Structure In Reciprocal Space Research Articles

Enhancing thermoelectric performance of AB2Sb2-type Zintl phase through band shaping and lattice distortion

AB2Sb2-type Zintl phases, especially Mg3Sb2, have garnered widespread attention in virtue of their nontoxic constituent elements and outstanding thermoelectric performance in n-type materials. However, the p-type counterpart of AB2Sb2 still struggles with a low figure of merit zT, primarily because of the limited valence band degeneracy, large band effective mass, and relatively high thermal conductivity. Here we simultaneously optimize the electrical and thermal transports of p-type AB2Sb2-based materials through shaping the band edge and distorting the crystal lattice. By alloying Zn at Mg2 site and Ca at Mg1 site, we successfully decorate the crystal lattice of Mg3Sb2 in real space and thereby modify the band structure in reciprocal space. Zn alloying promotes the band degeneracy while Ca alloying reduce the single band effective mass, which largely promotes the transport of charge carriers. Moreover, the distorted crystal lattice effectively blocks the heat-carry phonons to reduce the lattice thermal conductivity κL. A high zT of 0.81 is finally realized at 750 K in our Zn/Ca co-alloyed samples, which is three times larger than that of Mg3Sb2 matrix. Our work opens new possibilities for the synthetic optimization of both electrical and thermal transports in Zintl phase and other thermoelectric systems.

Optics and quantitative assessment of corneal transparency

While some optical properties of cornea like its stromal birefringence had been known since the early nineteenth century, theories explaining its transparency remained inconclusive until the publication of David Maurice' emblematic paper in 1957 Maurice made a model calculation of light scattering by a single collagen fibril as a “dielectric needle”. The volume density of the fibrils being known, Maurice could demonstrate that in the absence of any order among the fibrils the cornea would be opaque. Maurice postulated the hexagonal structure which since has been confirmed by electron microscopy. In modern terms, the order among the collagen fibrils within a stromal collagen lamella is described by a “structure factor” in the radiative transfer equation. This term corresponds to a description of the structure in reciprocal space and is thus equivalent to the representation of crystalline lattices in the first Brillouin zone used for structural modelling in crystallography.In a recent study we further analysed the collagen structure of healthy and oedematous corneas on a nanometric level observed by transmission electron microscopy (TEM). By automated image analysis we could retrieve the so‐called “pair correlation function” and calculate light attenuation within the corresponding cornea. Our findings show that the 1/e light penetration depth within healthy cornea could be calculated as 7300 ± 800 μm, which explains well the transparency of healthy cornea. However, the photon mean free path of oedematous cornea turned out to be from two to three times that value. In oedematous cornea, the “dilution” effect of the scatters is in fact stronger than the degradation of transparency by the perturbation of the structure. The actual loss of transparency in oedematous cornea may therefore be attributed to scattering processes at the micrometric level induced by the augmented optical contrast at the sites of the keratocytes between the collagen lamellae.Perturbations in corneal transparency do not only hamper vision, they also impact optical imaging methods on cornea. This, in turn, permits to study the transparency properties of the corneal tissue. One of the main research interests of our group is to analyse corneal images recorded by optical coherence tomography (OCT) and to extract quantitative parameters like photon mean path length and the fraction of coherently transmitted light.Our partners could show that in the light scattering regime, a term in the radiation transfer equation may be neglected and the attenuation of the full field OCT signal in the volume of the tissue reduces to a Lambert–Beer law. We developed a data analysis algorithm based on Bayesian inference to extract the attenuation coefficient and used it for the analysis of ex vivo corneal high‐resolution FF‐OCT images.In a second step, we developed an image processing algorithm for clinical OCT images removing typical artefacts and compensating for inhomogeneous illumination and corneal curvature. We presently are able to extract corneal transparency parameters from routine clinical OCT images and to provide post‐op follow‐up after keratoplasty or other surgical interventions.

