Reciprocal Vectors Research Articles

New stable decagonal quasicrystal in the system Al–Ir–Os

A stable decagonal quasicrystal with nominal composition Al 73Ir 14.5Os 12.5 has been grown from the aluminum-rich melt. It has an incongruent melting temperature of T m = 1283 °C and does not transform into a periodic phase within the maximum applied annealing time of 8 weeks at 1000 °C. The phase equlibria in the aluminum-rich part of the system Al–Ir–Os have been explored. The X-ray diffraction patterns can be indexed by five reciprocal basis vectors with lengths a i * = 0.2588 ( 2 ) Å − 1 , i = 1, …, 4, and a 5 * = 0.05945 ( 3 ) Å − 1 . This corresponds to a quasilattice parameter (i.e. the edge length of the related Penrose tiling) of 2.501(2) Å and a translation period along the 10-fold axis of 16.821(8) Å. The 5D space group is P10 5 mc. The structure of the decagonal phase has been solved by two different approaches, the 3D pseudo-approximant technique and the 5D charge-flipping method. It is closely related to those of decagonal Al–Os–Pd and Al–Ni–Ru and can be described as quasiperiodic packing of columnar clusters with 20.150 Å diameter. The clusters consist of eight quasiperiodic layers with stacking sequence …ABCDA′B′C′D ′…, where the prime (′) denotes the layers generated by a 10 5 operation.

Coupling of reciprocal vectors and corresponding degeneracy effect in a dual-periodic optical superlattice

A dual-periodic structure for quasi-phase matching cascaded optical parametric interactions is proposed. Due to the coupling of reciprocal vectors between the original and imposed periodic sequence, the reciprocal vectors and the corresponding effective nonlinear coefficients is no longer the simple combination of two periodic structures. The new analytical expression of the effective nonlinear coefficients is deduced and given. The degeneracy phenomena and the novel extinction rule resulting from the coupling of reciprocal vectors are found and investigated. The corresponding physical nature is also discussed.

Enhanced optical transmission through metal films with rotation-symmetrical hole arrays

The transmission of light through metal surface with subwavelength holes are influenced by many factors, and the rotational symmetry of hole arrays can be one of them. In this paper, we fabricated the hole lattices in metal films with different symmetry and measured the transmission spectra from the visible to near-infrared region. It is found that both the spectrum shape and the transmission efficiency are strongly dependent on the rotational symmetry. The spectrum shape is governed by the reciprocal vectors. And the higher is the symmetry order, the larger the peak efficiency. The results provide us with new insight into the unusual effect.

Spin dynamics of half-dopedLa3∕2Sr1∕2NiO4

We report polarized- and unpolarized-neutron inelastic-scattering measurements of the magnetic excitation spectrum in the spin-charge ordered phase of ${\mathrm{La}}_{3∕2}{\mathrm{Sr}}_{1∕2}\mathrm{Ni}{\mathrm{O}}_{4}$. Up to energies of $\ensuremath{\sim}30\phantom{\rule{0.3em}{0ex}}\mathrm{meV}$ we observe broad magnetic modes characteristic of a near checkerboard ordering. A linear spin-wave model for an ideal checkerboard ordering with a single antiferromagnetic exchange interaction ${J}^{\ensuremath{'}}=5.8\ifmmode\pm\else\textpm\fi{}0.5\phantom{\rule{0.3em}{0ex}}\mathrm{meV}$ between next-nearest-neighbor spins on ${\mathrm{Ni}}^{2+}$ sites, together with a small $XY$-like single-ion anisotropy, provides a reasonable description of the measured dispersion. Above $30\phantom{\rule{0.3em}{0ex}}\mathrm{meV}$ the excitations are not fully consistent with the linear spin-wave model, with modes near the two-dimensional reciprocal space wave vector (0.5, 0.5) having an anomalously large intensity. Furthermore, two additional dispersive modes not predicted by spin-wave theory were observed, both of which are probably magnetic. One disperses away from (0.5, 0.5) in the energy range between $50--56\phantom{\rule{0.3em}{0ex}}\mathrm{meV}$, and the other appears around $(h,k)$ type positions $(h,k=\text{integer})$ in the energy range $31--39\phantom{\rule{0.3em}{0ex}}\mathrm{meV}$. We propose a model in which these anomalous features are explained by the existence of discommensurations in the checkerboard ordering. At low energies there is additional diffuse scattering centred on the magnetic ordering wave vector. We associate this diffuse scattering with dynamic antiferromagnetic correlations between spins attached to the doped holes.

