Cauchy Relation Research Articles

Magneto-Electronic, Thermodynamic, and Thermoelectric Properties of 5f-Electron System BaBkO3

In the present work, the structural, electronic, magnetic, mechanical, thermodynamic, and thermoelectric properties of cubic perovskite BaBkO3 were investigated within the framework of density functional theory using generalized gradient approximation, onsite Coulomb interaction, and modified Becke-Johnson methods. BaBkO3 is stable in the ferromagnetic phase with a magnetic moment of 7 μB. The spin-polarized electronic band structure reveals 100% spin polarization of the material with metallic character in spin-up state and semiconducting in spin-down state. Pugh and Cauchy relation displays that the material is highly ductile. The quasi-harmonic Debye model was used to determine the thermodynamic properties of BaBkO3 under high pressure and temperature. BaBkO3 presents high value of Seebeck coefficient and power factor at room temperature with a value of 135 μVK− 1 and 1.5 × 1012μWcm− 1 K− 2 s− 1, respectively. The possession of half metallicity and high Seebeck coefficient makes BaBkO3 suitable for spintronic and thermoelectric device applications.

Modified analytic embedded atom method potential for chromium

In the present paper, we have developed a modified analytic embedded atom method potential for chromium. To solve the problem of a negative Cauchy relation for body-centered cubic metal at zero Kelvin temperature, a modification term is added to the total energy. Compared with available interatomic potentials, the present one is easier to be extended because of its analytic form. The fitted or input parameters are all well reproduced, and the properties beyond the fitting range (such as Cauchy relation, melting point and self-interstitial atom formation energies) are also well predicted. We have also studied the relation between the stacking faults energy and screw dislocation core structure with different potentials. Especially, only our potential predicts a compact core structure, which agrees with the results obtained by density functional theory calculations.

Third-order elastic constants and anharmonic properties of three fcc high-entropy alloys from first-principles

Third-order elastic constants (TOECs) are very important for understanding the nonlinear mechanical response of materials and evaluating the anharmonicity of crystal lattices. Here, we are concerned with investigating the six independent TOECs and related anharmonic properties of three face-centered cubic (fcc) high-entropy alloys (HEAs), namely CrFeCoNi, CrMnFeCoNi, and Cr10Mn40Fe40Co10, using density-functional simulations. To benchmark computational accuracy, three ab initio codes are used to obtain the complete set of TOECs for fcc Ni. For the HEAs, we observe that the TOECs C123 and C456 are positive, and C123 is particularly large. The Cauchy relations for the TOECs are partially satisfied for the three studied HEAs. With the help of the derived TOECs, the average TOECs for an isotropic polycrystal are estimated. Using the obtained TOECs, we reveal the pressure derivatives of the effective second-order elastic constants and polycrystalline moduli as well as derive the nonlinearity constant δ. The obtained pressure derivative of bulk modulus agrees very well with the available experimental data for CrMnFeCoNi. For the three considered HEAs, δ along high-symmetry directions orders as δ[011]>δ[111]>δ[100].

Influence of manganese doping on elastic and structural properties of silica borotellurite glass

The quaternary glass system tellurium dioxide, boron trioxide, silicon dioxide and manganese dioxide where x = 0.00 to 0.05 M fraction has been successfully prepared via conventional melt and quench-casting technique. The density was determined by Archimedes method with distilled water as buoyant liquid. The results showed that the density decreases and molar volume increases with respect to manganese concentration. FTIR spectral studies suggest that glass network is mainly built up of boron trioxide, tetrahedral boron, tellurium trioxide, trioxotellurate and silicon dioxide structural units. The ultrasonic velocities and elastic moduli were recorded using a pulse-echo method at frequency 5 MHz at room temperature. Some other physical properties such as Debye temperature, softening temperature, microhardness, Poisson's ratio, fugacity, fractal bond connectivity and Cauchy relation were determined. Results indicate that these parameters depend upon the composition of the glass system and the dopant concentration, MnO2 inside the glass system.

