Positions In Reciprocal Space Research Articles

Treatment of multiple-beam X-ray diffraction in energy-dependent measurements.

During X-ray diffraction experiments on single crystals, the diffracted beam intensities may be affected by multiple-beam X-ray diffraction (MBD). This effect is particularly frequent at higher X-ray energies and for larger unit cells. The appearance of this so-called Renninger effect often impairs the interpretation of diffracted intensities. This applies in particular to energy spectra analysed in resonant experiments, since during scans of the incident photon energy these conditions are necessarily met for specific X-ray energies. This effect can be addressed by carefully avoiding multiple-beam reflection conditions at a given X-ray energy and a given position in reciprocal space. However, areas which are (nearly) free of MBD are not always available. This article presents a universal concept of data acquisition and post-processing for resonant X-ray diffraction experiments. Our concept facilitates the reliable determination of kinematic (MBD-free) resonant diffraction intensities even at relatively high energies which, in turn, enables the study of higher absorption edges. This way, the applicability of resonant diffraction, e.g. to reveal the local atomic and electronic structure or chemical environment, is extended for a vast majority of crystalline materials. The potential of this approach compared with conventional data reduction is demonstrated by the measurements of the Ta L3 edge of well studied lithium tantalate LiTaO3.

Chiral phonons in quartz probed by X-rays

The concept of chirality is of great relevance in nature, from chiral molecules such as sugar to parity transformations in particle physics. In condensed matter physics, recent studies have demonstrated chiral fermions and their relevance in emergent phenomena closely related to topology1–3. The experimental verification of chiral phonons (bosons) remains challenging, however, despite their expected strong impact on fundamental physical properties4–6. Here we show experimental proof of chiral phonons using resonant inelastic X-ray scattering with circularly polarized X-rays. Using the prototypical chiral material quartz, we demonstrate that circularly polarized X-rays, which are intrinsically chiral, couple to chiral phonons at specific positions in reciprocal space, allowing us to determine the chiral dispersion of the lattice modes. Our experimental proof of chiral phonons demonstrates a new degree of freedom in condensed matter that is both of fundamental importance and opens the door to exploration of new emergent phenomena based on chiral bosons.

Heterogeneous Ice Growth in Micron-Sized Water Droplets Due to Spontaneous Freezing

Understanding how ice nucleates and grows into larger crystals is of crucial importance for many research fields. The purpose of this study was to shed light on the phase and structure of ice once a nucleus is formed inside a metastable water droplet. Wide-angle X-ray scattering (WAXS) was performed on micron-sized droplets evaporatively cooled to temperatures where homogeneous nucleation occurs. We found that for our weak hits ice grows more cubic compared to the strong hits that are completely hexagonal. Due to efficient heat removal caused by evaporation, we propose that the cubicity of ice at the vicinity of the droplet’s surface is higher than for ice formed within the bulk of the droplet. Moreover, the Bragg peaks were classified based on their geometrical shapes and positions in reciprocal space, which showed that ice grows heterogeneously with a significant population of peaks indicative of truncation rods and crystal defects. Frequent occurrences of the (100) reflection with extended in-planar structure suggested that large planar ice crystals form at the droplet surface, then fracture into smaller domains to accommodate to the curvature of the droplets. Planar faulting due to misaligned domains would explain the increased cubicity close to the droplet surface.

Ferromagnetic Resonance with Magnetic Phase Selectivity by Means of Resonant Elastic X-Ray Scattering on a Chiral Magnet.

Cubic chiral magnets, such as Cu_{2}OSeO_{3}, exhibit a variety of noncollinear spin textures, including a trigonal lattice of spin whirls, the so-called skyrmions. Using magnetic resonant elastic x-ray scattering (REXS) on a crystalline Bragg peak and its magnetic satellites while exciting the sample with magnetic fields at gigahertz frequencies, we probe the ferromagnetic resonance (FMR) modes of these spin textures by means of the scattered intensity. Most notably, the three eigenmodes of the skyrmion lattice are detected with large sensitivity. As this novel technique, which we label REXS FMR, is carried out at distinct positions in reciprocal space, it allows us to distinguish contributions originating from different magnetic states, providing information on the precise character, weight, and mode mixing as a prerequisite of tailored excitations for applications.

