Dynamical X-ray Diffraction Research Articles

Dynamical and kinematical X-ray diffraction in a bent crystal

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.

Double-slit asymmetrical dynamical diffraction of X-rays in ideal crystals.

The theoretical investigation of double-slit asymmetrical dynamical diffraction of X-rays in perfect crystals establishes that Young's interference fringes on the exit surface are formed. The position of the fringes in the cross section of the beam depends on deviation from the Bragg exact orientation and asymmetry angle. An equation for the period of the fringes is presented, according to which the period is polarization sensitive. The period increases with increasing the absolute value of the asymmetry angle. In its turn, the size of the interference region also increases with increasing the absolute value of the asymmetry angle. However, the ratio of interference region size to period, i.e. the number of observed fringes, decreases with increasing the absolute value of the asymmetry angle. The size of the interference region can be of the order of a few tens of mm, which can be used for obtaining Fourier dynamical diffraction holograms of a large size. This type of diffraction can also be used for obtaining double-slit dynamical diffraction contrast of defects and deformations. Due to the phase difference information, in comparison with single-slit diffraction, double-slit diffraction is more sensitive to the existence of objects and deformations in the path of the wave.

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.

Динамическая теория дифракции рентгеновских лучей на основе двумерных рекуррентных соотношений

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.

Dynamical X-ray diffraction of high-quality protein crystals

Dynamical X-ray diffraction of high-quality protein crystals

Introduction to Python Dynamic Diffraction Toolkit (PyDDT): structural refinement of single crystals via X-ray phase measurements.

PyDDT is a free Python package of computer codes for exploiting X-ray dynamic multiple diffraction in single crystals. A wide range of tools are available for evaluating the usefulness of the method, planning feasible experiments, extracting phase information from experimental data and further improving model structures of known materials. Graphical tools are also useful in analytical methodologies related to the three-dimensional aspect of multiple diffraction. For general X-ray users, the PyDDT tutorials provide the insight needed to understand the principles of phase measurements and other related methodologies. Key points behind structure refinement using the current approach are presented, and the main features of PyDDT are illustrated for amino acid and filled skutterudite single crystals.

Double-slit X-ray dynamical diffraction in curved crystals.

The theoretical investigation of double-slit X-ray dynamical diffraction in curved crystals shows that Young's interference fringes are formed. An expression for the period of the fringes has been established which is polarization sensitive. The position of the fringes in the cross section of the beam depends on the deviation from the Bragg exact orientation for a perfect crystal, on the curvature radius and on the thickness of the crystal. This type of diffraction can be used for determination of the curvature radius by measuring the shift of the fringes from the centre of the beam.

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 x-ray diffraction analysis of strain and interdiffusion in β-Ga2O3 superlattices of μ-Fe2O3 and β-(AlxGa1−x)2O3

Superlattices composed of either monoclinic μ-Fe2O3 or β-(AlxGa1−x)2O3 with β-Ga2O3 spacers are grown on (010) β-Ga2O3 substrates using plasma-assisted molecular beam epitaxy. High-resolution x-ray diffraction data are quantitatively fit using commercial dynamical x-ray diffraction software (LEPTOS) to obtain layer thicknesses, strain, and compositions. The strain state of β-(AlxGa1−x)2O3 and μ-Fe2O3 superlattices as characterized using reciprocal space maps in the symmetric (020) and asymmetric (420) diffraction conditions indicates coherent growths that are strained to the (010) β-Ga2O3 lattice. β-(AlxGa1−x)2O3 and μ-Fe2O3 superlattices grown at hotter substrate temperatures result in crystal structures with better coherency and reduced defects compared to colder growths. The growth rate of μ-Fe2O3 is ∼2.6 nm/min at Tsub = 700 °C and drops to ∼1.6 nm/min at Tsub = 800 °C due to increased Fe interdiffusion at hotter substrate temperatures. Scanning transmission electron microscopy data of a μ-Fe2O3 superlattice grown at Tsub = 700 °C confirm that there is significant diffusion of Fe atoms into β-Ga2O3 layers.

Time-dependent dynamical Bragg diffraction in deformed crystals by the beam propagation method.

This paper describes how to efficiently solve time-dependent X-ray dynamic diffraction problems in distorted crystals with a fast Fourier transform based beam propagation method. Examples are given of using the technique to simulate the propagation of X-ray beams in deformed crystals in space and time domains relevant to the cavity-based X-ray free-electron lasers and X-ray free-electron laser self-seeding systems.

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.

Experimental evidence of auxeticity in ion implanted single crystal calcite

We report initial experimental evidence of auxeticity in calcite by ion implanting (1010) oriented single crystalline calcite with Ar+ at room temperature using an ion energy of 400 keV and a dose of 1 × 1014 cm−2. Lattice compression normal to the substrate surface was observed, which is an atypical result for ion implanted materials. The auxetic behavior is consistent with predictions that indicate auxeticity had been predicted along two crystallographic directions including [1010]. Materials with a positive Poisson’s ratio experience lattice expansion normal to the substrate surface when ion implanted, whereas lattice contraction normal to the surface is evidence of auxetic behavior. Triple-axis X-ray diffraction measurements confirmed the auxetic strain state of the implanted calcite substrates. Reciprocal space maps for the symmetric 3030 and asymmetric 1450 reflections revealed that the implanted region was fully strained (pseudomorphic) to the bulk of the substrate, as is typical with implanted single crystals. A symmetric (3030) ω:2θ line scan was used with X-ray dynamical diffraction simulations to model the strain profile and extract the variation of compressive strain as a function of depth normal to the substrate surface. SRIM calculations were performed to obtain a displacement-per-atom profile and implanted Ar+ concentration profile. It was found that the strain profile matches the displacement-per-atom profile. This study demonstrated the use of ion implantation and X-ray diffraction methods to probe mechanical properties of materials and to test predictions such as the auxeticity.

