Dynamical theory of diffraction applicable to crystals with any kind of small distortion

An alternative method to the Takagi-Taupin equations for studying dark-field X-ray microscopy of deformed crystals

This study introduces an alternative method to the Takagi-Taupin equations for investigating the dark-field X-ray microscopy (DFXM) of deformed crystals. In scenarios where dynamical diffraction cannot be disregarded, it is essential to assess the potential inaccuracies of data interpretation based on the kinematic diffraction theory. Unlike the Takagi-Taupin equations, this new method utilizes an exact dispersion relation, and a previously developed finite difference scheme with minor modifications is used for the numerical implementation. The numerical implementation has been validated by calculating the diffraction of a diamond crystal with three components, wherein dynamical diffraction is applicable to the first component and kinematic diffraction pertains to the remaining two. The numerical convergence is tested using diffraction intensities. In addition, the DFXM image of a diamond crystal containing a stacking fault is calculated using the new method and compared with the experimental result. The new method is also applied to calculate the DFXM image of a twisted diamond crystal, which clearly shows a result different from those obtained using the Takagi-Taupin equations.

Formulation of perfect-crystal diffraction from Takagi-Taupin equations: numerical implementation in the crystalpy library

The Takagi-Taupin equations are solved in their simplest form (zero deformation) to obtain the Bragg-diffracted and transmitted complex amplitudes. The case of plane-parallel crystal plates is discussed using a matrix model. The equations are implemented in an open-source Python library crystalpy adapted for numerical applications such as crystal reflectivity calculations and ray tracing.

Surface polarization profile of ferroelectric thin films probed by X-ray standing waves and photoelectron spectroscopy

Understanding the mechanisms underlying a stable polarization at the surface of ferroelectric thin films is of particular importance both from a fundamental point of view and to achieve control of the surface polarization itself. In this study, we demonstrate that the X-ray standing wave technique allows the surface polarization profile of a ferroelectric thin film, as opposed to the average film polarity, to be probed directly. The X-ray standing wave technique provides the average Ti and Ba atomic positions, along the out-of-plane direction, near the surface of three differently strained [Formula: see text] thin films. This technique gives direct access to the local ferroelectric polarization at and below the surface. By employing X-ray photoelectron spectroscopy, a detailed overview of the oxygen-containing species adsorbed on the surface is obtained. The different amplitude and orientation of the local ferroelectric polarizations are associated with surface charges attributed to different type, amount and spatial distribution of the oxygen-containing adsorbates.

Phase-contrast visualization of human tissues using superimposed wavefront imaging of diffraction-enhanced x-rays

BACKGROUND

Phase-contrast computed tomography (CT) using high-brilliance, synchrotron-generated x-rays enable three-dimensional (3D) visualization of microanatomical structures within biological specimens, offering exceptionally high-contrast images of soft tissues. Traditional methods for phase-contrast CT; however, necessitate a gap between the subject and the x-ray camera, compromising spatial resolution due to penumbral blurring. Our newly developed technique, Superimposed Wavefront Imaging of Diffraction-enhanced x-rays (SWIDeX), leverages a Laue-case Si angle analyzer affixed to a scintillator to convert x-rays to visible light, capturing second-order differential phase contrast images and effectively eliminating the distance to the x-ray camera. This innovation achieves superior spatial resolution over conventional methods.

PURPOSE

In this paper, the imaging principle and CT reconstruction algorithm based on SWIDeX are presented in detail and compared with conventional analyzer-based imaging (ABI). It also shows the physical setup of SWIDeX that provides the resolution preserving second-order differential images for reconstruction. We compare the spatial resolution and the sensitivity of SWIDeX to conventional ABI.

METHODS

To demonstrate high-spatial resolution achievable by SWIDeX, the internal structures of four human tissues-ductal carcinoma in situ, normal stomach, normal pancreas, and intraductal papillary mucinous neoplasm of the pancreas-were visualized using an imaging system configured at the Photon Factory's BL14B beamline under the High Energy Accelerator Research Organization (KEK). Each tissue was thinly sliced after imaging, stained with hematoxylin and eosin (H&E) for conventional microscope-based pathology.

