Three-dimensional domain identification in a single hexagonal manganite nanocrystal.

The three-dimensional domain structure of ferroelectric materials significantly influences their properties. The ferroelectric domain structure of improper multiferroics, such as YMnO3, is driven by a non-ferroelectric order parameter, leading to unique hexagonal vortex patterns and topologically protected domain walls. Characterizing the three-dimensional structure of these domains and domain walls has been elusive, however, due to a lack of suitable imaging techniques. Here, we present a multi-peak Bragg coherent x-ray diffraction imaging determination of the domain structure in single YMnO3 nanocrystals. We resolve two ferroelectric domains separated by a domain wall and confirm that the primary atomic displacements occur along the crystallographic c-axis. Correlation with atomistic simulations confirms the Mexican hat symmetry model of domain formation, identifying two domains with opposite ferroelectric polarization and adjacent trimerization, manifesting in a clockwise arrangement around the hat's brim.

Hidden k-Space Magnetoelectric Multipoles in Nonmagnetic Ferroelectrics.

In condensed matter systems, the electronic degrees of freedom are often entangled to form complex composites, known as hidden orders, which give rise to unusual properties, while escaping detection in conventional experiments. Here we demonstrate the existence of hidden k-space magnetoelectric multipoles in nonmagnetic systems with broken space-inversion symmetry. These k-space magnetoelectric multipoles are reciprocal to the real-space charge dipoles associated with the broken inversion symmetry. Using the prototypical ferroelectric PbTiO_{3} as an example, we show that their origin is a spin asymmetry in momentum space resulting from the broken space inversion symmetry associated with the ferroelectric polarization. In PbTiO_{3}, the k-space spin asymmetry corresponds to a pure k-space magnetoelectric toroidal moment, which can be detected using magnetic Compton scattering, an established tool for probing magnetism in ferromagnets or ferrimagnets with a net spin polarization, which has not been exploited to date for nonmagnetic systems. In particular, the k-space magnetoelectric toroidal moment combined with the spin-orbit interaction manifest in an antisymmetric magnetic Compton profile that can be reversed using an electric field. Our work suggests an experimental route to directly measuring and tuning hidden k-space magnetoelectric multipoles via specially designed magnetic Compton scattering measurements.

Anti-symmetric Compton scattering in LiNiPO 4: Towards a direct probe of the magneto-electric multipole moment.

BACKGROUND: Magnetoelectric multipoles, which break both space-inversion and time-reversal symmetries, play an important role in the magnetoelectric response of a material. Motivated by uncovering the underlying fundamental physics of the magnetoelectric multipoles and the possible technological applications of magnetoelectric materials, understanding as well as detecting such magnetoelectric multipoles has become an active area of research in condensed matter physics. Here we employ the well-established Compton scattering effect as a possible probe for the magnetoelectric toroidal moments in LiNiPO 4. METHODS: We employ combined theoretical and experimental techniques to compute as well as detect the antisymmetric Compton profile in LiNiPO 4. For the theoretical investigation we use density functional theory to compute the anti-symmetric part of the Compton profile for the magnetic and structural ground state of LiNiPO 4. For the experimental verification, we measure the Compton signals for a single magnetoelectric domain sample of LiNiPO 4, and then again for the same sample with its magnetoelectric domain reversed. We then take the difference between these two measured signals to extract the antisymmetric Compton profile in LiNiPO 4. RESULTS: Our theoretical calculations indicate an antisymmetric Compton profile in the direction of the t y toroidal moment in momentum space, with the computed antisymmetric profile around four orders of magnitude smaller than the total profile. The difference signal that we measure is consistent with the computed profile, but of the same order of magnitude as the statistical errors and systematic uncertainties of the experiment. CONCLUSIONS: While the weak difference signal in the measurements prevents an unambiguous determination of the antisymmetric Compton profile in LiNiPO 4, our results motivate  further theoretical work to understand the factors that influence the size of the antisymmetric Compton profile, and to identify materials exhibiting larger effects.

Spatially resolved mapping of phase transitions in liquid-crystalline materials by X-ray birefringence imaging.

The X-ray Birefringence Imaging (XBI) technique, first reported in 2014, is a sensitive method for spatially resolved mapping of the local orientational properties of anisotropic materials. We report the first application of the XBI technique to characterize molecular orientational ordering in a liquid crystalline material, demonstrating significant potential for exploiting XBI measurements to advance structural understanding of liquid crystal phases.