Interpretation of diffuse scattering using superspace crystallography

Single crystal diffuse scattering is generally interpreted using correlation parameters that describe probabilities for certain configurations on a local scale. In this paper we present a novel interpretation of diffuse maxima using a disordered superspace approach. In (D+d)-dimensional superspace two modulation functions are disordered along the superspace axis $\vec{a}_{s,i}$ for i = 1,..,D, while the periodicity along the internal dimensions is maintained. This simple approach allows the generation of diffuse maxima of any width at any position in reciprocal space. The extinction rules that are introduced by superspace symmetry are also fulfilled by the diffuse maxima from structures generated using the disordered superspace approach. In this manuscript we demonstrate the disordered superspace approach using a simple a simple two-dimensional binary disordered system. The extension of the approach to (3+D)-dimensional superspace is trivial. The treatment of displacement and magnetic disorder in a similar approach is straight forward.

Strain engineering for controlled growth of thin-film FeNi L10

FeNi thin films in the L10 phase were successfully grown by magnetron sputtering on HF-etched Si substrates on Cu/Cu100−xNix buffers. The strain of the FeNi layer, , was varied in a controlled manner by changing the Ni content of the Cu100−xNix buffer layer from to , which influenced the common in-plane lattice parameter of the CuNi and FeNi layers. The presence of the L10 phase was confirmed by resonant x-ray diffraction measurements at various positions in reciprocal space. The uniaxial magnetocrystalline anisotropy energy is observed to be smaller (around 0.35 MJ m−3) than predicted for a perfect FeNi L10 sample, but it is larger than for previously studied films. No notable variation in with strain state is observed in the range achieved (), which is in agreement with theoretical predictions.

On the emergence of structure and transport properties in (Al50Mn50)100 −xCux

In this contribution we report on emerging structure and transport properties of thin films of (Al50Mn50)100−xCux, evaporated at low temperature. Their structures have been investigated by electron diffraction, their transport properties and thermal stability by resistivity measurements, showing global resonance effects between two global subsystems, as there is the Fermi gas as one and the forming static structure of the remaining ions as the other one. The global resonances are self-organizing by the exchange of characteristic momentum, trigger density anomalies and hybridization effects, both reported in this contribution. We show strong indications for a combined action of density and an effective-valence anomalies, to keep the peak position on a resonance-preferred position in reciprocal space. In real space it causes the spherical structural periodicity of nearest-neighbor shells at medium-range distances. The spherical-periodic atomic order of an amorphous phase is analog to planar order in a crystal. The characteristic atomic distances between nearest neighbor shells are determined by the Friedel wavelength, half of the Fermi wavelength, indicating resonance effects. They influence all the properties including phase stability and electronic transport. The electronic transport properties and thermal stability data strongly correlate with the structural determination under global resonances.

Spin excitations used to probe the nature of exchange coupling in the magnetically ordered ground state ofPr0.5Ca0.5MnO3

We have used time-of-flight inelastic neutron scattering to measure the spin wave spectrum of the canonical half-doped manganite Pr$_{0.5}$Ca$_{0.5}$MnO$_{3}$, in its magnetic and orbitally ordered phase. The data, which cover multiple Brillouin zones and the entire energy range of the excitations, are compared with several different models that are all consistent with the CE-type magnetic order, but arise through different exchange coupling schemes. The Goodenough model, i.e. an ordered state comprising strong nearest neighbor ferromagnetic interactions along zig-zag chains with antiferromagnetic inter-chain coupling, provides the best description of the data, provided that further neighbor interactions along the chains are included. We are able to rule out a coupling scheme involving formation of strongly bound ferromagnetic dimers, i.e. Zener polarons, on the basis of gross features of the observed spin wave spectrum. A model with weaker dimerization reproduces the observed dispersion but can be ruled out on the basis of discrepancies between the calculated and observed structure factors at certain positions in reciprocal space. Adding further neighbor interactions results in almost no dimerization, i.e. recovery of the Goodenough model. These results are consistent with theoretical analysis of the degenerate double exchange model for half-doping, and provide a recipe for how to interpret future measurements away from half-doping, where degenerate double exchange models predict more complex ground states.

