CsCl Lattice Research Articles

Determination of flexoelectric coefficients of higher-order continuum theories from CsCl lattice model

Flexoelectricity is an electromechanical coupling phenomenon between inhomogeneous deformation and electric polarization. The classical theory, which relies on the strain, is insufficient for representing flexoelectricity. Instead, a higher-order continuum theory that employs a measure capable of expressing local inhomogeneous deformation is necessary. In the present work, we concatenate higher-order continuum theories with a CsCl lattice structure consisting of charged particles and determine the resulting constitutive coefficients for higher-order continuum theories. First, we derived continuum internal energy density functions for dielectrics from the conservation laws. We consider two different higher-order continuum theories: coupled stress and strain gradient theories. Second, we present a procedure for determining the coefficients of higher-order continuum theories by using a discrete lattice model under the assumption that the internal energies of the lattice and continuum models are the same for identical deformations. To validate the coefficients, three sample problems are examined by presenting their analytical solutions. The results of sample problems demonstrate that the two higher-order theories can provide quantitative information about the size and flexoelectric effects. In particular, the strain gradient theory predicts the deformation of the lattice structure more accurately.

Stabilization of the ordered structure of thin condensed foil of the Pd-Cu solid solution in a hydrogen environment

Structural transformations of thin (≈4 µm) Pd-Cu alloy foil fabricated by magnetic sputtering of a target with the composition corresponding to the highest temperature of the existence of the ordered structure (β phase, a CsCl lattice) have been studied in the course of heating and cooling in vacuum or in a hydro� gen atmosphere. Initial foil has an ordered structure (β phase), a gradient grain substructure, and domi� nant 〈211〉 texture. In vacuum, phase disordering (the β → α transi� tion) occurs within the range 773-873 K, i.e., within the boundaries of the equilibrium phase diagram of the system. This is accompanied by collective recrystalli� zation that eliminates the gradient of the grain sub� structure and by appearance of 〈110〉 texture of the β phase. Annealing in a hydrogen environment led to an increase of the existence range of a singlephase structure (β phase) by nearly 200 K: at 1123 K, the intensity of the 111α reflections was ≈60% of the 110β intensity. In this paper, we report for that first time that annealing of the Pd-Cu alloy in a hydrogen environ� ment extends the temperature range of the existence of the ordered structure with a CsCltype lattice. The Pd-Cu solid solutions are among alloys rec� ommended for fabrication of selective membranes for deep purification of hydrogen (1, 2). In this aspect, the possibility of solid solution ordering to form the β phase (CsCl lattice), shown by the phase diagram (3), is of special interest since the activation energy of hydrogen diffusion in the ordered Pd-Cu alloy (0.035 eV) is almost one order of magnitude lower than in the disordered solid solution (0.325 eV) and in palladium (0.23 eV) (4). As follows from the studies of kinetics of phase transformations that accompany the formation of the ordered structure of the Cu-49 at % Pd alloy (5-7), the degree of ordering essentially depends on the preliminary deformation of the samples and their subsequent heat treatment. Longterm heat treatment after intensive plastic deformation afforded a nearly singlephase ordered structure (7). It has been demonstrated that singlephase ( β phase) condensed foil can be fabricated by magnetic sputtering of a Pd- Cu alloy target of the corresponding composition (8, 9). Studying the hydrogen permeability of Pd-Cu foil has confirmed the expected result of ordering: specific hydrogen permeability of ordered alloy membranes is ten or more times higher than that of membranes made of the alloy with a disordered structure (10). In this context, it is of interest to study the effect of the major component of the membrane environment (hydrogen gas) on the temperature range in which the ordered structure of a thin Pd-Cu alloy foil persists.

Assembling hierarchical cluster solids with atomic precision.