RESULTS

A comparison of SWIDeX-CT and pathological images visually demonstrates the effectiveness of SWIDeX-CT for biological tissue imaging. SWIDeX could generate clearer 3D images than existing analyzer-based phase-contrast methods and accurately delineate tissue structures, as validated against histopathological images.

CONCLUSIONS

SWIDeX can visualize important 3D structures in biological soft tissue with high spatial resolution and can be an important tool for providing information between the disparate scales of clinical and pathological imaging.

Development of an innovative diffraction scattering theory of X-rays and electrons in imperfect crystals

Fundamental equations describing the X-ray and electron diffraction scattering in imperfect crystals have been derived in the form of the matrix Fredholm-Volterra integral equation of the second kind. A theoretical approach has been developed using the perfect-crystal Green function formalism. In contrast, another approach utilizes the wavefield eigenfunctions related to the diagonalized matrix propagators of the conventional Takagi-Taupin and Howie-Whelan equations. Using the Liouville-Neumann-type series formalism for building up the matrix Fredholm-Volterra integral equation solutions, the general resolvent function solutions of the X-ray and electron diffraction boundary-valued Cauchy problems have been obtained. Based on the resolvent-type solutions, the aim is to reveal the features of the diffraction scattering onto the crystal lattice defects, including the mechanisms of intra- and interbranch wave scattering in the strongly deformed regions in the vicinity of crystal lattice defect cores. Using the two-stage resolvent solution of the second order, this approach has been supported by straightforward calculation of the electron bright- and dark-field contrasts of an edge dislocation in a thick foil. The results obtained for the bright- and dark-field profiles of the edge dislocation are discussed and compared with analogous ones numerically calculated by Howie & Whelan [Proc. R. Soc. A (1962), 267, 206].

Simulations of dislocation contrast in dark-field X-ray microscopy

Dark-field X-ray microscopy (DFXM) is a full-field imaging technique that non-destructively maps the structure and local strain inside deeply embedded crystalline elements in three dimensions. In DFXM, an objective lens is placed along the diffracted beam to generate a magnified projection image of the local diffracted volume. This work explores contrast methods and optimizes the DFXM setup specifically for the case of mapping dislocations. Forward projections of detector images are generated using two complementary simulation tools based on geometrical optics and wavefront propagation, respectively. Weak and strong beam contrast and the mapping of strain components are studied. The feasibility of observing dislocations in a wall is elucidated as a function of the distance between neighbouring dislocations and the spatial resolution. Dislocation studies should be feasible with energy band widths of 10-2, of relevance for fourth-generation synchrotron and X-ray free-electron laser sources.

Modelling structural colour from helicoidal multi-layer thin films with natural disorder

A coupled mode theory based on Takagi-Taupin equations describing electromagnetic scattering from distorted periodic arrays is applied to the problem of light scattering from beetles. We extend the method to include perturbations in the permittivity tensor to helicoidal arrays seen in many species of scarab beetle and optically anisotropic layered materials more generally. This extension permits analysis of typical dislocations arising from the biological assembly process and the presence of other structures in the elytra. We show that by extracting structural information from transmission electron microscopy data, including characteristic disorder parameters, good agreement with spectral specular and non-specular reflectance measurements is obtained.

Three-dimensional micro-X-ray topography using focused sheet-shaped X-ray beam

X-ray topography is a powerful method for analyzing crystal defects and strain in crystalline materials non-destructively. However, conventional X-ray topography uses simple X-ray diffraction images, which means depth information on defects and dislocations cannot be obtained. We have therefor developed a novel three-dimensional micro-X-ray topography technique (3D μ-XRT) that combines Bragg-case section topography with focused sheet-shaped X-rays. The depth resolution of the 3D μ-XRT depends mainly on the focused X-ray beam size and enables non-destructive observation of internal defects and dislocations with an accuracy on the order of 1 μm. The demonstrative observation of SiC power device chips showed that stacking faults, threading screw, threading edge, and basal plane dislocations were clearly visualized three-dimensionally with a depth accuracy of 1.3 μm. 3D μ-XRT is a promising new approach for highly sensitive and non-destructive analysis of material crystallinity in a three-dimensional manner.

