Friedel's Law Research Articles

Diffraction enhancement of symmetry and modular structures.

Diffraction enhancement of symmetry (DES) is the phenomenon according to which the symmetry of the diffraction pattern of a crystal can be higher than the point symmetry of the structure that has produced it. The most well known example is that of Friedel's law, which is however violated in the presence of resonant scattering. This phenomenon is addressed in monoarchetypal modular structures and it is shown that a sufficient condition for DES is that both the module and the family of stacking vectors are invariant under an isometry that is not a symmetry operation for the structure. In general, DES is still observed in the presence of significant resonant scattering. The example of SiC polytypes, where the phenomenon has been confirmed experimentally, is studied in detail but the conclusions depend only on the symmetry of the layer and on the stacking vectors: the same applies therefore to all compounds sharing these features, like ZnS.

Tailored diffraction asymmetries from spatially odd-symmetric phase gratings

Phase gratings, each period of which consists of even numbers of equal-width and equal-thickness elements, can be devised to attain asymmetric diffraction patterns. We show that engineering of offset refractive indices in different elements exhibiting a spatially odd symmetry leads to the elimination of a single diffraction order (directional elimination), while further manipulations on offset refractive indices may lead to the elimination of all odd or even orders (grouped elimination) or all orders but one or two selected (directional selection). These intriguing effects arise from destructive interference between diffracted amplitudes contributed by paired or successive elements, and violation of Friedel's law in such transparent gratings is an effect of higher order multiple scattering.

HfB2 ceramic polycrystals: A low-temperature metal-like ceramic at high temperatures?

Hafnium diboride (HfB2) is a highly refractory (melting above 3000°C) ceramic with many potential applications at high temperatures. To enable its use at temperature for extended periods its high-temperature plasticity must be known. This paper examines the mechanical response at temperatures between 900°C and 2000°C in air and in a reducing atmosphere, interpreting the data in the frame of classical models for the plasticity of compact-packed metals at low temperatures. In particular, the Friedel law and the principle of similitude for dislocation patterning are assessed. This reveals that HfB2 is a singular example of a ceramic material with “metal” mechanical behaviour.

Coherent Lattice Wobbling and Out-of-Phase Intensity Oscillations of Friedel Pairs Observed by Ultrafast Electron Diffraction.

The inspection of Friedel's law in ultrafast electron diffraction (UED) is important to gain a comprehensive understanding of material atomic structure and its dynamic response. Here, monoclinic gallium telluride (GaTe), as a low-symmetry, layered crystal in contrast to many other 2D materials, is investigated by mega-electronvolt UED. Strong out-of-phase oscillations of Bragg peak intensities are observed for Friedel pairs, which does not obey Friedel's law. As evidenced by the preserved mirror symmetry and supported by both kinematic and dynamic scattering simulations, the intensity oscillations are provoked by the lowest-order longitudinal acoustic breathing phonon. Our results provide a generalized understanding of Friedel's law in UED and demonstrate that by designed misalignment of surface normal and primitive lattice vectors, coherent lattice wobbling and effective shear strain can be generated in crystal films by laser pulse excitation, which is otherwise hard to achieve and can be further utilized to dynamically tune and switch material properties.

Imaging Polarity in Two Dimensional Materials by Breaking Friedel's Law

Friedel's law guarantees an inversion-symmetric diffraction pattern for thin, light materials where a kinematic approximation or a single-scattering model holds. Typically, breaking Friedel symmetry is ascribed to multiple scattering events within thick, non-centrosymmetric crystals. However, two-dimensional (2D) materials such as a single monolayer of MoS2 can also violate Friedel's law, with unexpected contrast between conjugate Bragg peaks. We show analytically that retaining higher order terms in the power series expansion of the scattered wavefunction can describe the anomalous contrast between hkl and hkl¯peaks that occurs in 2D crystals with broken in-plane inversion symmetry. These higher-order terms describe multiple scattering paths starting from the same atom in an atomically thin material. Furthermore, 2D materials containing heavy elements, such as WS2, always act as strong phase objects, violating Friedel's law no matter how high the energy of the incident electron beam. Experimentally, this understanding can enhance diffraction-based techniques to provide rapid imaging of polarity, twin domains, in-plane rotations, or other polar textures in 2D materials.

