Dynamical Diffraction Theory Research Articles

Reconstruction of Z-pinch emission spectra in the wavelength range of less than 10Å using a crystal X-ray spectrograph.

This work is devoted to the reconstruction of Z-pinch plasma emission spectra in the wavelength range of less than 10Å recorded by using a crystal x-ray spectrograph at the Angara 5-1 mega-ampere facility. The spectrograph JA-1 used in experiments has a cylindrical mica crystal with dimensions of 50 × 40mm2 and radius of curvature of 100mm. Registration of spectra is performed on the photographic film UF-4 with dimensions of 30 × 10mm2. To reconstruct the spectra, the previously developed method based on iterative approximation of a true spectrum shape while minimizing a residual between experimental and calculated spectrograms is used. The calculated spectrogram was obtained taking into account the instrumental function of the spectrograph. To define the instrumental function a virtual Monte-Carlo model in the Geant4 toolkit has been developed. This model takes into account the interaction of radiation with the mica crystal using dynamical theory of diffraction. A true spectrum of Z-pinch plasma radiation is reconstructed for a 16mm high load made of two nested cylindrical wire liners. External liner with a diameter of 12mm has 40 Al wires with a diameter of 18 μm. The internal liner with a diameter of 5mm has 4W wires with a diameter of 6 μm. The W wires have a sputtered layer of Rethat is 0.5 μm thick.

Coherent X-ray radiation excited by a relativistic electron in a periodic layered medium

The work is devoted to the development of the theory of coherent X-ray radiation near the direction of Bragg scattering of a beam of relativistic electrons crossing a target with a periodic layered structure with three different amorphous layers per period. The coherent X-ray radiation is considered within the two-wave approximation of the dynamic theory of diffraction as a sum of parametric X-ray radiation and diffracted transition radiation excited in the target by a relativistic electron. Expressions describing the spectral-angular and angular densities of PXR and DTR have been obtained and studied.

Violation of a Leggett-Garg Inequality Using Ideal Negative Measurements in Neutron Interferometry.

Leggett-Garg inequalities (LGIs) have been proposed in order to assess how far the predictions of quantum mechanics defy "macroscopic realism." With LGIs, correlations of measurements performed on a single system at different times are described. We report on an experiment that demonstrates the violation of an LGI with neutrons. The final measured value of the Leggett-Garg correlator K=1.120±0.007(stat)±0.019(sys), obtained in a neutron interferometric experiment, is clearly above the limit K=1 predicted by macrorealistic theories. The experimental results are analyzed within the framework of dynamical theory of neutron diffraction, evidently reproducing the obtained values.

Diffraction contrast of ferroelectric domains in DPC STEM images.

Differential phase contrast scanning transmission electron microscopy (DPC STEM) is a powerful technique for directly visualizing electromagnetic fields inside materials at high spatial resolution. Electric field observation within ferroelectric materials is potentially possible by DPC STEM, but concomitant diffraction contrast hinders the quantitative electric field evaluation. Diffraction contrast is basically caused by the diffraction-condition variation inside a field of view, but in the case of ferroelectric materials, the diffraction conditions can also change with respect to the polarization orientations. To quantitatively observe electric field distribution inside ferroelectric domains, the formation mechanism of diffraction contrast should be clarified in detail. In this study, we systematically simulated diffraction contrast of ferroelectric domains in DPC STEM images based on the dynamical diffraction theory, and clarify the issues for quantitatively observing electric fields inside ferroelectric domains. Furthermore, we conducted experimental DPC STEM observations for a ferroelectric material to confirm the influence of diffraction contrast predicted by the simulations.

