Kinematical Approximation Research Articles

Modeling and experimental validation of dynamical effects in Bragg coherent x-ray diffractive imaging of finite crystals

Bragg coherent diffractive imaging (BCDI) is a noninvasive microscopy technique that can visualize the shape and internal lattice deviations of crystals with nanoscale spatial resolution and picometer deformation sensitivity. Its strain imaging capability relies on Fourier transform--based iterative phase retrieval algorithms, which are mostly developed under the kinematical approximation. Such approximation prohibits the application of BCDI on larger crystals, which are commonly seen in most emerging functional materials. Understanding the dynamical effect in BCDI, as well as developing a validated method for modeling BCDI at the dynamical diffraction limit, is crucial for applying BCDI to hierarchical systems that contain micron-sized crystals and grains. Here we report a comparative study on the impact of dynamical diffraction effects by comparing the reconstruction results from two measurements of the same crystal. Forward simulation is implemented to show subtle changes of interference fringes in the diffraction pattern due to the dynamical diffraction, and is compared directly with the experimental data.

A C++ code for simulations of RHEED intensity oscillations within the kinematical approximation

This paper proposes a new software implementation for the rapid development and deployment of computer simulations of reflection high-energy electron diffraction (RHEED) intensity oscillations during the growth of thin epitaxial films. The calculations are based on the kinematical approximation. In this model the RHEED intensity oscillations from growing layers is considered as an effect of changes of surface roughness. New version program summaryProgram Title: growth22CPC Library link to program files:https://doi.org/10.17632/zkpwhrrfng.1Licensing provisions: GNU General Public License 3Programming language: C++Journal reference of previous version: Computer Physics Communications 170 (2005) 265–286Does the new version supersede the previous version?: YesReasons for the new version:The previous version of the program was written in Object Pascal [1]. This programming language is no longer popular in the scientific community. Responding to user's feedback, the current distribution of the software has been implemented in C++. This version implements improvements for ergonomics, computational performances and code readability. In addition, the output generated by the growth22 (saved in the coverageProfiles.dat file) can be used directly by program [2].Summary of revisions:The growth22 is written in standard C++, therefore one can use the GNU g++ compiler or any ANSI C++ compliant compiler to build the executable program. The code is distributed in the form of one growth22.cpp source file. A Table 1 lists the retired Object Pascal procedures and shows replacements by C++ functions. The present version of the program enables the user to easily manage the input data. Input data are loaded from a manually prepared inputData.dat text file where the user provides information on the parameters of the simulation, i.e. the growth model, the number of growing layers, the upper limit of growth time interval, the values of the phenomenological parameters that measure the net rate of transfer from one layer to the next, and the values of the growth rate for each layer. The user can edit this file as per the requirement of the problem domain. Details of the parameters in the sample inputData.dat file are described in the readme.pdf file provided with the software.The nature of problem:Generally, in order to explain the experimental data two theories—kinematical and dynamical—associated with high-energy electron diffraction in epitaxial layers are used. The dynamical theories take into account multiple scattering within the crystal and are used whenever diffraction from a perfect epitaxial layers is considered [3]. On the other hand, the kinematical theory treats the scattering from each plane in the crystal sample as being independent of others [1,4].In the kinematical theory of RHEED it is assumed that electron wave scatters from atom in the surface layer only once and then interferes with waves scattered on other atoms. This model can be used to describe diffraction effects for small angles of incidence, when the electron beam does not penetrate the deeper layers of the crystal. In the kinematical model the intensity oscillations of specularly reflected electron beam from growing layer is considered as an effect of changes of surface roughness.In this paper we present a time-efficient computer model for kinematical calculations of RHEED intensity oscillations during the growth of thin epitaxial films.The method of solution:According to kinematical approximation the diffracted-beam intensity is given by [4]:(1)I(S‖→=0,Sz)=|∑n=0∞(θn−θn+1)exp(−iSznd)|2 where S→ is the scattering vector, d is the interlayer spacing and n is an integer. θn−θn+1 is the exposed layer coverage which is treated as the atomic distribution function gives the position of surface atoms. For Sz=π/d Eq. (1) simplifies to:(2)I=|∑n=0∞(θn−θn+1)cos⁡(nπ)|2.Variation of the coverage ratio θn of the nth surface layer is modelled by a set of non-linear differential equations [1,4]. The Cash-Karp fifth-order Runge-Kutta method [5] with error estimation and adaptive stepsize control was used for solving initial value problem for non-linear differential equations [6]. The absolute and relative error tolerances and number of integration intervals are fully user-configurable.The proposed code approaches the modelling of RHEED from two different points of view, i.e. the data and functional perspectives. The data perspective (i.e. calculating the layer coverages as a function of their growth time) is used to characterize the data sources of the program whereas the functional perspective includes services provided by the program. It should be noted that the perspective of the data can be expressed by a completely different model introduced by the user.The coverage evolution curves for the diffusive growth model, corresponding kinematical diffracted intensity and total roughness of the surface are shown in Fig. 1. During the numerical calculations showed in Fig. 1 it was assumed that the number of growing layers (numLayers) equals 15, the upper limit of growth time interval (tMax) equals 12, the growth rate equals 1 MLs−1 (for all layers), and the values of the parameter kn (that determines the amount of interlayer diffusion) changing according to the linear relation:(3)kn=k1−(k1−kN)n−1N−1, where N is the total number of growing layers.

