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Algorithms in Diffraction Profile Analysis

This chapter introduces several approaches to crystallographic structure refinement from diffraction profile data including the Rietveld method, global optimization methods, and Bayesian modeling. First, the Rietveld method and related nonlinear least squares approaches to analyzing diffraction profiles are briefly discussed. We then review the most common algorithms within global optimization as alternative methods of structure solution: grid search, simulated annealing, and genetic algorithms. As the Rietveld method is the standard tool for analyzing diffraction profile data, we catalogue its challenges and limitations. A Bayesian approach to diffraction profile analysis is presented as an alternative, along with several sampling algorithms including Gibbs sampling, variants of Metropolis sampling, Hamiltonian Monte Carlo, and approximate Bayesian computing. These sampling algorithms are applied to the fitting of a neutron diffraction profile from a National Institute of Standards and Technology silicon standard reference material, and their results are compared.

Singh, Susheela↗

To image, or not to image: class-specific diffractive cameras with all-optical erasure of undesired objects

Abstract Privacy protection is a growing concern in the digital era, with machine vision techniques widely used throughout public and private settings. Existing methods address this growing problem by, e.g., encrypting camera images or obscuring/blurring the imaged information through digital algorithms. Here, we demonstrate a camera design that performs class-specific imaging of target objects with instantaneous all-optical erasure of other classes of objects. This diffractive camera consists of transmissive surfaces structured using deep learning to perform selective imaging of target classes of objects positioned at its input field-of-view. After their fabrication, the thin diffractive layers collectively perform optical mode filtering to accurately form images of the objects that belong to a target data class or group of classes, while instantaneously erasing objects of the other data classes at the output field-of-view. Using the same framework, we also demonstrate the design of class-specific permutation and class-specific linear transformation cameras, where the objects of a target data class are pixel-wise permuted or linearly transformed following an arbitrarily selected transformation matrix for all-optical class-specific encryption, while the other classes of objects are irreversibly erased from the output image. The success of class-specific diffractive cameras was experimentally demonstrated using terahertz (THz) waves and 3D-printed diffractive layers that selectively imaged only one class of the MNIST handwritten digit dataset, all-optically erasing the other handwritten digits. This diffractive camera design can be scaled to different parts of the electromagnetic spectrum, including, e.g., the visible and infrared wavelengths, to provide transformative opportunities for privacy-preserving digital cameras and task-specific data-efficient imaging.

47 OTHER INSTRUMENTATION↗

Controlling the sign of chromatic dispersion in diffractive optics with dielectric metasurfaces

Diffraction gratings disperse light in a rainbow of colors with the opposite order than refractive prisms, a phenomenon known as negative dispersion. While refractive dispersion can be controlled via material refractive index, diffractive dispersion is fundamentally an interference effect dictated by geometry. Here we show that this fundamental property can be altered using dielectric metasurfaces, and we experimentally demonstrate diffractive gratings and focusing mirrors with positive, zero, and hyper-negative dispersion. These optical elements are implemented using a reflective metasurface composed of dielectric nano-posts that provide simultaneous control over phase and its wavelength derivative. In addition, as a first practical application, we demonstrate a focusing mirror that exhibits a five-fold reduction in chromatic dispersion, and thus an almost three-times increase in operation bandwidth compared with a regular diffractive element. In conclusion, this concept challenges the generally accepted dispersive properties of diffractive optical devices and extends their applications and functionalities.

42 ENGINEERING↗

Materials Characterization Techniques Lecture Series: X-Ray Diffraction, A Summary of LANL's Materials Capabilities in MPA-11 [Slides]

This presentation provides an overview on X-Ray Diffraction (XRD) to cover background, fundamentals, limitations, best practices, applications, analysis, software, and resources. To summarize: XRD is a great tool for characterizing solid crystalline samples. X-Rays are E-Mag radiation and generate diffraction profiles based on the phase of the sample. Diffraction patterns are unique to the phase of the sample. Powder diffraction gives us better diffraction profiles. We can obtain both qualitative and quantitative data. Analysis requires us to know a little about our sample and a lot about our diffractometer.