Formulas for periodic dislocations in general interfaces

The present work has advanced the calculation method for determining the configuration of periodic dislocations in general interfaces, especially for three sets of dislocations that are usually unsolvable by other methods. In terms of a newly introduced reciprocal Burgers vector (bi*), the configuration and the Burgers vector of the interfacial dislocations are related by a simple expression with general validity. Existing formulas for spacing of the dislocations in various special interfaces can be integrated in a concise form of D=1∕∣Δbi*∣.

Photoemission from surface-localized structures on vicinal surfaces: Initial- or final-state superlattice effect?

Strong superlattice effects are reported on E ( k ‖ ) band dispersions measured by angle-resolved photoemission on W(3 3 1) and W(5 5 1), and graphene nanostripes on Ni(7 7 1). A splitting of the dispersions with reciprocal superlattice vector G = 2 π / L is observed for a surface resonance on W(3 3 1) and W(5 5 1) and for π -states of graphene/Ni(7 7 1). Photon-energy-dependent measurements and comparison to LEED show that electrons are confined to the one-dimensional terraces and stripes, respectively, and that the observed superlattice effects are due to diffraction in the final state of the photoemission transitions.

Quantum-well states and lateral superlattice effect in Ag and Au stripes on stepped W(1 4 5)

Au and Ag on the stepped W(1 4 5) surface with step width L ∼ 11.6 Å form quantum stripes up to a thickness of 2 and 3 monolayers (ML), respectively. The 2 ML stripes show a one-dimensional E ( k ‖ ) dispersion which perpendicular to the stripes is determined by a repetition with the reciprocal superlattice vector G = 2 π / L .

Distorted Hexagonal Phase Studied by Neutron Diffraction: Lipid Components Demixed in a Bent Monolayer

The recent discovery of a distorted hexagonal phase in 1,2-dioleoyl-sn-glycero-3-phosphatidylethanolamine/1,2-dioleoyl-sn-glycero-3-phosphatidylcholine (DOPE/DOPC) mixtures raised the intriguing question as to whether lipid mixtures demix in a bent monolayer. We performed neutron diffraction on a mixture of headgroup deuterated DOPC-d(13) and nondeuterated DOPE to study the lipid distribution in the distorted hexagonal phase. The 1:1 lipid mixture in full hydration and 25 degrees C was in a homogeneous lamellar phase. Upon dehydration the mixture transformed to a rhombohedral phase, then to a distorted inverted hexagonal phase, and finally to a regular inverted hexagonal phase. In the distorted hexagonal phase, the diffraction pattern showed a two-dimensional monoclinic lattice with two reciprocal vectors of equal length (1.5 nm(-)(1)) forming an angle 53 degrees between them. Diffraction intensities measured while varying the D(2)O/H(2)O ratio in the humidity was used to solve the phase problem. The neutron scattering length density distribution of the distorted hexagonal phase was constructed. The constant density contours are approximately elliptical. The difference in the eccentricities of the contours between the water and lipid distributions indicates that the DOPE/DOPC ratio is not uniform around the elliptical lipid tube in the unit cell. DOPE is preferentially distributed at the vertex regions where the curvature is the highest. Thus for the first time it is shown that when a monolayer of a homogeneous lipid mixture is bent, the lipid components are partially demixed in reaching the free energy minimum.

An elementary introduction to superspace crystallography

Abstract Aperiodic crystals are defined as a crystalline state of matter, that has atomic structures with long-range order but without translational symmetry. Experimentally, they are characterized by sharp Bragg reflections in the X-ray diffraction, that can be indexed by integers, if four or more reciprocal basis vectors are used. An introduction is given to the basic concepts of the superspace theory for structural analysis of incommensurately modulated crystals and incommensurate composite crystals [De Wolff, Janner and Janssen, Acta Crystallogr. 37 (1981) 625–636]. It is concluded that major challenges of the crystallography of incommensurate phases lie in the determination of the precise shapes of modulation functions, and in finding the relations between physical properties of incommensurate crystals and their atomic structures.