Hierarchical solids created from the binary assembly of cobalt chalcogenide and iron oxide molecular clusters are reported. Six different molecular clusters based on the octahedral Co6E8 (E = Se or Te) and the expanded cubane Fe8O4 units are used as superatomic building blocks to construct these crystals. The formation of the solid is driven by the transfer of charge between complementary electron-donating and electron-accepting clusters in solution that crystallize as binary ionic compounds. The hierarchical structures are investigated by single-crystal X-ray diffraction, providing atomic and superatomic resolution. We report two different superstructures: a superatomic relative of the CsCl lattice type and an unusual packing arrangement based on the double-hexagonal close-packed lattice. Within these superstructures, we demonstrate various compositions and orientations of the clusters.

Multi-Objective Materials Design by Genetic Algorithms—Generalized for B1 and B2 Ionic Structures

The design of NaCl(B1) and CsCl(B2) lattices are carried out in this work using multi-objective genetic algorithms, generalizing it for all the compounds possessing B1 and B2 crystal lattices. Three objectives, fracture toughness, density and energy of the system, were taken into the purview of this investigation, where the first objective is maximized and the remaining two are minimized. The interionic potential model used is a combination of Born Mayer and Coulomb potentials. The objectives are computed as a function of the ionic masses along with the inter-ionic distance. The fracture toughness studies are carried out by molecular dynamics studies (in which dependence on loading rate, crack size and electronegativity are also studied) and multi-objective optimization problem is solved by the well known multi-objective optimization algorithm NSGA II.

Unusual B1–B2 transition in PbSe under high pressure: evidence for two intermediate phases; transport, structural, and optical properties

AbstractWe report the results of a study of the thermoelectric power (Seebeck effect), the electrical resistivity, X‐ray diffraction, and Raman spectroscopy under high pressure (P) to 15 GPa for binary and tin‐doped lead selenide (PbSe and Pb1–x Snx Se (x = 0.125)). The gathered data were inconsistent with previous models suggesting only one intermediate phase between the ambient NaCl (B1) lattice and a high‐pressure metal CsCl (B2) lattice. Furthermore, the data hint at the existence of two competing intermediate phases. From the thermopower we established that a phase that is characterized by n‐type conductivity is dominating, while the other one – of p‐type conductivity is probably optional. The electrical conductivity in both of the phases is semiconducting. In the X‐ray diffraction studies in one of the three investigated samples of Pb0.875Sn0.125Se we could refine a pattern of the intermediate phase taken at ∼9.5 GPa in the Cmcm space group. This corresponds to the symmetry of earlier proposed structural types of TlI and CrB. The patterns of the intermediate phase in the two other samples show additional strong reflexes that are not related to either the NaCl or the CsCl lattices. This presents evidence of the presence of one more undetermined phase. The Raman spectra of the intermediate phase showed a dependence on a pressure‐transmitting medium (methanol‐ethanol or KCl). The pressure‐driven NaCl → CsCl transition and a pressure dependence of the semiconductor energy gap of PbSe are compared with those of other lead salts, namely, PbTe and PbS (galena). It is surmised that there is no interrelation between the closure of the energy gap at the NaCl phase and the phase transition to the intermediate phase. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On Bessel series expressions for some lattice sums: II

In part I (Kanemitsu S et al 2003 J. Northwest University) we have made explicit use of the Mellin–Barnes integrals to prove the Chowla–Selberg-type Bessel series expressions for zeta-functions associated with lattice structures. In this paper we shall make implicit use of Mellin–Barnes integrals, as embedded in our theory of modular relations and functional equations, to reveal relationships between the structure of Madelung constants of the NaCl and CsCl lattices. Namely, we shall elucidate the relation between the structures of the NaCl lattice and those of the CsCl lattice, so to speak using the symmetry of the zeta-function, i.e. using their functional equations. Thus we shall emphasize the symmetry properties of the zeta-functions, restoring the Schlomilch series and Hardy's theory of K-Bessel functions, to prove the functional equations, and then to prove the recurrence relations for the lattice zeta-functions.

