Dynamical Theory Of X-ray Diffraction Research Articles

Investigation of dynamical X-ray back diffraction at grazing incidence

We report a theoretical investigation of X-ray back diffraction at grazing incidence. Based on the framework of the dynamical theory of X-ray diffraction, the grazing incidence for Si (12 4 0) back diffraction is taken as an example to resolve the eigenvalue problem inside the crystal. The dispersion surface and the resulting diffraction intensities are strongly affected by the miscut angle as well as the diffraction geometry of grazing incidence. The kinematical relationship between the incident angle and the miscut angle is well explained by Snell’s law. While only the two-beam diffraction is considered, our treatment can be further extended to include the cases for multiple diffractions as well.

X-ray dynamical diffraction by quasi-monolayer graphene

We study the processes of dynamical diffraction of the plane X-ray waves on the graphene film/SiC substrate system in the case of the Bragg diffraction geometry. The statistical dynamical theory of X-ray diffraction in imperfect crystals is applied to the case of real quasi-two-dimensional systems. The necessity of the taking into account of the variability of the lattice parameter of multilayer graphene, as well as the influence of thickness on the thermal Debye–Waller factor at the calculation of the complex structural factors and Fourier components of polarizability, is demonstrated. It is shown that the change of the structural characteristics of the 3-layer graphene/substrate system, as well as its strained state, leads to a significant change in the diffraction profiles, which makes it possible to determine the characteristics by the X-ray diffraction method.

Intensity correlation properties of x-ray beams split with Laue diffraction

Beam splitting is one of the main approaches to achieving x-ray ghost imaging, and the intensity correlation between diffraction beam and transmission beam will directly affect the imaging quality. In this paper, we investigate the intensity correlation between the split x-ray beams by Laue diffraction of stress-free crystal. The analysis based on the dynamical theory of x-ray diffraction indicates that the spatial resolution of diffraction image and transmission image are reduced due to the position shift of the exit beam. In the experimental setup, a stress-free crystal with a thickness of hundred-micrometers-level is used for beam splitting. The crystal is in a non-dispersive configuration equipped with a double-crystal monochromator to ensure that the dimension of the diffraction beam and transmission beam are consistent. A correlation coefficient of 0.92 is achieved experimentally and the high signal-to-noise ratio of the x-ray ghost imaging is anticipated. Results of this paper demonstrate that the developed beam splitter of Laue crystal has the potential in the efficient data acquisition of x-ray ghost imaging.

Dynamical theory of X-ray diffraction by crystals with different surface relief profiles.

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.

Crystal optics simulations for delineation of the three-dimensional cellular nuclear distribution using analyzer-based refraction-contrast computed tomography

Refraction-contrast computed tomography (RCT) using a refractive angle analyzer of Si perfect crystal can reconstruct the three-dimensional structure of biological soft tissue with contrast comparable to that of stained two-dimensional pathological images. However, the blurring of X-ray beam by the analyzer has prevented improvement of the spatial resolution of RCT, and the currently possible observation of tissue structure at a scale of approximately 20 µm provides only limited medical information. As in pathology, to differentiate between benign and malignant forms of cancer, it is necessary to observe the distribution of the cell nucleus, which is approximately 5–10 µm in diameter. In this study, based on the X-ray dynamical diffraction theory using the Takagi–Taupin equation, which calculates the propagation of X-ray energy in crystals, an analyzer crystal optical system depicting the distribution of cell nuclei was investigated by RCT imaging simulation experiments in terms of the thickness of the Laue-case analyzer, the camera pixel size and the difference in spatial resolution between the Bragg-case and Laue-case analyzers.

Quantitative Analysis of Dislocations in 4H-SiC Wafers Using Synchrotron X-Ray Topography with Ultra-High Angular Resolution

The wide band gap semiconductor, 4H-SiC, is a promising material for high-power and high-temperature devices due to its outstanding physical properties. Unlike Silicon, defects in 4H-SiC still cannot be completely eliminated even using the latest crystal growth techniques. These defects can be detrimental to the devices and it is important to investigate their properties. The non-destructive characterization technique, X-ray topography (XRT), has been developed over several decades and widely applied to study the crystallographic features of various materials. The utilization of a Si (331) beam conditioner together with a Si (111) double-crystal monochromator (DCM) enables the angular resolution of XRT to be increased by an order of magnitude compared to grazing incidence topography (GIT) or back reflection topography conducted with the DCM alone. This improved technique with extremely small beam divergence is referred to as synchrotron X-ray plane-wave topography (SXPWT). In this study, we will show that the rocking curve width of 4H-SiC 0008 in PWT is only 2.5ʹʹ and thus the lattice distortion at the scale of 1ʹʹ will significantly affect the diffracted intensity. Herein, we report the ultra-high angular resolution in SXPWT enables the detailed probe of the lattice distortion outside the dislocation core in 4H-SiC where the sign of the Burgers vector can be readily determined (Fig. 1). Additionally, dynamical theory of X-ray diffraction was applied to calculate the rocking curve (reflectivity) in SXPWT, and the ray tracing simulation was modified accordingly to calculate the dislocation images under any diffraction conditions. Figure 1