New findings in phase transitions of spinel [formula omitted] studied by transmission electron microscopy

Phase transitions and the related phase identifications at different temperatures for spinel Li 1.07 Mn 1.93 O 4 − δ crystals have been studied by transmission electron microscopy. The preliminary results clearly show the existence of at least two types of low-temperature phase: monoclinic and orthorhombic. Oxygen deficiency may raise the phase-transition temperature for the high-temperature (HT) cubic to low-temperature (LT) tetragonal, monoclinic, or orthorhombic phases. When the oxygen deficiency δ is very close to zero, the transition temperature is below room temperature (RT). Therefore, only the HT cubic spinel is observed at RT. When δ=0.182, the transition temperature is higher than RT, so the structures of the LT phases can be studied at RT. This study reveals the structural relationships between the LT phase and the HT phase. These relationships can be summarized as follows: (1) Two orthogonal cubic [ 440 ] Cubic * and [ 4 4 ¯ 0 ] Cubic * reciprocal vectors are divided into two parts by {220} and { 2 2 ¯ 0 } super reflections, respectively. This situation resembles the structures caused by correlated tilting and/or distortion of the octahedra in perovskites. (2) One of the cubic 〈 440 〉 Cubic * reciprocal vectors is divided into three parts by weak super-reflections. This situation resembles the modulated structure caused by the charge- and orbital-ordering in the perovskite La 1/3Ca 2/3MnO 3 of the space group Pnma. (3) One of the cubic 〈 440 〉 Cubic * reciprocal vectors, e.g. [ 440 ] Cubic * , is divided into three parts by stronger super-reflections, at the same time some strong reflections of the cubic spinel, e.g. ( 1 1 ¯ 1 ) Cubic = ( 320 ) Ortho / 2 disappear, indicating that there is a phase transition containing atomic movement. Items (2) and (3) are new findings in the present work.

Optical harmonic generation in a Fibonacci dielectric superlattice of LiNbO3

Quasi-periodic structure can be introduced into nonlinear optical materials for realizing optical frequency conversion. The reciprocal vectors of the structure are used to compensate for the phase-mismatch between the interacting waves due to material dispersion. Second-harmonic and direct third-harmonic generations by second-order nonlinear optical process in a Fibonacci lithium niobate dielectric superlattice have been treated by using a transfer matrix method. Taking advantage of a numerical technique, we investigate the dependence of the second-harmonic and third-harmonic on the fundamental wavelength and structural parameters. Under typical structural parameters, which are designed by transfer matrix method, high conversion efficiency of the second-harmonic and third-harmonic can be obtained at some special fundamental wavelength.

Atomic displacement parameters and anisotropic thermal ellipsoid lengths and angles.

Relations between direct and reciprocal basis vectors and the atomic displacement matrix and anisotropic temperature-factor matrix are provided for generic dimension n, introducing a few new symbols. The moments of the probability distribution of the anisotropic thermal ellipsoids are provided as functions of dimension n. For specific dimensions n = 2 and n = 3, new explicit formulas for the semi-axis lengths and rotation angles of the thermal ellipsoids are derived from the atomic displacement matrix, taking into account certain special cases. For dimension n = 3, the Euler angles are defined. The resulting formulas can be used, for example, to display the thermal ellipsoids of atoms or for further analysis of molecular vibrations.

Flat Broad-Band Wavelength Conversion Based on Sinusoidally Chirped Optical Superlattices in Lithium Niobate

We report the enhancement of the bandwidth and the response flatness of wavelength conversion through cascaded second-order nonlinear processes by sinusoidally chirped optical superlattices (SCOSs) in lithium niobate crystal. The conversion bandwidth of the SCOS is significantly larger than that of uniform and segmented gratings. The SCOS shows very flat conversion response in the 1.5-/spl mu/m region. The physical origin of the bandwidth enhancement is ascribed to the plentiful reciprocal wave vectors of the SCOS, which provide enough momenta to simultaneously satisfy quasi-phase matching conditions for both second-harmonic generation and difference-frequency generation.

