Ewald Sphere Research Articles

Crystallography without crystals. I. The common-line method for assembling a three-dimensional diffraction volume from single-particle scattering

It is demonstrated that a common-line method can assemble a three-dimensional oversampled diffracted intensity distribution suitable for high-resolution structure solution from a set of measured two-dimensional diffraction patterns, as proposed in experiments with an X-ray free-electron laser (XFEL) [Neutze et al. (2000). Nature (London), 406, 752-757]. Even for a flat Ewald sphere, it is shown how the ambiguities due to Friedel's law may be overcome. The method breaks down for photon counts below about 10 per detector pixel, almost three orders of magnitude higher than expected for scattering by a 500 kDa protein with an XFEL beam focused to a 0.1 microm diameter spot. Even if 10(3) orientationally similar diffraction patterns could be identified and added to reach the requisite photon count per pixel, the need for about 10(6) orientational classes for high-resolution structure determination suggests that about 10(9) diffraction patterns must be recorded. Assuming pulse and readout rates of approximately 100 Hz, such measurements would require approximately 10(7) s, i.e. several months of continuous beam time.

On the intersection of grating truncation rods with the Ewald sphere studied by grazing-incidence small-angle X-ray scattering

In this work a grating made of lines having a height of 55 nm and a period of 450 nm has been characterized by grazing-incidence small-angle X-ray scattering (GISAXS). The GISAXS patterns are characterized by a series of spots corresponding to the intersection of the Ewald sphere with the grating truncation rods (GTRs). When the grating lines are almost parallel to the direct beam, the location of these spots is very sensitive to any change in the azimuthal angle. The precise location of the intersection of the GTRs with the Ewald sphere can be calculated for any azimuthal angle. From this analysis we can estimate the statistical coverage of the grating, its period and the width of the lines. In addition, the anisotropy of the width of the spots in theqzdirection is interpreted in terms of wavelength spread and angular divergence of the incident beam.

Holographic scattering in SiO_2 nanoparticle-dispersed photopolymer films

We describe an experimental study of holographic (coherent) scattering due to parasitic noise gratings recorded in SiO2 nanoparticle-dispersed photopolymer films. Dependences of film thickness and nanoparticle concentration on holographic scattering losses are evaluated. It is shown that the geometric feature of the holographic scattering pattern in the two-beam recording setup can be explained by the Ewald sphere construction. It is found that holographic scattering becomes noticeable when a film with nanoparticle concentrations higher than 10 vol.% is thicker than 100 microm. The significance of holographic scattering in the characterization of a volume grating recorded in a thick (>100 microm) nanoparticle-dispersed photopolymer film is also discussed.

Angular directivity of diffracted wave in Bragg-mismatched readout of volume holographic gratings

We investigated theoretically and experimentally angular directivity of a diffracted beam in volume holographic gratings. We measured the angular direction of the diffracted beam as a function of Bragg-angle deviation of the read beam and showed that the experimental result agrees well with the Ewald sphere vector model (ESVM). We also showed that the Kogelnik’s coupled-wave theory (CWT) is correct in predicting the diffraction efficiency, but is incomplete in its description of the direction of the diffracted wave. We show that the ESVM and the CWT theories taken together produce a self-consistent mathematical model of wave propagation inside the gratings that is confirmed with experimental results. The proper model for the direction of the output beam as presented here is important in developing theoretical models of image propagation through thick gratings for holographic imaging and correlation applications.

Structure reconstruction with Ewald sphere curvature considerations

An approach is proposed to determine the structure of a three-dimensional non-crystallineobject from a single two-dimensional x-ray scattering image. If x-ray absorption is ignored,this method, unlike what has been reported in the literature, always generates a structureof real electron density, even in the case that the Ewald sphere curvature is large. Inaddition, if the original object is thin, the reconstructed object is the integrated electrondensity of the original one along a path which makes an angle with the incoming x-rays(z-axis) in proportion to the maximum x-ray wavevector transfer magnitude along the axis,revealing the object’s depth information. It is demonstrated mathematically that the blursor ripples associated with a reconstructed object are due to a resolution function effect.Reconstructed structures of a hypothetical three-dimensional object are shown to agreewith the theoretical predictions.

