Reciprocal Vectors Research Articles

Electromagnetic modes of corrugated thin films and surfaces with a transition layer. II. Minigaps.

Our recent calculation [Phys. Rev. B 36 , 1007 (1987)] of electromagnetic modes that may propagate parallel to a thin corrugated film is applied to the study of the ``minigap'' regions. Two polariton branches of the planar film, when displaced by a reciprocal vector of the corrugation period, may intersect at values of the propagation constant (quasimomentum) that do not coincide with a Brillouin-zone boundary. In the vicinity of an intersection a coupling of modes occurs, and our first-order perturbational calculation shows that the behavior is determined by the signs of two parameters B and G, giving rise to four cases. B is a coupling constant, proportional to ${h}^{2}$ (where h is the height of the corrugation), and G is positive (negative) if the slopes of the unperturbed modes at the intersection have opposite (equal) signs. For B>0 and G>0, an energy gap occurs; in general, the minigap is ``indirect;'' that is, the minimum of the upper band and the maximum of the lower band are not located at the same value of the momentum. Within the gap we find solutions with complex values of the momentum. In the case B>0 and G0, a ``repulsion'' of modes takes place; there is no gap. If B0 and G>0, then we get a momentum gap that is also indirect, in general. Within this gap there exist solutions with a complex value of the frequency. In the last case, B0 and G0, a frequency gap and a momentum gap are obtained simultaneously. The widths of the minigaps are proportional to \ensuremath{\Vert}f^(i-j)\ensuremath{\Vert} (hence to h), the Fourier coefficient of the corrugation profile corresponding to unperturbed branches i and j. Special attention is given to the case that two branches of the smooth film intersect symmetrically at a Brillouin-zone boundary. In this case, B>0 and G>0 are applicable; the energy gap then becomes direct. We have also investigated the amplitudes of the partial, diffracted waves in the minigap regions. All our results are directly applied to a surface with a transition layer, as well as to a sharp surface, and to guided waves with either TE or TM polarization.

Solid-liquid phase transition in quasi 1D compounds containing iodine trimers

By a combination of transfer matrix method and Landau-type analysis, we study the thermodynamic behavior of triiodide chains embedded in an organic matrix. We emphasize the major role played by the internal structure of the iodine trimers and by the interchain Coulomb couplings at higher order reciprocal wave vectors. We discuss the high temperature fluctuations and the different possible phase transitions in relationship with recent X ray scattering measurements.

Localization, equivalence and transferability of non-orthogonal localized molecular orbitals

By introducing the definition of the reciprocal basis vector, the unit operator can be established, allowing vectors and operators to be represented in matrix form, and the solution of equations can be reduced to a series of basis vector transformations. By using this method and discarding the constraint of orbital orthogonality, non-orthogonal localized molecular orbitals (NLMOs) with optimum localization, equivalence and transferability can be obtained through non-singular linear transformations of canonical molecular orbitals. Some theorems and examples are introduced to explain the equivalence and transferability of NLMOs.

Matrix representation of vectors and operators in nonorthogonal basis vectors

The bases used in quantum mechanics, in general, are orthonormal. The relations in orthonormal basis vectors are simpler than those in nonorthogonal basis vectors. Also, the operators that represent different physical quantities are Hermitian and the eigenvectors of these operators belonging to different eigenvalues are orthonormal to each other. But, in many cases, such as sets of screened hydrogenic wavefunctions, nonorthogonal sets of basis vectors occur. It is, therefore, necessary to consider the representation of vectors and operators in nonorthogonal bases. Making use of the set of vectors reciprocal to the given set of nonorthogonal base vectors, the expression for the norms of vectors and for the matrix elements has been found. Also, adjoint and Hermitian matrices in nonorthogonal bases have been worked out using a set of reciprocal basis vectors.

A Numerical Approach to the Inverse Toeplitz Eigenproblem

A Toeplitz matrix $T = {\operatorname{Toep}}(t_0 \cdots t_{n - 1} ) = [t_{ij} ]$ is a matrix with the property that $t_{ij} = t_{i - j} $, i, $j = 1, \cdots ,n$. The inverse Toeplitz eigenproblem is: construct a real symmetric Toeplitz matrix having a given set of n real numbers as its eigenvalues. This paper presents a numerical approach to the problem. At each stage, a Toeplitz matrix $T_k $ is constructed having the given set of numbers as its Rayleigh quotients with respect to the current matrix $Q_k $ of approximate eigenvectors. The new $Q_{k + 1} $ is found by diagonalizing $T_k $. Convergence could not be proved. Numerical evidence indicates that the algorithm behaves like a typical quadratically convergent algorithm. In the course of testing the algorithm, a counterexample was found to the conjecture of Delsarte and Genin that the eigenvectors of a symmetric Toeplitz matrix, corresponding to eigenvalues arranged in decreasing order, alternate between reciprocal and antireciprocal vectors.