A von Hámos spectrometer for diamond anvil cell experiments at the High Energy Density Instrument of the European X-ray Free-Electron Laser

A von Hámos spectrometer has been implemented in the vacuum interaction chamber 1 of the High Energy Density instrument at the European X-ray Free-Electron Laser facility. This setup is dedicated, but not necessarily limited, to X-ray spectroscopy measurements of samples exposed to static compression using a diamond anvil cell. Si and Ge analyser crystals with different orientations are available for this setup, covering the hard X-ray energy regime with a sub-eV energy resolution. The setup was commissioned by measuring various emission spectra of free-standing metal foils and oxide samples in the energy range between 6 and 11 keV as well as low momentum-transfer inelastic X-ray scattering from a diamond sample. Its capabilities to study samples at extreme pressures and temperatures have been demonstrated by measuring the electronic spin-state changes of (Fe0.5Mg0.5)O, contained in a diamond anvil cell and pressurized to 100 GPa, via monitoring the Fe Kβ fluorescence with a set of four Si(531) analyser crystals at close to melting temperatures. The efficiency and signal-to-noise ratio of the spectrometer enables valence-to-core emission signals to be studied and single pulse X-ray emission from samples in a diamond anvil cell to be measured, opening new perspectives for spectroscopy in extreme conditions research.

Crystal bending in triple-Laue X-ray interferometry. Part I. Theory

The measured value of the (220) lattice-plane spacing of silicon 28 using scanning X-ray interferometry is essential to realize the kilogram by counting 28Si atoms. An assumption made is that the measured lattice spacing is the bulk value of an unstrained crystal forming the analyser of the interferometer. However, analytical and numerical studies of the X-ray propagation in bent crystals suggest that the measured lattice spacing might refer to the analyser surface. To confirm the result of these studies and to support experimental investigations of the matter by phase-contrast topography, a comprehensive analytical model is given of the operation of a triple-Laue interferometer having the splitting or recombining crystal bent.

A compact von Hámos spectrometer for parallel X-ray Raman scattering and X-ray emission spectroscopy at ID20 of the European Synchrotron Radiation Facility

A compact spectrometer for medium-resolution resonant and non-resonant X-ray emission spectroscopy in von Hámos geometry is described. The main motivation for the design and construction of the spectrometer is to allow for acquisition of non-resonant X-ray emission spectra while measuring non-resonant X-ray Raman scattering spectra at beamline ID20 of the European Synchrotron Radiation Facility. Technical details are provided and the performance and possible use of the spectrometer are demonstrated by presenting results of several X-ray spectroscopic methods on various compounds.

Physical optics simulations for synchrotron radiation sources

We describe approaches to high-accuracy physical optics calculations used for the development of x-ray beamlines at synchrotron radiation sources, as well as simulation of experiments and processing of experimental data at some of these beamlines. We pay special attention to the treatment of the partial coherence of x rays, a topic of high practical importance for modern low-emittance high-brightness synchrotron radiation facilities. The approaches are based, to a large extent, on the works of Emil Wolf and co-authors, including the basic scalar diffraction theory and the coherent mode decomposition method. The presented simulation examples are related to the case of the novel Coherent Diffractive Imaging beamline that is currently under development at the National Synchrotron Light Source II at the Brookhaven National Laboratory.

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.

Simulating dark-field X-ray microscopy images with wavefront propagation techniques

The simulation of a dark-field X-ray microscopy experiment using wavefront propagation techniques and numerical integration of the Takagi–Taupin equations is shown. The approach is validated by comparing with measurements of a near-perfect diamond crystal containing a single stacking-fault defect.

Dark-field X-ray microscopy is a diffraction-based synchrotron imaging technique capable of imaging defects in the bulk of extended crystalline samples. Numerical simulations are presented of image formation in such a microscope using numerical integration of the dynamical Takagi–Taupin equations and wavefront propagation. The approach is validated by comparing simulated images with experimental data from a near-perfect single crystal of diamond containing a single stacking-fault defect in the illuminated volume.

