Reciprocal Vectors Research Articles

E(K) OF Ni ABOVE THE VACUUM LEVEL AS MEASURED BY VLEED ON RELAXED (110) SURFACE

Varying incidence angle, VLEED spectra R(E) and dR/dE were measured on Ni(110) using target current technique. The results were interpreted with the reference spectra calculated on the basis of ground-state band structure. Some critical points (CPs) of E( k ) displayed themselves as dR/dE extremums while others remained invisible. Changing of incidence K || by a surface reciprocal vector switched manifesting CPs. Experimental CPs were determined by comparing energies of experimental dR/dE extremums with. their reference counterparts. Having mapped CPs along a variety of non-symmetry directions, bands along ΓX were constructed. Experimental E( k ) is consistent with other experimental data and APW calculations. Some peculiarities are attributed to the influence of surface relaxation. Capabilities of VLEED to provide PES with upper bands are discussed.

Layer by layer growth of epitaxial films of Bi2Sr2CuO6: Structural and electrical properties related to the oxydation level

Bi 2 Sr 2 CuO 6 thin films were deposited on SrTiO 3 (100) single crystals by atomic layer epitaxy. The layer by layer deposition process is controled by RHEED oscillations. Bi and Sr are evaporated from Knudsen cells while Cu is evaporated using an electron gun. An atomic oxygen plasma source was used to oxidize the films. The substrate temperature is as low as 470°C. Atomic concentrations are obtained from RBS and EDX measurements. The obtained films are stoechiometric, oriented and crisallized. AFM measurements show a typical roughness of +/− 5Å. The modulation structure periodicity is studied from the RHEED pattern. The correlations between the oxidation level, the modulation reciprocal vector q ∗ , the c axis lattice parameter and the room temperature of the films is described. The films exhibit a metallic behaviour at low or high oxidation level, while they are insulating in the medium oxidation level. Possible origins of this behaviour are discussed.

Misfit Layer Compounds Based on Double Layers MX and Sandwiches TX2 (M=Sn, Pb, Sb, Bi, Ln; T=Ti, V, Cr, Nb, Ta; X=S, Se)

Misfit layer compounds have a composite structure with two interpenetrating subsystems, an MX and an TX2 or (TX2)2 subsystem. Several structure types occur, the main difference coming from the different type of coordination of transition metal T by X atoms of the TX2 sandwiches. All structures show the presence of a common (b*, c*) plane in reciprocal space which comes from space fitting and from the interaction between the two subsystems. A classification of structures can be given based on the symmetries of the two subsystems. Since the diffraction pattern of the composite crystal is given by four reciprocal vectors, the symmetry can be described in (3+1)D superspace. The composition, which is generally non-stoichiometric, is determined by the ratio of the unit cell volumes of the two sublattices; the temperature dependence of the composition is governed by the thermal expansion. The physical properties are discussed in relation with chemical bonding. The electrical resistivity for compounds with NbX2, TaX2 or TiX2 sandwiches is metallic, the inplane resistivity being about 10-4 smaller than the resistivity along c. Hall effect and thermopower seem to indicate that the conduction takes place within the sandwich layers. For compounds with M=Sn, Pb, Sb, Bi, the intralayer M-X distances are shorter than the interlayer distances indicating a weak interaction; Hall effect and photoelectron spectroscopy indicate no or hardly any electron transfer from MX to TX2 for such compounds. The interaction between layers LnX and TX2 is strong; intra- and interlayer Ln-X distances are about equal and correspond to those for Ln3+; the magnetic properties also point to Ln3+. Hall effect and thermopower are in a greement with electron transfer and conduction in the TX2 sandwiches. Photoelectron spectroscopy confirms that Ln is trivalent; indications for charge transfer are less clear. Reflection spectra in the range 2000–20,000 cm-1 are in agreement with charge transfer.

On the band-structure contribution to the stability of the icosahedral quasicrystals and Frank-Kasper phases in the non-transition metal alloys

The electronic contribution to the energy of icosahedral quasicrystals and related crystalline phases is discussed within a nearly-free-electron approximation. The Penrose approximant is used through the Elser and Henley scheme (1985) for modelling the Frank-Kasper phase atomic structure. The relative position of the Fermi sphere and the wide-gap reciprocal vectors for both structures is analysed. It is shown that the Jones-like arguments concerning the electronic concentration explain the stability of these phases in systems exhibiting the stable quasicrystal formation. Some assumptions about quasicrystalline alloys of other simple metals are suggested.

