Reciprocal Space Analysis Research Articles

Deep learning for symmetry classification using sparse 3D electron density data for inorganic compounds

We report a novel deep learning (DL) method for classifying inorganic compounds using 3D electron density data. We transform Density Functional Theory (DFT)-derived CHGCAR files from the Materials Project (MP) and experimental data from the Inorganic Crystal Structure Database (ICSD) into point clouds and sparse tensors, optimized for use in DL models such as PointNet and Sparse 3D CNN. This approach effectively overcomes the limitations of handling the dense 3D data, a common challenge in DL. Contrasting with traditional 1D or 2D X-ray diffraction (XRD) patterns that necessitate complex reciprocal space analysis, our method utilizes 3D density data for direct interpretation in real lattice space. This shift significantly enhances classification accuracy, outperforming traditional XRD-driven DL methods. We achieve accuracies of 97.28%, 90.77%, and 90.10% for crystal system, extinction group, and space group classifications, respectively. Our 3D electron density-based DL approach not only showcases improved accuracy but also contributes a more intuitive and effective framework for materials discovery.

Pinpointing lattice-matched conditions for wurtzite ScxAl1−xN/GaN heterostructures with x-ray reciprocal space analysis

Using comprehensive x-ray reciprocal space mapping, we establish the precise lattice-matching composition for wurtzite ScxAl1−xN layers on (0001) GaN to be x = 0.14 ± 0.01. 100 nm thick ScxAl1−xN films (x = 0.09–0.19) were grown in small composition increments on c-plane GaN templates by plasma-assisted molecular beam epitaxy. The alloy composition was estimated from the fit of the (0002) x-ray peak positions, assuming the c-lattice parameter of ScAlN films coherently strained on GaN increases linearly with Sc-content determined independently by Rutherford backscattering spectrometry [Dzuba et al., J. Appl. Phys. 132, 175701 (2022)]. Reciprocal space maps obtained from high-resolution x-ray diffraction measurements of the (101¯5) reflection reveal that ScxAl1−xN films with x = 0.14 ± 0.01 are coherently strained with the GaN substrate, while the other compositions show evidence of relaxation. The in-plane lattice-matching with GaN is further confirmed for a 300 nm thick Sc0.14Al0.86N layer. The full-width-at-half-maximum of the (0002) reflection rocking curve for this Sc0.14Al0.86N film is 106 arc sec and corresponds to the lowest value reported in the literature for wurtzite ScAlN films.

Static and dynamic atom displacements in nanocrystalline powders from diffraction analysis at finite temperature

We evaluate the accuracy of x-ray diffraction analysis performed by Rietveld refinement in extracting the static and dynamic atomic displacement parameters of spherical gold nanocrystals with sizes of 30 nm and below and within a temperature interval of 10–300 K. A self-consistent computational workflow composed of real- and reciprocal- space analyses is designed for this purpose. We obtain average lattice parameters, static and thermal atomic displacements as well as their temperature and particle size dependence in an ideally monodispersed and realistic nanocrystalline powder generated by molecular dynamics simulations. Then, we compare crystallographic parameters obtained from real- and reciprocal- space analyses to evaluate the accuracy of diffraction analysis. We also extend the comparison to local variations of crystallographic parameters in the nanocrystals and show how Rietveld refinement relates to the average parameters of the core and surface regions separately. We show that 1) average lattice parameters obtained from Rietveld refinement are greater than the mean lattice parameter of small nanocrystals and converge to the core lattice parameter with increasing particle size, 2) Rietveld refinement significantly underestimates static atom displacements of surface atoms in small nanocrystals, 3) dynamic atomic displacements in nanocrystalline gold powders are reliably obtained within 5 % error by Rietveld refinement, provided that thermal diffuse scattering is adequately removed from the diffraction data, 4) the discrepancy between real- and reciprocal-space analysis increases for the smallest gold nanocrystal.

Compact 2D electron analyzer CoDELMA: Simultaneous wide reciprocal and real space analysis using wide-angle deceleration lens, CMA and projection lens

A compact wide-acceptance angle (around ± 50°) high-energy-resolution 2D electron analyzer CoDELMA (Compact DELMA) is proposed, constructed and tested. CoDELMA is composed of a recently-proposed variable-deceleration-ratio wide-acceptance-angle electrostatic lens (VD-WAAEL), a cylindrical mirror analyzer (CMA) and a projection lens. After a brief review of VD-WAAEL, the use of a plane grid in the lens is presented as a new design option, which allows keeping the focusing angle almost constant irrespective of the deceleration ratio. A detailed consideration is given to the combination of CMA and a projection lens, which allows simultaneous analysis of wide reciprocal ( ± 8°) and real space ( ± 5 mm) at the entrance of CMA. CoDELMA is the successor to the previously developed DELMA instrument including a concentric hemispherical analyzer. The whole system of CoDELMA becomes much more compact than that of DELMA owing to the use of VD-WAAEL and CMA, and this results in considerable reduction of the production cost. Results of test measurement of energy spectrum and angular distribution using an electron gun as an excitation source confirmed that the energy resolution and the acceptance angle are almost the same as the designed values.