Conical light scattering in strontium barium niobate crystals related to an intrinsiccomposition inhomogeneity

Conical light scattering is uncovered in poly- and mono-domain, nominally pure andEu-doped strontium barium niobate (SBN) crystals over a wide temperature regime. Theappearance of two scattering cones, a scattering line and a corona is observed and can beexplained comprehensively within the Ewald sphere concept. Photorefraction, scatteringfrom domain boundaries or from growth striations can be excluded from explaining theorigin of the scattering. It is shown that the temperature-persistent scattering process isrelated to a growth-induced seeding rod, i.e. a composition inhomogeneity primarilylocalized at the centre of the SBN sample. The rod is directed parallel to thec axis and yields a refractive-index inhomogeneity with spatial frequencies on the micro-scale.

Inelastic x-ray scattering from phonons under multibeam conditions

We report on an experimental observation of a previously neglected multibeam contribution to the inelastic x-ray scattering cross section. Its manifestation is a substantial modification of the apparent phonon selection rules when two (or more) reciprocal lattice points are simultaneously intercepted by the Ewald sphere. The observed multibeam contributions can be treated semi-quantitatively in the frame of Renninger's 'simplest approach'. A few corollaries, relevant for experimental work on inelastic scattering from phonons, are presented.

Structure of framboidal pyrite: a single crystal X-ray diffraction study

The crystallographic nature of pyrite framboids, self-assembled multicrystalline pyrite, was investigated by single crystal X-ray diffraction. The results show that even morphologically well-organised framboids are not extreme examples of single crystals but are microcrystalline aggregates. Two dimensional diffraction patterns obtained from well-organised framboids composed of relatively few (∼1000) microcrystals, are single-crystal like but detailed analysis shows that they are actually produced by the superimposition of XRD reflections from similarly, but not identically, oriented microcrystals. The number of coherent crystallographic orientations of pyrite crystals estimated from a complete Ewald sphere data set obtained from a larger single framboid, were found to be approximately four to five times fewer than the number of the microcrystals that are actually contained in the framboid. The result shows the presence of multiple single crystal-like domains that are arranged in similar crystallographic orientations. The results show that framboids are not formed by a process in which crystallographic control is a primary factor. Framboid self-assembly is probably caused by the aggregation of a large number of similarly sized and shaped pyrite microcrystals that are nucleated in a limited volume. Self-organisation occurs as these microcrystals adopt a minimum energy configuration.

Geometric aspects of lattice contrast visibility in nanocrystalline materials using HAADF STEM

In conventional high-resolution transmission electron microscopy (HRTEM) imaging on nanopolycrystalline materials , the surprising abundance of lattice fringe contrast has been readily explained by the geometric relationship between the Ewald sphere and the streaking of reciprocal lattice spots from small crystals. However, in high-angle annular dark field (HAADF) imaging in scanning transmission electron microscopy (STEM) the same interference considerations do not apply. We present a new method to quantify the probability of seeing lattice contrast from a set of random crystals by estimating as a function of instrumental resolution, the range of angular misorientations from each crystallographic pole, which can give rise to lattice contrast. The maximum angular deviation for common HAADF imaging conditions has been experimentally determined by tilting perfect silicon off pole. The visibility of single and multiple sets of fringes has been successfully predicted for a range of microscope resolutions and for two materials of different unit cell sizes. As many as 70% of randomly oriented nanocrystals can be expected to show fringes in an aberration-corrected STEM.

Holographic scattering in photopolymer-dispersed liquid crystals

Strong polarization-conserving holographic scattering was observed in a photopolymer-dispersed liquid crystal film fabricated from the UV curable mixture of commercially available constituents. During the photopolymerization process a bright corona of diffracted light evolves around the pump beam. The intensity of the rotationally symmetric light distribution increases upon exposure. By rotating the sample, two characteristic diffraction rings appear which can be explained by the Ewald sphere construction. Our results demonstrate that the associated parasitic holograms are very pronounced. Hence, their presence must be accounted for whenever preparing and utilizing holographic polymer-dispersed liquid crystals in any application.

Resonant soft X-ray diffraction –in extremis

The use of softer-energy X-rays produced by synchrotron radiation for diffraction is an area of current interest. In this paper, experiments exploiting resonant scattering at the L absorption edges of 3d transition metal elements are reported. Such energies, typically 500-1000 eV, are at the extreme limit of soft X-ray diffraction where absorption effects are so severe that the sample and diffractometer must be placed in a windowless high-vacuum vessel. In addition, the Ewald sphere is so small as to likely contain, at most, only a single Bragg reflection. Advantages of using such radiation for the study of weak diffraction effects such as anomalous scattering, charge ordering, magnetic diffraction and orbital ordering are reported.

