Calculated Diffraction Patterns Research Articles

A new approach for the numerical calculation of diffraction patterns using starshades

We studied the imaging of exoplanetary systems using starshades, which are externally occulted coronagraphs in space. We provide a new method for precisely evaluating the stray light due to the star and a rapid calculation of the point spread functions in the presence of vignetting effects from the external occulter. Our study used shaped occulter configurations published in the literature, in particular, the SISTER NI2 and NW2 systems. The wavefront at the telescope aperture was computed using the classic Fresnel filtering method. The Fourier transform of the occulter was obtained with the highest possible precision using an approach initially developed for radio antennas, known as the polygonal shape factor. We show that the Fresnel diffraction for a finite spatial field operates at very low frequencies only, and that it is sufficient to calculate the Fourier transforms there. Diffraction patterns computed numerically fully agree with theoretical predictions. The central parts of diffractions of petal and apodized occulters are identical over a large central area that increases in size with the number of petals. These diffraction patterns are used to compute the point spread functions. We computed the stray light for a non-point source star; this shows that starshades are not sensitive to star leakage, with a star diameter limit for a given configuration. We also computed signal-to-noise ratios for a perfect experiment limited by photon noise.

Verification of the McStas code using two double axis neutron diffractometers

The Monte Carlo ray-tracing simulation package McStas is verified by modeling two double axis neutron diffractometers, KARL and KANDI-II, located at the Israel Research reactors IRR-1 and IRR-2, respectively. The advantage of simulating these instruments is their simple design and small number of components. We focus on the simulation of a Soller slit collimator as well as Pyrolytic Graphite and Copper single crystal monochromators. The simulation is compared to neutron flux measurements, performed via activation analysis along the neutron beam as well as observed diffraction patterns from different powder samples. Calculated diffraction patterns are found to agree with observations to within 20%–50% in reflection peak width, while calculated attenuation of the neutron beam agrees to within 20% when the entire instrument is considered, and can show a discrepancy of up to 60% for individual components. Finally, a possible optimization of the neutron flux in the KARL diffractometer is presented.

Structural Aspects of “Memory Effect” for MgGa LDHs: New Data Obtained by Simulation of XRD Patterns for 1D Disordered Crystals

Simulation of diffraction patterns for 1D disordered crystals was used to investigate the structure of the initial CO32− containing MgGa LDHs with a different Mg2+/(Mg2+ + Ga3+) ratio equal to 0.67, 0.75, and 0.80; mixed oxides obtained by calcination of LDHs at a temperature of 550 °C; and the hydroxide obtained by hydration of MgGa oxide with the Mg2+ content of 0.80. The initial LDHs contain lamellar inclusions of manasseite structure (polytype 2H1) in the hydrotalcite structure (3R1). A loss of water at 200 °C leads to the formation of a metastable dehydrated phase where layers are packed, as in polytypes 3R2 and 1H, with turbostratic disorder. The structure of mixed oxides is also layered and consists of periclase-like octahedral layers and spinel-like octahedral-tetrahedral layers. Hydration of the oxides results in restoring the initial layered hydrotalcite structure (polytype 3R1) for Mg2+ mole fractions 0.67 and 0.75. For the Mg2+ content of 0.80, the phase composition is represented by the hydroxide with hydrotalcite structure and the layered mixed hydroxide with the alternation of hydrotalcite and brucite lamellar domains, which was also revealed by calculation of diffraction patterns using models of 1D disordered crystals.

Measurements of elastic light-scattering patterns and images of single, oriented, optically trapped particles

We demonstrate a method for recording 2D forward-scattering patterns from optically trapped single airborne particles at multiple angular orientations. We collected images of trapped particles simultaneously with their scattering patterns to observe changes in the patterns with orientation. We present measurements of particles with wide-ranging morphologies from nearly spherical to highly irregular and with various compositions. The highly irregularly shaped particles have scattering patterns with spots of high intensity whose locations change with particle orientation. Additionally, measurements of a manufactured polystyrene spheroid of a known size are compared with calculated diffraction patterns. Such controlled-orientation measurements have the potential to help develop algorithms to characterize airborne aerosol particles using morphological information inferred from remote-sensing or other elastic light scattering measurements.