Nature of interfacial Dzyaloshinskii-Moriya interactions in graphene/Co/Pt(111) multilayer heterostructures

Density-functional theory calculations utilizing the generalized Bloch theorem show that interfacial Dzyaloshinskii-Moriya interactions (DMI) at both interfaces of graphene/${\mathrm{Co}}_{n}$/Pt(111) multilayer heterostructures are decoupled for $n\ensuremath{\ge}3$. Unlike the property of magnetocrystalline anisotropy, DMI in this system is not strongly affected by stacking defects in the Co layer. The effect of graphene (Gr) is to invert the chirality of the vaccum/Co interfacial DMI, overall reducing the DMI of the heterostructure, which is nevertheless dominated by the strong spin-orbit coupling (SOC) of Pt. A spectral analysis in reciprocal space shows that DMI at both the Gr/Co and Co/Pt interfaces have similar contributions from the strongly hybridized SOC-split $d$ bands. Our analysis proves that the DMI in this family of Gr-capped $3d/5d$ metal heterostructures displays richer physics than what is expected from the prediction of single-band models, such as the Rashba DMI model.

Reciprocal Space Study of Brownian Yet Non-Gaussian Diffusion of Small Tracers in a Hard-Sphere Glass

The simultaneous presence of normal (Brownian) diffusion and non-Gaussian statistics of particle displacements has been identified as a recurring motif for a broad spectrum of physical and biological systems. While not yet fully understood, it is generally accepted that a key ingredient for observing this Brownian yet non-Gaussian (BNG) diffusion is that the environment hosting the particles appears stationary and homogenous on the small length and time scales, while displaying significant fluctuations on larger distances and/or longer time scales. To date, most of the experimental studies on systems displaying BNG diffusion have been performed in direct space, usually via a combination of optical microscopy and particle tracking to quantify the particle’s self-diffusion. Here, we demonstrate that a reciprocal space analysis of the density fluctuations caused by the particle motion as a function of the wave vector enables the investigation of BNG diffusion in situations where single-particle tracking is impossible. To accomplish this aim, we use confocal differential dynamic microscopy (ConDDM) to study the BNG dynamics of diluted sub-resolution tracers diffusing in a glassy matrix of larger hard spheres. We first elucidate the nontrivial connection between the tracer self-diffusion and collective relaxation of the resulting density fluctuations. We find that the experimentally determined intermediate scattering functions are in excellent agreement with the recent predictions of a “diffusing diffusivity” model of BNG diffusion, whose analytical predictions are available only in reciprocal space. Our results show that studying BNG diffusion in reciprocal space can be an invaluable strategy to access the fast, anomalous dynamics occurring at very small scales in crowded environments.

Nematic state of the FeSe superconductor

We study the crystal structure of the tetragonal iron selenide FeSe and its nematic phase transition to the low-temperature orthorhombic structure using synchrotron x-ray and neutron scattering analyzed in both real and reciprocal space. We show that in the local structure the orthorhombic distortion associated with the electronically driven nematic order is more pronounced at short length scales. It also survives up to temperatures above 90 K where reciprocal-space analysis suggests tetragonal symmetry. Additionally, the real-space pair distribution function analysis of the synchrotron x-ray diffraction data reveals a tiny broadening of the peaks corresponding to the nearest Fe-Fe, nearest Fe-Se, and the next-nearest Fe-Se bond distances as well as the tetrahedral torsion angles at a short length scale of 20 angstr\"om. This broadening appears below 20 K and is attributed to a pseudogap. However, we did not observe any further reduction in local symmetry below orthorhombic down to 3 K. Our results suggest that the superconducting gap anisotropy in FeSe is not associated with any symmetry-lowering short-range structural correlations.

Synchrotron total-scattering data applicable to dual-space structural analysis.