X-ray Raman Scattering: A Hard X-ray Probe of Complex Organic Systems

This paper provides a review of the characterization of organic systems via X-ray Raman scattering (XRS) and a step-by-step guidance for its application. We present the fundamentals of XRS required to use the technique and discuss the main parameters of the experimental set-ups to optimize spectral and spatial resolution while maximizing signal-to-background ratio. We review applications that target the analysis of mixtures of organic compounds, the identification of minor spectral features, and the spatial discrimination in heterogeneous systems. We discuss the recent development of the direct tomography technique, which utilizes the XRS process as a contrast mechanism for assessing the three-dimensional spatially resolved carbon chemistry of complex organic materials. We conclude by exposing the current limitations and provide an outlook on how to overcome some of the existing challenges and advance future developments and applications of this powerful technique for complex organic systems.

High-resolution X-ray Bragg diffraction in Al thermomigrated Si channels

Si(111) wafers patterned with an array of vertical 120 µm-wide Al-doped (1 × 1019 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.

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.

X-ray free-electron laser based dark-field X-ray microscopy: a simulation-based study

Dark-field X-ray microscopy (DFXM) is a nondestructive full-field imaging technique providing three-dimensional mapping of microstructure and local strain fields in deeply embedded crystalline elements. This is achieved by placing an objective lens in the diffracted beam, giving a magnified projection image. So far, the method has been applied with a time resolution of milliseconds to hours. In this work, the feasibility of DFXM at the picosecond time scale using an X-ray free-electron laser source and a pump–probe scheme is considered. Thermomechanical strain-wave simulations are combined with geometrical optics and wavefront propagation optics to simulate DFXM images of phonon dynamics in a diamond single crystal. Using the specifications of the XCS instrument at the Linac Coherent Light Source as an example results in simulated DFXM images clearly showing the propagation of a strain wave.

Fourier ptychographic dark field x-ray microscopy

Dark-field x-ray microscopy (DFXM) is an x-ray imaging technique for mapping three-dimensional (3D) lattice strain and rotation in bulk crystalline materials. At present, these maps of local structural distortions are derived from the raw intensity images using an incoherent analysis framework. In this work, we describe a coherent, Fourier ptychographic approach that requires little change in terms of instrumentation and acquisition strategy, and may be implemented on existing DFXM instruments. We demonstrate the method experimentally and are able to achieve quantitative phase reconstructions of thin film samples and maps of the aberrations in the objective lens. The method holds particular promise for the characterization of crystalline materials containing weak structural contrast.

X-ray focusing by bent crystals: focal positions as predicted by the crystal lens equation and the dynamical diffraction theory

The location of the beam focus when monochromatic X-ray radiation is diffracted by a thin bent crystal is predicted by the `crystal lens equation'. This equation is derived in a general form valid for Bragg and Laue geometries. It has little utility for diffraction in Laue geometry. The focusing effect in the Laue symmetrical case is discussed using concepts of dynamical theory and an extension of the lens equation is proposed. The existence of polychromatic focusing is considered and the feasibility of matching the polychromatic and monochromatic focal positions is discussed.

Imaging Ultrafast Dynamical Diffraction Wave Fronts in Strained Si with Coherent X Rays

Dynamical diffraction effects in thin single crystals produce highly monochromatic parallel x-ray beams with a mutual separation of a few microns and a time delay of a few femtoseconds-the so-called echoes. This ultrafast diffraction effect is used at X-Ray Free Electron Lasers in self-seeding schemes to improve beam monochromaticity. Here, we present a coherent x-ray imaging measurement of echoes from Si crystals and demonstrate that a small surface strain can be used to tune their temporal delay. These results represent a first step toward the ambitious goal of strain tailoring new x-ray optics and, conversely, open up the possibility of using ultrafast dynamical diffraction effects to study strain in materials.

X-ray Laue diffraction by sectioned multilayers. I. Pendellösung effect and rocking curves

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.