On integrating the techniques of direct methods and isomorphous replacement. A new probabilistic formula for triplet invariants

The mathematical technique recently used [Hauptman (1982). Acta Cryst. A38, 289-294] for integrating direct methods and isomorphous replacement techniques is reconsidered. The atomic positions are assumed to be the primitive random variables instead of the reciprocal vectors. A new probabilistic formula for estimating three-phase invariants given six magnitudes has been obtained which differs from the corresponding result of Hauptman. The first applications suggest that the formula is quite efficient for picking up triplet cosines near 0 or π.

A Coupled-Mode Approach to the Analysis of Fields in Space-Curved and Twisted Waveguides

A coupled-mode approach for solving Maxwell's equations in terms of unitary and reciprocal unitary vectors is deduced in a twisted space-curved coordinate system. Application of this method to the analysis of fields in a space-curved and twisted single-mode optical fiber and a twisted rectangular microwave waveguide is presented and compared with the results from existing literature.

Electronic structure of a quasiperiodic system.

The general solution of a Schr\"odinger equation with a quasiperiodic potential in $n$ dimensions is obtained. A boost technique is presented, which will transform the problem to the solution of a periodic pseudo-Schr\"odinger equation in $n+m$ dimensions, to which Floquet-Bloch theory is applicable. We show that the eigenfunctions of the original problem and the boosted problem are related to each other by a simple radon transform, and the eigenvalues are exactly equal. We identify the hierarchical gap structure in the energy spectrum observed in numerical simulations and we show that the location of gaps can be indexed by the reciprocal wave vector given by the diffraction pattern of the quasicrystal. The position and the magnitude of the gaps so predicted are in qualitative agreement with numerical simulations.

Angle resolved ultraviolet photoemission spectroscopy study of the electronic structure of InSb(111) surfaces along the [110] azimuth

The analysis of non-normal photoemission spectra of the polar A(111) and B (1̄1̄1̄) surfaces of InSb along the [110] azimuth indicates strong contribution from bulk features. Transitions common to both surfaces around 1.4, 2.6, 3.3, and 6.2 eV are associated with critical points in the bulk band structure. Umklapp scattering of photoelectrons with reciprocal vectors of the InSb(1̄1̄1̄)(3×3) surface is identified for transitions around 3.3 eV. Modifications in the spectra of this surface after Sn coverage suggest contribution from a surface resonance band around 1.4 eV; bands around 4.3 and 4.9 eV dispersing within a gap of the projected bulk band structure calculated by a realistic empirical tight-binding method are tentatively assigned to backbond states.

Estimation of quartet phase sums from a new joint probability distribution of normalized structure factors

A new joint probability distribution of normalized structure factors is derived for equal-atom structures in space group P1. From this general distribution, a series expansion, the conditional joint probability distribution of the quartet phase sum is obtained, when restrictive conditions among the reciprocal vectors are imposed. The main difference from existing quartet distributions is the possibility of enclosing higher-order terms to any given order of N, although an approximation employed in the derivation limits the number of them considerably. The higher-order terms present are easily employed in the series since the determination of their explicit appearance has been automated: a computer program derives the terms up to a desired order and stores them in a coded form. In general, the incorporation of selective terms up to order N-3 appears to yield sufficient convergence. Only high |E| values or a low N value may necessitate the use of higher-order terms. Test results show that, in contrast to results from the quartet distributions of Hauptman and Giacovazzo, systematic estimation errors are hardly present, while absolute estimation errors are reduced as well.

Shear-free homogeneous cosmological model with heat flux

We present the condition of vanishing shear in a spatially homogeneous spacetime in terms of the Ricci rotation co-efficients corresponding to an orthonormal tetrad (ν α. A η α) (whereν α is the unit vector along the time axis and A η α are the three independent reciprocal group vectors). Assuming that the velocity vector can be expanded in the direction ofν α and any one of the A η α’s it is shown that shear-free motion is possible only in case of some special Bianchi types, and these cases are studied assuming the velocity vector to be geodetic and that there may be a nonvanishing heat flux term.

Neutron diffraction study of Al-Mn quasicrystal

Powder neutron diffraction measurements have been carried out on the Al-Mn quasicrystal at room temperature. Nine diffraction peaks of the quasicrystal phase were observed, and indexed on the basis of six independent reciprocal vectors pointing to the vertices of an icosahedron.