Polymorph and Polytype Identification from Individual Mica Particles Using Selected Area Electron Diffraction

AbstractDioctahedral micas are composed of two tetrahedral sheets and one octahedral sheet to form TOT or 2:1 layers. These minerals are widespread and occur with structures differing by (1) the layer stacking mode (polytypes), (2) the location of vacancies among non-equivalent octahedral sites (polymorphs), and (3) the charge-compensating interlayer cation and isomorphic substitutions. The purpose of the present study was to assess the potential of parallel-illumination electron diffraction (ED) to determine the polytype/polymorph of individual crystals of finely divided dioctahedral micas and to image their morphology. ED patterns were calculated along several zone axes close to the c*- and c-axes using the kinematical approximation for trans- and cis-vacant varieties of the four common mica polytypes (1M, 2M1, 2M2, and 3T). When properly oriented, all ED patterns have similar geometry, but differ by their intensity distribution over hk reflections of the zero-order Laue zone. Differences are enhanced for ED patterns calculated along the [001] zone axis. Identification criteria were proposed for polytype/polymorph identification, based on the qualitative distribution of bright and weak reflections. A database of ED patterns calculated along other zone axes was provided in case the optimum [001] orientation could not be found. Various polytype/polymorphs may exhibit similar ED patterns depending on the zone axis considered.

What does kinematical target mass sensitivity in DIS reveal about hadron structure?

We study the role of purely external kinematical approximations in inclusive deep-inelastic lepton--hadron scattering within QCD factorization, and consider factorization with an exact treatment of the target hadron mass. We discuss how an observed phenomenological improvement obtained by accounting for target mass kinematics could be interpreted in terms of general properties of target structure, and argue that such an improvement implies a hierarchy of nonperturbative scales within the hadron.

Deterministic Bragg Coherent Diffraction Imaging

A deterministic variant of Bragg Coherent Diffraction Imaging is introduced in its kinematical approximation, for X-ray scattering from an imperfect crystal whose imperfections span no more than half of the volume of the crystal. This approach provides a unique analytical reconstruction of the object’s structure factor and displacement fields from the 3D diffracted intensity distribution centred around any particular reciprocal lattice vector. The simple closed-form reconstruction algorithm, which requires only one multiplication and one Fourier transformation, is not restricted by assumptions of smallness of the displacement field. The algorithm performs well in simulations incorporating a variety of conditions, including both realistic levels of noise and departures from ideality in the reference (i.e. imperfection-free) part of the crystal.

Structure refinement using precession electron diffraction tomography and dynamical diffraction: tests on experimental data.

The recently published method for the structure refinement from three-dimensional precession electron diffraction data using dynamical diffraction theory [Palatinus et al. (2015). Acta Cryst. A71, 235-244] has been applied to a set of experimental data sets from five different samples - Ni2Si, PrVO3, kaolinite, orthopyroxene and mayenite. The data were measured on different instruments and with variable precession angles. For each sample a reliable reference structure was available. A large series of tests revealed that the method provides structure models with an average error in atomic positions typically between 0.01 and 0.02 Å. The obtained structure models are significantly more accurate than models obtained by refinement using kinematical approximation for the calculation of model intensities. The method also allows a reliable determination of site occupancies and determination of absolute structure. Based on the extensive tests, an optimal set of the parameters for the method is proposed.

Extremely asymmetric diffraction as a method of determining magneto-optical constants for X-rays near absorption edges

The spectral dependence of the Bragg peak position under conditions of extremely asymmetric diffraction has been analyzed in the kinematical and dynamical approximations of the diffraction theory. Simulations have been performed for the L3 absorption edge of yttrium in a single-crystal YFe2 film; they have shown that the magneto-optical constants (or, equivalently, the dispersion corrections to the atomic scattering factor) for hard X-rays can be determined from this dependence. Comparison with the experimental data obtained for a Nb(4 nm)/YFe2(40 nm〈110〉)/Fe(1.5 nm)/Nb(50 nm)/sapphire sample at the European Synchrotron Radiation Facility has been made.