Mössbauer-Effect Study of Metastable c-FeSi Synthesized by Molecular Beam Epitaxy (MBE) and by Ion Implantation

Conversion electron Mössbauer spectroscopy (CEMS) has been applied to the study of the metastable c-FeSi phase (i.e. an iron silicide with CsCl lattice structure) that was synthesized by implantation of Si + ions of 50 keV in energy into f -Fe (95% 57 Fe) near room temperature with a nominal dose of 5 2 10 17 cm m 2 , and by molecular beam epitaxy (MBE). Iron silicide layers with different stoichiometry (FeSi 0.85 , FeSi, Fe 0.85 Si) were grown by codeposition of 57 Fe and Si onto an Fe buffer layer on MgO(100). For all FeSi layers the defective CsCl structure was observed after annealing at different temperatures. X-ray diffraction measurements were performed to determine the structure and epitaxial relationship of the c-FeSi films. The lattice parameter perpendicular to the film plane was found to be 2.77(5) Å. CEMS measurements revealed a lower than cubic site symmetry of the iron atoms for both the c-FeSi layers synthesized by ion implantation and by MBE. The formation of nearly undistorted c-FeSi after annealing is favored by excess Fe atoms in the deposited film.

Atomic-scale three-dimensional kinetic Monte Carlo simulation of organometallic vapor-phase epitaxy of ordered films

We present an atomistic three-dimensional method for simulating growth of ordered films during organometallic vapor-phase epitaxy (OMVPE). Epitaxial film evolution during growth is studied under typical OMVPE reactor conditions by using a kinetic Monte Carlo technique that incorporates important surface chemical reactions occurring in the reactor. The reactor model consists of a temperature-dependent deposition reaction and a surface etching reaction that depends on the local atomic environment. As a representative ordered film, we study the evolution of an AB film on a CsCl lattice. The growth of the epitaxial film is simulated on homoepitaxial, elemental, and disordered substrates with {011} and {001} orientations. Under typical OMVPE conditions, single-crystal homoepitaxial films are observed on homoepitaxial substrates. On elemental and disordered substrate, the film morphology showed domains of opposite orientations separated by antiphase boundaries. In all cases, the growth rate shows an Arrhenius dependence on temperature. Film quality as characterized by the short-range order decreases with increasing temperature. Surface roughness of the epitaxial films corresponds to a staggered surface consisting of a few (2--3) monolayers.

Structure and bonding in crystalline and molten Li-Sn alloys: A first-principles density-functional study

Ab-initio local-density-functional investigations of crystalline and molten Li-Sn alloys at equiatomic compositions and a stoichiometry close to ${\mathrm{Li}}_{4}\mathrm{Sn}$ (allowing to complete the octet-shell of the Sn-ions if complete charge-transfer is assumed) have been performed. The aim of these studies is to investigate the influence of the formation of direct covalent Sn-Sn bonds on the structure and stability of the alloys. At the equiatomic composition we find that the monoclinic LiSn structure with the Sn-atoms forming rumpled layers (like those in crystalline As) is more stable than a simple ionic CsCl lattice; in the molten phase, this is reflected by the formation of a dense disordered network of Sn-Sn bonds. At the octet composition, an ionic structure with no direct Sn-Sn neighbors and a partially covalent structure with ${\mathrm{Sn}}_{2}$ dimers are of comparable stability. In the molten phase, the importance of covalent Sn-Sn bonding is reduced and only a small fraction of the dimers present in the crystal structure survive. Detailed ab-initio molecular dynamics studies for the structural, dynamic, and electronic properties of the liquid alloys are presented. The results are discussed in relation to a generalized Zintl principle for structure and bonding in such strongly interacting alloy systems.

Structural characterization of metastable FeSi films and of [formula omitted] multilayers

The structural properties of molecular beam epitaxy (MBE)-grown metastable CsCl-type FeSi (i.e. an iron silicide with the CsCl lattice structure) on Si (1 1 1) and of Fe FeSi multilayers grown with MBE on MgO (0 0 1) are reported. Rutherford backscattering and channeling spectrometry (RBS-C) combined with X-ray diffraction are used to determine the structure, epitaxy and crystalline quality of the layers. Single metastable CsClFeSi layers could be stabilized epitaxially up to a thickness of 62 nm and are found to be twinned with respect to the Si (1 1 1) substrate. Moreover, an extensive study on antiferromagnetically (AF) coupled ( Fe FeSi ) 15 MgO multilayers indicates that the Fe layers are slightly expansively strained while the FeSi spacers are fully relaxed. Further, we could prove that the FeSi spacers in the multilayers have a cubic CsCl lattice structure with a lattice parameter of 0.28 nm.