Investigation of multiple Bragg reflections and their possible exploitation

This paper summarizes recent results of an exhaustive experimental study of multiple Bragg reflections accompanying allowed as well as forbidden reflections. The multiple reflection observations were carried out in the frame of Bragg diffraction optics experiments on cylindrically bent perfect single crystals. It has been found that depending on the thickness and curvature of the crystal slabs and the diffraction geometry (reflection, transmission), many strong multiple reflections can be excited which can also be used as a source of highly monochromatic and highly collimated beams for further experiments requiring extremely high resolution.

The Band-Gap Studies of Short-Period CdO/MgO Superlattices

Trends in the behavior of band gaps in short-period superlattices (SLs) composed of CdO and MgO layers were analyzed experimentally and theoretically for several thicknesses of CdO sublayers. The optical properties of the SLs were investigated by means of transmittance measurements at room temperature in the wavelength range 200-700 nm. The direct band gap of {CdO/MgO} SLs were tuned from 2.6 to 6 eV by varying the thickness of CdO from 1 to 12 monolayers while maintaining the same MgO layer thickness of 4 monolayers. Obtained values of direct and indirect band gaps are higher than those theoretically calculated by an ab initio method, but follow the same trend. X-ray measurements confirmed the presence of a rock salt structure in the SLs. Two oriented structures (111 and 100) grown on c- and r-oriented sapphire substrates were obtained. The measured lattice parameters increase with CdO layer thickness, and the experimental data are in agreement with the calculated results. This new kind of SL structure may be suitable for use in visible, UV and deep UV optoelectronics, especially because the energy gap can be precisely controlled over a wide range by modulating the sublayer thickness in the superlattices.

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.

X-ray microbeam diffraction in a crystal

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.

Imaging defects in vanadium(iii) oxide nanocrystals using Bragg coherent diffractive imaging

Here, Fohtung and colleagues capture nanoscale three-dimensional defects in vanadium(iii) oxide nanocrystals using X-ray Bragg coherent diffractive imaging.

General method to calculate the elastic deformation and X-ray diffraction properties of bent crystal wafers

Toroidally and spherically bent single crystals are widely employed as optical elements in hard X-ray spectrometry at synchrotron and free-electron laser light sources, and in laboratory-scale instruments. To achieve optimal spectrometer performance, a solid theoretical understanding of the diffraction properties of such crystals is essential. In this work, a general method to calculate the internal stress and strain fields of toroidally bent crystals and how to apply it to predict their diffraction properties is presented. Solutions are derived and discussed for circular and rectangular spherically bent wafers due to their prevalence in contemporary instrumentation.

X-ray dark-field phase-contrast imaging: Origins of the concept to practical implementation and applications

The basic idea of X-ray dark-field imaging (XDFI), first presented in 2000, was based on the concepts used in an X-ray interferometer. In this article, we review 20 years of developments in our theoretical understanding, scientific instrumentation, and experimental demonstration of XDFI and its applications to medical imaging. We first describe the concepts underlying XDFI that are responsible for imparting phase contrast information in projection X-ray images. We then review the algorithms that can convert these projection phase images into three-dimensional tomographic slices. Various implementations of computed tomography reconstructions algorithms for XDFI data are discussed. The next four sections describe and illustrate potential applications of XDFI in pathology, musculoskeletal imaging, oncologic imaging, and neuroimaging. The sample applications that are presented illustrate potential use scenarios for XDFI in histopathology and other clinical applications. Finally, the last section presents future perspectives and potential technical developments that can make XDFI an even more powerful tool.

Improving contrast and spatial resolution in crystal analyzer-based x-ray dark-field imaging: Theoretical considerations and experimental demonstration

PURPOSE

This paper describes and experimentally validates a methodology for improving contrast and spatial resolution of the x-ray dark-field imaging (XDFI) by cutting the monochromator-collimator asymmetrically and thinning the Laue angle analyzer.

METHODS

We measure the spatial resolution of our XDFI setup using a test object consisting of wolfram tungsten meshes and compare it with the theoretical prediction. Using x-ray dark-field computed tomography of breast cancer specimens (lobular carcinoma in situ), we demonstrate that the resolution of XDFI is sufficient for histopathologic analysis.