Modeling of monolayer charge-stabilized colloidal crystals with static hexagonal crystal lattice

The mathematical model of monolayer colloidal crystals of charged hard spheres in liquid electrolyte is proposed. The particles in the monolayer are arranged into the two-dimensional hexagonal crystal lattice. The model enables finding elastic constants of the crystals from the stress-strain dependencies. The model is based on the nonlinear Poisson-Boltzmann differential equation. The Poisson-Boltzmann equation is solved numerically by the finite element method for any spatial configuration. The model has five geometrical and electrical parameters. The model is used to study the crystal with particles comparable in size with the Debye length of the electrolyte. The first- and second-order elastic constants are found for a broad range of densities. The model crystal turns out to be stable relative to small uniform stretching and shearing. It is also demonstrated that the Cauchy relation is not fulfilled in the crystal. This means that the pair effective interaction of any kind is not sufficient to proper model the elasticity of colloids within the one-component approach.

The atomistic representation of first strain-gradient elastic tensors

We derive the atomistic representations of the elastic tensors appearing in the linearized theory of first strain-gradient elasticity for an arbitrary multi-lattice. In addition to the classical second-Piola) stress and elastic moduli tensors, these include the rank-three double-stress tensor, the rank-five tensor of mixed elastic moduli, and the rank-six tensor of strain-gradient elastic moduli. The atomistic representations are closed-form analytical expressions in terms of the first and second derivatives of the interatomic potential with respect to interatomic distances, and dyadic products of relative atomic positions. Moreover, all expressions are local, in the sense that they depend only on the atomic neighborhood of a lattice site. Our results emanate from the condition of energetic equivalence between continuum and atomistic representations of a crystal, when the kinematics of the latter is governed by the Cauchy–Born rule. Using the derived expressions, we prove that the odd-order tensors vanish if the lattice basis admits central-symmetry. The analytical expressions are implemented as a KIM compliant algorithm to compute the strain gradient elastic tensors for various materials. Numerical results are presented to compare representative interatomic potentials used in the literature for cubic crystals, including simple lattices (fcc Al and Cu and bcc Fe and W) and multi-lattices (diamond-cubic Si). We observe that central potentials exhibit generalized Cauchy relations for the rank-six tensor of strain-gradient elastic moduli. In addition, this tensor is found to be indefinite for many potentials. We discuss the relationship between indefiniteness and material stability. Finally, the atomistic representations are specialized to central potentials in simple lattices. These expressions are used with analytical potentials to study the sensitivity of the elastic tensors to the choice of the cutoff radius.

Ab initio Theory of Elastic Properties of Rare-Gas Crystals Under High Pressure

The quantum mechanical model of deformable and polarizable atoms has been developed for the research of the elastic properties of rare-gas crystals Ne, Ar, Kr, and Xe over a wide range of pressure. It is shown that it is impossible to reproduce the observed deviation from the Cauchy relation δ(p) for Ne, Kr, Xe adequately taking into account the many-body interaction only. The individual dependence δ(p) for each of the crystals is the result of two competing interactions, namely, the many-body interaction and the electron-phonon interaction, which manifests itself in a quadrupole deformation of atoms electron shells due to displacements of the nuclei. The contributions of these interactions to Ne, Kr, and Xe compensated each other with high precision that provides δ with a positive value which is weakly dependent on pressure. In case of Ar the many-body interaction prevails. The compressed Ar has a negative deviation from the Cauchy relation the absolute value of which increases with the rise of pressure. The consideration of the quadrupole deformation is of great importance for heavy rare-gas crystals Kr and Xe. The represented ab initio calculated dependences of Birch elastic moduli Bij(p) and δ(p) are in good agreement with the experiment.