Synchrotron powder X-ray diffraction (PXRD) is a well established technique for investigating the atomic arrangement of crystalline materials. At modern beamlines, X-ray scattering data can be collected in a total-scattering setting, which additionally opens up the opportunity for direct-space structural analysis through the atomic pair distribution function (PDF). Modelling of PXRD and PDF data is typically carried out separately, but employing a concurrent structural model to both direct- and reciprocal-space data has the possibility to enhance total-scattering data analysis. However, total-scattering measurements applicable to such dual-space analyses are technically demanding. Recently, the technical demands have been fulfilled by a MYTHEN microstrip detector system (OHGI), which meets the stringent requirements for both techniques with respect to Q range, Q resolution and dynamic range. In the present study, we evaluate the quality of total-scattering data obtained with OHGI by separate direct- and reciprocal-space analysis of Si. Excellent agreement between structural parameters in both spaces is found, demonstrating that the total-scattering data from OHGI can be utilized in dual-space structural analysis e.g. for in situ and operando measurements.

Exciton dissociation in correlated molecular photocells

Electron-hole separation is a crucial process for the photocurrent generation in organic solar cells, where the interaction of molecular photocells and its influence on the energy conversion efficiency are studied by means of an attractive Hubbard model on an excitonic-state lattice, whose impurity sites originated from the local electron-hole interaction inhibit a reciprocal-space analysis. Alternatively, an independent channel transformation followed by a real-space renormalization approach were used to address a large lattice of excitonic states. The results show a remarkable growth of the quantum efficiency (Q) when a charge transfer between photocells is considered, and such high Q remains even in the strong electron-hole interaction region. The calculation of Q is based on the local density of states at such impurity sites, whose third and fourth spectral moments have been analytically confirmed. Finally, this study suggests an optimal distance between photocells to achieve high Q in accordance with experimental results.

Prediction of large magnetic anisotropy for non-rare-earth based permanent magnet of Fe16 − xMnxN2 alloys

Exploring the metastable magnetic nanostructures of Mn substituted α″-Fe16N2 with large saturation magnetization μ0MS, high Curie temperature TC and giant magnetic anisotropy are of technological merit as promising candidates for non-rare-earth based permanent magnets. Here, we present in-depth analysis for the structural and magnetic properties of Fe16 − xMnxN2 using first-principles calculations. We predict a large magnetocrystalline anisotropy energy (MAE) constant of K1 = 2.02 MJ/m3 for the Fe14Mn2N2 alloy, which is more than twice that of pristine Fe16N2. The underlying mechanism associated with boosting K1 is attributed to the local distortion of orbitals induced by Mn substitution. The MAE is also carefully analyzed in terms of reciprocal space analysis by employing the magnetic force theorem, revealing the regions in the Brillouin zone that are prominent for giving rise to MAE.

Analysis in reciprocal space of the band-pass filter effect in uniform and non-uniform grating couplers

A detailed study on the origin of the bandwidth reduction in the Coupling- Efficiency (CE) spectrum for non-uniform grating couplers (nu-GCs) due to a band-pass filter effect is reported. This effect is deeply investigated through Finite Difference Time Domain simulations (FDTD) in order to look at the spatial Fourier Transform (sFT) of the nu-GC’s emission, which gives the energies and the wavevectors sustained by the overall structure. We show: how to evaluate the bandwidth from the sFT and how the band-pass filter effect arises from the sFT affecting the CE spectrum giving a physical insight on the bandwidth’s reduction. Moreover, we also point out how the concept of photonic crystal bandstructure can be applied for such nu-GCs and used as a powerful tool to describe the dispersion of the light inside these non periodic structures. Finally, both the sFT and the bandstructure approaches are shown to give the same outcomes in terms of the bandwidth evaluation and to be in agreement with the CE spectra’s behaviours. The entire analysis is proposed also for uniform-GC in order to directly compare the two different types of structures.

Temperature dependent dielectric function and direct bandgap of Ge

The dielectric function of bulk Ge is determined between 0.5 and 6.3 eV in a temperature range of 10–738 K using spectroscopic ellipsometry. The authors provide the data in a tabulated format that can be interpolated as a function of photon energy and temperature using commercial software. Another focus of this paper lies on the analysis of critical points, in particular, on the investigation of the temperature dependence of the direct bandgap E0 and the critical point E0+Δ0, where Δ0 is the spin–orbit splitting. To explore the temperature dependence of critical points, the parameters that characterize their line shapes are calculated using three different techniques. First, the common method of numerically calculating and analyzing the second derivatives of the dielectric function works well for critical points at higher energies. Second, an analysis in reciprocal space by performing a discrete Fourier transform and analyzing the resulting Fourier coefficients yields values for the energies of E0 and E0+Δ0. Third, the energy determined from a parametric semiconductor model is shown as a function of temperature. The authors observe a temperature dependent redshift of the E0 and E0+Δ0 critical point energies as well as an increase in the broadening of E0 with temperature.