A novel approach to the identification of microcrystals with electrons

Measurement of conventional-cell data requires observations along several specific zone axes in most cases. On the contrary, as shown in and outlined below, primitive-cell data can be extracted ab-initio, without trial-and-error, from single CBED patterns exposed along any zone axis, and displaying at least ZOLZ and first-order Laue-zone (FOLZ) spots.The CRYSTDAT database, which is up-to-date and accessible on-line, contains over 180,000 entries, each with cell data, chemistry and reference to published work for a compound. It can be searched for primitive cell data and partial chemistry. In combination with the previous remark, this shows that crystallites can be identified from single CBED patterns, and from partial chemical information obtained on them by Energy-Dispersive X-ray spectroscopy.CBED patterns are analyzed as follows. Two-dimensional (xP.vP) cartesian coordinates are measured for centers of all diffraction spots on the CBED film from an arbitrary origin. To each Bragg spot P on the film corresponds a scaied reciprocal vector FS with projections on the three axes:

Refinement of incommensurate structures against diffraction data from a twinned crystal

Twinning can lead to a diffraction pattern with additional reflections that are incommensurate with the reflections of a crystal with only one orientation of the structure. The integer indexing of such a diffraction pattern involves more than three reciprocal-basis vectors. Analogously, for incommensurate crystals, the original number of (3+d) reciprocal vectors should be extended to a larger set for a twinned incommensurate crystal. In this paper, it is shown that the diffraction symmetry for a twinned crystal can be analyzed in a way analogous to the treatment of the symmetry of an incommensurate structure. The theory is implemented in a refinement program for X-ray and neutron diffraction data and allows all intensity data from isolated and overlapping reflections to be taken into account. The method can also be applied to the refinement of ordinary crystal structures. The program has been used to deter-mine the modulated structure of the inorganic misfit layer compound (HoS)1.23NbS2.

Symmetry aspects in spin-polarized relativistic electronic structure calculations based on the Green function method

The consideration of spin-orbit coupling in spin-polarized electronic structure calculations leads to magnetic anisotropies and reduces the symmetry of the problem. The theory of magnetic groups is required to make full use of the remaining symmetry. A systematic method is presented to treat this problem in the framework of the Green function method. An application to the magnetocrystalline anisotropy energy of Fe fails to converge numerically but reveals where this effect originates as a function of reciprocal space vector and energy.

The identification of general pseudoreciprocal vector fields

A reciprocal vector field is a vector field for which the linearization of the vector field is reciprocal. A vector field is called pseudoreciprocal if it can be written as the composition of a matrix with a reciprocal vector field. The main types of pseudoreciprocal vector fields studied here are those where the matrix is invertible. The identification of such vector fields is completed for the cases when the matrix is either invertible, invertible symmetric, symmetric positive definite, diagonal positive definite, or diagonal invertible. In the process of such identification, a decomposition of the original vector field as the composition of a matrix and a reciprocal vector field will ensue. A definitive answer is given to the following question: given a state equation dx/dt =f(x), does there exist a nonlinear circuit made of only two-terminal, and/or reciprocal n terminal resistors, capacitors and inductors, which realizes this equation?.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Dynamical theory for strongly correlated two-dimensional electron systems.

We describe a method for treating the time-dependent behavior of the electron cloud surrounding each electron in a two-dimensional electron gas. The method is based on a conserving solution of the quantum kinetic equation for the relaxation function of the system using the Mori formalism with two distinct memory functions, one for the collective degrees of freedom and one for the single-particle modes. At lower electron densities these dynamic effects are shown to become increasingly important and to play a part in the eventual solidification into a Wigner crystal. We find that the plasmon resonance energy ${\mathrm{\ensuremath{\omega}}}_{\mathit{p}}$(q) is strongly depressed compared with the random-phase approximation, leading to a negative dispersion for large q in the low-density region. We have determined the width of the plasmon peak and find that as the Wigner-crystal transition point is approached, the plasmon peak remains well defined even for q comparable to the Fermi momentum. For ${\mathit{r}}_{\mathit{s}}$\ensuremath{\gtrsim}20, we observe a low-energy peak in the excitation spectrum of the electron liquid for values of q matching the reciprocal vector of the Wigner lattice. The occurrence of this peak is matched by the appearance of a large peak in the static susceptibility at that q value.