RESULTS

Our experimental results show that the overall spatial resolution of XDFI can be improved by approximately a factor of 2 when these modifications are implemented. The reconstructed images of breast cancer specimens provide sufficient details for radiologic histopathology.

CONCLUSIONS

By cutting the monochromator-collimator and thinning the Laue angle analyzer, XDFI can achieve the resolution sufficient for radiologic histopathology.

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.

A study on the limit of application of kinematical theory of X-ray diffraction

In this work, the limit of application of the kinematical theory of X-ray diffraction was calculate integrated intensities was evaluated as a function of perfect crystal thickness, when compared with the Ewald–Laue dynamical theory. The percentual difference between the dynamical and kinematical integrated intensities was calculated as a function of unit cell volume, Bragg angle, wavelength, module, and phase of structure factor and linear absorption coefficient. A critical thickness was defined to be the value for which the intensities differ 5%. We show that this critical thickness is 13.7% of the extinction length, which a specific combination of the parameters mentioned before. Also, we find a general expression, for any percentage of the difference between both theories, to determine the validity of the application of the kinematical theory. Finally, we also showed that the linear absorption decreases this critical thickness.

Takagi-Taupin dynamical X-ray diffraction simulations of asymmetric X-ray diffraction from crystals: the effects of surface undulations

Dynamical X-ray diffraction simulations from crystals with surface undulations are reported. The Takagi-Taupin equations are applied and used to derive results in good agreement with experimental data reported in a separate paper [Macrander, Pereira, Huang, Kasman, Qian, Wojcik & Assoufid (2020). J. Appl. Cryst. 53, 789-792]. The development of Uragami [J. Phys. Soc. Jpn, (1969), 27, 147-154] is followed. Although previous work by Olekhnovich & Olekhnovich [Acta. Cryst. (1980), A36, 22-27] treated a crystal in the shape of a round cylinder, there do not seem to be any reports of previous dynamical X-ray diffraction treatments specifically for surface undulations. The significance of the present work is that it bridges the diffraction treatment of more classical dynamical diffraction theory, which assumes a flat surface, and the simple kinematic diffraction theory. The kinematic theory has, to date, been the primary means of simulating X-ray diffraction from surfaces.

New kind of interference in the case of X-ray Laue diffraction in a single crystal with uneven exit surface under the conditions of the Borrmann effect. Analytical solution

The analytical solution of the problem of X-ray spherical-wave Laue diffraction in a single crystal with a linear change of thickness on the exit surface is derived. General equations are applied to a specific case of plane-wave Laue diffraction in a thick crystal under the conditions of the Borrmann effect. It is shown that if a thickness increase takes place at the side of the reflected beam, the related reflected wave amplitude is calculated as a sum of three terms, two of which are complex. If all three terms have a comparable modulus, it can lead to an increase in the reflected beam intensity by up to nine times due to interference compared with the value for a plane parallel shape of the crystal. The equation for the related transmitted wave amplitude contains only two terms. Therefore, the possibility to increase intensity is smaller compared with the reflected beam. The analytical solution is obtained after a solution of the integral equations by means of the Laplace transformation. A general integral form of the Takagi equations derived earlier is used. The results of relative intensity calculations by means of analytical equations coincide with the results of direct computer simulations.

Towards a solution of the inverse X-ray diffraction tomography challenge: theory and iterative algorithm for recovering the 3D displacement field function of Coulomb-type point defects in a crystal

The theoretical framework and a joint quasi-Newton-Levenberg-Marquardt-simulated annealing (qNLMSA) algorithm are established to treat an inverse X-ray diffraction tomography (XRDT) problem for recovering the 3D displacement field function f_Ctpd(r - r₀) = h · u(r - r₀) due to a Coulomb-type point defect (Ctpd) located at a point r₀ within a crystal [h is the diffraction vector and u(r - r₀) is the displacement vector]. The joint qNLMSA algorithm operates in a special sequence to optimize the XRDT target function {\cal F}\{ {\cal P} \} in a χ² sense in order to recover the function f_Ctpd(r - r₀) [{\cal P} is the parameter vector that characterizes the 3D function f_Ctpd(r - r₀) in the algorithm search]. A theoretical framework based on the analytical solution of the Takagi-Taupin equations in the semi-kinematical approach is elaborated. In the case of true 2D imaging patterns (2D-IPs) with low counting statistics (noise-free), the joint qNLMSA algorithm enforces the target function {\cal F} \{ {\cal P} \} to tend towards the global minimum even if the vector {\cal P} in the search is initially chosen rather a long way from the true one.

