Takagi-Taupin Equations Research Articles

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.

Computer simulation of x-ray section topography of gas pores in a silicon carbide crystal

The results of computer simulation of images of gas pores in a silicon carbide crystal in sectional topograms, that is, during diffraction of a narrow beam of X-rays in the crystal, are presented for the first time. For this purpose, a special module of the universal computer program XRWP was used. This program is developing by the author to calculate the effects of coherent X-ray optics. The calculation method combines two methods, previously known, namely, Fourier transform methods (Kato method), and the method of solving the Takagi-Taupin equations. It is shown that gas pores can produce a wide variety of images, depending on the experimental conditions and the position of the gas pore inside the crystal.

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.

Revisiting the ‘magic condition’ on the basis of the Takagi–Taupin theory

A numerical framework based on the integral solution of the Takagi–Taupin equations has been developed for cylindrically bent Laue crystals. On the basis of this framework, diffraction geometries that satisfy the `magic condition' have been studied from the perspective of dynamical theory. The numerical findings indicate that, in certain diffraction geometries, the focusing behaviour of cylindrically bent Laue crystals will be notably influenced by dynamical effects and the foci of different energies will not converge as predicted by the `magic condition', which is derived from geometric optics theory. These dynamical effects are further explained through a direct numerical analysis of the influence function.

Асимметричная рентгеновская дифракция ограниченных пучков в кристаллах

Asymmetric dynamic diffraction of confined X-ray beams in crystals has been theoretically studied using Takagi–Taupin equations, including numerical solution using nodal grid and calculations of X-ray fields in Fourier space. We have fulfilled simulation of scattering intensity distribution maps from silicon crystal near reciprocal lattice site depending on size of X-ray beams.

Towards to Theory of the X-ray Diffraction Tomography of Crystals with Nano-Sized Defects

X-ray diffraction tomography is an innovative method that is widely used to obtain 2D-phase-contrast diffraction images and their subsequent 3D-reconstruction of structural defects in crystals. The most frequent objects of research are linear and helical dislocations in a crystal, for which plane wave diffraction images are the most informative, since they do not contain additional interference artifacts unrelated to the images of the defects themselves. In this work the results of modeling and analysis of 2D plane wave diffraction images of a nano-dimensional Coulomb-type defect in a Si(111) thin crystal are presented based on the construction of numerical solutions of the dynamic Takagi-Taupin equations. An adapted physical expression for the elastic displacement field of the point defect, which excludes singularity at the defect location in the crystal, is used. A criterion for evaluating the accuracy of numerical solutions of the Takagi-Taupin equations is proposed and used in calculations. It is shown that in the case of the Coulomb-type defect elastic displacement field, out of the two difference algorithms for solving the Takagi-Taupin equations used in their numerical solution, only the algorithm for solving the Takagi-Taupin equations where the displacement field function enters in exponential form is acceptable in terms of the required accuracy-duration of the calculations.

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].

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.

Comprehensive Recovery of Point Defect Displacement Field Function in Crystals by Computer X-ray Diffraction Microtomography

In the case of the point defect in a crystal, the inverse Radon’s problem in X-ray diffraction microtomography has been solved. As is known, the crystal-lattice defect displacement field function f(r) = h·u(r) determines phases − (±h)-structure factors incorporated into the Takagi–Taupin equations and provides the 2D image patterns by diffracted and transmitted waves propagating through a crystal (h is the diffraction vector and u(r) is the displacement field crystal-lattice-defects vector). Beyond the semi-kinematical approach for obtaining the analytical problem solution, the difference-equations-scheme of the Takagi–Taupin equations that, in turn, yield numerically controlled-accuracy problem solutions has been first applied and tested. Addressing the inverse Radon’s problem solution, the χ2-target function optimization method using the Nelder–Mead algorithm has been employed and tested in an example of recovering the Coulomb-type point defect structure in a crystal Si(111). As has been shown in the cases of the 2D noise-free fractional and integrated image patterns, based on the Takagi–Taupin solutions in the semi-kinematical and difference-scheme approaches, both procedures provide the χ2-target function global minimum, even if the starting-values of the point-defect vector P1 is chosen rather far away from the reference up to 40% in relative units. In the cases of the 2D Poisson-noise image patterns with noise levels up to 5%, the figures-of-merit values of the optimization procedures by the Nelder–Mead algorithm turn out to be high enough; the lucky trials number is 85%; and in contrast, for the statistically denoised 2D image patterns, they reach 0.1%.