Thermal Degradation of Polycrystalline Ni-Rich Cathodes, New Insights from Total Scattering and Machine-Learning Assisted Tomography

Alkali metal-based batteries (e.g., lithium batteries) are considered the most effective energy storage candidate for portable electronics, power grids, and electric vehicles due to their considerable energy density, power density, cycle life, safety, and competitive prices. The Ni-rich Co-less ternary LiNi x Mn y Co1-x-y O2 (NMC) (Ni>80%) is one of the candidate materials due to its high discharge capacity (>200 mAh g-1) and high energy density (>750 Wh kg-1). However, the thermal stability of charged Ni-rich NMCs is inferior, following a general negative correlation between Ni content and stability. Such severer degradation in Ni-rich NMCs might lead to dysfunctionality, thermal runaway, and eventually catastrophic failure and explosion of batteries, raising enormous safety concerns. Therefore, it is critical to understand the thermal behavior of charged Ni-rich NMC cathode from different perspectives in order to conduct material design for better thermal stability.After literature survey of typical and widely used probes by this community (e.g., diffraction, spectroscopy, imaging), we found these approaches subject to compromise to some degree in the experimental design, resulting in an inadequate description of the thermal degradation mechanism. Here we offer two novel approaches, neutron total scattering and energy-resolved 3D nano-tomography, to probe the reaction and evolution of the charged NMCs under non-ambient conditions in real-time. Such in situ measurements were only accomplished by deliberate experimental design at beamlines and deep integrations of neutron or synchrotron facilities with battery science. In situ heating neutron scattering captures the transition of crystal structure in reciprocal space (by Rietveld refinements), as well as of the local atomic framework in real space (by pair distribution function analysis). Moreover, machine learning algorithms were applied to the 3D nano-tomography datasets to reveal the detailed microstructural and oxidation state progression inside the charged particles. Thereafter, we discussed the key factors which have previously been overlooked by the community and offer constructive suggestions on particle design for thermally stable cathodes from our unique perspective.The work was supported by the National Science Foundation under Grant no. DMR-1832613 (F.L.). The machine-learning assisted tomography analysis was partially supported from NASA under 80NSSC21M0333 and NSF under NSF-2119688 (X.-D.Z)

LiquidDiffract: software for liquid total scattering analysis

LiquidDiffract is an open source, Python-based graphical application for X-ray total scattering analysis of liquids and disordered solids. The software implements procedures to obtain information on macroscopic bulk properties and local atomic-scale structure of monatomic or polyatomic samples from X-ray total scattering data. LiquidDiffract provides an easy to use interface with tools to perform background subtraction; calculation, normalisation, and refinement of the reciprocal-space structure factor and real-space correlation functions; and the extraction of structural information such as bond lengths, coordination number, and bulk density. The software is well suited to investigations of amorphous materials at extreme conditions, such as studies of high-pressure melt structure, polyamorphic phase transitions, and liquid equations of state. The open-source distribution and graphical interface will be of particular benefit to researchers who are new to the field. In this article we describe the distribution, system requirements, and installation of LiquidDiffract, and detail the data processing workflow and underlying numerical methods.

Topological magnon band structure of emergent Landau levels in a skyrmion lattice.

The motion of a spin excitation across topologically nontrivial magnetic order exhibits a deflection that is analogous to the effect of the Lorentz force on an electrically charged particle in an orbital magnetic field. We used polarized inelastic neutron scattering to investigate the propagation of magnons (i.e., bosonic collective spin excitations) in a lattice of skyrmion tubes in manganese silicide. For wave vectors perpendicular to the skyrmion tubes, the magnon spectra are consistent with the formation of finely spaced emergent Landau levels that are characteristic of the fictitious magnetic field used to account for the nontrivial topological winding of the skyrmion lattice. This provides evidence of a topological magnon band structure in reciprocal space, which is borne out of the nontrivial real-space topology of a magnetic order.

Dissolving salt is not equivalent to applying a pressure on water

Salt water is ubiquitous, playing crucial roles in geological and physiological processes. Despite centuries of investigations, whether or not water’s structure is drastically changed by dissolved ions is still debated. Based on density functional theory, we employ machine learning based molecular dynamics to model sodium chloride, potassium chloride, and sodium bromide solutions at different concentrations. The resulting reciprocal-space structure factors agree quantitatively with neutron diffraction data. Here we provide clear evidence that the ions in salt water do not distort the structure of water in the same way as neat water responds to elevated pressure. Rather, the computed structural changes are restricted to the ionic first solvation shells intruding into the hydrogen bond network, beyond which the oxygen radial-distribution function does not undergo major change relative to neat water. Our findings suggest that the widely cited pressure-like effect on the solvent in Hofmeister series ionic solutions should be carefully revisited.