Icosahedral symmetry by an incommensurately modulated crystal structure model

A crystal structure model of an incommensurately modulated structure is presented. Although six different reciprocal vectors are used to describe the model, all calculations are done in three dimensions making calculation of the real-space structure trivial. Using this model, it is shown that both the positions of the bragg reflections and information about the relative intensities of these reflections are in full accordance with the diffraction patterns reported for microcrystals of the rapidly quenched Al86Mn14 alloy. It is also shown that at least the local structure possesses full icosahedral symmetry with distances which are in agreement with the nearest-neighbor distances between Al-Mn and Al-Al atoms in the crystalline compound Al6Mn.

On the reciprocal vector optimization problems

A necessary and sufficient condition is established for an optimal solution of a primal vector optimization problem to be an optimal solution of its reciprocal. Such a condition is developed and analyzed in the Pareto case, the strong case, and the lexicographic case. We detail these results for ordinary (i.e., scalar) optimization problems.

Recording diffuse X-ray reflections with continuous synchrotron radiation. An application to type la diamond

Abstract When photographing the diffraction pattern of a single crystal using a collimated incident beam of continuous-spectrum X-rays, the two-dimensional diffuse-scattering pattern received on the photograph is a projection of the three-dimensional distribution of diffuse-scattering power, D(R), from reciprocal vectors g + R surrounding the reciprocal lattice vector g of the Bragg reflection. By using a synchrotron radiation source, useful diffuse patterns from selected small volumes of crystal can be recorded in a few minutes. Some three-dimensional information in D(R) lost in projection can be retrieved by partially eclipsing D(R) with an absorption edge of a beam filter. In an application to diamond, new ‘spike’ diffuse reflections are observed which are interpreted as evidence for the presence of sub-microscopic {111} faceted cavities known as ‘voidites’.

The rhombohedral phase produced in partially-stabilized zirconia

It has recently been reported that the rhombohedral phase (r-phase) is produced on the abraded surface of partially-stabilized zirconia (PSZ) [1]. The r-phase found in yttria-stabilized PSZ (Y-PSZ) has a different lattice parameter and composition with the ordered phase Zr3Y4012, which is an equilibrium phase in the ZrO2-Y203 system [2, 3]. The r-phase is considered to be a metastable phase which is induced by mechanical polishing, although the stability of the phase has not been clarified yet. This paper aims to report that the r-phase is formed not only on abraded surfaces but also in bulk Y-PSZ of a particular composition. Five ZrO2--Y203 alloys with Y203 content 2.0, 3.1, 4.0, 5.0 and 6.2mo1% were prepared from zirconia and yttria powders with 99.9% purity produced by Rare Metallic Co. Ltd. The oxide powders were mixed in a ball mill followed by pelleting in a steel die. The pellets were sintered at 1400 ° C for 1 h, and then arc-melted. The average weight of arc-melted samples was about 1.5 g. As reported previously [4], the arc-melted samples were black due to oxygen deficiency, but annealing at 1400 ° C for a few hours restored the usual colour. The oxygen deficiency was estimated to be about 1 at% from the increase in weight of a pure zirconia sample by the annealing. EPMA analysis has shown that the samples have a uniform concentration of Y203 on the cross-sectional plane within an accuracy of less than 0.5mo1%. The structures of the alloys were examined by X-ray diffraction, optical and electron microscopy. X-ray diffraction studies have revealed that the arc-melted alloys with 5.0 and 6.2mo1% Y2Oz were composed of cubic and tetragonal phases. Fig. 1 shows an electron micrograph of ZrO25.0molY203 alloy in which the two-phase structure is seen. It was found that the diffraction pattern from the matrix in the two-phase structure was indexed by an fc c lattice with a fluorite structure. However, the forbidden reflections in the fc c lattice often appeared from the lath-like phase in Fig. 1 [5]. From X-ray and electron diffraction data, the lath-like phase was recognized to be a tetragonal phase (t-phase) embedded in the cubic matrix (c-phase). On the other hand, the structure of ZrO2-2 .0mol%Y203 alloy was composed of monoclinic and tetragonal phases. It was found that a phase with a characteristic structure was developed in ZRO2-3.1 and 4.0 mol%Y203 alloys together with cand t-phases. Fig. 2 is an electron micrograph of the phase with a "herring-bone" appearance, which has a different microstructure to the c-phase or t-phase. The structure in Fig. 2 resembles the rhombohedral /3-phase in ZrO2-Sc203 alloys [6, 7], which is believed to be an equilibrium phase [7, 8]. An examination of the diffraction pattern in Fig. 2 has revealed that the pattern is slightly distorted from that of the f c c lattice with a zone axis close to [100] . The angle between 0 2 0 and 0 0 2 reflections is about 91 °, and the reciprocal vector 0 22 is slightly larger than 02 2. Such a distortion is expected from the rhombohedral distortion in the (111) direction of the fluorite lattice. The