Molecular packing analysis of the crystal smectic E phase of a benzothieno-benzothiophene derivative by a combined experimental / computational approach

ABSTRACT The molecule 2-decyl-7-phenyl[1]benzothieno[3,2-b][1]benzothiophene has gained a lot of attention since high charge carrier mobility was observed in thin film transistors. Its thermotropic liquid crystalline states may play an important role in the thin film formation since the smectic A and the crystal smectic E phase (SmE) are claimed to be pre-stages of the final bulk structure. To understand the phase diversity, structural characterisation of solution processed thin films is performed by X-ray diffraction in the complete temperature range up to the isotropic state at 240°C. The diffraction pattern of the SmE phase is analysed in detail. Peak broadening analysis reveals that the crystallographic order across the smectic layers is larger than the order along the smectic layers. A combined experimental and computational approach is used to determine the molecular packing within the SmE phase. It leads to a number of different packing motifs. A comparison of the calculated diffraction pattern with the experimental results reveals that nano-segregation is present within the SmE phase. Energy consideration clearly favours a herringbone arrangement of the aromatic units. The nano-segregation within the SmE phase with herringbone packing of the aromatic units is accompanied with interdigitation of side chains from neighbouringherringbone layers.

Diffraction by multipoles in a 5d2 rhenium double perovskite

A recent polarized neutron diffraction experiment on the $5{d}^{2}$ rhenium double perovskite ${\mathrm{Ba}}_{2}\mathrm{YRe}{\mathrm{O}}_{6}$ held at a low temperature uncovered weak magnetic diffraction peaks. Data analysis inferred a significantly reduced Re dipole moment, and long-range order compatible with an antiferromagnetic, noncollinear motif. To interpret the experimental findings, we present a model wave function for Re ions derived from the crystal field potential, Coulomb interaction, and spin-orbit coupling that fully respects the symmetry of the low-temperature ordered state. It is used to calculate in analytic form all multipole moments visible in neutron and resonance enhanced x-ray diffraction. A minimal model consistent with available neutron diffraction data predicts significant multipolar moments up to the hexadecapole and, in particular, a dominant chargelike quadrupole moment. Calculated diffraction patterns embrace single crystal x-ray diffraction at the Re $L$ edge, and renewed neutron diffraction, to probe the presumed underlying multipolar order.

Generalized far-field integral.

The propagation of light in homogeneous media is a crucial technology in optical modeling and design as it constitutes a part of the vast majority of optical systems. Any improvements in accuracy and speed are therefore helpful. The far-field integral is one of the most widely used tools to calculate diffraction patterns. As a general rule, this approximate method requires the observation plane located in the far-field region, i.e., a very considerable propagation distance. Only in the well-designed (namely aberration-free) optical system does the far-field integral not suffer from the limitation of the large distance. Otherwise, the far-field integral cannot provide accurate results. In the present work, we generalize the far-field integral to a more general concept with a much more flexible application scope, which allows for the inclusion of aberrations as well. Finally, as an essential part of this generalization, the propagation to arbitrarily oriented planes is also taken into account.

Diffraction on a Microbubble and the Morphology of the Silicon Surface Irradiated through Glycerol by a Pair of Femtosecond Laser Pulses

The effect of two successive laser pulses on silicon placed in glycerol has been studied experimentally and numerically with electromagnetic, hydrodynamic, and atomistic simulation programs. It has been shown that a microbubble in the liquid is formed on the surface after the first pulse; then, the second pulse whose width is comparable with the diameter of the microbubble is diffracted on this microbubble. The calculated diffraction pattern and light intensity distribution on the silicon surface indicate that the maximum intensity at the diffraction peaks can be noticeably higher than the intensity on the axis of the incident Gaussian beam. An increase in the intensity concentrated in one bright narrow ring around the microbubble results in the formation of a characteristic groove surrounded by ridges on silicon. The molecular dynamics simulation has shown that intense heating at the diffraction peak is responsible for the melting and displacement of the melt from the center of heating. This leads to the formation of grooves with ridges having a profile similar to that measured in the experiment.