Elastic properties of static charge-stabilized colloidal crystal with two-dimensional hexagonal lattice

Two-dimensional charge-stabilized colloidal crystal with a hexagonal crystal lattice and constant electric potential on the particles is studied numerically. The properties of the crystal are described by the theory based on the Poisson-Boltzmann non-linear differential equation. The force constants and the equilibrium pressure in the crystals are calculated for a broad range of the lattice parameters. The numerical procedures of the force constant and pressure determination is briefly described. Elastic constants of the first and second order are also calculated on the base of the force constants to obtain information about the many-body effective interactions in the system. It was shown that the Cauchy relation between the elastic constants breaks down in the whole range of the lattice parameter especially at higher densities. This can be interpreted as an effect of the many-body effective interaction between the particles in the crystal.

Irreducible decompositions of the elasticity tensor under the linear and orthogonal groups and their physical consequences

We study properties of the fourth rank elasticity tensor C within linear elasticity theory. First C is irreducibly decomposed under the linear group into a “Cauchy piece” S (with 15 independent components) and a “non-Cauchy piece” A (with 6 independent components). Subsequently, we turn to the physically relevant orthogonal group, thereby using the metric. We find the finer decomposition of S into pieces with 9+5+1 and of A into those with 5+1 independent components. Some reducible decompositions, discussed earlier by numerous authors, are shown to be inconsistent. — Several physical consequences are discussed. The Cauchy relations are shown to correspond to A = 0. Longitudinal and transverse sound waves are basically related by S and A, respectively.

Adiabatic potential and elastic properties of compressed rare‐gas crystals in the model of deformable atoms

AbstractThe quantum‐mechanical model of deformable and polarizable atoms has been developed for the purpose of investigating the elastic properties of crystals of rare gases Ne, Kr, and Xe over a wide range of pressures. The inclusion of the deformable electron shells in the analysis is particularly important for the shear moduli of heavy rare‐gas crystals. It has been shown that the observed deviation from the Cauchy relation δ(p) for Kr and Xe cannot be adequately reproduced when considering only the many‐body interaction. The individual dependence δ(p) for each of the rare‐gas crystals is the result of two competitive interactions, namely, the many‐body and electron–phonon interactions, which manifests itself in a quadrupole deformation of the electron shells of the atoms due to displacements of the nuclei. The contributions of these interactions in Ne, Kr, and Xe are compensated with good accuracy, which provides a weakly pressure‐dependent value for the parameter δ. The ab initio calculated dependences δ(p) for the entire series Ne–Xe are in good agreement with the experiment.

Auxetic properties of cubic metal single crystals

The behaviour of the Poisson's ratio is investigated on the basis of derived by the authors explicit equations determining the components of the compliance tensor in an arbitrarily rotated coordinate frame. Clear analytical expressions describing the angular dependencies of the auxetic properties have been found and the crystallographic regions possessing such auxetic behaviour have been revealed for single crystals of elemental metals. This was completed with the straightforward expressions for calculation of the maximum values of auxeticity. The connection between the auxetic properties and the level of anisotropy of single crystals as well as the Cauchy relation has been analysed for various metals with BCC and FCC lattices.

Tailoring the Microstructural, Optical, and Electrical Properties of Nanocrystalline WO3 Thin Films Using Al Doping

We have studied the influence of Al doping on the microstructural, optical, and electrical properties of spray-deposited WO3 thin films. XRD analyses confirm that all the films are of polycrystalline WO3 in nature, possessing monoclinic structure. EDX profiles of the Al-doped films show aluminum peaks implying incorporation of Al ions into WO3 lattice. On Al doping, the average crystallite size decreases due to increase in the density of nucleation centers at the time of film growth. The observed variation in the lattice parameter values on Al doping is attributed to the incorporation of Al ions into WO3 lattice. Enhancement in the direct optical band gap compared to the undoped film has been observed on Al doping due to decrease in the width of allowed energy states near the conduction band edge. The refractive indices of the films follow the Cauchy relation of normal dispersion. Electrical resistivity compared to the undoped film has been found to increase on Al doping.