Reduction methods for nonlinear steady‐state thermal analysis

AbstractHybrid analysis techniques based on the combined use of finite elements and the classical Bubnov–Galerkin approximation are presented for predicting nonlinear steady‐state temperature distributions in structures and solids. In these hybrid techniques the modelling versatility of the finite element method is preserved and a substantial reduction in the number of degrees‐of‐freedom is achieved by expressing the vector of nodal temperatures as a linear combination of a small number of global‐temperature modes, or basis vectors. The Bubnov–Galerkin technique is then used to compute the coefficients of the linear combination (i.e. the amplitudes of the global–temperature modes).The basis vectors chosen are the path derivatives commonly used in perturbation techniques, namely, the derivatives of the nodal–temperature vector with respect to a preselected control (or path) parameter(s). The vectors are generated by using the finite element model of the initial discretization. Also, the performance of alternate sets of basis vectors is investigated. In the alternate sets, only a few path derivatives are generated, and they are augmented by a constant vector representing a uniform temperature rise (or drop), and by reciprocal vectors with nonzero components equal to the reciprocals of the nonzero components of the path derivatives. A problem‐adaptive computational algorithm is presented for efficient evaluation of global approximation vectors and generation of the reduced system of equations and for monitoring the accuracy of the reduced system of equations.The potential of the proposed reduction methods for the solution of large‐scale, nonlinear steady‐state thermal problems is also discussed. The effectiveness of these methods is demonstrated by means of four numerical examples, including conduction, convection and radiation modes of heat transfer.This study shows that the use of the uniform‐temperature mode and the path derivatives as global approximation vectors significantly increases the accuracy of the solutions obtained by reduction methods, thereby enhancing the effectiveness of these methods for the solution of large‐scale, nonlinear thermal problems.

Anisotropy effects and optical excitation of acoustic phonons in n-i-p-i doping superlattices

The effect of impurity charges associated with a microstructured one-dimensional periodic n- and p-doping sequence in an otherwise homogeneous semiconductor on its lattice dynamical properties is studied. The Coulomb interaction does not remove the degeneracy of the back-folded acoustic phonons at the “mini-Brillouin zone” boundaries, introduced by the superstructure. Some of these phonon branches are, however, slightly affected at the Γ-point. The most striking result is the possibility to excite by electromagnetic waves, longitudinal and transverse acoustic phonos whose wavevectors are odd multiples of the smallest reciprocal superlattice vector.

AN ANALYTICAL METHOD FOR INDEXING X-RAY POWDER PATTERNS OF MONOCLINIC SUBSTANCES

In this paper a more simple and flexible analytical method for indexing X-ray powder patterns of monoclinic substances by using computer is proposed. The lattice parameter b can be obtained from the repeat number m of δ value of every pair of diffraction lines in δ-m chart and their linear relationship. By the reducing procedure the reciprocal lattice is projected on the a*-c*plane and the angle of each pair of reciprocal vectors can be calculated from the Qnkl data of each triplet of reciprocal points. According to the repeat number M in β-M chart the monoclinic angle β can be obtained and then the lattice parameters a and c can be calculated. Finally the indexing of the entire diffraction patterns is made by iterative correction method for indexing powder patterns. The computer program of the analytical method is given. The indexing of two monoclinic substances nickel acetate hydrate (C4H6NiO)4·4H2O) and potassium tungstate (K2WO4) has been made by using this method, the result proves that the method is fairly applicable.

Syntheses of optimal actuators for oscillatory and non-oscillatory systems with a scalar input

Abstract Syntheses of optimal actuators which minimize an energy are considered for oscillatory and non-oscillatory systems with a scalar input from the viewpoint of terminal control with an arbitrary terminal time. Under a certain assumption for the latter systems, hut no assumptions for the former systems, an optimal actuator matrix is shown to be given by the vector directed to the centre of gravity of the convex polyhedron which is constructed by the reciprocal vectors of the eigenvectors of a transition matrix. In the case where the above assumption is not satisfied for the non-oscillatory systems, a sequential approximation method, whose starting point is the vector directed to the centroid of the convex polyhedron, is proposed to obtain a suboptimal solution.