A unified formulation of dichroic signals using the Borrmann effect and twisted photon beams.

Dichroic X-ray signals derived from the Borrmann effect and a twisted photon beam with topological charge l = 1 are formulated with an effective wavevector. The unification applies for non-magnetic and magnetic materials. Electronic degrees of freedom associated with an ion are encapsulated in multipoles previously used to interpret conventional dichroism and Bragg diffraction enhanced by an atomic resonance. A dichroic signal exploiting the Borrmann effect with a linearly polarized beam presents charge-like multipoles that include a hexadecapole. A difference between dichroic signals obtained with a twisted beam carrying spin polarization (circular polarization) and opposite winding numbers presents charge-like atomic multipoles, whereas a twisted beam carrying linear polarization alone presents magnetic (time-odd) multipoles. Charge-like multipoles include a quadrupole, and magnetic multipoles include a dipole and an octupole. We discuss the practicalities and relative merits of spectroscopy exploiting the two remarkably closely-related processes. Signals using beams with topological charges l ≥ 2 present additional atomic multipoles.

Determining Molecular Orientations in Disordered Materials from X-ray Linear Dichroism at the Iodine L1-Edge.

To demonstrate that measurements of X-ray linear dichroism are effective for determining bond orientations in disordered materials, we report the first observation of X-ray linear dichroism at the iodine L1-edge. The iodine-containing molecular solid studied in this work was the inclusion compound containing 4,4'-diiodobiphenyl guest molecules in the perhydrotriphenylene host structure. In this material, the guest substructure does not exhibit three-dimensional ordering, and thus diffraction-based techniques do not provide insights on the orientational properties of the guest molecules. Iodine L1-edge X-ray absorption spectra, recorded as a function of orientation of a single crystal of the material, exhibit significant dichroism (whereas no dichroism is observed at the iodine L2- and L3-edges). From quantitative analysis of the X-ray dichroism, the orientational properties of the C-I bonds within this material are established. The results pave the way for applying X-ray dichroism to determine molecular orientational properties of other materials, especially for partially ordered materials such as liquid crystals, confined liquids, and disordered crystalline phases, for which diffraction techniques may not be applicable.

Resonant X-ray emission with a standing wave excitation.

The Borrmann effect is the anomalous transmission of x-rays in perfect crystals under diffraction conditions. It arises from the interference of the incident and diffracted waves, which creates a standing wave with nodes at strongly absorbing atoms. Dipolar absorption of x-rays is thus diminished, which makes the crystal nearly transparent for certain x-ray wave vectors. Indeed, a relative enhancement of electric quadrupole absorption via the Borrmann effect has been demonstrated recently. Here we show that the Borrmann effect has a significantly larger impact on resonant x-ray emission than is observable in x-ray absorption. Emission from a dipole forbidden intermediate state may even dominate the corresponding x-ray spectra. Our work extends the domain of x-ray standing wave methods to resonant x-ray emission spectroscopy and provides means for novel spectroscopic experiments in d- and f-electron systems.

X-ray Birefringence Imaging of Materials with Anisotropic Molecular Dynamics.

The X-ray birefringence imaging (XBI) technique, reported very recently, is a sensitive tool for spatially resolved mapping of the local orientational properties of anisotropic materials. In this paper, we report the first XBI measurements on materials that undergo anisotropic molecular dynamics. Using incident linearly polarized X-rays with energy close to the Br K-edge, the X-ray birefringence is dictated by the orientational properties of the C-Br bonds in the material. We focus on two materials (urea inclusion compounds containing 1,8-dibromooctane and 1,10-dibromodecane guest molecules) for which the reorientational dynamics of the brominated guest molecules (and hence the reorientational dynamics of the C-Br bonds) are already well characterized by other experimental techniques. The XBI results demonstrate clearly that, for the anisotropic molecular dynamics in these materials, the effective X-ray optic axis for the X-ray birefringence phenomenon is the time-averaged resultant of the orientational distribution of the C-Br bonds.

Identification of inversion domains in KTiOPO4 via resonant X-ray diffraction.