Ether tails make a large difference for the structural dynamics of imidazolium-based ionic liquids

Ionic Liquids are often characterized by multiple length scales and these tend to be well separated. As an example, for 1-methyl-3-octylimidazolium bis(trifluoromethylsulfonyl)imide apolar pockets or domains are separated by a characteristic distance that is different from that of positive-negative charge alternation along a charge network. Accordingly, these structural features have their own well separated signatures in the reciprocal space structure function. Each of these well separated structural features of prototypical ionic liquids relaxes on its own time scale and we have explored this in a set of recent articles. The overarching goal of the current work is to explore what happens when the structural motifs are still present, but the length scale difference between these is small. Specifically, for ionic liquids containing ether tail functionalities such as 1-(2-(2-ethoxyethoxy)ethyl)-3-methylimidazolium bis(trifluoromethylsulfonyl)imide, the separation between cationic head subspecies intercalated by anions (or anions sharing a cationic head counterion) is not that different from that of cationic heads or anions intercalated by tail regions and the time scale for relaxation of these structural motifs are also similar. For the ether-tail containing system, the slowest liquid relaxation scale associated with tail region-charge network correlations is about an order of magnitude faster than for its alkyl-tail isoelectronic analog.

Correlation between crystal structure and magnetism in PLD grown epitaxial films of ε-Fe2O3 on GaN

ABSTRACT In the present paper we discuss correlations between crystal structure and magnetic properties of epitaxial ε-Fe2O3 films grown on GaN. The large magnetocrystalline anisotropy and room temperature multiferroic properties of this exotic iron oxide polymorph, make it a perspective material for the development of low power consumption magnetic media storage devices. Extending our recent progress in PLD growth of ε-Fe2O3 on the surface of technologically important nitride semiconductors, we apply reciprocal space tomography by electron and x-ray diffraction to investigate the break of crystallographic symmetry occurring at the oxide-nitride interface resulting in the appearance of anisotropic crystallographic disorder in the sub-100 nm ε-Fe2O3 films. The orthorhombic-on-hexagonal nucleation scenario is shown responsible for the development of a peculiar columnar structure observed in ε-Fe2O3 by means of HRTEM and AFM. The complementary information on the direct and reciprocal space structure of the columnar ε-Fe2O3 films is obtained by various techniques and correlated to their magnetic properties. The peculiar temperature dependence of magnetization studied by the small-field magnetization derivative method and by neutron diffraction reveals the existence of a magnetic softening below 150 K, similar to the one observed earlier solely in nanoparticles. The magnetization reversal in ε-Fe2O3 films probed by X-ray magnetic circular dichroism is found different from the behavior of the bulk averaged magnetization measured by conventional magnetometry. The presented results fill the gap between the numerous studies performed on randomly oriented ε-Fe2O3 nanoparticles and much less frequent investigations of epitaxial epsilon ferrite films with lattice orientation fixed by the substrate.

Monolayer MoS2 on sapphire: an azimuthal reflection high-energy electron diffraction perspective

Molybdenum disulfide (MoS2) on the c-plane sapphire has been a very popular system to study in the two-dimensional (2D) materials community. Bottom-up synthesis of monolayer (ML) MoS2 with excellent electrical properties has been achieved on sapphire by various methods, making it a very promising candidate to be used in the next generation nano-electronic devices. However, large-area ML MoS2 with comparable quality as the relatively small size exfoliated ML remains quite a challenge. To overcome this bottle neck, a comprehensive understanding of the structure of the as-grown ML material is an essential first step. Here, we report a detailed structural characterization of wafer-scale continuous epitaxial ML MoS2 grown by metalorganic chemical vapor deposition on sapphire using an azimuthal reflection high-energy electron diffraction (ARHEED) technique. With ARHEED we can map not only 2D but also 3D reciprocal space structure of the ML statistically. From the oscillation in the ARHEED intensity profile along the vertical direction of the ML, we derived a real space distance of ~3 Å at the interface of ML and sapphire. Quantitative diffraction spot broadening analyses of the 3D reciprocal space map reveals low density defects and a small angular misalignment of orientation domains in ML MoS2. Based on atomic force microscopy height distribution analysis, cross-section scanning transmission electron microscopy, and density functional theory calculations, we suggest that there exists a passivation layer between MoS2 ML and sapphire substrate. This ARHEED methodology also has been applied to ML WS2 and is expected to be applicable to other ML transition metal dichalcogenides on arbitrary crystalline or non-crystalline substrates.