Towards a full retrieval of the deformation tensor F using convergent beam electron diffraction

A new method to retrieve the local lattice parameters and rotations in a crystal from off-axis convergent beam electron diffraction (CBED) patterns is presented and validated using Bloch wave dynamical simulations. The originality of the method is to use both the diffracted and transmitted beams and to use kinematical approximations in the fitting algorithm. The study is based on the deformation gradient tensor F which includes rotation and strain. Working on simulated images it is shown that (i) from a single direction of observation, seven parameters out of the nine parameters of F can be determined with an accuracy of 3×10−4 for the normal strain parameters εxx, εyy, and εzz, (ii) the unit cell volume can only be retrieved if the diffracted and transmitted beams are both included in the fitting and (iii) all the nine parameters of F can be determined by combining two directions of observation separated by about 20°.

Accurate structure refinement from 3D electron diffraction data

Structure determination from electron diffraction data has seen an enormous progress over the past few years. At present, complex structures with hundreds of atoms in the unit cell can be solved from electron diffraction using the concept of electron diffraction tomography (EDT), possibly combined with precession electron diffraction (PED) [1]. Unfortunately, the initial model is typically optimized using the kinematical approximation to calculate model diffracted intensities. This approximation is quite inaccurate for electron diffraction and leads to high figures of merit and inaccurate results with unrealistically low standard uncertainties. The obvious remedy to the problem is the use of dynamical diffraction theory to calculate the model intensities in structure refinement. This technique has been known and used before, but it has not become very popular, because good fits could be obtained only for sufficiently perfect and sufficiently thin crystals. It has been shown recently on several zone-axis patterns [2] that the quality of the refinement can be improved by using precession electron diffraction. In the present contribution we demonstrate that the same approach can be successfully used to refine crystal structures against non-oriented patterns acquired by EDT combined with PED (PEDT in short). Because the PEDT technique provides three-dimensional diffraction information, it can be used for a complete structure refinement. Several test examples demonstrate that the dynamical structure refinement yields better figures of merit and more accurate results than the refinement using kinematical approximation.

Electron Diffraction vs aperiodic crystal: from indexation to structure solution

Can we solve aperiodic structures using intensities from electron diffraction? Yes! How? No mystery about it: the data analysis and the tools used for structure solution are essentially the same as the ones used in X-ray crystallography. The trick actually lies in new approaches used in electron crystallography. In analogy to X-ray diffraction, the so-called Electron Diffraction Tomography (EDT) [1] corresponds to a phi-scan data collection on a single crystal. There lies one major advantage of this technique: a powder sample is easily converted to infinitely large number of single crystals for electron diffraction. In case of aperiodic crystals this makes the difference over X-ray or neutron powder diffraction where, often, the lack of peaks clearly assignable to satellite reflections prevents any indexation and further analysis of the structure [2]. EDT allows for an accurate estimation of the modulation vector and a good guess of the super space group. These informations can be advantageously used as an input for X-ray or neutron powder diffraction. Not limited to indexation, EDT combined with Precession Electron Diffraction (PED) [3], offers a unique tool for solving modulated structures when crystals suitable for X-ray diffraction are missing. By limiting the paths for multiple scattering, PED makes the diffracted intensities closer to kinematical approximation so that they can be used efficiently for structure solution. Regarding aperiodic crystals, the superspace electron density map, generated as an output of the charge flipping algorithm used in Superflip, can be interpreted to obtain a structural model. This will be illustrated on a series of layered materials closely related to the Aurivillius phases belonging to the pseudo-binary Bi5Nb3O15-ABi2Nb2O9 (A=Pb, Sr, Ca, Ba). Limitations and possible combination with powder diffraction patterns will be discussed.