X-ray diffraction study of the structural phase transitions in RbCN under high pressure

Pressure-induced structural changes in RbCN were studied by X-ray diffraction. A single crystal sample was loaded in a miniature diamond-anvil cell and diffraction patterns were recorded at room temperature by the precession method on a laboratory radiation source. It was confirmed that a two-step structural transition takes place up to a pressure of 2 GPa: from the NaCl-type to the CsCl-type structure and further to the monoclinic structure. The CsCl-type structure forms with its (111) plane parallel to the three equivalent {100} planes of the NaCl-type structure. The monoclinic structure with the space group Cm forms as a result of preferential alignment of the CN molecules in the direction of one of the body-diagonals of the CsCl lattice with its (010) plane parallel to the {110} plane of the lattice. No definite conclusion was obtained regarding the head-and-tail ordering of the CN molecules.

Applications of the quadratic Korringa-Kohn-Rostoker band-theory method to complex lattices.

The quadratic Korringa-Kohn-Rostoker band-theory method is generalized to treat crystals with many atoms per unit cell. It is applied to palladium hydride with the NaCl structure and a test case in which two niobium atoms are placed on the sites of a CsCl lattice. We demonstrate that this is a fast and reliable band-theory method for compounds.

Evaluation of lattice sums by convergence accelerating transforms

A new method for the evaluation of lattice sums, based on applying convergence accelerating transforms to successive direct partial sums over suitable expanding cells is proposed. The method is applied to the evaluation of Madelung constants for NaCl and CsCl lattices. The Coulomb potentials at the site of a defect for distorted lattices of NaCl and CsCl are also calculated.

Electrical and thermoelectric transport properties of RZn and RCd compounds (R = Tb, Gd, Nd, Pr)

We present the results of an extended investigation of the transport properties of the CsCl structure intermetallic compounds GdZn, TbZn, GdCd, TbCd, PrCd and NdCd (all single crystals), using high resolution measurements of the electrical resistivity (ϱ, dϱ dT ) and the thermoelectric power (S, dS dT ) , from 4.2 up to 300 K. Both the resistivity and the thermoelectric power are separated in their magnetic and nonmagnetic contributions, and relevant information and parameters are obtained for the different scattering mechanisms. A detailed analysis of the data is presented in the vicinity of the different phase transitions present along the series: of magnetic ( T c , T N), structural ( T ∗, T ∗ L) and spin reorientation types ( T sr). A through comparison of the data with the available theoretical treatment to describe transport properties in magnetic metals is presented. For the case of the thermopower we show that the existing theories are inadequate to describe quantitatively, and in some cases even qualitatively, the main features associated with the magnetic transitions. Finally, new experimental evidence is given on the CsCl lattice instability in RCd intermetallic compounds.

Orthorhombic distortion of the CsCl and NaCl lattices expansion of the Madelung parameters in terms of strain tensor components

The Madelung parameters of orthorhombically distorted CsCl and NaCl structures are expanded in terms of ( b a −1) and ( c a −1) . The results are illustrated by contour plots. The calculational approach can also be applied to obtain similar expressions for other cubic or orthorhombic structures. Expansions of this kind are particularly useful in the simulation of phase transitions of ionic crystals.