Impact of frequency-dependent spherical aberration in the focusing of ultrashort pulses.

In this paper, the temporal and spatial intensity pulse distributions are calculated around the focal region of an optical system using a combination of ray tracing and a wave propagation method. We analyze how to measure the width of the intensity pulse distributions to estimate pulse duration and spot size in order to study the impact of the variation of spherical aberration with frequency in a pulse on the intensity distributions. Two experimental techniques used in the laboratory are also modeled: the knife-edge test to measure spatial distribution and the intensity autocorrelation technique to measure the temporal distribution. We use two measuring criteria, the full-width half-maximum (FWHM) and standard deviation (σ), to compare the spatial and temporal intensity distributions of the calculated diffraction patterns and those obtained from the simulated experimental techniques. We show that the FWHM is not a good criterion, since it gives different results in the measured intensity distributions in time and space when they are measured directly from the theoretical modeling and when they are measured from the modeled experimental techniques used in the laboratory. The standard deviation, however, is a consistent criterion, giving the same results for the calculated intensity distributions and the modeled experiments.

Synthesis and Characterization of Polycrystalline Fe<sub>1.025</sub>Se Prepared Using Mechanical Alloying Followed by Powder in a Sealed Tube Method

In this work, preparation of Fe1.025Se samples was done using a solid-state reaction. Iron and selenium powder were used as raw materials. The raw materials were milled using a shaker mill for 5 hours after weighed in an atomic ratio of Fe:Se=1.025:1. To find out the formation of polycrystalline Fe1.025Se, the milled powder was then compacted in a stainless-steel tube and sintered at a temperature of 355°C (for sample 1) and 745°C (for sample 2) for 6 hours. All samples were sintered in the air atmosphere and cooled rapidly in the air. XRD was used to analyze the phase formation. Based on the diffraction pattern calculation, the sintered sample at a temperature of 745°C has a higher mass fraction of the tetragonal phase than the sample sintered at 355°C. The lattice parameters of the sample sintered at a temperature of 745°C are a = 3.7688 Å and c = 5.5236 Å. According to the resistivity measurement results, it appears that the sample sintered at 745°C has a critical temperature onset of ~13.28 K.

Fast and exact diffraction integral calculus: A comparison with fresnel approximation

In Lens-less Digital Holographic Microscopy1, 2, 3, amplitude and phase of an object can be recovered from the irradiance of a diffraction pattern. To do this, an iterative algorithm with constrictions is applied, which is based on the calculation of diffraction patterns between two fixed planes. In one dimension and for N pixels on each straight line segment, we must calculate N2 values of the impulse response function, h, in order to evaluate the optical field on N pixels along another straight line segment. By using translation and permutation symmetries of h, and without any approximation, in this paper we will show that to calculate the diffraction pattern over N pixels we need only N values of h. Adding to this, if iterations are applied between two pixel lines, the N values of h are calculated only one time, and then we achieved a significative reduction in calculation time. Finally, our exact calculations were compared with Fresnel diffraction patterns reported by Goodman4, and we found that when distance between object and diffraction planes diminishes then the diffraction pattern irradiance differences are increased up to 53 % with respect to Fresnel approximation.

Universal Scherrer equation for graphene fragments

Graphene fragments spanning a wide range of size and shape were studied computationally using the Debye scattering equation. The calculated diffraction patterns were analysed using the Scherrer equation to infer the fragment size, La. Comparison with the known fragment sizes reveals a strong affine relationship between La and the Scherrer quantity λ/(Bcosθ). To preserve this relationship, we propose modifying the Scherrer equation to include an empirical additive constant. Our approach solves the well-known problem of size-dependence in the shape factor and yields a universal expression by defining La as the square-root of the fragment area. The relationship between observed diffraction peak positions and unit cell parameters is also discussed.