X-Ray Diffraction Tomography Recovery of the 3D Displacement-Field Function of the Coulomb-Type Point Defect in a Crystal

A successive approach to the solution of the inverse problem of the X-ray diffraction tomography (XRDT) is proposed. It is based on the semi-kinematical solution of the dynamical Takagi–Taupin equations for the σ-polarized diffracted wave amplitude. Theoretically, the case of the Coulomb-type point defect in a single crystal Si(111) under the exact conditions of the symmetric Laue diffraction for a set of the tilted X-ray topography 2D-images (2D projections) is considered provided that the plane-parallel sample is rotated around the diffraction vector [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{{\bf{2}}}$$\end{document}2¯20]. The iterative simulated annealing (SA) and quasi-Newton gradient descent (qNGD) algorithm codes are used for a recovery of the 3D displacement-field function of the Coulomb-type point defect. The computer recovery data of the 3D displacement-field function related to the one XRDT 2D projection are presented. It is proved that the semi-kinematical approach to the solution of the dynamical Takagi–Taupin equations is effective for recovering the 3D displacement-field function even for the one XRDT 2D projection.

X-ray dynamical diffraction analogues of the integer and fractional Talbot effects

The X-ray integer and fractional Talbot effect is studied under two-wave dynamical diffraction conditions in a perfect crystal, for the symmetrical Laue case of diffraction. The fractional dynamical diffraction Talbot effect is studied for the first time. A theory of the dynamical diffraction integer and fractional Talbot effect is given, introducing the dynamical diffraction comb function. An expression for the dynamical diffraction polarization-sensitive Talbot distance is established. At the rational multiple depths of the Talbot depth the wavefield amplitude for each dispersion branch is a coherent sum of the initial distributions, shifted by rational multiples of the object period and having its own phases. The simulated dynamical diffraction Talbot carpet for the Ronchi grating is presented.

Dependence of X-ray asymmetrical Bragg case plane-wave rocking curves on the deviation from exact Bragg orientation in and perpendicular to the diffraction plane

For the asymmetrical Bragg case the X-ray plane-wave reflection coefficients and rocking-curve dependences on the deviation angles from the exact Bragg orientation in the diffraction plane and in the direction perpendicular to the diffraction plane are analysed. The region of total reflection and its size dependence on two deviation angles are analysed as well. New peculiarities of the rocking curves are obtained.

Johann-type laboratory-scale x-ray absorption spectrometer with versatile detection modes

We present a low-cost laboratory X-ray absorption spectrometer that uses a conventional X-ray tube source and bent Johann-type crystal monochromators. The instrument is designed for X-ray absorption spectroscopy studies in the 4-20 keV range which covers most K edges of 3d transition metals and L edges of 5d transition metals and actinides. The energy resolution is typically in the range of 1-5 eV at 10 keV depending on the crystal analyser and the Bragg angle. Measurements can be performed in transmission, fluorescence, and imaging modes. Due to its simple and modular design, the spectrometer can be modified to accommodate additional equipment and complex sample environments required for in situ studies. A showcase of various applications is presented.

X-ray topo-tomography studies of linear dislocations in silicon single crystals

This article describes complete characterization of the polygonal dislocation half-loops (PDHLs) introduced by scratching and subsequent bending of an Si(111) crystal. The study is based on the X-ray topo-tomography technique using both a conventional laboratory setup and the high-resolution X-ray image-detecting systems at the synchrotron facilities at KIT (Germany) and ESRF (France). Numerical analysis of PDHL images is performed using the Takagi–Taupin equations and the simultaneous algebraic reconstruction technique (SART) tomographic algorithm.