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.

Long-lasting deformation potential effect in Ge induced by UV photoexcitation

Understanding ultrafast response of a semiconductor is necessary for next-generation optoelectronic device applications. Here, we investigate the ultrafast response of an archetypal semiconductor Ge [111] crystal upon photoexcitation at two pump wavelengths, 800 nm (fundamental) and 400 nm (second harmonic, UV radiation), using time-resolved x-ray diffraction (TXRD). The simulated TXRD profiles using a proposed four-layer model and Takagi–Taupin equations reveal that the strain propagation is primarily due to electron diffusion. Intriguingly, the initial (<100 ps) fast decrease in the induced strain for second harmonic excitation suggests that the strain originates from the deformation potential (DP) effect. The higher photon energy (second harmonic pump) excites the electrons to higher conduction band valleys, resulting in DP-induced strain compared to the fundamental pump, which only generates thermo-elastic strain. The DP strain, estimated from the DP coefficient (simulated via density functional theory-based electronic structure simulations) and electron density (simulated from experimental results), shows an excellent agreement with the observed strain. The strain after ∼100 ps delay for second harmonic excitation and strain from fundamental excitation are attributed to the thermo-elastic effect as confirmed by the density-dependent two-temperature model. Our study reveals the long-lasting DP effect in Ge [111], which provides an opportunity to avoid heating in optoelectronic devices due to the thermo-elastic effect at the initial time scale.

Towards to solution of the fractional Takagi–Taupin equations. The Green function method

Developing the comprehensive theory of the X-ray diffraction by distorted crystals remains to be topical of the mathematical physics. Up to now, the X-ray diffraction theory grounded on the Takagi–Taupin equations with the first-order partial derivatives over the two coordinates within the X-ray scattering plane. In the work, the theoretical approach based on the first-order fractional Takagi–Taupin equations with the ‘quasi-time variable’ of the order $$\alpha \in (0,1]$$ along the crystal depth has been suggested and the corresponding X-ray Cauchy issue is formulated. Accordingly, using the Green function method in the scope of the Cauchy issue, the fractional Takagi–Taupin equations in the integral form have been derived. In the case of the inhomogeneous incident X-ray beam, the solution of the Cauchy issue of the X-ray diffraction by perfect crystal has been obtained and compared with the corresponding one based on the solution of the conventional Takagi–Taupin equations, $$\alpha =1.$$ In turn, notice that the value of order $$\alpha $$ may be adjusted from the experimental X-ray diffraction data.

COMPUTER SIMULATION OF THE EFFECT OF COHERENT DYNAMICAL DIFFRACTION OF SYNCHROTRON RADIATION IN CRYSTALS OF ARBITRARY SHAPE AND STRUCTURE

A new scheme for the numerical solution of Takagi–Taupin equations, which makes it possible to simulate the effect of synchrotron radiation diffraction in crystals of arbitrary structure, is described in detail. The new scheme is convenient to perform calculations for crystals of arbitrary shape. The rectangular coordinate system and the algorithm for calculating derivatives at half of step have proven their efficiency and are used, but the recurrence equations of this algorithm have been modified towards simplification. The boundary conditions are in no way related to the crystal boundaries. A computer program is developed, and two examples are considered for the cases of diffraction in the Laue and Bragg geometries, for which the analyticl solutions are known. The calculation results are in complete agreement with these solutions.