Temperature and magnetic field dependence of spin-ice correlations in the pyrochlore magnetTb2Ti2O7

We present a parametric study of the diffuse magnetic scattering at (1/2,1/2,1/2) positions in reciprocal space, ascribed to a frozen antiferromagnetic spin ice state in single crystalline Tb2Ti2O7. Our high-resolution neutron scattering measurements show that the elastic (-0.02 meV < E < 0.02 meV) (1/2,1/2,1/2) scattering develops strongly below ~275 mK, and correlates with the opening of a spin gap of ~0.06-0.08 meV over most of the Brillouin zone. The concomitant low-lying magnetic spin excitations are weakly dispersive and appear to soften near the (1/2,1/2,1/2) wave vector at 80 mK. The nature of the transition at 275 mK has many characteristics of spin glass behavior, consistent with ac-susceptibility measurements. The application of a magnetic field of 0.075 T applied along the [1-10] direction destroys the (1/2,1/2,1/2) elastic scattering, revealing the fragility of this short-range ordered ground state. We construct a refined H-T phase diagram for Tb2Ti2O7 and [1-10] fields which incorporates this frozen spin ice regime and the antiferromagnetic long-range order previously known to be induced in relatively large fields. Specific heat measurements on the same crystal reveal a sharp anomaly at Tc~450 mK and no indication of a transition near ~275 mK. We conclude that the higher temperature specific heat peak is not related to the magnetic ordering but is likely a signal of other, nonmagnetic, correlations.

Antiferromagnetic spin ice correlations at (12,12,12) in the ground state of the pyrochlore magnet Tb2Ti2O7

We present high-resolution single crystal time-of-flight neutron scattering measurements on the candidate quantum spin liquid pyrochlore Tb2Ti2O7 at low temperature and in a magnetic field. At ~70 mK and in zero field, Tb2Ti2O7 reveals diffuse magnetic elastic scattering at (1/2,1/2,1/2) positions in reciprocal space, consistent with short-range correlated regions based on a two-in, two-out spin ice configuration on a doubled conventional unit cell. This elastic scattering is separated from very low-energy magnetic inelastic scattering by an energy gap of ~0.06-0.08 meV. The elastic signal disappears under the application of small magnetic fields and upon elevating temperature. Pinch-point-like elastic diffuse scattering is observed near (1,1,1) and (0,0,2) in zero field at ~70 mK, in agreement with Fennell et al. (Ref. 1), supporting the quantum spin ice interpretation of Tb2Ti2O7.

Inelastic neutron and x-ray scattering from incommensurate magnetic systems

Neutrons and x-rays are powerful probes for studying magnetic and lattice excitations instrongly correlated materials over very wide ranges of momentum and energy transfers. Inthe focus of the present work are the incommensurate magnetic systems MnSi and Cr.Under application of a magnetic field, helically ordered MnSi transforms intoa weak itinerant ferromagnet. Using polarized neutrons we demonstrate thatthe Stoner excitations are spin-flip excitations. The amplitude (longitudinal)fluctuations associated with the magnon modes are already strong far away fromTc. Interestingly, even the non-spin-flip excitations associated with the Stoner modes areobservable. In Cr, we have observed Kohn anomalies in the phonon spectrum at thoseincommensurate positions in reciprocal space, where the spin density wave is observed. Thecorresponding phonon and magnon modes are not coupled. In addition, an anomaloussoftening of a transverse phonon branch along the N–H zone boundary line is observed thatis caused by strong electron–phonon coupling. High resolution neutron scattering indicatesthat the low energy Fincher–Burke excitations may rather correspond to localized modes inmomentum and energy and not to propagating collective modes. Finally, we demonstratethat in the near future it may become feasible to investigate excitations in verysmall samples, thus allowing us to measure the dynamics of strongly correlatedmaterials under extreme conditions and in the vicinity of quantum phase transitions.

Anomalous High-Energy Spin Excitations in the High-TcSuperconductor-Parent AntiferromagnetLa2CuO4

Inelastic neutron scattering is used to investigate the collective magnetic excitations of the high-temperature superconductor-parent antiferromagnet La2CuO4. We find that while the lower energy excitations are well described by spin-wave theory, including one- and two-magnon scattering processes, the high-energy spin waves are strongly damped near the (1/2, 0) position in reciprocal space and merge into a momentum dependent continuum. This anomalous damping indicates the decay of spin waves into other excitations, possibly unbound spinon pairs.