Application of the paracrystal model to GISAXS analysis of the 3D self-assembled nanoparticle crystals

An original diffraction model for the evaluation of the grazing-incidence small-angle X-ray scattering (GISAXS) of the three-dimensional (3D) supported self-assembled nanoparticle crystals is presented. The model utilizes the formalism of traditional crystallography that is applied to nanoparticles as scattering entities. The infinite paracrystal concept was adopted in order to include cumulative disorder that is typical for the self-assembled nanoparticle structures. Azimuthal averaging around a chosen crystallographic direction allows incorporation of texture effects. A transformation of the SAXS paracrystal equations to comply with the GISAXS geometry was employed to describe the scattering process in the limit of the kinematical approximation which facilitates calculations. The model was developed for the face-centered cubic (FCC) symmetry, however, its extension to any other cubic symmetry is straightforward. The model was tested on a 3D nanoparticle assembly formed on silicon from a dried drop-cast colloidal solution of monodisperse silver nanoparticles of 5.8 ± 0.6 nm diameter. Presence of the FCC nanoparticle crystal with the [111] texture along the substrate normal was evidenced from evaluation of the synchrotron GISAXS data. The degree of paracrystal disorder is 2.5% and 3.6% parallel and perpendicular to the substrate, respectively, suggesting the anisotropic character of the self-assembling process. A SAXS measurement of the nanoparticle crystal formed from the same solution in a capillary revealed a randomly oriented FCC structure with a slightly larger paracrystal disorder of 4%. Obviously, different conditions of the nanoparticle crystal formation affect its azimuthal and translational disorder.

Darwin's approach to X-ray diffraction on lateral crystalline structures

Darwin's dynamical theory of X-ray diffraction is extended to the case of lateral (i.e., having a finite length in the lateral direction) crystalline structures. This approach allows one to calculate rocking curves as well as reciprocal-space maps for lateral crystalline structures having a rectangular cross section. Numerical modelling is performed for these structures with different lateral sizes. It is shown that the kinematical approximation is valid for thick crystalline structures having a small length in the lateral direction.

Kinematical limit in the theory of X-ray magnetic reflectivity

A kinematical approximation has been derived from the general formulas for calculating the reflectivity from a multilayer anisotropic structure. An explicit form of reflection matrices at the boundary of layers and the refraction tensors has been obtained for a particular case where the layer magnetization is oriented in the surface plane and for an arbitrary form of X-ray susceptibility tensors. Calculations of the reflectivity from a Nb(4 nm)/[Dy(5 nm)/Lu(3 nm)]420 model structure with helicoidal ordering of Dy magnetic layers show that kinematical approximation can be applied for grazing angles that are larger than the critical total-reflection angle and that the “magnetic” refraction must be taken into account.

Structure refinement from precession electron diffraction data

Electron diffraction is a unique tool for analysing the crystal structures of very small crystals. In particular, precession electron diffraction has been shown to be a useful method for ab initio structure solution. In this work it is demonstrated that precession electron diffraction data can also be successfully used for structure refinement, if the dynamical theory of diffraction is used for the calculation of diffracted intensities. The method is demonstrated on data from three materials - silicon, orthopyroxene (Mg,Fe)(2)Si(2)O(6) and gallium-indium tin oxide (Ga,In)(4)Sn(2)O(10). In particular, it is shown that atomic occupancies of mixed crystallographic sites can be refined to an accuracy approaching X-ray or neutron diffraction methods. In comparison with conventional electron diffraction data, the refinement against precession diffraction data yields significantly lower figures of merit, higher accuracy of refined parameters, much broader radii of convergence, especially for the thickness and orientation of the sample, and significantly reduced correlations between the structure parameters. The full dynamical refinement is compared with refinement using kinematical and two-beam approximations, and is shown to be superior to the latter two.

Interference amplification of the magneto-optic Kerr effect in multilayer structures in the X-ray range

The problem of X-ray reflection from multilayer structures with magnetic layers has been solved. The matrix for reflection of electromagnetic waves from such structures has been calculated analytically in kinematical and dynamical approximations. It is shown that not only reflection coefficients, but also magneto-optic effects are enhanced near Bragg peak. The possibility of measuring magneto-optic effects in multilayer structures without using synchrotron radiation has been estimated.

Physical origin of in-plane lattice spacing oscillations measured by reflection high-energy electron diffraction during epitaxial growth

Intensity oscillations of the specular $(0,0)$ RHEED spot during layer-by-layer growth are well known, whereas the associated oscillation of the position of the RHEED streaks remains more controversial. We revisit the problem of the origin of the peak-to-peak oscillations observed in RHEED spectra during layer-by-layer epitaxial growth. For this purpose we perform solid-on-solid KMC simulations to describe the growth, and we use the kinematical approximation to simulate RHEED intensity profiles. We show that the peak-to-peak oscillations result from the angular dispersion of the incident beam and the periodic oscillation of the size of the growing islands, without the need of invoking the strain relaxation in the islands.