Symmetry in the diffraction of beams carrying orbital angular momentum

Diffraction of orbital-angular-momentum--carrying Laguerre-Gauss vortex (LGV) beams by $N$-fold rotationally symmetric regular polygons is studied analytically and experimentally. The structure, symmetry, and dependence of the diffraction pattern on the angular momentum index of the LGV beam and the number of sides in the polygon are systematically investigated, and features and trends are identified. The evolution of the diffraction pattern and its symmetry with the aperture position relative to the waist is also studied, leading to a generalized Friedel's law for diffraction of LGV beams.

Imaging rotations and vibrations in polyatomic molecules with X-ray scattering.

An approach for calculating elastic X-ray scattering from polyatomic molecules in specific electronic, vibrational, and rotational states is presented, and is used to consider the characterization of specific states in polyatomic molecules using elastic X-ray scattering. Instead of the standard independent atom model (IAM) method, the X-ray scattering is calculated directly from ab initio wavefunctions. The role of molecular symmetry and Friedel's law is examined, with the molecules BF3, C5H5-, NF3, and 1,3-cyclohexadiene used as specific examples. The contributions to the elastic X-ray scattering from the electronic, vibrational, and rotational portions of the molecular wavefunction are examined in CS2. In particular, it is observed that the rotational states give rise to distinct signatures in the scattering signal.

Extension of Friedel's law to vortex-beam diffraction

Friedel's law states that the modulus of the Fourier transform of real functions is centrosymmetric, while the phase is antisymmetric. As a consequence of this, elastic scattering of plane-wave photons or electrons within the first-order Born-approximation, as well as Fraunhofer diffraction on any aperture, is bound to result in centrosymmetric diffraction patterns. Friedel's law, however, does not apply for vortex beams, and centrosymmetry in general is not present in their diffraction patterns. In this work we extend Friedel's law for vortex beams by showing that the diffraction patterns of vortex beams with opposite topological charge, scattered on the same two-dimensional potential, always are centrosymmetric to one another, regardless of the symmetry of the scattering object. We verify our statement by means of numerical simulations and experimental data. Our research provides deeper understanding in vortex-beam diffraction and can be used to design new experiments to measure the topological charge of vortex beams with diffraction gratings or to study general vortex-beam diffraction.

Interpretation of angular symmetries in electron nanodiffraction patterns from thin amorphous specimens.

The interpretation of angular symmetries in electron nanodiffraction patterns from thin amorphous specimens is examined. It is found that in general there are odd symmetries in experimental electron nanodiffraction patterns. Using simulation, it is demonstrated that this effect can be attributed to dynamical scattering, rather than other divergences from the ideal experimental conditions such as probe-forming lens aberrations and camera noise. The departure of opposing diffracted intensities from Friedel's law in the phase grating formalism is calculated using a general structure factor for disordered materials. On the basis of this, a simple correction procedure is suggested to recover the kinematical angular symmetries, and thus readily interpretable information that reflects the symmetries of the original projected object. This correction is numerically tested using both the phase object and multislice calculations, and is demonstrated to fully recover all the kinematical diffracted symmetries from a simulated atomic model of a metallic glass.

Electrical resistivity anomaly: A consequence of nanometric particles in a metal matrix

The additament of an impurity to a metal matrix generates a shift in the local electron density of states due to a change in the local electric potential so that an electron or electron beam will be scattered upon interaction with the ensuing potential, causing its wave function measured at a distance far removed from the source to show a quantum phase change. This phenomenon, reflected in Friedel's law, permits to relate the quantum phase shifts with both the density of states at the Fermi level and the transport properties of the system. This work calculates the electric resistivity of nanometric clusters in an aluminiumsupercell, emulating the formation of Guinier-Preston zones during pre-precipitation in an Al-Zn alloy. The density of states is evaluated each time a new atom of solute is added to the Al supercell using the ab initio method of the Density Functional Theory or DFT in its GGA and LDA approximations via the Dmol^{3} package of Materials Studio Software. A resistivity anomaly is obtained when, in concordance with literature reports for diluted AlZn alloys, the supercell contains 25 atoms, thus confirming that the electric resistivity anomaly during cluster formation is a consequence of both density of states variations and quantum transitions at the Fermi Level for a nanometric size in particular. The results are contrasted with those obtained via the transport equation.