Equilibrium Solvation, Electron-Transfer Reactions, and Stokes-Shift Dynamics in Ionic Liquids.

A microscopic theory of solvent response by room-temperature ionic liquids is formulated based on the dynamic longitudinal susceptibility of liquid's charge density. The susceptibility function combines the structural information in terms of reciprocal-space structure factors with the memory function responsible for solvation dynamics. The charge-density structure factors and corresponding intermediate scattering functions are analyzed here by molecular dynamics simulations. They show the existence of two drastically different time scales of charge-density fluctuations. Faster, stretched-exponential dynamics are consistent with dielectric measurements. It contributes to the Stokes-shift dynamics of coumarin-153 optical dye calculated with the new theory and compared to experimental reports. The second, much slower and exponential, relaxation shows the phenomenology of de Gennes narrowing: the relaxation time passes through a strong maximum at the wave vector representing the first peak of the structure factor. This peak, which is particularly sharp for the charge density, contributes significantly to the equilibrium free energy of solvation, thus invalidating dielectric theories of solvation for ionic liquids. Dynamics of charge density fluctuations at the length scale consistent with the sharp peak require long observation times. Electron-transfer reactions occurring on faster time-scales are not affected by these slow dynamics. Nonergodic reorganization energy of electron transfer, accounting for the observation window established by the reaction time, drops sharply when the reaction rate crosses the main peak in the Stokes-shift loss spectrum. The dependence of the reorganization energy on the reaction rate strongly affects the energy-gap law of electron transfer, with a tendency for a shallow or entirely disappearing inverted region.

Stacking, strain, and twist in 2D materials quantified by 3D electron diffraction

The field of two-dimensional (2D) materials has expanded to multilayered systems where electronic, optical, and mechanical properties change-often dramatically-with stacking order, thickness, twist, and interlayer spacing [1-5]. For transition metal dichalcogenides (TMDs), bond coordination within a single van der Waals layer changes the out-of-plane symmetry that can cause metal-insulator transitions [1, 6] or emergent quantum behavior [7]. Discerning these structural order parameters is often difficult using real-space measurements, however, we show 2D materials have distinct, conspicuous three-dimensional (3D) structure in reciprocal space described by near infinite oscillating Bragg rods. Combining electron diffraction and specimen tilt we probe Bragg rods in all three dimensions to identify multilayer structure with sub-Angstrom precision across several 2D materials-including TMDs (MoS2, TaSe2, TaS2) and multilayer graphene. We demonstrate quantitative determination of key structural parameters such as surface roughness, inter- & intra-layer spacings, stacking order, and interlayer twist using a rudimentary transmission electron microscope (TEM). We accurately characterize the full interlayer stacking order of multilayer graphene (1-, 2-, 6-, 12-layers) as well the intralayer structure of MoS2 and extract a chalcogen-chalcogen layer spacing of 3.07 +/- 0.11 Angstrom. Furthermore, we demonstrate quick identification of multilayer rhombohedral graphene.

Multi-orbital nature of the spin fluctuations in Sr2RuO4

The spin susceptibility of strongly correlated Sr2RuO4 is known to display a rich structure in reciprocal space, with a prominent peak at . It is still debated if the resulting incommensurate spin-density-wave fluctuations foster unconventional superconductivity at low temperature or compete therewith. By means of density functional theory combined with dynamical mean-field theory, we reveal the realistic multi-orbital signature of the (dynamic) spin susceptibility beyond existing weak-coupling approaches. The experimental fluctuation spectrum up to 80 meV is confirmed by theory. Furthermore, the peak at Qi is shown to carry nearly equal contributions from each of the Ru orbitals, pointing to a cooperative effect resulting in the dominant spin fluctuations.

Reflection high-energy electron diffraction measurements of reciprocal space structure of 2D materials

Knowledge on the symmetry and perfection of a 2D material deposited or transferred to a surface is very important and valuable. We demonstrate a method to map the reciprocal space structure of 2D materials using reflection high energy diffraction (RHEED). RHEED from a 2D material gives rise to ‘streaks’ patterns. It is shown that from these streaks patterns at different azimuthal rotation angles that the reciprocal space intensity distribution can be constructed as a function of momentum transfer parallel to the surface. To illustrate the principle, we experimentally constructed the reciprocal space structure of a commercial graphene/SiO2/Si sample in which the graphene layer was transferred to the SiO2/Si substrate after it was deposited on a Cu foil by chemical vapor deposition. The result reveals a 12-fold symmetry of the graphene layer which is a result of two dominant orientation domains with 30° rotation relative to each other. We show that the graphene can serve as a template to grow other materials such as a SnS film that follows the symmetry of graphene.