Two-dimensional recurrence relations and Takagi–Taupin equations. I. Dynamical X-ray diffraction by a perfect crystal

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 qx sections are similar to each other, and the qz sections completely coincide.

Performance of a cryo-cooled crystal monochromator illuminated by hard X-rays with MHz repetition rate at the European X-ray Free-Electron Laser.

Due to the high intensity and MHz repetition rate of photon pulses generated by the European X-ray Free-Electron Laser, the heat load on silicon crystal monochromators can become large and prevent ideal transmission in Bragg diffraction geometry due to crystal deformation. Here, we present experimental data illustrating how heat load affects the performance of a cryogenically cooled monochromator under such conditions. The measurements are in good agreement with a depth-uniform model of X-ray dynamical diffraction taking beam absorption and heat deformation of the crystals into account.

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.

New developments including X-ray standing waves in the dynamical Bragg diffraction program of X-ray Server.

X-ray Server (https://x-server.gmca.aps.anl.gov) is a collection of programs for online modelling of X-ray diffraction and scattering. The dynamical diffraction program is the second most popular Server program, contributing 34% of total Server usage. It models dynamical X-ray diffraction from strained crystals and multilayers for any Bragg-case geometry including grazing incidence and exit. This paper reports on a revision of equations used by the program, which yields ten times faster calculations in most use cases, on implementing calculations of X-ray standing waves and on adding new options for modelling diffraction from monolayers.

Emergent long-range magnetic order in ultrathin (111)-oriented LaNiO3 films

The emergence of ferromagnetism in materials where the bulk phase does not show any magnetic order demonstrates that atomically precise films can stabilize distinct ground states and expands the phase space for the discovery of materials. Here, the emergence of long-range magnetic order is reported in ultrathin (111) LaNiO3 (LNO) films, where bulk LNO is paramagnetic, and the origins of this phase are explained. Transport and structural studies of LNO(111) films indicate that NiO6 octahedral distortions stabilize a magnetic insulating phase at the film/substrate interface and result in a thickness-dependent metal–insulator transition at t = 8 unit cells. Away from this interface, distortions relax and bulk-like conduction is regained. Synchrotron x-ray diffraction and dynamical x-ray diffraction simulations confirm a corresponding out-of-plane unit-cell expansion at the interface of all films. X-ray absorption spectroscopy reveals that distortion stabilizes an increased concentration of Ni2+ ions. Evidence of long-range magnetic order is found in anomalous Hall effect and magnetoresistance measurements, likely due to ferromagnetic superexchange interactions among Ni2+–Ni3+ ions. Together, these results indicate that long-range magnetic ordering and metallicity in LNO(111) films emerges from a balance among the spin, charge, lattice, and orbital degrees of freedom.

X-ray dynamical diffraction Talbot effect behind a crystal in free space.

The dynamical diffraction Talbot effect takes place inside a crystal, when a periodic object is illuminated by a plane or spherical X-ray wave which then falls on the crystal at an angle close to the Bragg angle for some reflection. Both theoretical consideration and numerical calculations show that the dynamical diffraction Talbot effect also takes place behind the crystal. The effect is accompanied by the dynamical diffraction pendulum effect and wave focusing. Expressions are found for the dynamical diffraction Talbot distance for areas before and after focusing. The spatial Fourier spectrum of the periodic object is obtained on the focusing plane. Detailed analysis when the periodic object is illuminated by a plane wave has shown new features of this effect. The dynamical diffraction Talbot effect in free space can be used to determine the structure of a periodic object, to determine the structure of an arbitrary object placed before or after the periodic object, and to determine structural defects and deformations of the crystal.

Finite-element simulation for X-ray volume diffractive optics based on the wave optical theory.

We developed a novel numerical simulation method for volume diffractive optics based on the Takagi-Taupin (TT) dynamical theory of diffraction. A general integral system of equations with a powerful and convenient distortion function was developed for finite-element analysis (FEA). The proposed framework is promising with regard to flexibility, robustness, and stability and has potential for solving dynamical X-ray diffraction problems related to diffractive optical elements of arbitrary shape and deformation. This FEA method was used for evaluating laterally graded multilayer (LGML) mirrors, and a general coordinate system was introduced to make the geometric optimization simple and effective. Moreover, the easily implemented boundary conditions inherent in FEA, combined with the analysis of the energy resolution derived from the TT theory, can make the simulation of volume diffractive optics, including LGML mirrors, more accurate. Thus, a comprehensive and highly efficient computation of LGML mirror diffraction problems was performed. The evaluation of the effects of the figure errors can provide practical guidance for the fabrication of X-ray optical elements.

Simulation of X-ray diffraction in a cylindrical crystal

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.