A diagonalized multilevel fast multipole method with spherical harmonics expansion of the k-space Integrals

Diagonalization of the fast multipole method (FMM) for the Helmholtz equation is usually achieved by expanding the multipole representation in propagating plane waves. The resulting k-space integral over the Ewald sphere is numerically evaluated. Storing the k-space quadrature samples of the method of moments (MoM) basis functions constitutes a large portion of the overall memory requirements of the resulting algorithm for solving the integral equations of scattering and radiation problems. In this paper, it is proposed to expand the k-space representation of the basis functions by spherical harmonics in order to reduce the sampling redundancy introduced by numerical quadrature rules. Aggregations, plane wave translations, and disaggregations in the realized multilevel fast multipole method (MLFMM) are carried out using the k-space samples of a numerical quadrature rule. However, the incoming plane waves on the finest MLFMM level are expanded in spherical harmonics again. Thus, due to the orthonormality of spherical harmonics, the testing integrals for the individual testing functions are simplified into series over products of spherical harmonics expansion coefficients. Overall, the resulting MLFMM can save a considerable amount of memory without compromising accuracy and numerical speed.

Efficient MLFMA, RPFMA, and FAFFA Algorithms for EM Scattering by Very Large Structures

Based on the addition theorem, the principle of a multilevel ray-propagation fast multipole algorithm (RPFMA) and fast far-field approximation (FAFFA) has been demonstrated for three-dimensional (3-D) electromagnetic scattering problems. From a rigorous mathematical derivation, the relation among RPFMA, FAFFA, and a conventional multilevel fast multipole algorithm (MLFMA) has been clearly stated. For very large-scale problems, the translation between groups in the conventional MLFMA is expensive because the translator is defined on an Ewald sphere with many sampling k/spl circ/ directions. When two groups are well separated, the translation can be simplified using RPFMA, where only a few sampling k/spl circ/ directions are required within a cone zone on the Ewald sphere. When two groups are in the far-field region, the translation can be further simplified by using FAFFA where only a single k/spl circ/ is involved in the translator along the ray-propagation direction. Combining RPFMA and FAFFA with MLFMA, three algorithms RPFMA-MLFMA, FAFFA-MLFMA, and RPFMA-FAFFA-MLFMA have been developed, which are more efficient than the conventional MLFMA in 3-D electromagnetic scattering and radiation for very large structures. Numerical results are given to verify the efficiency of the algorithms.

Synchrotron X-Ray Topography of Lattice Undulation of Bonded Silicon-On-Insulator Wafers

Lattice undulation of Silicon-on-Insulator (SOI) layers of bonded SOI wafers was observed by synchrotron X-ray topography. Patterns observed on topographs depended on the thickness of the SOI layer, the camera distance between a specimen and an X-ray film, and the diffraction geometry of the Laue and Bragg cases. The dependence was interpreted as the effects of the geometrical relation in reciprocal space among the Ewald sphere, the reciprocal lattice vector, and the surface normal direction. To confirm the origin of the pattern formation, the topographic images were simulated in the framework of the kinematical diffraction theory. Based on the simulation, it was found that a granular pattern observed in the 115 Bragg case was due to the divergence/convergence effect of X-rays diffracted from the undulated SOI layer.

A sparse data fast fourier transform (SDFFT)

A multilevel algorithm that efficiently Fourier transforms sparse spatial data to sparse spectral data with controllable error is presented. The algorithm termed "sparse data fast Fourier transform" (SDFFT) is particularly useful for signal processing applications where only part of the k-space is to be computed - regardless of whether it is a regular region like an angular section of the Ewald sphere or it consists of completely arbitrary points scattered in the k-space. In addition, like the various nonuniform fast Fourier transforms, the O(NlogN) algorithm can deal with a sparse, nonuniform spatial domain. In this paper, the parabolic reflector antenna problem is studied as an example to demonstrate its use in the computation of far-field patterns due to arbitrary aperture antennas and antenna arrays. The algorithm is also promising for various applications such as backprojection tomography, diffraction tomography, and synthetic aperture radar imaging.

A Born scatterer in an acoustical waveguide.