Determination and Analysis of Structural and Optical Properties for Thermally Evaporated ZnO Thin Films

ZnO films have been deposited on glass slide substrates at room temperature by thermal evaporation technique. The prepared samples were annealed at temperature of 300°C and 400°C in air atmosphere. Optical and structural properties of as-deposited films have been compared by that of the annealed samples. X-ray diffraction (XRD) patterns of the obtained films showed that they have polycrystalline and exhibit wurtzite structure. Micro-structural properties such as mean crystallite size and micro-strain were discussed from XRD peak broadening. Optical properties were identified by measuring transmittance using UV-Vis spectrophotometer. The optical constants such as the refractive index n, extinction coefficient k as well as films’ thickness were calculated in the spectral range of 350-800 nm from transmittance data using a reverse engineering method (point-wise unconstrained minimization approach, PUMA). Dispersion of refractive index shows similar trend as Cauchy relation. Absorption coefficient depicts a maximal value around 3.33 eV for annealed samples. Using balance between electrical power and emissive power, the temperature of tungsten furnace was calculated under deposition condition. The connection between temperature and vapor pressure of ZnO was estimated by the Clausius-Clapyeron equation and thermochemical data.

On three facts of reticences in the classical mathematical modeling of elastic materials

Three facts of reticences (passing over in silence, an absence of comments) in the procedures of mathematical modeling of elastic materials are described and commented. The first fact consists in a reticence of one of the first steps in the mentioned above procedure – an assumption that the kinematics of deformation is described by the linear approximation of motion of material continuum, namely by gradients of deformation. In the paper, a novel nonlinear approach to this procedure is offered. The second and third facts are associated with constitutive relations. The second fact consists in the absence of necessary comments relative to determination of smallness of strains and gradients of displacements (absence of comments relative to a criterion of applicability of the linear model) because the criterion $ |u_{i,k}|\ll 1$ is sufficiently abstract. It is shown that there exists a based on the nonlinear Cauchy relations approximate procedure of determination of threshold values of strains and gradients of deformations starting with which a nonlinearity of process appears. The third fact consists in the absence of comments relative to essential differences between the nonlinear constitutive equations, which are written for the ordered pairs "Lagrange stress tensor – Cauchy-Green strain tensor" and "Kirchhoff stress tensor – gradients of displacements". It is shown on an example of the shear stress and the Murnaghan model of nonlinear elastic deformation that deviation from the corresponding straight lines of linear deformation for different pairs differs in many times in the range of small strains and small gradients of displacements. The general estimate of facts of reticences looks positive, because for one part of scientists-mechanicians the reticences form the comfort feeling of monolithic character of the classical theory of elasticity, whereas for another part the reticences form a space for development of the theory of elasticity.

Ab initio theory of the many-body interaction and elastic properties of rare-gas crystals under pressure

The short-range many-body forces induced by the overlap of electron shells of atoms have been investigated. The nonorthogonality of atomic orbitals of the nearest-neighbor atoms of a crystal leads to the appearance of terms in the potential energy, which depend on the coordinates of three, four, and more nearest-neighbor atoms. An expression has been obtained for the energy of the electron subsystem of the crystal in the Hartree–Fock approximation in the basis set of atomic orbitals exactly orthogonalized at different crystal sites. The behavior of the contributions from two-body and three-body interactions to the crystal energy under compression has been analyzed. The short-range three-body potential has been calculated from first principles and proposed in the simple form. The three-body forces, obtained change the behavior of the dispersion curves for all wave vectors, in particular, thus violating the Cauchy relation. The theoretical and experimental deviations from the Cauchy relation for Ne and Ar are in good agreement under pressure.