Real-Time Surface X-ray Scattering Study of Au(111) Electrochemical Dissolution

Direct in situ surface X-ray scattering studies of electrochemical dissolution at technologically relevant rates are presented using Au(111) electrodes in acidic chloride-containing solutions as an example. The dissolution process was investigated with high time resolution by monitoring the time-dependent X-ray intensity at selected positions in reciprocal space and by simultaneously recording high-quality electrochemical data. With increasing electrode potential, transitions from step-flow to layer-by-layer dissolution, manifesting as layering oscillations in the X-ray intensity as well as in electrochemical current−time transients, are observed prior to surface passivation. A quantitative analysis based on an atomic-scale structural model reveals that the dissolution process proceeds via progressive vacancy island nucleation in the whole active dissolution regime with next-layer nucleation occurring at comparatively low critical coverages between θc = 0.29 and θc = 0.44. This dissolution behavior shows the same characteristics as the intermediate “smooth multilayer growth” regime recently introduced in kinetic growth theories.

Phonon instabilities in uniaxially compressed fcc metals as seen in molecular dynamics simulations

We show that the generation of stacking faults in perfect face-centered-cubic (fcc) crystals, uniaxially compressed along [001], is due to transverse-acoustic phonon instabilities. The position in reciprocal space where the instability first manifests itself is not a point of high symmetry in the Brillouin zone. This model provides a useful explanation for the magnitude of the elastic limit, in addition to the affects of box size, temperature, and compression on the time scale for the generation of stacking faults. We observe this phenomenon in both simulations that use the Lennard-Jones potential and embedded atom potentials. Not only does this work provide fundamental insight into the microscopic response of the material but it also describes certain behavior seen in previous molecular dynamics simulations of single-crystal fcc metals shock compressed along the principal axis.

Resonant x-ray scattering investigation of the multipolar ordering inCa3Co2O6

We have performed a resonant x-ray scattering (RXS) study near the $\text{Co}\text{ }K$ edge on a single crystal of ${\text{Ca}}_{3}{\text{Co}}_{2}{\text{O}}_{6}$. In the magnetically ordered phase a new class of weak reflections appears at the magnetic propagation vector $\stackrel{P\vec}{\ensuremath{\tau}}\ensuremath{\simeq}(\frac{1}{3},\frac{1}{3},\frac{1}{3})$. These new reflections allow direct access to the dipolar-quadrupolar ${E}_{1}{E}_{2}$ scattering channel. The theoretical possibility of observing isolated ${E}_{1}{E}_{2}$ electromagnetic multipoles has attracted a lot of interest in recent years. Unfortunately in many systems of interest, parity even and parity odd tensor contributions occur at the same positions in reciprocal space. We demonstrate that in ${\text{Ca}}_{3}{\text{Co}}_{2}{\text{O}}_{6}$ it is possible to completely separate the parity even from the parity odd terms. The possibility of observing such terms even in globally centrosymmetric systems using RXS has been investigated theoretically; ${\text{Ca}}_{3}{\text{Co}}_{2}{\text{O}}_{6}$ allows a symmetry based separation of this contribution.

Spin gaps and magnetic structure ofNaxCoO2

We present two experiments that provide information on spin anisotropy and the magnetic structure of NaxCoO2. First, we report low-energy neutron inelastic scattering measurements of the zone-center magnetic excitations in the magnetically ordered phase of Na0.75CoO2. The energy spectra suggest the existence of two gaps, and are very well fitted by a spin-wave model with both in-plane and out-of-plane anisotropy terms. The gap energies decrease with increasing temperature and both gaps are found to have closed when the temperature exceeds the magnetic ordering temperature T_m~22 K. Secondly, we present neutron diffraction studies of Na0.85CoO2 with a magnetic field applied approximately parallel to the c axis. For fields in excess of ~8T a magnetic Bragg peak was observed at the (0,0,3) position in reciprocal space. We interpret this as a spin-flop transition of the A-type antiferromagnetic structure, and we show that the spin-flop field is consistent with the size of the anisotropy gap.

Small angle x-ray scattering metrology for sidewall angle and cross section of nanometer scale line gratings

High-volume fabrication of nanostructures requires nondestructive metrologies capable of measuring not only the pattern size but also the pattern shape profile. Measurement tool requirements will become more stringent as the feature size approaches 50nm and tolerances of pattern shape will reach a few nanometers. A small angle x-ray scattering (SAXS) based technique has been demonstrated to have the capability of characterizing the average pitch size and pattern width to subnanometer precision. In this study, we report a simple, modeling-free protocol to extract cross-section information such as the average sidewall angle and the pattern height of line grating patterns from the SAXS data. Diffraction peak intensities and reciprocal space positions are measured while the sample is rotated around the axis perpendicular to the grating direction. Linear extrapolations of peak positions in reciprocal space allow a precise determination of both the sidewall angle and the pattern height.