Preparation and measurement of an x-ray Laue-type monochromator based on a WSi2/Si multilayer

The Laue-type multilayer monochromator (LMM) is a promising optical element with a small size and high efficiency in a synchrotron radiation facility. By the dynamical diffraction theory, using DC magnetron sputtering technology, an LMM with a total thickness of 47 µm and a periodic thickness of 4.7 nm WSi2/Si multilayer at 26 keV is designed and fabricated. During the preparation, the total number of layers is up to 20000, and every 300th layer of Si is replaced by WSi2 as the marker, so the multilayer is divided into 67 areas. The cross section of the multilayer is measured by a scanning electron microscope (SEM), and the marker region thickness error is 0.28% (RMS). The diffraction test experiment of the LMM is carried out at the Shanghai synchrotron radiation facility (SSRF). The 1st-order peak angle is 5.05 mrad, and the efficiency is 75.0%, which is close to the theoretical calculation result of 5.1 mrad and 79.1%. The Darwin width of the LMM is 0.17 mrad which is equal to the theoretical calculation. Based on the Bragg’s diffraction equation, the energy resolution (ΔE/E) is 3.3%.

Mechanochemical Synthesis and Three-Dimensional Electron Diffraction Structure Solution of a Novel Cu-Based Protocatechuate Metal-Organic Framework.

Mechanochemical synthesis is a powerful approach to obtain new materials, limiting costs, and times. However, defected and submicrometrical-sized crystal products make critical their characterization through classical single-crystal X-ray diffraction. A valid alternative is represented by three-dimensional (3D) electron diffraction, in which a transmission electron microscope is used, like a diffractometer. This work matches a green water-based mechanochemical synthesis and 3D electron diffraction to obtain and characterize a Cu-based protocatechuate metal-organic framework (PC-MOF). Its structure has been fully refined through dynamical diffraction theory, and free water molecules could be detected in the channels of the framework. Thermal characterization, focused on the dehydration profile determination, leads to the formation of a novel high-temperature 2D coordination polymer, fully solved with 3D electron diffraction data. At last, the strong activity of the PC-MOF against cationic dyes like methylene blue has been reported.

Structure solution and refinement of beam-sensitive nano-crystals

Radiation sensitive materials are among the most difficult materials to study, even more so if they exist only as nanometer-sized particles, where their size is either intentional because of enhanced properties at the nano-scale or it is unintentional because it is impossible to obtain bigger particles of the same structure. In both cases characterization methods need to be optimized to get the most information out of these particles before the radiation damages them to a point where their structure is altered. When the particles are crystallized, both characteristics, the small size and the beam sensitivity, call for electron diffraction as a privileged investigation tool. The strong interaction of electrons (as compared to X-rays) with matter allows single crystal diffraction experiments on nanometer-sized crystals and for the same amount of beam damage, electron diffraction yields more information than X-rays. These inherent advantages of electron diffraction are optimized in the recently developed low-dose electron diffraction tomography (LD-EDT) by minimizing the necessary dose for a complete data collection. In this contribution we show that in some cases even doses as low as 2 e-/Ų can induce damage in crystal structures that inhibit a correct structure refinement. However, by LD-EDT we can obtain data using extremely low doses that don’t alter the structure which make it then possible not only to solve crystal structures but also to refine them using dynamical diffraction theory. Here a synthetic oxide containing volatile Na and a metal-organic framework are given as examples. A dynamical refinement of the structures is possible with data sets requiring a dose of less than 0.15 e-/Ų.

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.

Multi-exposure diffraction pattern fusion applied to enable wider-angle transmission Kikuchi diffraction with direct electron detectors

Diffraction pattern analysis can be used to reveal the crystalline structure of materials, and this information is used to nano- and micro-structure of advanced engineering materials that enable modern life. For nano-structured materials typically diffraction pattern analysis is performed in the transmission electron microscope (TEM) and TEM diffraction patterns typically have a limited angular range (less than a few degrees) due to the long camera length, and this requires analysis of multiple patterns to probe a unit cell. As a different approach, wide angle Kikuchi patterns can be captured using an on-axis detector in the scanning electron microscope (SEM) with a shorter camera length. These ‘transmission Kikuchi diffraction’ (TKD) patterns present a direct projection of the unit cell and can be routinely analysed using EBSD-based methods and dynamical diffraction theory. In the present work, we enhance this analysis significantly and present a multi-exposure diffraction pattern fusion method that increases the dynamic range of the detected patterns captured with a Timepix3-based direct electron detector (DED). This method uses an easy-to-apply exposure fusion routine to collect data and extend the dynamic range, as well as normalise the intensity distribution within these very wide (>95°) angle patterns. The potential of this method is demonstrated with full diffraction sphere reprojection and highlight potential of the approach to rapidly probe the structure of nano-structured materials in the scanning electron microscope.