Computer simulations of X-ray spherical wave dynamical diffraction in one and two crystals in the Laue case

This article reports computer simulations of X-ray spherical wave dynamical diffraction in one and two single crystals in the Laue case. An X-ray compound refractive lens (CRL) as a secondary radiation source of spherical waves was considered for the first time and in contrast to previous simulations with the assumption of the use of a slit. The main properties of the CRL as a secondary source are discussed and two focusing phenomena are analysed. The first one is the diffraction focusing effect for one single crystal in the reflected beam and in the case of a large source-to-detector distance. The second one is the same but for two single crystals and for the twice-reflected beam in the case of a short distance between the source and detector. The first effect is well pronounced in the case of strong absorption. However, it may also be used as an element of an energy spectrometer in the medium and even weak absorption case. The second effect will appear in the case of weak absorption. It is shown that it is not effective to use it in an energy spectrometer. In the case of weak absorption the transverse size of the diffraction focused beam will oscillate together with the reflected beam integral intensity. The oscillation period is close to the extinction length.

The Effect of Crystal Defects on 3D High-Resolution Diffraction Peaks: A FFT-Based Method

Forward modeling of diffraction peaks is a potential way to compare the results of theoretical mechanical simulations and experimental X-ray diffraction (XRD) data recorded during in situ experiments. As the input data are the strain or displacement field within a representative volume of the material containing dislocations, a computer-aided efficient and accurate method to generate these fields is necessary. With this aim, a current and promising numerical method is based on the use of the fast Fourier transform (FFT)-based method. However, classic FFT-based methods present some numerical artifacts due to the Gibbs phenomenon or “aliasing” and to “voxelization” effects. Here, we propose several improvements: first, a consistent discrete Green operator to remove “aliasing” effects; and second, a method to minimize the voxelization artifacts generated by dislocation loops inclined with respect to the computational grid. Then, we show the effect of these improvements on theoretical diffraction peaks.

Estimating the structure factors in X-ray diffraction

This article takes the concepts of the `new diffraction theory' [Fewster (2014). Acta Cryst. A70, 257-282] and examines the implications for the interpretation of experimental results and the estimation of structure factors. Further experimental evidence is included to justify the conclusions in the theory, showing that the residual intensity at twice the Bragg angle is a diffraction effect and not associated with the crystal shape. This `enhancement' effect is independent of whether kinematical or dynamical theories are applied and can lead to a clearer understanding of how the dynamical effects are suppressed in imperfect crystals. By applying the idea that the higher-order peaks are due to path lengths of nλ, it is shown that `systematically absent' reflections in the conventional theory may not be absent. Because this new theory considers the intensity to be more distributed, it suggests that the entire structure factor can be difficult to capture by experiment. This article suggests some routes to achieve a good approximation of the structure factors for typical methods of data collection. Any measurement of intensity with background removal will exclude some of the distributed intensity, again leading to an underestimate of the structure factors, and therefore the missing intensity needs to be estimated.

The nanodiffraction problem

The results of a systematic rigorous study on the accuracy of lattice parameters computed from X-ray diffraction patterns of ideally perfect nanocrystalline powder and thin-film samples are presented. It is shown that, if the dimensions of such samples are below 20 nm, the lattice parameters obtained from diffraction analysis will deviate from their true values. The relative deviation depends on the relevant size parameter through an inverse power law and, for particular reflections, depends on the angular peak positions. This size-dependent error, Δa/a, is larger than the precision of typical X-ray diffraction measurements for ∼20 nm-thick diffracting domains, and it can be several orders of magnitude larger for particles smaller than 5 nm.

Dependence of X-ray plane-wave rocking curves on the deviation from exact Bragg orientation in and perpendicular to the diffraction plane for the asymmetrical Laue case

For the asymmetrical Laue case the X-ray plane-wave transmission and reflection coefficients and rocking curves are analysed as a function of the deviation angles from the exact Bragg orientation in the diffraction plane and in the direction perpendicular to the diffraction plane. New peculiarities of the rocking curves are obtained. The peculiarities of both the effective absorption coefficient and rocking curves in thick crystals are also investigated.