Accurateab initiotight-binding Hamiltonians: Effective tools for electronic transport and optical spectroscopy from first principles

The calculations of electronic transport coefficients and optical properties require a very dense interpolation of the electronic band structure in reciprocal space that is computationally expensive and may have issues with band crossing and degeneracies. Capitalizing on a recently developed pseudo-atomic orbital projection technique, we exploit the exact tight-binding representation of the first principles electronic structure for the purposes of (1) providing an efficient strategy to explore the full band structure $E_n({\bf k})$, (2) computing the momentum operator differentiating directly the Hamiltonian, and (3) calculating the imaginary part of the dielectric function. This enables us to determine the Boltzmann transport coefficients and the optical properties within the independent particle approximation. In addition, the local nature of the tight-binding representation facilitates the calculation of the ballistic transport within the Landauer theory for systems with hundreds of atoms. In order to validate our approach we study the multi-valley band structure of CoSb$_3$ and a large core-shell nanowire using the ACBN0 functional. In CoSb$_3$ we point the many band minima contributing to the electronic transport that enhance the thermoelectric properties; for the core-shell nanowire we identify possible mechanisms for photo-current generation and justify the presence of protected transport channels in the wire.

Real Space Imaging of Nanoparticle Assembly at Liquid-Liquid Interfaces with Nanoscale Resolution.

Bottom up self-assembly of functional materials at liquid-liquid interfaces has recently emerged as method to design and produce novel two-dimensional (2D) nanostructured membranes and devices with tailored properties. Liquid-liquid interfaces can be seen as a "factory floor" for nanoparticle (NP) self-assembly, because NPs are driven there by a reduction of interfacial energy. Such 2D assembly can be characterized by reciprocal space techniques, namely X-ray and neutron scattering or reflectivity. These techniques have drawbacks, however, as the structural information is averaged over the finite size of the radiation beam and nonperiodic isolated assemblies in 3D or defects may not be easily detected. Real-space in situ imaging methods are more appropriate in this context, but they often suffer from limited resolution and underperform or fail when applied to challenging liquid-liquid interfaces. Here, we study the surfactant-induced assembly of SiO2 nanoparticle monolayers at a water-oil interface using in situ atomic force microscopy (AFM) achieving nanoscale resolved imaging capabilities. Hitherto, AFM imaging has been restricted to solid-liquid interfaces because applications to liquid interfaces have been hindered by their softness and intrinsic dynamics, requiring accurate sample preparation methods and nonconventional AFM operational schemes. Comparing both AFM and grazing incidence X-ray small angle scattering data, we unambiguously demonstrate correlation between real and reciprocal space structure determination showing that the average interfacial NP density is found to vary with surfactant concentration. Additionally, the interaction between the tip and the interface can be exploited to locally determine the acting interfacial interactions. This work opens up the way to studying complex nanostructure formation and phase behavior in a range of liquid-liquid and complex liquid interfaces.

X-ray equipment for laboratory and synchrotron studies of thin-film crystalline structures

The capabilities of X-ray equipment produced by Radicon Ltd. for the high-resolution structural investigation of single crystals and thin films using different methods including high-resolution methods, are demonstrated. A double-crystal DSO-1T X-ray diffractometer with a large sample-to-detector distance and a narrow slit in front of the detector allows one to perform high-quality mapping of the GaAs/GaP/Si heteroepitaxial structure in reciprocal space. A new double-crystal PDP console for the DRON-3 diffractometer provides the mapping of large wafers through the use of rocking curves. A new specialized X-ray DSO-2-01 diffractometer equipped with a motorized console for linear displacement of detector and rotary slit changer in front of the detector ensures automatic crystal orientation and precise measurements of the crystal lattice. All X-ray diffractometers manufactured by Radicon have rather flexible software including a custom macros editor.