A novel method is presented for the identification of the absolute crystallographic structure in multi-domain polar materials such as ferroelectric KTiOPO4. Resonant (or 'anomalous') X-ray diffraction spectra collected across the absorption K edge of Ti (4.966 keV) on a single Bragg reflection demonstrate a huge intensity ratio above and below the edge, providing a polar domain contrast of ∼270. This allows one to map the spatial domain distribution in a periodically inverted sample, with a resolution of ∼1 µm achieved with a microfocused beam. This non-contact, non-destructive technique is well suited for samples of large dimensions (in contrast with traditional resonant X-ray methods based on diffraction from Friedel pairs), and its potential is particularly relevant in the context of physical phenomena connected with an absence of inversion symmetry, which require characterization of the underlying absolute atomic structure (such as in the case of magnetoelectric coupling and multiferroics).

X-ray investigation of lateral hetero-structures of inversion domains in LiNbO3, KTiOPO4 and KTiOAsO4.

In this paper periodically domain-inverted (PDI) ferroelectric crystals are studied using high-resolution X-ray diffraction. Rocking curves and reciprocal-space maps of the principal symmetric Bragg reflections in LiNbO3 (LN) (Λ = 5 µm), KTiOPO4 (KTP) (Λ = 9 µm) and KTiOAsO4 (KTA) (Λ = 39 µm) are presented. For all the samples strong satellite reflections were observed as a consequence of the PDI structure. Analysis of the satellites showed that they were caused by a combination of coherent and incoherent scattering between the adjacent domains. Whilst the satellites contained phase information regarding the structure of the domain wall, this information could not be rigorously extracted without a priori knowledge of the twinning mechanism. Analysis of the profiles reveals strain distributions of Δd/d = 1.6 × 10(-4) and 2.0 × 10(-4) perpendicular to domain walls in KTP and LN samples, respectively, and lateral correlation lengths of 63 µm (KTP), 194 µm (KTA) and 10 µm (LN). The decay of crystal truncation rods in LN and KTP was found to support the occurrence of surface corrugations.

High-contrast sub-millivolt inelastic X-ray scattering for nano- and mesoscale science.

Photon and neutron inelastic scattering spectrometers are microscopes for imaging condensed matter dynamics on very small length and time scales. Inelastic X-ray scattering permitted the first quantitative studies of picosecond nanoscale dynamics in disordered systems almost 20 years ago. However, the nature of the liquid-glass transition still remains one of the great unsolved problems in condensed matter physics. It calls for studies at hitherto inaccessible time and length scales, and therefore for substantial improvements in the spectral and momentum resolution of the inelastic X-ray scattering spectrometers along with a major enhancement in spectral contrast. Here we report a conceptually new spectrometer featuring a spectral resolution function with steep, almost Gaussian tails, sub-meV (≃620 µeV) bandwidth and improved momentum resolution. The spectrometer opens up uncharted space on the dynamics landscape. New results are presented on the dynamics of liquid glycerol, in the regime that has become accessible with the novel spectrometer.

X-ray birefringence imaging.

The polarizing optical microscope has been used since the 19th century to study the structural anisotropy of materials, based on the phenomenon of optical birefringence. In contrast, the phenomenon of x-ray birefringence has been demonstrated only recently and has been shown to be a sensitive probe of the orientational properties of individual molecules and/or bonds in anisotropic solids. Here, we report a technique-x-ray birefringence imaging (XBI)-that enables spatially resolved mapping of x-ray birefringence of materials, representing the x-ray analog of the polarizing optical microscope. Our results demonstrate the utility and potential of XBI as a sensitive technique for imaging the local orientational properties of anisotropic materials, including characterization of changes in molecular orientational ordering associated with solid-state phase transitions and identification of the size, spatial distribution, and temperature dependence of domain structures.

Size-dependent shape and tilt transitions in In2O3 nanoislands grown on cubic Y-stabilized ZrO2(001) by molecular beam epitaxy.