Vapour-assisted roll-to-roll nanoimprinting of micropillars on nanocellulose films

Here we demonstrate a straightforward vapour-assisted roll-to-roll nanoimprinting lithography (R2RVANIL) patterning method to biobased nanocellulosic films with the aim to fabricate microstructured films with diffraction effect. In the R2RVANIL technique a roll with pillar structures is softly pressed against nanocellulose film in the presence of hot water vapour replicating the pattern onto the film. The printed pillar structures are compared with the structures produced using traditional thermal R2RNIL. The manufacturing method is demonstrated using optically transparent TEMPO-oxidised cellulose nanofibrils (TEMPO CNF). TEMPO CNF films possess high tensile strength and modulus whereas the tensile strain is low, they are very hygroscopic - a property which improves material processing at elevated humidities, thus overcoming a lack of a clear softening point. In this context, the water molecules are acting as plasticising molecules thereby softening the relative brittle nanocellulose film structures. Patterned pillars were achieved due to the swelling and softening of the film which was contacted with moist air. R2RVANIL method appears to be more gentle and surface properties such as roughness are different from the ones achieved via conventional thermal NIL method. The diffraction grating is demonstrated using laser beam compared to calculated diffraction patterns. Water vapour assisted patterning methods are beneficial, for example, in the future optics and electronics where films cannot be processed under high pressure and, on the other hand, high production volumes are needed.

A routine for the determination of the microstructure of stacking-faulted nickel cobalt aluminium hydroxide precursors for lithium nickel cobalt aluminium oxide battery materials.

The microstructures of six stacking-faulted industrially produced cobalt- and aluminium-bearing nickel layered double hydroxide (LDH) samples that are used as precursors for Li(Ni1-x-yCo x Al y )O2 battery materials were investigated. Shifts from the brucite-type (AγB)□(AγB)□ stacking pattern to the CdCl2-type (AγB)□(CβA)□(BαC)□ and the CrOOH-type (BγA)□(AβC)□(CαB)□ stacking order, as well as random intercalation of water molecules and carbonate ions, were found to be the main features of the microstructures. A recursive routine for generating and averaging supercells of stacking-faulted layered substances implemented in the TOPAS software was used to calculate diffraction patterns of the LDH phases as a function of the degree of faulting and to refine them against the measured diffraction data. The microstructures of the precursor materials were described by a model containing three parameters: transition probabilities for generating CdCl2-type and CrOOH-type faults and a transition probability for the random intercalation of water/carbonate layers. Automated series of simulations and refinements were performed, in which the transition probabilities were modified incrementally and thus the microstructures optimized by a grid search. All samples were found to exhibit the same fraction of CdCl2-type and CrOOH-type stacking faults, which indicates that they have identical Ni, Co and Al contents. Different degrees of interstratification faulting were determined, which could be correlated to different heights of intercalation-water-related mass-loss steps in the thermal analyses.

Rietveld refinement, morphological, vibrational, Raman, optical and electrical properties of Ca/Mn co-doped BiFeO3