36 MATERIALS SCIENCE↗

Diffraction of waves by a perfect electromagnetic conductor half-plane between isorefractive media

In this study, diffraction of waves by a perfectly electromagnetic conductor half-plane which is lying at the boundary of isorefractive media is investigated. The diffracted field expressions are constructed by using the relation between scattered geometrical optics fields and the transition boundaries. The high frequency asymptotic expressions of diffracted fields reach infinite magnitude values at the transition boundaries. The uniform expressions of fields are obtained by using the uniform theory of diffraction method in order to obtain the finite magnitude values at the transition boundaries and the results are analyzed numerically for different set of parameters.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

A Planar Biaxial Experiment Platform for In Situ High-Energy Diffraction Studies

In this work, an experimental platform for multiscale studies of materials subjected to plane stress loads is presented. Specifically, a planar biaxial mechanical load frame with four independent hydraulic actuators capable of applying arbitrary loading paths and ratios of tension and compression was designed and built for in situ diffraction experimentation. The load frame is integrated for use at the Argonne National Laboratory Advanced Photon Source (APS) synchrotron, Sector 1, 1-ID-E endstation and the Los Alamos Neutron Science Center (LANSCE) spallation neutron source, Spectrometer for Materials Research at Temperature and Stress (SMARTS) instrument. Cruciform specimen geometries were designed to experience loading ratios in the gauges commensurate with those applied at the grips, and to minimize interference with diffracted X-rays and neutrons. The finite element models used to design the cruciform specimen geometries were experimentally validated using stereo digital image correlation measurements. This complete planar biaxial in situ diffraction platform provides a new capability for studying multiaxial micromechanics of crystalline materials (e.g., elastic, slip, twinning, phase transformation) and their dependencies on grain size, location, texture, and neighborhood characteristics. The coupling with far-field high energy diffraction microscopy (ff-HEDM) for grain-by-grain measurements provides an additional benefit; provided crystal elastic constants of a material are known, it enables the direct measurement of local and average stress tensors in the gauge sections of cruciform specimens, resolving a long outstanding challenge of planar biaxial mechanical testing.

36 MATERIALS SCIENCE↗

Structure resolution by electron diffraction tomography of the complex layered iron-rich Fe-2234-type Sr{sub 5}Fe{sub 6}O{sub 15.4}

The crystal structure of the strontium ferrite Sr{sub 5}Fe{sub 6}O{sub 15.4}, was solved by direct methods on electron diffraction tomography data acquired on a transmission electron microscope. The refined cell parameters are a=27.4047(3) Å, b=5.48590(7) Å and c=42.7442(4) Å in Fm2m symmetry. Its structure is built up from the intergrowth sequence between a quadruple perovskite-type layer with a complex rock-salt (RS)-type block. In the latter iron atoms are found in two different environments : tetragonal pyramid and tetrahedron. The structural model was refined by Rietveld method based on the powder X-ray diffraction pattern. - Highlights: • Complex structure of Sr{sub 5}Fe{sub 6}O{sub 15.4} solved by electron diffraction tomography. • Observed Fourier maps allow determining missing oxygen atoms in the structure. • Structural model refined from powder X-ray diffraction data. • Intergrowth between quadruple perovskite layer with double rock-salt-type layer.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Multibeam Electron Diffraction

One of the primary uses for transmission electron microscopy (TEM) is to measure diffraction pattern images in order to determine a crystal structure and orientation. In nanobeam electron diffraction (NBED), we scan a moderately converged electron probe over the sample to acquire thousands or even millions of sequential diffraction images, a technique that is especially appropriate for polycrystalline samples. However, due to the large Ewald sphere of TEM, excitation of Bragg peaks can be extremely sensitive to sample tilt, varying strongly for even a few degrees of sample tilt for crystalline samples. In this paper, we present multibeam electron diffraction (MBED), where multiple probe-forming apertures are used to create multiple scanning transmission electron microscopy (STEM) probes, all of which interact with the sample simultaneously. We detail designs for MBED experiments, and a method for using a focused ion beam to produce MBED apertures. We show the efficacy of the MBED technique for crystalline orientation mapping using both simulations and proof-of-principle experiments. We also show how the angular information in MBED can be used to perform 3D tomographic reconstruction of samples without needing to tilt or scan the sample multiple times. Finally, we also discuss future opportunities for the MBED method.