Nanometer Scale Assessment of Mechanical Strain Induced in Silicon by a Periodic Line Array

Measuring stress and strain, induced by nanostructures, at the nanometer scale is still a challenge. In this work, we investigate the strain induced by sub-micrometric periodic line arrays deposited on single crystal (001) Si substrate. We study the influence of the lines width and the spacing between the lines for two sets of samples: a silicon nitride lines array and a poly-silicon line array capped with a Si3N4 stressor layer. The periodic strain field in mono-crystalline silicon is investigated by High Resolution X-ray Diffraction which is very sensitive to local strain (< 10(-4)), has the required resolution, and is non-destructive. X-ray reciprocal space maps (RSM) are measured on a 4 circles goniometer with a laboratory source. The line arrays induce a periodic strain field in silicon, which gives rise to distinct satellites in reciprocal space. The intensity envelope of these satellites is related to the strain field in one cell. In order to assess this strain field in silicon, mechanical modeling is necessary. Elastic calculations are performed with a Finite Element Modeling (FEM) code in order to extract the displacement field that is used for structure factor calculations within kinematical approximation. The calculated reciprocal space map is compared to the experimental results in order to validate the strain field. We show that for capped poly arrays, the diffracted intensity envelope is influenced by the spacing between the lines. This area is filled with silicon nitride which induces a noticeable change in displacement and strain field. While for bare stressor arrays the nitride line width is responsible of change in displacement field and thus on the RSM intensity envelope.

Local strain induced in silicon by Si3N4 lines: Modeling and experimental investigation via X-ray diffraction

The influence of local strain fields on electrical properties such as the mobility of electrons and holes in silicon is of growing concern. In this work, we consider the displacement field generated in a semi-infinite Si substrate by a periodic array of lines located at the surface. We focus on a model system: an array of silicon nitride lines deposited on a monocrystalline silicon substrate. Measurements with X-ray diffraction (XRD) need mechanical modeling to validate the strain field. For that purpose we use finite element modeling (FEM) for simulating the displacement field, which is then used for calculating the diffracted intensity in the kinematical approximation. Agreement between measured and calculated intensity allows for a validation of the calculated strain field in the silicon.

Pawley and Rietveld refinements using electron diffraction from L12-type intermetallic Au3Fe1−x nanocrystals during their in-situ order–disorder transition

During the in-situ order-disorder transition of intermetallic L1(2)-type Au(3)Fe(1-x) nanocrystals, structural information has been retrieved from their electron diffraction patterns based on the Pawley refinement that is unrelated to the electron kinematical or dynamical scattering nature as well as the Rietveld refinement using a kinematical approximation. At room temperature, it was found that the nanocrystals contain approximately x=40% vacancies at the Fe site. Based on in-situ heating this phase displayed an irreversible order-disorder transition, with the transition temperature between 553 and 593 K. A sudden increase in lattice parameter was detected during the first heating from the ordered phase, while the second heating of the disordered phase showed only a linear relationship with temperature. From the lattice parameter measurement of the disordered phase, the coefficient of thermal expansion was estimated as 1.462 × 10(-5)K(-1). The long-range order parameter S was determined by the refined site occupancies, as well as the integrated intensities of the superlattice (100) and fundamental (220) reflections using the Pawley and Rietveld refinements during the order-disorder transition. Considering the dynamical scattering effect, Blackman two-beam approximation theory was applied to corrected S, which slightly attenuated after the correction. A comparison of the electron diffraction with X-ray diffraction data was made. It was demonstrated that elemental and structural information could be retrieved through quantitative electron diffraction studies of the nanomaterials. Since the Pawley refinement algorithm does not include the electron scattering event, it is especially useful to refine the electron diffraction data regardless of the sample thickness.

Polytype and polymorph identification of finely divided aluminous dioctahedral mica individual crystals with SAED. Kinematical and dynamical electron diffraction

This work investigates the potential of selected-area electron diffraction (SAED) for the polytype and polymorph identification of finely divided K-bearing aluminous dioctahedral mica. Individual mica crystals may indeed differ by their layer-stacking sequence and by the inner structure of their octahedral sheets (polytypic and polymorphic variants, respectively). This diversity of natural mica is commonly considered to be responsible for their morphological variety. The present article thus analyzes the intensity distribution between hk0 beams as a function of the crystal structure and thickness. The comparison of ED calculations with experimental diffraction data shows that predicted dynamical effects are not observed for finely divided dioctahedral mica. The influence of different structure defects on calculated intensities is analyzed, and their widespread occurrence in natural mica is hypothesized to be responsible for the limitation of dynamical diffraction effects. SAED may thus be used to identify the structure of individual dioctahedral mica crystals using the kinematical approximation to simulate and qualitatively interpret the observed intensities.