Ca and Mn co-substituted Bi1-xCaxFe1-xMnxO3 (x = 0 and 0.10) polycrystalline ceramic samples were prepared using conventional solid state reaction method. The structural, morphological, vibrational, Raman, optical and electrical properties of co-doped and pristine compounds were carried out. The formation of the compounds was analyzed using RT (room temperature) X-ray diffraction patterns. These diffraction patterns were further refined using Rietveld refinement via FULLPROF software and were found to fit in rhombohedral crystalline structure with space group R3c. The observed and calculated diffraction patterns are in good agreement with each other. The average crystallite size has been calculated using Scherrer formula. The same has also been calculated using W–H plot along with the strain developed in the lattice due to substitution. The average crystallite size has further been used to calculate the dislocation density and number of crystallites in a given pellet of area 38.56 mm2 and 63.92 mm2 for Bi1-xCaxFe1-xMnxO3 with x = 0 and 0.10 respectively. The 2D and 3D surface morphology of prepared samples were analyzed using Atomic Force Microscopy (AFM). The average roughness and average crystallite size was also calculated using SPIP software. FTIR spectra in transmittance mode give a brief idea of Vibrational modes around Fe–O–Fe stretching bonds. The appearance of active modes in Raman spectroscopy at RT also confirmed the rhombohedral structure with R3c space group. The Optical property of the materials has been studied using UV-VIS-NIR spectroscopy. The optical energy band gap of the pristine compound was found to be 2.04eV which is comparable to that of the literature value but the co-doping of Ca and Mn in pristine compound lowered (1.86eV) the optical band gap. The dc conductivity was analyzed and the activation energies were calculated using Arrhenius relation. The Current-Voltage (I–V) characteristic as a function of temperature was also analyzed which shows the semiconducting behaviour of the compounds. The results are discussed in detail.

Enhanced surface sensitivity of X-ray photoelectron holography through the example of Bi2Te3(1 1 1) surface

X-ray photoelectron holography (XPH) is new tool in surface science capable to visualize in real space coordination of the atom of interest (emitter). To study surface phenomena the ultimate and variable (in the range of 1–5 ML) surface sensitivity is needed. Here we discuss possibilities to improve surface sensitivity of XPH by the example of Bi2Se3 (1 1 1) surface where different coordinations of the same kind of atoms are present in different layers. To reach ultimate surface sensitivity we used both lower electron kinetic energy and partial analysis of hologram related to grazing angle photoemission. Using calculations of diffraction patterns based on EDAC code and reverse reconstruction of the real space within SPEA-MEM formalism we evaluate the contribution of each atomic layer into diffraction pattern and found that the surface relaxation could be directly visualised at the kinetic energy less than 100 eV, whereas the contribution of the second quintuple layer can be vanished at 200 eV or below. The same effect can be achieved by partial analysis of hologram at grazing detection angles more than 55° to the surface normal. This approach enables broader practical applications of the XPH to various materials in applied surface science.

Three methods for determining the type of LPSO-phase in an Mg–Zn–Y alloy by transmission electron microscopy

ABSTRACTThree methods for determining the type of LPSO-phases in an Mg–Zn–Y alloy are described: electron diffraction and two methods based on high-resolution image analysis. The diffraction method consists in counting the diffraction spot between (000)Mg and (00.1)Mg reflections; comparison with the calculated diffraction patterns allows determining the mixed types of LPSO-phase. The routine operations with high-resolution imaging using Fourier filtering and building intensity profiles in Digital Micrograph are described in detail. Recommendations on the preferred orientation of the sample for each method are given. The procedure for identifying (00.2)Mg layers using template figures and a simplified search for stacking faults in an image using a Fourier mask for reflection is described. The advantages, possibilities and limitations of each method are discussed.

Fraunhofer diffraction of a Gaussian beam by a four-sector binary grating with a half period fringes shift between adjacent sectors

The study of Fraunhofer diffraction of a Gaussian laser beam by a computer-generated, binary four-sector grating (FSG) is presented. In the four equal angular sectors of the FSG, parts of a binary rectilinear grating are embedded in such a way that, two neighboring parts are shifted by a half spatial rectilinear grating period. Analytical expressions, describing the diffracted wave field amplitude and intensity distributions in the focal plane of a converging lens, are derived. The zeroth-diffraction-order beam is a Gaussian beam whose transverse cross section has a reduced Gaussian radius. The higher-diffraction-order beam transverse intensity profiles have four bright spots divided by dark crossed lines and a central dark, non-vortex core. They can be described as cosine-Laguerre–Gaussian Kummer beams. The numerically calculated diffraction patterns, as well as the interference patterns of the diffracted beam with a slightly inclined plane wave, are supported by experimentally obtained ones. In both investigations (theoretical and experimental) the interferograms confirm the non-vortex nature of the diffraction spots.