Derivation of exponential joint probability distributions of structure factors in P1 with maximum entropy and irreducible cluster integrals

Recently the application of the maximum-entropy method to direct methods has been initiated for a priori uniformly and independently distributed atoms, introducing non-uniformity in direct space by putting constraints on the expected values of the distribution [Bricogne (1984). Acta Cryst. A40, 410–445; (1988). Acta Cryst. A44, 517–545]. In this paper a start is made in using the maximum-entropy principle for deriving exponential joint probability distributions of structure factors for a chemically more realistic model of a priori non-uniformly and non-independently distributed atoms. The maximum-entropy equations are obtained by treating the atomic positions as well as the reciprocal vectors as random variables and applying constraints on the maximum of the distribution. The interdependence of the Lagrange multipliers leads to inequalities which may be compared with the Karle–Hauptman inequalities. The radial interatomic correlations such as minimal interatomic distances lead to integrals whose evaluation via the cluster integral mechanism is shown to be equivalent to those of the classical hard-sphere gas in an external field [Van Kampen (1961). Physica (Utrecht), 27, 783–792]. The Debye scattering equation results from these calculations. The exponential multiplet terms are expressed as cluster integrals. From the distribution of the single structure factor the influence of the interatomic correlations on the normalization procedure is assessed. The exponential triplet distribution up to order N−3/2 is derived and is shown to be in agreement with the exponential Edgeworth result [Karle & Gilardi (1973). Acta Cryst. A29, 401–407]. The effect of the radial interatomic correlations on the triplet distribution is discussed. The exponential quartet distribution up to order N−1 is derived, and found to be equal to the well known result [Hauptman (1975). Acta Cryst. A31, 617–679, 680–687] except for some normalization terms resulting from the interatomic correlations.

Anomalous X-Ray Diffraction Line Shiftin Heterogeneous Langmuir-Blodgett Multilayer Films

The X-ray diffraction pattern of Langmuir-Blodgett films formed with alternately deposited Cd stearate and Cd behenate exhibits anomalous shifts in the diffraction angles when they have the deposition order with dimerized Cd layers. The magnitude of the reciprocal vector of the n -th order is represented by G n = n G a -(-1) n g n , where G a is for averaged unit vector and g n is the shift that changes gradually with the index of diffraction and the multiplicity of the layers.

Rational approximant structures and phason strain in icosahedral quasicrystalline phases

The degeneracy of the reciprocal space with respect to the icosahedral symmetry has been demonstrated by experimental diffraction patterns obtained from rapidly solidified AlMnSi, MgAlAg, MgAlZn alloys. The distortion in diffraction patterns has been attributed to rational approximant (RA) structures. The distorted pattern obtained from MgAlZn system has been indexed based on 3 2 RA structure. The shift parameter has been quantified and shown to vary linearly with the pseudospace counterpart of the corresponding quasicrystalline reciprocal vector, which is the characteristic feature of phason strain.

First-principles study of short range order and instabilities in Au [sbnd]Cu, Au [sbnd]Ag and Au [sbnd]Pd alloys

Interaction energies derived from first-principles total energy calculation are used together with the Cluster Variation Method in order to study phase diagrams, ordering instabilities and the Fourier spectrum of the Warren-Cowley short range order parameters for Au Cu, Au Ag and Au Pd alloys. The present study is carried out using both the tetrahedron and the tetrahedron-octahedron approximations of the CVM. It is shown that the tetrahedron approximation predicts the wrong Fourier spectrum for the SRO parameters although it gives qualitatively correct phase diagrams. In particular, the tetrahedron approximation fails to reproduce the expected maxima in the fluctuation spectrum at the 〈1,0,0〉-type reciprocal space vectors, characteristics of L1 0 and L1 2 ordering systems. The correct features of the k-space fluctuation spectrum is obtained in the tetrahedron-octahedron approximation.

On the variety of electron diffraction patterns from a face-centred icosahedral quasicrystal

The discovery of the icosahedral phase by Shechtman et al. (i) has led us into an exciting world of quasicrystals. Chattopadhyay et al. (2) reported on the variety of electron diffraction patterns from a primitive icosahedral phase of the AI-Mn system. They identified important zone axes by choosing a set of strong reflections. From the stereogram, six important zone axes were identified by the combination of relrods corresponding to the vertex, edge and face vectors of an icosahedron. Intensities were compared with those calculated from the Landau generation scheme of Nelson and Sachdev (3), where a combination of star vectors pointing to the vertices of an icosahedron sequentially generate all the reciprocal vectors, with a decrease in intensity with increasing generation number.