External removal of endpoint-discontinuity artifacts in the reciprocal-space analysis of spectra

We present a systematic method of removing endpoint-discontinuity artifacts in the Fourier analysis of spectral segments, enabling the more accurate extraction of information. This principal-component-removal approach differs from a previous version by using extrapolated (or extended) data outside rather than inside the spectral range. This not only allows coefficients to be accessed to the white-noise limit with no distortion of the segment, but also generates interpolated coefficients for improved analytic insight. Examples are provided.

X-ray characterization of the micromechanical response ahead of a propagating small fatigue crack in a Ni-based superalloy

The small fatigue crack (SFC) growth regime in polycrystalline alloys is complex due to the heterogeneity in the local micromechanical fields, which result in high variability in crack propagation directions and growth rates. In this study, we employ a suite of techniques, based on high-energy synchrotron-based X-ray experiments that allow us to track a nucleated crack, propagating through the bulk of a Ni-based superalloy specimen during cyclic loading. Absorption contrast tomography is used to resolve the intricate 3D crack morphology and spatial position of the crack front. Initial near-field high-energy X-ray diffraction microscopy (HEDM) is used for high-resolution characterization of the grain structure, elucidating grain orientations, shapes, and boundaries. Cyclic loading is periodically interrupted to conduct far-field HEDM to determine the centroid position, average orientation, and average lattice strain tensor for each grain within the volume of interest. Reciprocal space analysis is used to further examine the deformation state of grains that plasticize in the vicinity of the crack. Analysis of the local micromechanical state in the grains ahead of the crack front is used to rationalize the advancing small crack path and growth rate. Specifically, the most active slip system in a grain, determined by the maximum resolved shear stress, aligns with the crack growth direction; and the degree of microplasticity ahead of the crack tip helps to identify directions for potential occurrences of crack arrest or propagation. The findings suggest that both the slip system level stresses and microplasticity events within grains are necessary to get a complete description of the SFC progression. Further, this detailed dataset, produced by a suite of X-ray characterization techniques, can provide the necessary validation, at the appropriate length-scale, for SFC models.

Probing orientation information using 3-dimensional reciprocal space volume analysis.

The crystallographic texture of polycrystalline materials is the result of how these materials are processed and what external forces materials have experienced. Neutron and X-ray diffraction are standard methods to characterize global crystallographic textures. However, conventional neutron and X-ray texture analyses rely on pole figure inversion routines derived from intensity analysis of individual reflections or powder Rietveld analysis to reconstruct and model the orientation distribution from slices through reciprocal space. In this work, we describe an original approach to directly probe the crystallographic texture information of rolled aluminum from the intensity distribution in 3-dimensional reciprocal space volumes measured simultaneously. Using the TOPAZ time-of-flight Laue neutron diffractometer, reciprocal space analysis allowed determination of "pole spheres" with <1° angular resolution. These pole spheres are compared with reconstructed pole figures from classic texture analysis.

Thermodynamic and crystallographic model for anion uptake by hydrated calcium aluminate (AFm): an example of molybdenum

Amongst all cement phases, hydrated calcium aluminate (AFm) plays a major role in the retention of anionic species. Molybdenum (Mo), whose 93Mo isotope is considered a major steel activation product, will be released mainly under the form of MoO42− in a radioactive waste repository. Understanding its fate is of primary importance in a safety analysis of such disposal. This necessitates models that can both predict quantitatively the sorption of Mo by AFm and determine the nature of the sorption process (i.e., reversible adsorption or incorporation). This study investigated the Cl−/MoO42− exchange processes occurring in an AFm initially containing interlayer Cl in alkaline conditions using flow-through experiments. The evolution of the solid phase was characterized using an electron probe microanalyzer and synchrotron high-energy X-ray scattering. All data, together with their quantitative modeling, coherently indicated that Mo replaced Cl in the AFm interlayer. The structure of the interlayer is described with unprecedented atomic-scale detail based on a combination of real- and reciprocal-space analyses of total X-ray scattering data. In addition, modeling of several independent chemical experiments elucidated that Cl−/OH− exchange processes occur together with Cl−/MoO42− exchange. This competitive effect must be considered when determining the Cl−/MoO42− selectivity constant.