Mechanical stretching of a perfect single crystal: a routeway to a d-spacing tunable X-ray monochromator

The feasibility of using traction forces to produce mechanical stretching/shrinkage in a perfect Si 400 single crystal is explored theoretically (by finite element analysis and dynamical theory of X-ray diffraction) and experimentally in order to use it as an angular static interplanar-distance (d-spacing) tunable monochromator, in the Bragg case. It is shown that, with traction forces of ∼100 N and an appropriate pre-load, it is possible to scan, with a linear response and reproducibility of ∼5%, a 30 eV bandwidth, which corresponds to a d-spacing scan of 7 × 10−3 Å.

Large-area total-thickness imaging and Burgers vector analysis of dislocations in β -Ga2O3 using bright-field x-ray topography based on anomalous transmission

Using bright-field x-ray topography based on anomalous transmission (AT), we have demonstrated the first large-area total-thickness imaging of dislocations in β-Ga2O3 at the substrate scale. The dislocation images were acquired from the entire 10 mm × 15 mm × 680 μm (001)-oriented substrate prepared by edge-defined film-fed growth (EFG) by stitching together hundreds of topographic images, each recorded with the forward-diffracted beam in the Laue geometry for g = 020, 0–20, 022, and 400, under the conditions in which AT occurred. Dislocations distributed over the entire crystal volume were imaged as long as their Burgers vectors (b) were not orthogonal to the g-vectors. The results of the g·b analysis of the dislocation contrasts clearly revealed three major dislocation types that were numerically dominant in the EFG crystal: (i) b-axis screw-type dislocations with b∥ξ∥[010] (ξ is the unit vector of line direction), (ii) b-axis edge-type dislocations with b∥[001] and ξ∥[010], and (iii) curved mixed-type dislocations lying on the (001) planes with b∥[010]. Based on their b- and ξ-vectors, types (i) and (ii) were attributed to dislocations that propagated during EFG pulling up along the [010] direction, while type (iii) was attributed to dislocations generated through glide in the [010](001) slip system under stress. The extent to which AT can manifest itself is explained by using the effective absorption coefficient calculated for the above g-vectors based on dynamical x-ray diffraction theory.

Cекционные методы рентгеновской дифракционной топографии (обзор)

X-ray topography is a group of methods for obtaining diffraction images of structural defects in crystals. Among them, section topography techniques are distinguished by their abilities in acquiring quantitative information about defects based on the analysis of the images. For this purpose, special applications of the dynamic theory of X-ray diffraction are being developed. The interference of wave fields excited in a crystal by an X-ray beam is the basis of the section methods. Their sensitivity to weak lattice distortions is much higher than that of other X-ray methods. This review describes the physical foundations and implementation of the section topography techniques, as well as the results of computer simulating the wave field in a crystal. We present some examples of solving materials science and microelectronics problems and briefly describe the section topography using synchrotron radiation.

Section methods of X-Ray diffraction topography

X-ray topography is a group of methods for obtaining diffraction images of structural defects in crystals. Among them, section topography techniques are distinguished by their abilities in acquiring quantitative information about defects based on the analysis of the images. For this purpose, special applications of the dynamic theory of X-ray diffraction are being developed. The interference of wave fields excited in a crystal by an X-ray beam is the basis of the section methods. Their sensitivity to weak lattice distortions is much higher than that of other X-ray methods. This review describes the physical foundations and implementation of the section topography techniques, as well as the results of computer simulating the wave field in a crystal. We present some examples of solving materials science and microelectronics problems and briefly describe the section topography using synchrotron radiation. Keywords: X-ray topography, single crystals, defects in crystals, silicon, diamond, indium antimonide.

Beam splitting characteristics of crystal X-ray Laue diffraction

The beam splitter is an optical element that divides a beam of light into two or more subbeams. It is an essential component in many optical experiments. X-ray has the characteristics of short wavelength and strong penetration ability, making it hard to use the optical elements in the visible-light region. Therefore, it is necessary to develop optical elements suitable for X-rays. The atomic layer spacing of the perfect crystal is of the same order of magnitude as the X-ray wavelength, so the crystal diffraction effect can be used to achieve the X-ray modulation. In this paper, the beam splitting characteristics of Laue crystal are analyzed based on X-ray diffraction dynamics and the influences of crystal absorption and incident light angular divergence on the rocking curves of transmission and diffraction are simulated. The modulation of the crystal diffraction in-plane angle and crystal thickness to Laue diffraction beam-splitting ratio is presented quantitatively. The results show that the kinematical theory of X-ray diffraction is not enough to analyze the beam splitting characteristics of the crystal. It is necessary to consider the interaction between the wave fields in the crystal and use the Pendellӧsung effect in the dynamical theory of X-ray diffraction to explain the change of the crystal beam-splitting ratio quantitatively. The influence of angular divergence and crystal absorption are considered in the simulation. The angular divergence broadens the bandwidth of the diffraction, thereby reducing diffraction intensity. The crystal absorption results in asymmetry and peak shift of the transmission curve and affects the intensity of diffraction and the intensity of transmission beam. The experimental results show that the non-dispersive (+n, –n) configuration can effectively eliminate the influence of angle divergence. The beam-splitting ratio can be adjusted in a small range (±2%) by changing the in-plane angle and adjusted in a wide range (±75%) by changing the crystal thickness, thereby realizing the quantitative modulation of the intensity of transmission and diffraction beam.