Infrared Spectra of Sodium Salts in CsCl Disks

The use of CsCl as a matrix for the preparation of disks for infrared spectra of sodium salts was investigated. During the grinding of the disks, sodium salts form a substitutional solid solution of hydrated Na+ in a CsCl lattice. The IR spectrum of the substitutional hydrated solid solution has four characteristic HOH absorption bands which are very sharp, whereas in general, the spectra of hygroscopic salts have broad HOH absorption bands. The spectra of sodium salts in a CsCl matrix were compared to those of (1) potassium halides and LiCl in KBr, CsCl, and CsI matrices; (2) various cesium salts in NaCl and NaBr matrices; (3) various sodium salts in a CsI matrix. From the various spectra it was concluded that the formation of the substitutional solid solution of hydrated Na+ in CsCl crystals is a mechanochemical process. In the first step, NaCl is formed because of a double decomposition reaction between the sodium salt and CsCl. In the second step, a solid solution of NaCl in CsCl crystals is obtained because of a mechanochemical process in which planes of NaCl crystals and atmospheric water combine with planes or fractures of CsCl crystals due to sliding of crystal planes under the effect of shearing. Substitutional solid solutions of hydrated Na+ in CsBr and CsI are obtained when NaBr or NaI is ground with CsBr or CsI.

The madelung constant for the tetragonal distortion of the cscl lattice

The Madelung constant is calculated for the tetragonal distortion of the CsCl lattice using Ewald's method over a large c a range. The results deviate considerably from the analytical expressions for the Madelung constant in the literature. As a checkpoint for our calculation, the Madelung constant for the ideal cubic AuCuI structure ( c a = √2) is also computed by direct lattice summation using Evjen's method.

Electrostatic energy in ionic crystals

The problem of calculating electrostatic energies in large, finite and arbitrarily shaped pieces of ionic crystal is analysed. The electrostatic energy of a unit cell of the crystal deep within the interior of the piece of crystal is shown to be composed of a shape-independent part depending on the structure of the crystal lattice concerned and the distribution of ions within a unit cell, and a shape-dependent part which depends on the shape of the piece of crystal and the dipole moment of a unit cell. The shape-dependent part is zero if this dipole moment is zero. The electrostatic energy of the whole piece of crystal is shown to be the unit cell energy multiplied by the number of unit cells in the piece of crystal, plus corrections proportional to the surface area of the piece of crystal. These surface corrections are calculated explicitly for a finite cube of simple cubic crystal. Different descriptions of the same crystal structure are shown to lead to different bulk energies. This disagreement is discussed for the CsCl lattice, and is shown to arise from the way the different descriptions of the lattice imply different surface structures on the surface of a cube of crystal. The energy of a test charge at the surface of a cube of simple cubic crystal, and then the energy of layers of charges on the surfaces of a cube of simple cubic crystal, are analysed. The analyses confirm the origin of the disagreement in bulk energies for the two descriptions of the CsCl lattice. The role of the energies of surface layers in the bulk electrostatic energy of a piece of ionic crystal, and the relation of this bulk energy to a shape-independent Madelung constant are discussed. Some conjectures on the role of bulk energies in surface reconstruction are also discussed.

NMR and structural properties of CsAu and RbAu

NMR and X-ray diffraction studies of granular samples of the intermetallic compounds, CsAu and RbAu, are reported. Studies of X-ray line intensities indicate that the CsCl lattice structure is well ordered for both materials. For the semiconducting compound CsAu, excess Cs in the lattice is shown to be the primary source of conduction electrons. X-ray studies show an expansion of lattice constant with addition of excess Cs. Measurements of NMR line shapes, Knight shifts, and relaxation times for CsAu confirm that the excess Cs is atomically dispersed in the samples. NMR measurements show our RbAu samples to have a metallic nature. However, the metallic characteristics may well derive from excess Rb known to be present in the samples.

On the Elasticity of Ionic Compounds under Hydrostatic Pressure

AbstractThe pressure derivatives of the elastic constants of CsCl, NaCl, and ZnS lattices were calculated as functions of the difference in sizes of ions and the range of the repulsive forces. General conditions for the negative pressure dependence of the shear constants are examined. The agreement with the experimental results is reasonable, which indicates that the ion‐size difference will have to be taken into account in all future considerations of the elastic properties of compounds.