Finite-element simulation for crystals with surface undulations based on Takagi–Taupin theory

A finite-element method based on a weak formulation of the Takagi–Taupin equations was adopted to study the X-ray diffraction of crystals with surface undulations. A general diffraction geometry was simulated to investigate the diffraction features caused by the surface undulations. The numerical results reveal that the effects of surface undulations on Bragg diffraction are local for those limited in the low-frequency range and physically result from the refractive effect, brought on by the variation of the local asymmetry angle. Thus, a formula based on the local perfect flat crystal approximation was introduced to efficiently evaluate the change in the direction of the diffracted wave caused by surface undulations.

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

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.

A finite difference scheme for integrating the Takagi-Taupin equations on an arbitrary orthogonal grid.

Calculating dynamical diffraction patterns for X-ray diffraction imaging techniques requires numerical integration of the Takagi-Taupin equations. This is usually performed with a simple, second-order finite difference scheme on a sheared computational grid in which two of the axes are aligned with the wavevectors of the incident and scattered beams. This dictates, especially at low scattering angles, an oblique grid of uneven step sizes. Here a finite difference scheme is presented that carries out this integration in slab-shaped samples on an arbitrary orthogonal grid by implicitly utilizing Fourier interpolation. The scheme achieves the expected second-order convergence and a similar error to the traditional approach for similarly dense grids.

Generalizing the quantum information model for dynamic diffraction

The development of novel neutron optics devices that rely on perfect crystals and nano-scale features are ushering a new generation of neutron science experiments, from fundamental physics to material characterization of emerging quantum materials. However, the standard theory of dynamical diffraction (DD) that analyzes neutron propagation through perfect crystals does not consider complex geometries, deformations, and/or imperfections which are now becoming a relevant systematic effect in high precision interferometric experiments. In this work, we expand upon a quantum information (QI) model of DD that is based on propagating a particle through a lattice of unitary quantum gates. We show that the model output is mathematically equivalent to the spherical wave solution of the Takagi-Taupin equations when in the appropriate limit, and that the model can be extended to the Bragg as well as the Laue-Bragg geometry where it is consistent with experimental data. The presented results demonstrate the universality of the QI model and its potential for modeling scenarios that are beyond the scope of the standard theory of DD.

Evolution of laser-induced strain in a Ge crystal for the [111] and [100] directions probed by time-resolved X-ray diffraction

Ultra-short laser-pulse-induced strain propagation in a Ge crystal is studied in the [111] and [100] directions using time-resolved X-ray diffraction (TXRD). The strain propagation velocity is derived by analysis of the TXRD signal from the strained crystal planes. Numerical integration of the Takagi–Taupin equations is performed using open source code, which provides a very simple approach to estimate the strain propagation velocity. The present method will be particularly useful for relatively broad spectral bandwidths and weak X-ray sources, where temporal oscillations in the diffracted X-ray intensity at the relevant phonon frequencies would not be visible. The two Bragg reflections of the Ge sample, viz. 111 and 400, give information on the propagation of strain for two different depths, as the X-ray extinction depths are different for these two reflections. The strain induced by femtosecond laser excitation has a propagation velocity comparable to the longitudinal acoustic velocity. The strain propagation velocity increases with increasing laser excitation fluence. This fluence dependence of the strain propagation velocity can be attributed to crystal heating by ambipolar carrier diffusion. Ge is a promising candidate for silicon-based optoelectronics, and this study will enhance the understanding of heat transport by carrier diffusion in Ge induced by ultra-fast laser pulses, which will assist in the design of optoelectronic devices.

Modeling of polygonal half–loops dislocations in silicon single crystal using X–ray diffraction topo–tomography data

Topo-tomographic methods for identifying defects in semiconductors are the most effective among X-ray methods. Combining experimental data with model calculations makes it possible to determine various parameters and properties of crystal structures. In this work, using X-ray diffraction topo-tomography, images of the dislocation half-loop in Si (111) crystal were obtained. The Takagi-Taupin equations have been used to modeling the topograms. A quantitative comparison of the images made it possible to determine the direction of the Burgers vector.