Direct Observation of the Polar State in the Relaxor Ba(Ti1−xZrx)O3 by Transmission Electron Microscopy

In the ferroelectric mixed-oxide system Ba(Ti1−xZrx)O3 (BTZ), an increase in Zr content results in the crossover from the ferroelectric MA-type state with monoclinic symmetry to the relaxor state around x = 0.28. As a result of the crossover, BTZ is identified as the relaxor for x > 0.28. To understand the nature of the relaxor in BTZ, direct observations of the paraelectric (PE), MA, and relaxor states for 0.17 ≤ x ≤ 0.40 was carried out between 400 K and 87 K by transmission electron microscopy with the help of the failure of Friedel's law. The observations indicated that polar nanometer-sized regions with <110>PE and <001>PE components were separately observed in the PE state above both Tc and Tm , which are, respectively, a transition temperature of the direct (PE→MA) transition and a maximum temperature of the real dielectric permittivity for the relaxor. During cooling from the paraelectric state above Tm in the relaxor, in the relaxor state below it, <001>PC-component regions locally formed a 180° domain structure, together with no coalescence of <110>PC regions. It is thus understood that the MA-to-relaxor crossover is characterized by the complete suppression of the coalescence of polar nanometer-sized regions with <110>PE components. The relaxor state below Tm for 0.29 ≤ x ≤ 0.40 could be, as a result, identified as an assembly of polar nanometer-sized domains, presumably with monoclinic symmetry.

Non-centro-symmetric electron diffraction pattern of icosahedral quasicrystal induced by combination of linear phason strain and curvature of Ewald sphere

Non-centro-symmetric characteristics are observed in the experimental electron diffraction patterns (EDPs) from the icosahedral quasicrystalline precipitates in ZrAlNiCuNb alloys. Different from the well-known breaking of the Friedel's law, where a strong dynamical effect will reveal in EDPs the concealed non-centro-symmetry originated from the crystal structures themselves, the current results can be interpreted in terms of changes in deviation parameters due to a delicate combination of the linear phason strain characteristic of quasicrystals and the curvature of Ewald sphere. After taking this effect into consideration, the corresponding simulated EDPs fit quite well to the experimental data.

Common arc method for diffraction pattern orientation

Very short pulses of X-ray free-electron lasers opened the way to obtaining diffraction signal from single particles beyond the radiation dose limit. For three-dimensional structure reconstruction many patterns are recorded in the object's unknown orientation. A method is described for the orientation of continuous diffraction patterns of non-periodic objects, utilizing intensity correlations in the curved intersections of the corresponding Ewald spheres, and hence named the common arc orientation method. The present implementation of the algorithm optionally takes into account Friedel's law, handles missing data and is capable of determining the point group of symmetric objects. Its performance is demonstrated on simulated diffraction data sets and verification of the results indicates a high orientation accuracy even at low signal levels. The common arc method fills a gap in the wide palette of orientation methods.

Failure of Friedel's Law in Electron Diffraction Scattering in Bilayer Graphene and Graphene with Defects

Extended abstract of a paper presented at Microscopy and Microanalysis 2011 in Nashville, Tennessee, USA, August 7–August 11, 2011.

Interpretation of the Nano-Electron-Diffraction Patterns along the Five-Fold Axis of Decahedral Gold Nanoparticles

The transition from 10-fold to 5-fold symmetry was observed during the analysis of nanodiffraction patterns of a gold decahedral multiple twinned nanoparticle of 15 nm in diameter. The analysis shows that as the convergence of the beam is increased, the rotational symmetry of the diffraction pattern shifts from 10- to 5-fold. The 10-fold symmetry predicted by Friedel's law is lost by the asymmetric shift of the diffraction spots, an effect that becomes more noticeable as the electron beam convergence increases. Dynamical and kinematical diffraction calculations indicate this decrease in symmetry is the result of a double refraction effect coupled with the variation of the dynamical diffraction conditions arising from a varying electron beam convergence.

Room‐Temperature Synthesis of the Highly Polar Cluster Compound Sn[SnCl][W3Cl13

AbstractA new synthetic strategy is employed to synthesize a tungsten trinuclear cluster compound with an ionic liquid (IL) as the reaction medium. Dark brown block‐shaped crystals of Sn[SnCl][W3Cl13] were obtained at room temperature by reacting tin and WCl6 in Lewis acidic ionic liquid [BMIM]Cl/AlCl3 ([BMIM]+ = 1‐butyl‐3‐methylimidazolium). Compound Sn[SnCl][W3Cl13] crystallizes in the polar hexagonal space group P63 with cell parameters a = 959.8(1) and c = 1190.3(2) pm. The crystallographic point group of the [W3(μ3‐Cl)(μ‐Cl)3Cl9]3– cluster anions is C3, but C3v pseudosymmetry is observed within twice the standard deviation of the bond lengths. The W–W bond length is 272.18(6) pm. The tin(II) cations, which are in trigonal pyramidal and in distorted trigonal‐antiprismatic coordination spheres that comprise chloride ions, connect the clusters into layers that are stacked according to the 63 screw axis. One of the chloride ions is bound only to a single tin cation resulting in a [SnIICl]+ group with a Sn–Cl bond length of 236.1(5) pm. The pronounced polarity of the crystal structure means that its diffraction pattern exhibits huge deviations from Friedel's law. Most remarkable is that the intensity (Fc2) of reflection 004 is 175 % of the intensity of the 00$\bar {4}$ reflection.