Rietveld Analysis on X-Ray Diffraction Of South Kalimantan Kaolin Clays

At least 13 million tons of kaolin claystone lie in several regencies of South Kalimantan covering Banjar, Tapin, Hulu Sungai Utara and Kotabaru regencies. This paper reports an attempt to explore their crystalline state characteristics, projecting their potential use for geopolymer. Sungai Tabuk, Cintapuri and Tatakan, due to their largest kaolin claystone deposits, were chosen as the sampling sites. The kaolin samples were prepared by syphoning method prior to X-ray diffraction (XRD) characterizations in determining their crystalline phases. X’Pert HighScore Plus and Rietica software were respectively responsible for the qualitative and quantitative phase analyses. The qualitative analysis used search and match method at peak position and peak height between measured and calculated diffraction patterns. Our study revealed the existence of two main phases in the sample, i.e. quartz (SiO2) and kaolinite (Al2Si2O5(OH)4). In addition, the Quantitative analysis used the Rietveld method with the least squares method approach. Rietveld refinement was based on a goodness of fit score of less than 4% by minimizing the difference in the character of the diffraction pattern (position, height, width and peak shape) between the observed and the calculated XRD patterns. The Rietveld quantitative analysis shows, Tatakan is an area with kaolinite-richest deposit (±84%), followed by Cintapuri (±76%) and Tabuk (±70%); quartz is found in reverse.

Direct derivation (DD) of weight fractions of individual crystalline phases from observed intensities and chemical composition data: incorporation of the DD method into the whole-powder-pattern fitting procedure

A formula for quantitative phase analysis (QPA), called the intensity–composition (IC) formula, can be used for deriving weight fractions of individual crystalline phases in a mixture from sets of observed integrated intensities, measured in a wide 2θ range, with chemical composition data [Toraya (2016). J. Appl. Cryst. 49, 1508–1516; Toraya (2017). J. Appl. Cryst. 50, 820–829]. In this study, the IC formula has been incorporated into the whole-powder-pattern fitting (WPPF) procedure to conduct QPA. The fitting function for calculating the profile intensity at each step of the scattering angle consists of three sub-functions that represent the individual component diffraction patterns. The first sub-function calculates the diffraction pattern using a set of integrated intensities, the parameter values of which are determined by the least-squares fitting of the whole-powder pattern as is usually done by the whole-powder-pattern decomposition (WPPD) method. The second sub-function uses a set of integrated intensity parameters, which are preliminarily prepared by WPPD or may be calculated from a crystal structure model. These intensity parameters, multiplied by a scale factor, are fixed at their original values while the scale factor is adjusted in WPPF. The third sub-function uses an observed or calculated diffraction pattern multiplied by a scale factor. This diffraction pattern can be fitted directly by adjusting the scale factor. Therefore, one can fit patterns consisting of heavily broadened and degraded diffraction lines, like those of clay minerals, without being concerned with the problem of peak overlap in decomposing the diffraction pattern. The IC formula uses the total sums of the intensities under the diffraction patterns of individual phases as observed data sets; therefore, it can equally treat these intensity data sets irrespective of differences in the profile models used by the three sub-functions. The three sub-functions can arbitrarily be chosen and linearly combined, and then they can simultaneously be fitted to the observed diffraction pattern of a target mixture. The capability of the above method has been demonstrated with QPA of mixtures consisting of α-quartz, albite and kaolinite. Theories of currently used QPA techniques are reviewed from a viewpoint of the present theory and they can be interpreted as being based on the same principle, whereby the total observed intensities of individual phases are divided by the standard reference intensity per unit weight.