Jahn‐Teller distortion ordering in single‐crystal Nd 1/2 Sr 1/2 MnO 3

Charge, spin and orbital degrees of freedom play important roles in the electrical and magnetic properties of manganites. Nd0.5Sr0.5MnO3 is a paramagnetic insulator with Imma orthorhombic symmetry at room temperature. Upon cooling it becomes a ferromagnetic metal below 240 K and at 160 K undergoes a first-order phase transition. Below this temperature it is a CE-type antiferromagnetic insulator. In this paper we report synchrotron X-ray scattering investigations of superlattice reflections observed in the low temperature phase of Nd0.5Sr0.5MnO3. Upon cooling below 160 K we observed additional peaks occurring at non-integer positions in reciprocal space. These reflections had a wavevector (1/2, 0, 0) and are due to the structural modulation arising from the Jahn-Teller distortion associated with charge ordering of Mn3+ and Mn4+ ions and orbital ordering in the sample. The peaks observed below 160 K have an intensity of approximately 10−3 that of the Bragg reflections. The intensity profiles of these reflections show that the transition into the ordered phase is strongly first-order. The structural phase transition was observed to display a hysteresis width of 10 K. The measured widths of these peaks show that the charge modulation is correlated primarily in the ac plane rather than the K-direction (long-axis). (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Generic representation and evaluation of properties as a function of position in reciprocal space

A generalized approach is described for evaluating arbitrary functions of position in reciprocal space. This is a generalization which subsumes a whole range of calculations that form a part of almost every crystallographic software application. Examples include scaling of structure factors, the calculation of structure-factor statistics, and some simple likelihood calculations for a single parameter. The generalized approach has a number of advantages: all these calculations may now be performed by a single software routine which need only be debugged and optimized once; the existing approach of dividing reciprocal space into resolution shells with discontinuities at the boundaries is no longer necessary; the implementation provided makes employing the new functionality extremely simple and concise. The calculation is split into three standard components, for which a number of implementations are provided for different tasks. A `basis function' describes some function of position in reciprocal space, the shape of which is determined by a small number of parameters. A `target function' describes the property for which a functional representation is required, for example \langle |F|^2\rangle. An `evaluator' takes a basis and target function and optimizes the parameters of the basis function to fit the target function. Ideally the components should be usable in any combination.

Effect of chromium addition on the ordering behaviour of Ni–Mo alloy: experimental results vs. electronic structure calculations

The ordering behaviour of a Ni-Mo alloy in the presence of a ternary additive, viz. Cr, has been studied. The sequence of ordering transformations in binary Ni–Mo alloys has been shown earlier to be controlled by a competition between several f.c.c.-based superlattices, viz. Ni 2Mo (Pt 2Mo type), Ni 3Mo (DO 22), Ni 4Mo (D1 a ) and the so-called short range ordered (SRO) structure characterized by the presence of diffraction intensity maxima at the 1 1/ 20 f.c.c. positions in reciprocal space. The effect of ternary addition of chromium in the selection of the superlattice structure has been examined in this paper in an alloy of composition Ni–24 at.% Mo–6 at.% Cr. The presence of Cr has been experimentally found to favour the formation of Ni 2(Mo,Cr) (Pt 2Mo-type) phase in preference to the Ni 3Mo (DO 22) and Ni 4Mo (D1 a ) superlattices. This leads to a sequence of transformation different from that obtained in binary alloys. The effect of Cr on the ground state phase stability is determined, in this alloy, using the first-principles local-density based full-potential augmented plane wave (FP-LAPW) method. From the differences in the electronic structures, densities of states and total energies of binary Ni 2Cr, Ni 2Mo (Pt 2Mo type), Ni 3Mo (DO 22) and Ni 4Mo (D1 a ) phases and the corresponding ternary superlattice structures, an attempt has been made to predict the hierarchy of relative phase stabilities. Theoretical predictions based on these electronic structure calculations have been found to be consistent with the experimental microstructural observations of the evolutionary stages of ordering in the ternary Ni–24 at.% Mo–6 at.% Cr alloy.