Bloch Oscillations in Complex Crystals withPTSymmetry

Bloch oscillations in complex lattices with PT symmetry are theoretically investigated with specific reference to optical Bloch oscillations in photonic lattices with gain or loss regions. Novel dynamical phenomena with no counterpart in ordinary lattices, such as nonreciprocal Bloch oscillations related to violation of the Friedel's law of Bragg scattering in complex potentials, are highlighted.

The Mystery of Perpendicular Fivefold Axes and the Fourth Dimension in Intermetallic Structures

The structures of eight related known intermetallic structure types are the impetus to this paper: Li21Si5, Mg44Rh7, Zn13(Fe,Ni)2, Mg6Pd, Na6Tl, Zn91Ir11, Li13Na29Ba19, and Al69Ta39. All belong to the F43m space group, have roughly 400 atoms in their cubic unit cells, are built up at least partially from the gamma-brass structure, and exhibit pseudo-tenfold symmetric diffraction patterns. These pseudo-tenfold axes lie in the {110} directions, and thus present a paradox. The {110} set is comprised of three pairs of perpendicular directions. Yet no 3D point group contains a single pair of perpendicular fivefold axes (by Friedel's Law, a fivefold axis leads to a tenfold diffraction pattern). The current work seeks to resolve this paradox. Its resolution is based on the largest of all 4D Platonic solids, the 600-cell. We first review the 600-cell, building an intuition discussing 4D polyhedroids (4D polytopes). We then show that the positions of common atoms in the F43m structures lie close to the positions of vertices in a 3D projection of the 600-cell. For this purpose, we develop a projection method that we call intermediate projection. The introduction of the 600-cell resolves the above paradox. This 4D Platonic solid contains numerous orthogonal fivefold rotations. The six fivefold directions that are best preserved after projection prove to lie along the {110} directions of the F43m structures. Finally, this paper shows that at certain ideal projected cluster sizes related to one another by the golden mean (tau=(1+ radical 5)/2), constructive interference leading to tenfold diffraction patterns is optimized. It is these optimal values that predominate in actual F43m structures. Explicit comparison of experimental cluster sizes and theoretically derived cluster sizes shows a clear correspondence, both for isolated and crystalline pairs of projected 600-cells.

Crystallography without crystals. I. The common-line method for assembling a three-dimensional diffraction volume from single-particle scattering

It is demonstrated that a common-line method can assemble a three-dimensional oversampled diffracted intensity distribution suitable for high-resolution structure solution from a set of measured two-dimensional diffraction patterns, as proposed in experiments with an X-ray free-electron laser (XFEL) [Neutze et al. (2000). Nature (London), 406, 752-757]. Even for a flat Ewald sphere, it is shown how the ambiguities due to Friedel's law may be overcome. The method breaks down for photon counts below about 10 per detector pixel, almost three orders of magnitude higher than expected for scattering by a 500 kDa protein with an XFEL beam focused to a 0.1 microm diameter spot. Even if 10(3) orientationally similar diffraction patterns could be identified and added to reach the requisite photon count per pixel, the need for about 10(6) orientational classes for high-resolution structure determination suggests that about 10(9) diffraction patterns must be recorded. Assuming pulse and readout rates of approximately 100 Hz, such measurements would require approximately 10(7) s, i.e. several months of continuous beam time.

Finite-size effects in dislocation glide through random arrays of obstacles: Line tension simulations

The stress required to force a dislocation to glide through a random environment of obstacles depends on both the dislocation length and the glide distance because of the statistical nature of the obstacle configurations. In order to study this finite size effect, we employ a line tension model in conjunction with an evolution algorithm inspired from larger scale dislocation dynamics simulations. We show that with finite arrays, the estimated critical resolved shear stress is larger than its infinite array limit. Moreover, the lower the resistance and the density of the obstacles, the larger the overestimation. The controlling parameters, estimated from Friedel's law, are the average number of obstacles along the dislocation line and the average number of obstacle configurations met by the dislocation in its glide. We analyze this effect through a model based on an analogy with branching processes.