Inter-particle correlations in a hard-sphere colloidal suspension with polymer additives investigated by Spin Echo Small Angle Neutron Scattering (SESANS)

Using a neutron scattering technique that measures a statistically-averaged density correlation function in real space rather than the conventional reciprocal-space structure factor, we have measured correlations between poly(methyl-methacrylate) (PMMA) colloidal particles of several sizes suspended in decalin. The new method, called Spin Echo Small Angle Neutron Scattering (SESANS) provides accurate information about particle composition, including the degree of solvent penetration into the polymer brush grafted on to the PMMA spheres to prevent aggregation. It confirms for particles, between 85 nm and 150 nm in radius that inter-particle correlations closely follow the Percus-Yevick hard-sphere model when the colloidal volume-fraction is between 30% and 50% provided the volume-fraction is used as a fitted parameter. No particle aggregation occurs in these systems. When small amounts of polystyrene are added as a depletant to a concentrated suspension of PMMA particles, short-range clustering of the particles occurs and there is an increase in the frequency of near-neighbor contacts. Within a small range of depletant concentration, near-neighbor correlations saturate and large aggregates with power law density correlations are formed. SESANS clearly separates the short- and long-range correlations and shows that, in this case, the power-law correlations are visible for inter-particle distances larger than roughly two particle diameters. In some cases, aggregate sizes are within our measurement window, which can extend out to 16 microns in favorable cases. We discuss the advantages of SESANS for measurements of the structure of concentrated colloidal systems and conclude that the method offers several important advantages.

MAIN software for density averaging, model building, structure refinement and validation.

MAIN is software that has been designed to interactively perform the complex tasks of macromolecular crystal structure determination and validation. Using MAIN, it is possible to perform density modification, manual and semi-automated or automated model building and rebuilding, real- and reciprocal-space structure optimization and refinement, map calculations and various types of molecular structure validation. The prompt availability of various analytical tools and the immediate visualization of molecular and map objects allow a user to efficiently progress towards the completed refined structure. The extraordinary depth perception of molecular objects in three dimensions that is provided by MAIN is achieved by the clarity and contrast of colours and the smooth rotation of the displayed objects. MAIN allows simultaneous work on several molecular models and various crystal forms. The strength of MAIN lies in its manipulation of averaged density maps and molecular models when noncrystallographic symmetry (NCS) is present. Using MAIN, it is possible to optimize NCS parameters and envelopes and to refine the structure in single or multiple crystal forms.

Reciprocal-space structure and dispersion of the magnetic resonant mode in the superconducting phase of RbxFe2−ySe2single crystals

Inelastic neutron scattering is employed to study the reciprocal-space structure and dispersion of magnetic excitations in the normal and superconducting states of single-crystalline Rb${}_{0.8}$Fe${}_{1.6}$Se${}_{2}$. We show that the recently discovered magnetic resonant mode in this compound has a quasi-two-dimensional character, similar to overdoped iron-pnictide superconductors. Moreover, it has a rich in-plane structure that is dominated by four elliptical peaks, symmetrically surrounding the Brillouin zone corner, without $\sqrt{5}\ifmmode\times\else\texttimes\fi{}\phantom{\rule{-0.16em}{0ex}}\sqrt{5}$ reconstruction. We also present evidence for the dispersion of the resonance peak, as its position in momentum space depends on energy. Comparison of our findings with the results of band structure calculations leads to a robust bulk-sensitive estimate of the electron count in the superconducting phase and provides strong support for the itinerant origin of the observed signal. It can be traced back to the nesting of electronlike Fermi pockets in the doped metallic phase of the sample in the absence of iron-vacancy ordering.

Order-disorder effects in structure and color relation of photonic-crystal-type nanostructures in butterfly wing scales

In order to study local and global order in butterfly wing scales possessing structural colors, we have developed a direct space algorithm, based on averaging the local environment of the repetitive units building up the structure. The method provides the statistical distribution of the local environments, including the histogram of the nearest-neighbor distance and the number of nearest neighbors. We have analyzed how the different kinds of randomness present in the direct space structure influence the reciprocal space structure. It was found that the Fourier method is useful in the case of a structure randomly deviating from an ordered lattice. The direct space averaging method remains applicable even for structures lacking long-range order. Based on the first Born approximation, a link is established between the reciprocal space image and the optical reflectance spectrum. Results calculated within this framework agree well with measured reflectance spectra because of the small width and moderate refractive index contrast of butterfly scales. By the analysis of the wing scales of Cyanophrys remus and Albulina metallica butterflies, we tested the methods for structures having long-range order, medium-range order, and short-range order.