The growth of In(2)O(3) on cubic Y-stabilized ZrO(2)(001) by molecular beam epitaxy leads to formation of nanoscale islands which may tilt relative to the substrate in order to help accommodate the 1.7% tensile mismatch between the epilayer and the substrate. High-resolution synchrotron-based X-ray diffraction has been used in combination with atomic force microscopy to probe the evolution in island morphology, orientation, and tilt with island size. Very small islands formed at low substrate coverage are highly strained but exhibit no tilt, while intermediate islands are tilted randomly in all directions, giving rise to distinctive doughnut-shaped structure in three-dimensional reciprocal space isosurfaces. The largest islands with lateral sizes on the order of 1 µm tilt away from the four equivalent in-plane <110> directions, giving three-dimensional scattering isosurfaces dominated by structure at the four corners of a square. Spatially resolved reciprocal space mapping using an X-ray beam with dimensions on the order of 1 µm suggests that the four-fold symmetry observed using a larger beam arises from averaging over an ensemble of islands, each with an individual tilt down one direction, rather than from the coexistence of differently tilted domains within a given island.

X-ray Birefringence: A New Strategy for Determining Molecular Orientation in Materials.

While the phenomenon of birefringence is well-established in the case of visible radiation and is exploited in many fields (e.g., through the use of the polarizing optical microscope), the analogous phenomenon for X-rays has been a virtually neglected topic. Here, we demonstrate the scope and potential for exploiting X-ray birefringence to determine the orientational properties of specific types of bonds in solids. Specifically, orientational characteristics of C-Br bonds in the bromocyclohexane/thiourea inclusion compound are elucidated from X-ray birefringence measurements at energies close to the bromine K-edge, revealing inter alia the changes in the orientational distribution of the C-Br bonds associated with a low-temperature order-disorder phase transition. From fitting a theoretical model to the experimental data, reliable quantitative information on the orientational properties of the C-Br bonds is determined. The experimental strategy reported here represents the basis of a new approach for gaining insights into the orientational properties of molecules in anisotropic materials.

Coherent X-ray diffraction investigation of twinned microcrystals.

Coherent X-ray diffraction has been used to study pseudo-merohedrally twinned manganite microcrystals. The analyzed compositions were Pr(5/8)Ca(3/8)MnO(3) and La(0.275)Pr(0.35)Ca(3/8)MnO(3). The prepared loose powder was thermally attached to glass (and quartz) capillary walls by gentle heating to ensure positional stability during data collection. Many diffraction data sets were recorded and some of them were split as expected from the main observed twin law: 180° rotation around [101]. The peak splitting was measured with very high precision owing to the high-resolution nature of the diffraction data, with a resolution (Δd/d) better than 2.0 × 10(-4). Furthermore, when these microcrystals are illuminated coherently, the different crystallographic phases of the structure factors induce interference in the form of a speckle pattern. The three-dimensional speckled Bragg peak intensity distribution has been measured providing information about the twin domains within the microcrystals. Research is ongoing to invert the measured patterns. Successful phase retrieval will allow mapping out the twin domains and twin boundaries which play a key role in the physical properties.

Quadrupole transitions revealed by Borrmann spectroscopy.

The Borrmann effect-a dramatic increase in transparency to X-ray beams-is observed when X-rays satisfying Bragg's law diffract through a perfect crystal. The minimization of absorption seen in the Borrmann effect has been explained by noting that the electric field of the X-ray beam approaches zero amplitude at the crystal planes, thus avoiding the atoms. Here we show experimentally that under conditions of absorption suppression, the weaker electric quadrupole absorption transitions are effectively enhanced to such a degree that they can dominate the absorption spectrum. This effect can be exploited as an atomic spectroscopy technique; we show that quadrupole transitions give rise to additional structure at the L(1), L(2) and L(3) absorption edges of gadolinium in gadolinium gallium garnet, which mark the onset of excitations from 2s, 2p(1/2) and 2p(3/2) atomic core levels, respectively. Although the Borrmann effect served to underpin the development of the theory of X-ray diffraction, this is potentially the most important experimental application of the phenomenon since its first observation seven decades ago. Identifying quadrupole features in X-ray absorption spectroscopy is central to the interpretation of 'pre-edge' spectra, which are often taken to be indicators of local symmetry, valence and atomic environment. Quadrupolar absorption isolates states of different symmetries to that of the dominant dipole spectrum, and typically reveals orbitals that dominate the electronic ground-state properties of lanthanides and 3d transition metals, including magnetism. Results from our Borrmann spectroscopy technique feed into contemporary discussions regarding resonant X-ray diffraction and the nature of pre-edge lines identified by inelastic X-ray scattering. Furthermore, because the Borrmann effect has been observed in photonic materials, it seems likely that the quadrupole enhancement reported here will play an important role in modern optics.