Scattering from a Born inhomogeneity in a homogeneous, acoustical waveguide is considered and results compared to the corresponding scattering in a homogeneous, unbounded medium. It is found that the Ewald sphere in the unbounded medium case is replaced by a Ewald "strip" in a waveguide, the strip consisting of many individual Ewald spheres embedded in a ball the radius of which is twice that of an individual sphere. The physics behind the Ewald strip is discussed along with the implications of waveguide Born data.

Improving triplet-phase accuracy by symmetry observations in reference-beam diffraction measurements.

The reference-beam X-ray diffraction technique allows parallel measurements of many three-beam interference profiles in a modified rotating-crystal experiment. Because of the unique symmetry relations that exist in the reference-beam geometry, a complete 360 degrees rotation leads to a fourfold redundancy of any given triplet phase delta(H) involving primary reflection H, reference reflection G and coupling reflection H-G, measured in a data set. Two such redundant measurements correspond to simple duplications with the reciprocal node H going into and coming out of the Ewald sphere, while the other two correspond to the Friedel mate G-H of the coupling reflection H-G passing through the Ewald sphere in a similar fashion. It is shown that substantial differences may exist between the H and the G-H cases, with an enhanced triplet-phase interference effect in one of them. These symmetry-related observations, coupled with inverse-beam triplet measurements, would improve the phase measurement accuracy of the reference-beam diffraction technique.

Structure and Dynamics — An Atomic View of Materials

Preface 1. Introduction 2. Structure of materials 3. Formal description of crystal structures 4. The reciprocal lattice 5. Atomic bonding in crystals 6. Diffraction 7. Physical properties 8. Lattice dynamics 10. Experimental methods for measurements of vibrational frequencies 11. Anharmonic interactions 12. Displacive phase transitions A. Real crystals! B. Fourier analysis C. Schoenflies representation of the point groups D. Rhombohedral, trigonal and hexagonal unit cells E. Space groups F. Lattice energy minimization G. Some note on the variational theorem H. Ewald Sphere I. The Wilson Plot J. Diffraction from isotropic materials K. Calculation of physical properties L. Partition function: some key results M. Lattice sums N. Mean-square atomic displacement and temperature factors

High‐Energy X‐Rays: A tool for Advanced Bulk Investigations in Materials Science and Physics

High‐energy X‐rays between 30 keV and 1MeV, such as provided by modern synchrotron radiation sources as the ESRF and HASYLAB, bear the advantage of high penetration into most materials. Even heavy element compositions can be accessed in their volume. The range of applications is huge and spreads from nuclear spectroscopy to the characterization of metal extrusion under industrial conditions. This article compiles an overview over the most common instrumental diffraction techniques.Modern two‐dimensional detectors are used to obtain rapid overviews in reciprocal space. For example, diffuse scattering investigations benefit from the very flat Ewald sphere as compared to low energies, which allow mapping of several Brillouin zones within one single shot. Diffraction profiles from liquids or amorphous materials can be recorded easily. For materials science purposes, whole sets of Debye–Scherrer rings are registered onto the detector, their diameters and eccentricities or their intensity distribution along the rings relating to anisotropic strain or texture measurements, respectively. At this point we stress the resolution of this technique which has to be carefully taken into account when working on a second generation synchrotron source.Energy‐dispersive studies of local residual strain can be studied by a dedicated three‐circle diffractometer which allows accurately to adjust the scattering angle from a defined gauge volume.Triple axis diffractometry and reciprocal space mapping is introduced and can be employed for highest resolution purposes on single crystal characterization, even under heavy and dense sample environments. Thus, the perfection of single crystals can be mapped and strain fields and superstructures as introduced by the modulation from ultrasonic waves into crystals or epitaxially grown Si/Ge layers can be investigated in detail. Phase transitions as magnetic ordering can be studied directly or through its coupling to the crystal lattice. Time resolved studies are performed stroboscopically from a sub‐nanosecond to a second time scale.The combination of these techniques is a strong issue for the construction and development of future instruments.

Geometry-independent intensity correction factors for grazing-incidence diffraction

A geometry-independent approach to a unified description of intensity corrections factors for the two most common surface scattering geometries is presented. Simple analytical formulas relate the representations of the Lorentz factor, polarization factor, area factor, and rod interception factor in a surface reference frame to the actual diffractometer circles. An analytical formula for the area correction for very large exit angles, as can be achieved on modern diffractometers, is derived. A test data set shows that the derived formulas provide a good description of the scattering intensity up to the edge of the Ewald sphere.