Microstructural modifications induced in Si+-implanted yttria-stabilised zirconia: a combined RBS-C, XRD and Raman investigation.

The structural differences in (100)-, (110)- and (111)-oriented cubic yttria-stabilised zirconia (YSZ) single crystals after implantation with 2 MeV Si+ ions at the fluences of 5 × 1015, 1 × 1016 and 5 × 1016 cm-2 were studied using Rutherford backscattering spectrometry in the channelling mode (RBS-C), X-ray diffraction (XRD) and Raman spectroscopy. The RBS-C results show that the damage accumulation in the 〈110〉 direction exhibits a lower level of disorder (<0.3) than the other orientations (<0.6) and it seems that the (110) crystallographic orientation is the most resistant to radiation damage. The experimental results from the RBS measurement were compared with the results from the XRD measurements. The XRD data were analysed using the standard two-beam dynamical X-ray diffraction theory and the pure isotropic strain was deduced from the fit for the fluence of 5 × 1015 cm-2. It was shown that the maximum value of the isotropic strain does not depend on the surface orientation. The increase in signal intensity at ∼689 cm-1 is probably related to an increase in implantation defects such as oxygen vacancies.

Laue X-ray diffraction studies of the structural perfection of Al-doped thermomigration channels in silicon

Si(111) wafers patterned with an array of vertical 100 µm-wide Al-doped (1 × 1019 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.

Multiple reflection optimization package for X-ray diffraction

New software for the simulation and fitting of 2θ–ω scans of symmetric and asymmetric reflections based on the dynamical theory of X-ray diffraction is presented.

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

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.

Evaluation of crystal quality of thin protein crystals based on the dynamical theory of X-ray diffraction.

Knowledge of X-ray diffraction in macromolecular crystals is important for not only structural analysis of proteins but also diffraction physics. Dynamical diffraction provides evidence of perfect crystals. Until now, clear dynamical diffraction in protein crystals has only been observed in glucose isomerase crystals. We wondered whether there were other protein crystals with high quality that exhibit dynamical diffraction. Here we report the observation of dynamical diffraction in thin ferritin crystals by rocking-curve measurement and imaging techniques such as X-ray topography. It is generally known that in the case of thin crystals it is difficult to distinguish whether dynamical diffraction occurs from only rocking-curve profiles. Therefore, our results clarified that dynamical diffraction occurs in thin protein crystals because fringe contrasts similar to Pendellösung fringes were clearly observed in the X-ray topographic images. For macromolecular crystallography, it is hard to obtain large crystals because they are difficult to crystallize. For thin crystals, dynamical diffraction can be demonstrated by analysis of the equal-thickness fringes observed by X-ray topography.

Dynamical effects in the integrated X-ray scattering intensity from imperfect crystals in Bragg diffraction geometry. I. Semi-dynamical model.

The analytical expressions for the coherent and diffuse components of the integrated reflection coefficient are considered in the case of asymmetric Bragg diffraction geometry for a single crystal of arbitrary thickness, which contains randomly distributed Coulomb-type defects. The possibility to choose the combinations of diffraction conditions optimal for characterizing defects of several types by accounting for dynamical effects in the integrated coherent and diffuse scattering intensities, i.e. primary extinction and anomalous absorption, has been analysed based on the statistical dynamical theory of X-ray diffraction by imperfect crystals. The measured integrated reflectivity dependencies of the imperfect silicon crystal on azimuthal angle were fitted to determine the diffraction parameters characterizing defects in the sample using the proposed formulas in semi-dynamical and semi-kinematical approaches.

The dynamical theory of diffraction in a crystal modulated by a surface acoustic wave in the case of spatially restricted X-ray beams

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.

Dynamical Theory of X-Ray Diffraction in a Crystal with a Surface Grating of Another Material

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 SiO2 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.

Polarization control with an X-ray phase retarder for high-time-resolution pump-probe experiments at SACLA.

Control of the polarization of an X-ray free-electron laser (XFEL) has been performed using an X-ray phase retarder (XPR) in combination with an arrival timing diagnostic on BL3 of the SPring-8 Angstrom Compact free-electron LAser (SACLA). To combine with the timing diagnostic, a pink beam was incident on the XPR crystal and then monochromated in the vicinity of samples. A high degree of circular polarization of ∼97% was obtained experimentally at 11.567 keV, which agreed with calculations based on the dynamical theory of X-ray diffraction. This system enables pump-probe experiments to be operated using circular polarization with a time resolution of 40 fs to investigate ultrafast magnetic phenomena.