Takagi Taupin Research Articles

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.

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%.

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.

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.

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.

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.

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.

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.

Effects of surface undulations on asymmetric X-ray diffraction: a rocking-curve topography study.

The results are reported of an X-ray diffraction study of an Si crystal designed and fabricated for very asymmetric diffraction from the 333 reflection for X-ray energies of 8.100 and 8.200 keV. A crystal with an asymmetry angle of 46 ± 0.1° between the surface and the (111) planes was studied. The grazing angles of incidence were near 1.08 and 0.33° for these two energies, respectively. Features arising from surface undulations were not observed at 8.100 keV, but were observed at 8.200 keV. The results at 8.100 keV allow an alternative explanation based on strain near the surface to be ruled out. Topographic images were obtained as a function of rocking angle, and in the case of 8.200 keV the surface morphology is evident. The results are found to be in agreement with dynamical X-ray diffraction calculations made with the Takagi-Taupin equations specialized to a surface having convex or concave features, as reported in the accompanying paper [Macrander (2020). J. Appl. Cryst. 53, 793-799].

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 fCtpd(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 fCtpd(r - r0) [{\cal P} is the parameter vector that characterizes the 3D function fCtpd(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.

Study of Properties of Materials by Absorption and Diffraction X-Ray Microtomography

Single crystals of silicon and gallstones are studied on a laboratory X-ray microtomograph with a spatial resolution of 10 μm. The tomographic experiment included the use of a monochromatic “parallel beam” with subsequent three-dimensional reconstruction based on two-dimensional projections. The topo-tomographic measurements were performed in the rotation mode of the samples around the normal to the reflection plane tuned by the Laue diffraction reflection geometry, which allowed identification and study of single dislocations in perfect silicon crystals. The dislocation loops were simulated by means of a numerical solution of the Takagi–Taupin equations. Human gallstones were studied in vitro with microtomography. We found that the stones have a layered formation, whose composition may involve calcium carbonate modifications. The internal structure of the stones is heterogeneous and contains numerous cavities and cracks formed during their growth. The assessment of the porosity of gallstones, however, is necessary, because it can affect the dissolution rate of the stone during litholytic treatment. Linear attenuation coefficients of X‑ray radiation of cholesterol-type gallstones were found from the results of measurements. A good agreement between the experimental and calculated absorption coefficients for pure cholesterol suggests that the tomographic method can be used for in vivo diagnosis of cholesterol-type gallstones.