Electrostatic Energy Calculation and Parameter Optimization in Computer Molecular Simulation

Molecular simulation is a powerful tool in studying properties of complex fluids composed of charged particles such as electrolyte solutions, room temperature ionic liquids and colloid solutions, where the long-range interactions play a determinative role. Several methods have been available for treating the long-range interactions between charged particles and point dipole. These include the Ewald sum (ES), the reaction field and particle-particle particle-mesh methods. Among these approaches, the ES is most commonly used. However, several ES versions have been occurred in the literatures of molecular simulation, and some times it is difficult for one to choose the right formula to use in molecular simulation study. The coulombic interaction energy between charged particles is divided into the sum of real space, reciprocal space and self energy, and theircal culation equations are obtained respectively using electrostatics theory and Fourier transformation method. The Ewald sum formulate have been derived and the clear physical picture involved has been depicted. A Monte Carlo computer simulation for electrostatic interaction energy of charged hard sphere system has been conducted at varying conditions, and a good agreement with MSA is obtained. On this basis the effects on the simulation accuracy and efficiency of real space cut distance r(cut), convergence parameter 1 and reciprocal maximum vector K(max) have been analyzed. From the theoretical and computer simulation presented here, the optimization parameters for rcut, 1 and K(max) are obtained as 0.5 L (i.e. half box length), 5.8 and 3～5, respectively.

Evolution of Polar and Dipolar Dislocation Density from Polychromatic Microdiffraction

ABSTRACTPolychromatic X-ray microdiffraction (PXM) is sensitive to the density and organization of the dislocations, which occurs at several structural levels. At the lowest level statistically stored (dipolar) individual dislocations can exist within a crystal. At a higher structural level dislocations can organize into strongly correlated arrangements including walls and sub-boundaries. After plastic deformation geometrically necessary (polar) dislocations as well as geometrically necessary boundaries may be formed in a crystal. These dislocations cause not only random deformation, but also strongly correlated long range rotations within the crystal, grain, or subgrain. Non homogeneous plastic deformation is observed even in single crystals at smaller scale. Polar dislocations spread the conditions for x-ray (or neutron) diffraction transverse to the reciprocal space vector of each reflection. Diffracted intensity depends on the second rank dislocation density tensor. The polar dislocations density is related to the incompatibility of the plastic deformation and to the local lattice curvature. Laue patterns are therefore sensitive to the ratio between polar and dipolar dislocations density. Different slip systems of polar dislocations population cause distinctly different streaking in Laue patterns. Examination of streaked patterns enables one to determine statistically stored (dipolar) dislocations and geometrically necessary (polar) dislocations, GNDs, and to quantitativly determine the dislocation patterning parameters. The co-evolution of the statistically stored (dipolar) and geometrically necessary dislocations (polar) may be analyzed, and the ratio between the two densities may be obtained from the analysis of the Laue spots intensity distribution. The microbeam technique is applied to analyze a dislocation structure in a Ni bicrystal under uniaxial pulling.

TEM study of superstructure in a perovskite lead lanthanum zirconate stannate titanate ceramic

The superstructure of an antiferroelectric Pb 0.97La 0.02(Zr 0.66Sn 0.25Ti 0.09)O 3 phase, whose composition is near the morphotropic-phase boundary, was characterized. Systematic selected-area diffraction revealed that there were two kinds of superlattice reflections in the pseudocubic reciprocal lattice, i.e. 1 2 hk l superlattice reflections ( h, k, l all odd), and g± 1 7.24 a ∗ +b ∗ one-dimensional incommensurate superlattice reflections, where g denotes the vectors of fundamental or 1 2 hk l superlattice reflections. Convergent-beam electron diffraction disclosed that the average structure of the phase was rhombohedral with space group of R 3m . Based on the rhombohedral reciprocal lattice, the 1 2 hk l reflections were no longer superlattice but fundamental reflections, and the reciprocal vector of the one-dimensional incommensurate reflections was re-expressed as H=ha ∗ +kb ∗ +lc ∗ ± 1 7.24 a ∗ +b ∗ , where h, k, l are integers and (− h + k + l) = 3 n. In the light of the average structure and the reflection condition, the superspace Bravais class of the phase with one-dimensional incommensurate structure was determined to be P 11 R 3m in a (3 + 1)-dimensional space. In addition, the origins of the superlattice reflections were also examined and discussed.