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.

Quantum information approach to the implementation of a neutron cavity

Using the quantum information model of dynamical diffraction we consider a neutron cavity composed of two perfect crystal silicon blades capable of containing the neutron wavefunction. We show that the internal confinement of the neutrons through Bragg diffraction can be modelled by a quantum random walk. Furthermore, we introduce a toolbox for modelling crystal imperfections such as surface roughness and defects. Good agreement is found between the simulation and the experimental implementation, where leakage beams are present, modelling of which is impractical with the conventional theory of dynamical diffraction. Analysis of the standing neutron waves is presented in regards to the crystal geometry and parameters; and the conditions required for well-defined bounces are derived. The presented results enable new approaches to studying the setups utilizing neutron confinement, such as the experiments to measure neutron magnetic and electric dipole moments.

EBSD patterns simulating multilayer twins based on dynamical theory of electron diffraction.

Electron backscatter diffraction (EBSD) can be employed to determine crystal structures but has not been used alone to identify defects at the atom scale due to the lack of understanding of the EBSD patterns generated by various structure defects. In the present work, the EBSD patterns of FCC-Fe with 9-layer, 6-layer and 3-layer twin structures are simulated, respectively, using the revised real space (RRS) method and compared with the counterpart of perfect crystals. Our results show that when the electron beam is incident along a direction parallel to the twin plane, the pattern appears symmetrical with respect to the corresponding Kikuchi band of the twin plane, and the diffraction details within the Kikuchi band also exhibit symmetry with respect to the middle line of the Kikuchi band. Moreover, the overall clarity of the patterns decreases, and the pattern becomes more blurred with increasing the distance from the Kikuchi band corresponding to the twin plane. By contrast, the incident electron beam along the direction perpendicular to the twin plane results in diffraction superposition of the matrix region and the shear region, which shows twofold rotational symmetry with respect to the Kikuchi pole corresponding to the normal to the twin plane. In addition, some extra Kikuchi bands appear in the EBSD patterns due to the long-period structures of the multilayer twins. As the number of multilayer twins decreases, the number of extra Kikuchi bands decreases and the area of the blurring pattern increases. The correlation between twin structures and EBSD patterns provides theoretical insights for identifying twin structures by the EBSD technique.

The Limits of X-ray Diffraction Theory

X-ray diffraction theory allows the interpretation of experiments to build a structural model that fits the collected data. As with any experimental science, the observations are subject to uncertainty through the instrument and user limitations. Similarly, the theory can never be perfectly complete; it will have limits, and therefore the resultant model will have uncertainties associated with it. This article discusses the limits of X-ray kinematical and dynamical diffraction theories. These are not the only theories, but are the most widely used. These theories are often extended to accommodate new findings, which can reach the stage at which their fundamental premise is clouded. At that point, the theory requires a rethink. There should be nothing sacrosanct about a theory; it should represent the best usable explanation that will allow a good interpretation of the data. Both kinematical and dynamical theories assume that the X-rays see an average structure, which is not what a photon experiences. The observed diffraction pattern is the average of the diffraction patterns created by all the photons, which is not the same as the diffraction pattern from the average structure. Accounting for this has a profound influence on the interpretation of the data.

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.

Origin of irregular X-ray mirage fringes from a bent, thin crystal.

The dynamical theory of diffraction is used to analyse irregular X-ray mirage interference fringes observed in Si220 X-ray reflection topography from a weakly bent, thin crystal due to gravity. The origin of the irregular fringes is attributed to the interference between mirage diffracted beams and a reflected beam from the back surface, which is a new type of interference fringe. The irregular fringes are reproduced by calculating the reflected intensities numerically. The effects of absorption and thermal vibration are quite important for the reproduction. The result shows that the interference fringes depend on the strain as well as the thickness of the crystal, which indicates that the fringes should be useful for analysing weak strain in a crystal as an application.