Electron-density distribution from X-ray powder data by use of profile fits and the maximum-entropy method

Following the profile decomposition of CeO2 X-ray powder data into individual structure factors, the maximum-entropy method (MEM) has been used to obtain an electron-density-distribution map. In the profile decomposition process, it is impossible to avoid the problems of overlapping peaks which have the same magnitude of reciprocal vectors, such as d*(511) and d*(333), for a cubic crystal, or very severely overlapping reflections. The formalism to treat such overlapping reflections in the MEM analysis is to introduce combined structure factors. The maximum value of the scattering vector, 4π(sinθ)/λ, which was used in the present analysis is small (about 7.8 Å−1) but the resulting electron-density-distribution map is of a high quality and much superior to the conventional map. As a consequence, the ionic charge of Ce and O ions can be obtained with reasonable accuracy from the MEM density map. Furthermore, the map reveals the existence of electrons around the supposedly vacant site surrounded by eight O atoms, which is probably related to the high ionic conductivity of this substance.

X-Ray Study on the Successive Phase Transitions of K2CoCl4

The structural phase transition of K 2 CoCl 4 is investigated by means of X-ray scattering. The transition temperatures are 558 K, 454 K and 142 K. With decreasing temperature, the symmetry of the crystal changes as follows: phase I, normal [ Pmcn ]; phase II, incommensurate [ q =(1/3-δ) c * ]; phase III, commensurate [ P 2 1 c n , q = c * /3]; and phase IV, commensurate [ C 1 c 1, q = a ^*/2+ b * /2+ c * /3]. Here the reciprocal vectors refer to the normal phase. In a narrow temperature range just above 142 K, the superreflection at (5/2 3/2 0) splits into two diffuse peaks. The general feature of the successive transitions is very similar to K 2 ZnCl 4 .

Random-tiling quasicrystal in three dimensions.

A three-dimensional random-tiling icosahedral quasicrystal is studied by a Monte Carlo simulation. The hypothesis of long-range positional order in the system is confirmed through analysis of the finite-size scaling behavior of phason fluctuations and Fourier peak intensities. By investigating the diffuse scattering we determine the phason stiffness constants. A finite-size scaling form for the Fourier intensity near an icosahedral reciprocal wave vector is proposed.

Measure of nonplanarity in conjugated organic molecules: which structurally characterized molecule displays the highest degree of pyramidalization?

A set of vectors reciprocal to those pointing along the internuclear axes from a conjugated carbon atom are introduced. The resultant of the reciprocal vectors is shown to define the π-orbital axis vector at a nonplanar conjugated atom. The quantities introduced in the transformation are sufficient to define all of the popular measures of pyramidalization in nonplanar conjugated organic molecules and to allow comparisons between the different schemes. The measures of nonplanarity are divided according to isotropy, and it is shown that there is a high degree of correlation between the different scales of pyramidalization, particularly in the case of the isotropic analyses

Harmonic generations in an optical Fibonacci superlattice.

An optical Fibonacci superlattice has been proposed to produce the second-harmonic generation and the third-harmonic generation, which is the sum frequency of the second-harmonic and the fundamental frequency in the same material. Because of the quasiperiodicity of the optical Fibonacci superlattice, the phase mismatches of the optical parametric processes caused by the frequency dispersion of the refractive index can be compensated with the reciprocal vectors which the optical Fibonacci superlattice provides. A theory which analyzes the second-harmonic generation and the third-harmonic generation processes in the material and the calculations applied to the optical Fibonacci superlattice made from a single ${\mathrm{LiNbO}}_{3}$ crystal is presented in detail. The calculations show that the efficiencies of the second-harmonic generation and the third-harmonic generation are comparable to, or even larger than, those obtained with commonly used phase-matching methods.

Electron diffraction studies of variable periodicity in decagonal quasicrystals in aluminium-cobalt alloys

Abstract The reciprocal space vectors of the decagonal phase can be generated by a slight distortion of the basis vectors of a regular icosahedron to induce commensuration in one dimension. In different alloys a multiple of this basic periodicity has been observed. In particular, in Al-Co alloys, apart from the basic periodicity, doubling, tripling and quadrupling of this basic underlying periodicity has been observed. Stereographic analysis, simulation and experimental observation of electron diffraction patterns have been used to explore the nature of this varying periodicity in detail.