Indexing schemes for quasilattices

Various schemes have been proposed and employed, which extend the method of indexing lattice vectors and reciprocal-lattice vectors, so that it can be used in the context of quasicrystals. The concept of the generalized inverse of a matrix provides an elegant unified approach to the vectors and reciprocal vectors of quasilattices, and their associated zone laws and inflation rules. We present a survey, from the viewpoint provided by the concept of the Moore–Penrose inverse, of the indexing problem for quasicrystals.

First results obtained by the Cluster STAFF experiment

Abstract. The Spatio Temporal Analysis of Field Fluctuations (STAFF) experiment is one of the five experiments, which constitute the Cluster Wave Experiment Consortium (WEC). STAFF consists of a three-axis search coil magnetometer to measure magnetic fluctuations at frequencies up to 4 kHz, a waveform unit (up to either 10 Hz or 180 Hz) and a Spectrum Analyser (up to 4 kHz). The Spectrum Analyser combines the 3 magnetic components of the waves with the two electric components measured by the Electric Fields and Waves experiment (EFW) to calculate in real time the 5 × 5 Hermitian cross-spectral matrix at 27 frequencies distributed logarithmically in the frequency range 8 Hz to 4 kHz. The time resolution varies between 0.125 s and 4 s. The first results show the capabilities of the experiment, with examples in different regions of the magnetosphere-solar wind system that were encountered by Cluster at the beginning of its operational phase. First results obtained by the use of some of the tools that have been prepared specifically for the Cluster mission are described. The characterisation of the motion of the bow shock between successive crossings, using the reciprocal vector method, is given. The full characterisation of the waves analysed by the Spectrum Analyser, thanks to a dedicated program called PRASSADCO, is applied to some events; in particular a case of very confined electromagnetic waves in the vicinity of the equatorial region is presented and discussed.Key words. Magnetospheric physics (magnetopause, cusp and boundary layer) – Space plasma physics (waves and instabilities; shock waves)

Unique Indexing Scheme of Decagonal Phases Based on a Six Dimensional Model

ABSTRACTMandal and Lele (1989) have proposed a six dimensional model for the structural description of the decagonal phases. The integral linear combination of six basis vectors for indicating a physical vector in their model, however, leaves the problem of redundancy in indexing. While revisiting their model, we have noted that the condition of a null vector in physical space permits the formulation of unique indexing scheme both in physical reciprocal and direct spaces, we will demonstrate that our scheme, unlike all previously discussed ones, relies only on the information contained in the model. It will also be shown that diffracted spot having equivalent indices possesses identical intensity. This aspect, though equally important, has been totally ignored in the past. We shall substantiate our claim by taking examples from the known decagonal phases. We shall also present parity condition on indices that will be helpful in discussing subtle features of diffraction patterns. The notion of weak and strong diffracting conditions is explained based on the zone rule in physical space in terms of Cartesian components of direct and reciprocal vectors.

Experimental determination of pore shape and size using q-space NMR microscopy in the long diffusion-time limit

The signal obtained with q-space NMR imaging applied to a confined liquid is directly related to the pore shape in the limit where all molecules have sampled the whole pore. We investigate the diffusion of water across a ∼50 μm thick film formed between planes of glass. The diffusion time t is changed almost three orders of magnitude. For short t, the root-mean-square displacement increases with a rate which is slightly less than for freely diffusing water. At t longer than 0.3 s, the displacement is constant at 24 μm which implies that the water is confined in the measuring direction defined by the applied gradient pulses. Perfectly smooth and aligned planes give rise to sharp diffraction-like features on the echo attenuation curve, i.e., NMR signal vs. the reciprocal space vector q. The experimental data with rather smooth local minima and maxima can be explained in terms of either surface roughness or a misalignment of the planes. We discuss the averaging effect of diffusion along a laterally inhomogeneous film and propose two model-free methods to determine the pore shape from the echo attenuation curve obtained in the long- t limit.