47 OTHER INSTRUMENTATION↗

Necessary Experimental Conditions for Single-Shot Diffraction Imaging of DNA-Based Structures with X-ray Free-Electron Lasers

It has been proposed that the radiation damage to biological particles and soft condensed matter can be overcome by ultrafast and ultraintense X-ray free-electron lasers (FELs) with short pulse durations. The successful demonstration of the “diffraction-before-destruction” concept has made single-shot diffraction imaging a promising tool to achieve high resolutions under the native states of samples. However, the resolution is still limited because of the low signal-to-noise ratio, especially for biological specimens such as cells, viruses, and macromolecular particles. Here, we present a demonstration single-shot diffraction imaging experiment of DNA-based structures at SPring-8 Angstrom Compact Free Electron Laser (SACLA), Japan. Through quantitative analysis of the reconstructed images, the scattering abilities of gold and DNA were demonstrated. Suggestions for extracting valid DNA signals from noisy diffraction patterns were also explained and outlined. To sketch out the necessary experimental conditions for the 3D imaging of DNA origami or DNA macromolecular particles, we carried out numerical simulations with practical detector noise and experimental geometry using the Linac Coherent Light Source (LCLS) at the SLAC National Accelerator Laboratory, USA. In conclusion, the simulated results demonstrate that it is possible to capture images of DNA-based structures at high resolutions with the technique development of current and next-generation X-ray FEL facilities.

59 BASIC BIOLOGICAL SCIENCES↗

Study of optical Laue diffraction

We present the study of the optical diffraction pattern of one and two-dimensional gratings with defects, designed using desktop pc and printed on OHP sheet using laser printer. Gratings so prepared, using novel low cost technique provides good visual aid in teaching. Diffraction pattern of the monochromatic light (632.8nm) from the grating so designed is similar to that of x-ray diffraction pattern of crystal lattice with point defects in one and two-dimensions. Here both optical and x-ray diffractions are Fraunhofer. The information about the crystalline lattice structure and the defect size can be known.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Modeling spatially localized photonic nanojets from phase diffraction gratings

We investigated numerically the specific spatially localized intense optical structure, a photonic nanojet (PNJ), formed in the near-field scattering of optical radiation at phase diffraction gratings. The finite-difference time-domain technique was employed to study the PNJ key parameters (length, width, focal distance, and intensity) produced by diffraction gratings with the saw-tooth, rectangle, and hemispheric line profiles. Our analysis showed that each type of diffraction gratings produces a photonic jet with unique characteristics. Based on the numerical calculations, we demonstrate that the PNJ could be manipulated in a wide range through the variation of period, duty cycle, and shape of diffraction grating rulings.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Innovative diffraction gratings for high-resolution resonant inelastic soft x-ray scattering spectroscopy

High-resolution Resonant Inelastic X-ray Scattering (RIXS) requires diffraction gratings with very exacting characteristics. The gratings should provide both very high dispersion and high efficiency which are conflicting requirements and extremely challenging to satisfy in the soft x-ray region for a traditional grazing incidence geometry. To achieve high dispersion one should increase the groove density of a grating; this however results in a diffraction angle beyond the critical angle range and results in drastic efficiency loss. The problem can be solved by use of multilayer coated blazed gratings (MBG). In this work we have investigated the diffraction characteristics of MBGs via numerical simulations and have developed a procedure for optimization of grating design for a multiplexed high resolution imaging spectrometer for RIXS spectroscopy to be built in sector 6 at the Advanced Light Source (ALS). We found that highest diffraction efficiency can be achieved for gratings optimized for 4{sup th} or 5{sup th} order operation. Fabrication of such gratings is an extremely challenging technological problem. We present a first experimental prototype of these gratings and report its performance. High order and high line density gratings have the potential to be a revolutionary new optical element that should have great impact in the area of soft x-ray RIXS.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

Subwavelength diffraction gratings in the visible spectral range

We report the results of computer calculations and measurements of subwavelength diffraction gratings in the visible range of the radiation spectrum. The influence of various grating parameters (duty cycle, microrelief shape and depth, material, angle of incidence, wavelength, and radiation polarisation) on the diffraction efficiency is studied. A distinctive feature of the subwavelength gratings in question is that the entire diffracted energy of the beam is redistributed into the zero and –1st orders. It is found that the zero order can be suppressed by choosing the depth and shape of the grating relief. The subwavelength gratings with a period of 400 nm are fabricated and measurements are performed using lasers and laser diodes emitting in the visible wavelength range. High diffraction efficiency into the –1st order (more than 70 %) is observed in a wide spectral range of 450 – 650 nm with an increase in the grating relief depth (at a depth of h = 80 nm). It is experimentally demonstrated that under certain conditions, the plasmon resonance effect arises, in which total absorption of incident radiation takes place. The optical elements considered can be used in image processing systems, projection displays, in the development of various sensors, etc. (paper)

