially restricted X-ray beams in a thick perfect crystal is studied using two-dimensional recurrence relations and the Takagi–Taupin (T-T) equations. It is shown that the two-dimensional recurrence relations are transformed into T-T equations when passing from a crystal with an array of discrete lattice planes to a model of continuous periodic electron density. The results of calculations of the X-ray diffraction field inside the crystal and the angular distribution of the scattering intensity in reciprocal space based on these two approaches are presented. It is shown that, when using the two-dimensional recurrence relations and T-T equations, the calculated contours of reciprocal-space maps and their q x sections are similar to each other, and the q z sections completely coincide.

Using the two-dimensional recurrence relations of X-ray dynamic diffraction, the scattering intensity distribution in elliptical crystals has been numerically simulated. This approach makes it possible to study diffraction in cylindrical structures, as well as in rectangular crystals.

Si(111) wafers patterned with an array of vertical 100 µm-wide Al-doped (1 × 10 19 cm −3 ) p-channels extending through the whole wafer were studied by X-ray Laue diffraction techniques. The X-ray techniques included projection topography, and measurement of rocking curves and cross sections in the vicinity of the 02\overline 2 reciprocal space node in the double- and triple-crystal geometry, respectively. The channels are uniform along the depth of the wafer, and their structural perfection is comparable to that of the silicon matrix between the channels. Simulation of the rocking curves was performed using the methods of the dynamical theory of X-ray diffraction. The rocking-curve calculations both taking into account and neglecting the effect of the instrumental function were carried out using the Takagi–Taupin equations. The calculated angular dependences of intensities of both diffracted and transmitted X-rays correspond well to the experimentally obtained rocking curves and demonstrate their high sensitivity to the structural distortions in the channel. An unambiguous reconstruction of strain and structural distortions in the Si(Al) channel using the Laue diffraction data requires further development of the theoretical model.

Using the formalism of dynamical scattering of spatially restricted X-ray fields, the diffraction of a microbeam in a crystal with boundary functions for the incident and reflected amplitudes was studied in the case of geometrical optics and the Fresnel approximation (FA). It is shown that, for a wide front of the X-ray field, the angular distributions of the scattered intensity in the geometrical optics approximation (GOA) and the FA are approximately the same. On the other hand, it is established that, for a narrow exit slit in the diffraction scheme, it is always necessary to take into account the X-ray diffraction at the slit edges. Reciprocal-space maps and the distribution of the diffraction intensity of the microbeam inside the crystal were calculated.

Using the Takagi-Taupin equations, X-ray Laue dynamical diffraction in flat and wedge multilayers is theoretically considered. Recurrence relations are obtained that describe Laue diffraction in structures that are inhomogeneous in depth. The influence of sectioned depth, imperfections and non-uniform distribution of the multilayer period on the Pendellösung effect and rocking curves is studied. Numerical simulation of Laue diffraction in multilayer structures W/Si and Mo/Si is carried out. It is shown that the determination of sectioned depths based on the period of the interference fringes of the experimental rocking curves of synchrotron radiation is not always correct.

A numerical simulation has been performed to study the effect of the mismatch of layer thicknesses of multilayer Laue lenses (MLLs) on X-ray focusing. The spatial distributions of X-ray intensities in the bulk of a lens have been calculated with different gradients of the variation of the period of a MLL. It has been shown that the focal length, as well as the focal spot size, decreases with an increase in the mismatch of layer thicknesses of the MLL. The calculated focal spot size are significantly different from the values reported in other works. A physical nature of the focusing by the MLL has been revealed. It is determined by the Bragg diffraction and is not related to the configuration of the Fresnel zone plate.

Using the two-dimensional recurrence relations of X-ray dynamical diffraction, a numerical simulation of reflection and transmission intensity in a cylindrical crystal has been performed. It is shown that for crystals with a small radius Bragg diffraction is realized. For crystals of large radius, Bragg–Laue diffraction occurs, which is characterized by Bragg diffraction on the upper part of the crystal, as well as the presence of Pendellösung oscillations inside the cylindrical crystal. The reciprocal space maps of dynamical and kinematical diffraction have been calculated.

The dynamical theory of X-ray diffraction in a crystal modulated by a surface acoustic wave (SAW) is developed for spatially restricted beams. It is shown that this approach is applicable to X-ray reciprocal space mapping. Rayleigh's surface-wave model is used to describe ultrasonic excitation. Based on the recurrent relations, a numerical simulation of the dynamical diffraction in a crystal modulated by a SAW is performed. Within the framework of the triple-axis diffraction scheme, the effect of the instrumental function on X-ray diffraction data is studied.

For solving inverse problem of X-ray diffraction on GaAs-AlAs superlattices with InAs quantum dots within the statistical theory of X-ray diffraction the Genetic Algorithm in the form of Differential Evolution method was used. The calculation results were compared with experimental data.

Dynamical X ray scattering in crystals modulated by an acoustic wave has been consistently considered. Diffraction in the cases of a spatially uniform amplitude of ultrasound in the crystal bulk and a surface Rayleigh wave has been investigated. It has been shown that the diffraction order profiles for a homogeneous acoustic wave correspond to the traditional Bragg peaks. The splitting of different diffraction maxima for large amplitudes of the surface acoustic wave has been observed. This splitting is caused by the X ray interference that is due to variations of elastic strains from the surface to the bulk of the crystal.

Abstract The dynamical theory of X-ray diffraction is developed to the case when the incident and reflected X-ray beams are spatially bounded. X-ray diffraction in complicated lateral periodic structures is considered. Effects, due to periodic regular elastic deformations of a crystal lattice on an angular distribution of the scattered intensity, are investigated. 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 a surface acoustic 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. 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.

A general theory is developed for dynamical X-ray diffraction in a crystal on the surface of which a lateral periodic structure of thin-film lines (strips) of another material is formed. On the basis of the model of edge forces, the fields of elastic lattice displacements in the substrate are calculated that arise as a result of formation of a lateral surface grating (SG). With the use of the formalism of diffraction of spatially restricted X-ray beams, solutions are obtained for the amplitudes of X-ray waves reflected from a crystal with an SG whose chemical composition differs from the composition of the substrate. A numerical simulation is carried out of X-ray diffraction in a silicon substrate with SGs of tungsten and SiO 2 oxide. It is shown that the angular distributions of the scattering intensity by a silicon crystal with tungsten and oxide lines of identical size differ significantly, and the physical nature of such difference is established.