Magnetocrystalline anisotropy of Laves phase Fe2Ta1−xWx from first principles: Effect of 3d−5d hybridization

The magnetic properties of Fe$_2$Ta and Fe$_2$W in the hexagonal Laves phase are computed using density functional theory in the generalised gradient approximation, with the full potential linearised augmented plane wave method. The alloy Fe$_2$Ta$_{1-x}$W$_x$ is studied using the virtual crystal approximation to treat disorder. Fe$_2$Ta is found to be ferromagnetic with a saturation magnetization of $\mu_0 M_\text{s} = 0.66~\mathrm{T}$ while, in contrast to earlier computational work, Fe$_2$W is found to be ferrimagnetic with $\mu_0 M_\text{s} = 0.35~\mathrm{T}$. The transition from the ferri- to the ferromagnetic state occurs for $x \leq 0.1$. The magnetocrystalline anisotropy energy (MAE) is calculated to $1.25~\mathrm{MJ/m^3}$ for Fe$_2$Ta and $0.87~\mathrm{MJ/m^3}$ for Fe$_2$W. The MAE is found to be smaller for all values $x$ in Fe$_2$Ta$_{1-x}$W$_x$ than for the end compounds and it is negative (in-plane anisotropy) for $0.1 \leq x \leq 0.9$. The MAE is carefully analysed in terms of the electronic structure. Even though there are weak 5d contributions to the density of states at the Fermi energy in both end compounds, a reciprocal space analysis, using the magnetic force theorem, reveals that the MAE originates mainly from regions of the Brillouin zone with strong 3d-5d hybridisation near the Fermi energy. Perturbation theory and its applicability in relation to the MAE is discussed.

Kinetic Monte Carlo simulations of GaN homoepitaxy on c- and m-plane surfaces.

The surface orientation can have profound effects on the atomic-scale processes of crystal growth and is essential to such technologies as GaN-based light-emitting diodes and high-power electronics. We investigate the dependence of homoepitaxial growth mechanisms on the surface orientation of a hexagonal crystal using kinetic Monte Carlo simulations. To model GaN metal-organic vapor phase epitaxy, in which N species are supplied in excess, only Ga atoms on a hexagonal close-packed (HCP) lattice are considered. The results are thus potentially applicable to any HCP material. Growth behaviors on c-plane (0001) and m-plane (011¯0) surfaces are compared. We present a reciprocal space analysis of the surface morphology, which allows extraction of growth mode boundaries and direct comparison with surface X-ray diffraction experiments. For each orientation, we map the boundaries between 3-dimensional, layer-by-layer, and step flow growth modes as a function of temperature and growth rate. Two models for surface diffusion are used, which produce different effective Ehrlich-Schwoebel step-edge barriers and different adatom diffusion anisotropies on m-plane surfaces. Simulation results in agreement with observed GaN island morphologies and growth mode boundaries are obtained. These indicate that anisotropy of step edge energy, rather than adatom diffusion, is responsible for the elongated islands observed on m-plane surfaces. Island nucleation spacing obeys a power-law dependence on growth rate, with exponents of -0.24 and -0.29 for the m- and c-plane, respectively.

Concurrent determination of nanocrystal shape and amorphous phases in complex materials by diffraction scattering computed tomography

Advanced functional materials often contain multiple phases which are (nano)crystalline and/or amorphous. The spatial distribution of these phases and their properties, including nanocrystallite size and shape, often drives material function yet is difficult to obtain with current experimental techniques. This article describes the use of diffraction scattering computed tomography, which maps wide-angle scattering information onto sample space, to address this challenge. The wide-angle scattering signal contains information on both (nano)crystalline and amorphous phases. Rietveld refinement of reconstructed diffraction patterns is employed to determine anisotropic nanocrystal shapes. The background signal from refinements is used to identify contributing amorphous phases through multivariate curve resolution. Thus it is demonstrated that reciprocal space analysis in combination with diffraction scattering computed tomography is a very powerful tool for the complete analysis of complex multiphase materials such as energy devices.

Neutron Study of Multilevel Structures of Diamond Gels

The structure of a hydrogel consisting of diamond nanoparticles formed by the explosion method has been studied. Small angle neutron scattering has been used as a method for characterization of the gel. Joint approaches for data analysis in reciprocal and direct space have been developed to restore a multilevel structure. The pristine hydrogel of positively charged diamond particles (~5 nm in size, concentration ~5 wt %), even by four-fold dilution below its formation critical point, (C* ~ 4 wt %) retains practically the original structure where single particles are joined into small groups integrated into chain fractal-type aggregates creating a network. This indicates a local stability of the gel and means a transformation of continuous gel into a system of micro-domains suspended in water. A perfection of the diamond crystals’ facets was revealed that is of principal importance for the configuration of potentials, inducing the diamonds’ electrostatic attraction due to different electric charges of facets. It is distinguished from the results for the suspensions of diamonds in graphene shells that showed a deviation of scattering from Porod’s law.