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

New capabilities in high-resolution neutron Larmor diffraction at ORNL

Using superconducting magnetic Wollaston prisms, high-resolution neutron Larmor diffraction has been implemented at the High-Flux Isotope Reactor of Oak Ridge National Laboratory (ORNL), Tennesse, USA. This technique allows the inverse relationship between the achievable diffraction resolution and the usable neutron flux to be overcome. Instead of employing physically tilted radio-frequency spin flippers, the method uses magnetic Wollaston prisms which are electromagnetically tuned by changing the field configurations in the device. As implemented, this method can be used to measure lattice-spacing changes induced, for example, by thermal expansion or strain with a resolution of Δd/d ≃ 10 –6 , and the splitting of sharp Bragg peaks with a resolution of Δd/d = 3 × 10 –4 . The resolution for discerning a change in the profile of a Bragg peak is Δd/d < 10 –5 . This is a remarkable degree of precision for a neutron diffractometer as compact as the one used in this implementation. Higher precision could be obtained by implementing this technique in an instrument with a larger footprint. In conclusion, the availability of this technique will provide an alternative when standard neutron diffraction methods fail and will greatly benefit the scientific communities that require high-resolution diffraction measurements.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Methods for operando coherent X-ray diffraction of battery materials at the Advanced Photon Source

Bragg coherent X-ray diffraction imaging has become valuable for visualization of the structural, morphological and strain evolution of crystals in operando electrode materials. As the electrode material particles (either in a single-crystal form or an aggregation form of single crystals) are evenly dispersed and randomly oriented in the electrode laminate, the submicrometer-sized coherent X-ray beam can be used to probe the local properties of electrode material crystals using two approaches. Coherent multi-crystal diffraction provides collective structural information of phase transitions in tens of crystals simultaneously as well as the individual behavior from single crystals, which are oriented at the Bragg condition in the X-ray illumination volume. Bragg coherent diffractive imaging enables one to monitor the evolution of the morphology and strain in individual crystals. This work explores and highlights the Bragg coherent X-ray diffraction measurements of battery electrode materials in operando conditions at the 34-ID-C beamline at the Advanced Photon Source. The experiment is demonstrated with NaNi 1/3 Fe 1/3 Mn 1/3 O 2 , a sodium-ion cathode material loaded in a half cell. Furthermore, the paper will discuss, in detail, the beamline setup, sample mounting and handling, alignment strategies and the data acquisition protocols.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Diffraction gratings with two-orders-of-magnitude-enhanced dispersion rates for sub-meV resolution soft X-ray spectroscopy

Diffraction gratings with large angular dispersion rates are central to obtaining high spectral resolution in grating spectrometers operating over a broad spectral range from infrared to soft X-ray domains. The greatest challenge is of course to achieve large dispersion rates in the short-wavelength X-ray domain. Here it is shown that crystals in non-coplanar asymmetric X-ray Bragg diffraction can function as high-reflectance broadband soft X-ray diffraction gratings with dispersion rates that are at least two orders of magnitude larger than those that are possible with state-of-the-art man-made gratings. Additionally, this opens new opportunities to design and implement soft X-ray resonant inelastic scattering (RIXS) spectrometers with spectral resolutions that are up to two orders of magnitude higher than what is currently possible, to further advance a very dynamic field of RIXS spectroscopy, and to make it competitive with inelastic neutron scattering. Examples of large-dispersion-rate crystal diffraction gratings operating near the 930 eV L 3 absorption edge in Cu and of the 2.838 keV L 3 -edge in Ru are presented.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Efficient modeling of Bragg coherent x-ray nanobeam diffraction

X-ray Bragg diffraction experiments that utilize tightly focused coherent beams produce complicated Bragg diffraction patterns that depend on scattering geometry, characteristics of the sample, and properties of the x-ray focusing optic. In this paper, we use a Fourier-transform-based method of modeling the 2D intensity distribution of a Bragg peak and apply it to the case of thin films illuminated with a Fresnel zone plate in three different Bragg scattering geometries. Finally, the calculations agree well with experimental coherent diffraction patterns, demonstrating that nanodiffraction patterns can be modeled at nonsymmetric Bragg conditions with this approach—a capability critical for advancing nanofocused x-ray diffraction microscopy.

36 MATERIALS SCIENCE↗

Double Bragg Diffraction of Bessel Light Beams on Ultrasound in Uniaxial Gyrotropic Crystals

Noncollinear double acousto-optical diffraction of Bessel light beams on ultrasound in uniaxial gyrotropic crystals is studied theoretically and experimentally. It is found that when a Bessel light beam propagates along the optic axis of a crystal, the diffracted light beams of the ±1 order are symmetric relative to the incident beam and have roughly the same intensity. Double Bragg diffraction is also observed when a Bessel light beam is incident at the Bragg angle and near the optic axis, but the intensities of the +1 and +2 order diffracted beams are significantly different.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