A charge-optimized many-body potential for the U–UO2–O2 system

Building on previous charge-optimized many-body (COMB) potentials for metallic α-U and gaseous O2, we have developed a new potential for UO2, which also allows the simulation of U–UO2–O2 systems. The UO2 lattice parameter, elastic constants and formation energies of stoichiometric and non-stoichiometric intrinsic defects are well reproduced. Moreover, this is the first rigid-ion potential that produces the correct deviation of the Cauchy relation, as well as the first classical interatomic potential that is able to determine the defect energies of non-stoichiometric intrinsic point defects in UO2 with an appropriate reference state. The oxygen molecule interstitial in the α-U structure is shown to decompose, with some U–O bonds approaching the natural bond length of perfect UO2. Finally, we demonstrate the capability of this COMB potential to simulate a complex system by performing a simulation of the α-U + O2 → UO2 phase transformation. We also identify a possible mechanism for uranium oxidation and the orientation of the resulting fluorite UO2 structure relative to the coordinate system of orthorhombic α-U.

The investigation of structural and optical properties of Ge30−xSe70Cdx thin films

Chalcogenide thin film glasses with compositions of Ge30−xSe70Cdx, where x=0 and 16, were prepared and studied in terms of their structures and optical properties. The short range order proposed was obtained by interpretation of the radial distribution function, deduced from the experimental data of X-ray diffraction. The interatomic distances results for the first and second neighbors (2–5Å) indicate that the structure changes drastically after Cd additive. The optical parameters and the thickness were determined from the extremes of the interference fringes of transmission spectra at normal incidence in the wavelength range of 200–2500nm using the envelope method. The calculated values of the refractive index were fitted using the two-term Cauchy relation. The dispersion of the refractive index is discussed in terms of the single-oscillator Wemple and DiDomenico model. The optical band gap was carried from the absorption coefficient values using Tauc's procedure and the estimated values were 2.05 and 1.86eV for x=0 and 16, respectively.

Elastic properties of heavy rare-gas crystals under pressure in the model of deformable atoms

The quantum-mechanical model of deformable and polarizable atoms has been developed for the purpose of investigating the elastic properties of crystals of rare gases Kr and Xe over a wide range of pressures. The inclusion of the deformable electron shells in the analysis is particularly important for the shear moduli of heavy rare-gas crystals. It has been shown that the observed deviation from the Cauchy relation δ(p) for Kr and Xe cannot be adequately reproduced when considering only the many-body interaction. The individual dependence δ(p) for each of the rare-gas crystals is the result of two competitive interactions, namely, the many-body and electron-phonon interactions, which manifests itself in a quadrupole deformation of the electron shells of the atoms due to displacements of the nuclei. The contributions of these interactions in Kr and Xe are compensated with good accuracy, which provides a weakly pressure-dependent value for the parameter δ. The ab initio calculated dependences δ(p) for the entire series Ne-Xe are in good agreement with the experiment.

Theoretical Predication Method of High Order Elastic Constants of Cubic and Tetragonal Crystal Material

The theoretical method of predicating second and third order elastic constants of cubic and tetragonal material are presented by using first-principles total-energy method combined with the means of homogeneous deformation. The predicted results of SrTiO3provide reasonable agreement with the reported experimental data, other theoretical results and Cauchy relations. Since high order elastic constants are very difficult to be measured, the methods presented here provide a valuable guidance for experiments and the investigation of high order elastic properties for cubic and tetragonal materials.

Development of semi-ab initio interionic potential for CaO and MgO

In this paper, we propose a novel method to derive the interionic potentials for CaO and MgO in conjunction with ab initio calculation and empirical three-body interaction. By using the Chen–Mobius lattice inversion, the pairwise interaction between cations and anions can be evaluated from multiple virtual structures. The quantum-chemistry calculation is carried out to derive the short-range potential for the same species of ions. Empirical three-body interactions are then adopted to heal the drawbacks arising from purely pairwise potential, such as Cauchy relation. The proposed potential is verified by molecular dynamics simulations of some primary properties, including pressure and temperature dependence of lattice constant, elastic constants and phase transition of CaO and MgO. Simulation results are in good agreement with the existing experimental data and ab initio calculations, showing that the developed potentials are valid over a wide range of interionic separations. It is believed that this approach can be readily extended into other materials.