The determination of stresses in thin films; modelling elastic grain interaction

X-ray diffraction is frequently employed for the analysis of mechanical stresses in polycrystalline specimens. To this end, suitable so-called diffraction elastic constants are needed for determining the components of the mechanical stress tensor from measured lattice strains. These diffraction elastic constants depend on the single-crystal elastic constants of the material considered and the so-called grain interaction, describing the distribution of stresses and strains over the crystallographically differently oriented crystallites composing the specimen. Well-known grain interaction models, as due to Voigt, to Reuss, to Neerfeld and Hill and to Eshelby and Kröner, may be applied to bulk specimens, but they are generally not suitable for thin films. In this paper, an average 'effective' grain interaction model is proposed that consists of a linear combination of basic extreme models including new models specially suited to thin films. Experimental verification has been achieved by X-ray diffraction strain measurements performed on a sputter-deposited copper film. This is the first time that anisotropic grain interaction has been analysed quantitatively.     

Prediction study of the structural and elastic properties for the cubic skutterudites LaFe4A12 (A = P,As and Sb) under pressure effect

We have performed accurate ab initio total energy calculations using the full-potential linear augmented plane wave plus local orbitals method with the local density approximation for the exchange–correlation potential to investigate the systematic trends for structural and elastic properties of the cubic LaFe₄A₁₂ skutterudites’ family depending on the type of A pnicogen atom (A stands for P, As and Sb). The calculated equilibrium lattice constants and internal free parameters are in good agreement with the experimental results. For the first time, the numerical estimates of the independent elastic constants and their pressure dependence are performed using the total energy variation as function of strain technique. Isotropic elastic parameters and related properties, namely bulk modulus, shear modulus, Young’s modulus, Poisson’s ratio, Lamé’s coefficients, average sound velocity and Debye temperature, are estimated in the framework of the Voigt–Reuss–Hill approximation for ideal polycrystalline LaFe₄A₁₂ aggregates.     

Ab initio investigation of the structural,elastic and electronic properties of the anti-perovskite TlNCa3

We have investigated the structural, elastic and electronic properties of the anti-perovskite TlNCa₃ using ab initio calculations within the generalized gradient approximation and the local density approximation for the exchange–correlation potential. The lattice constant, bulk modulus, elastic constants and their pressure dependence, energy band structures, density of states and charge density distribution are calculated and analyzed in comparison with the available experimental and theoretical data. The bulk modulus, shear modulus, Young’s modulus, Poisson’s ratio, Lamé’s coefficients, average sound velocity and Debye temperature are numerically estimated for ideal polycrystalline TlNCa₃ aggregates in the framework of the Voigt–Reuss–Hill approximation. This is the first theoretical prediction of the elastic constants and their related properties for TlNCa₃ that requires experimental confirmation.     

Do Reuss and Voigt bounds really bound in high-pressure rheology experiments?

Energy dispersive synchrotron x-ray diffraction is carried out to measure differential lattice strains in polycrystalline Fe(2)SiO(4) (fayalite) and MgO samples using a multi-element solid state detector during high-pressure deformation. The theory of elastic modelling with Reuss (iso-stress) and Voigt (iso-strain) bounds is used to evaluate the aggregate stress and weight parameter, α (0≤α≤1), of the two bounds. Results under the elastic assumption quantitatively demonstrate that a highly stressed sample in high-pressure experiments reasonably approximates to an iso-stress state. However, when the sample is plastically deformed, the Reuss and Voigt bounds are no longer valid (α becomes beyond 1). Instead, if plastic slip systems of the sample are known (e.g.?in the case of MgO), the aggregate property can be modelled using a visco-plastic self-consistent theory.     

Effect of many body forces on the photoelastic constants of the alkali halides

A simple phenomenological lattice theory based on the Lundqvist potential of ionic cohesion (many body theory of elastic dielectric), to explain photoelastic behaviour of the solids with rock-salt structure, is presented and shown to apply within reasonable errors to the known experimental constants. The predicted values of the photoelastic constants of the alkali halides are very close to the experimental values and are better than obtained by all earlier workers.     

Analysis of the photoelastic effect in ionic crystals

An analysis of the photoelastic effect in ionic crystals has been presented within the framework of Clausius-Mossotti theory of the dielectric constant. The values of the strain derivative of the electronic dielectric constant have been calculated in alkali halides and MgO crystals by taking into account the variation of electronic polarizabilities with compressive stress. The results obtained are found closer to the experimental values. The photoelastic behaviour of MgO crystal is predicted to be of opposite nature to that of alkali halides, in conformity with the experimental observations.     

Theory of strain derivative of electronic dielectric constant of ionic crysts

A theoretical method for evaluating the strain derivative of the electronic dielectric constant of ionic crystals has been developed. The analysis presented here is based on the shell model and takes account of the exchange charge polarizations. Values of strain derivative of the electronic constant dielectric calculated for 6 alkali halides and MgO show a remarkable agreement with experimental data on photoelastic constants.     

Structural,elastic, electronic and thermal properties of M2SbP (M = Ti,Zr and Hf)

The structural, elastic, electronic and thermal properties of M₂SbP (M = Ti, Zr and Hf) were studied by means of a pseudo-potential plane-wave method based on the density functional theory within both the local density approximation and the generalised gradient approximation. The optimised zero-pressure geometrical parameters, i.e. the two unit cell lengths (a, c) and the internal coordinate (z), were in good agreement with available experimental and theoretical data. The effect of high pressure, up to 20 GPa, on the lattice constants shows that the contractions along the a-axis were higher than along c-axis. The anisotropic independent elastic constants were calculated using the static finite strain technique. Numerical estimations of the bulk modulus, shear modulus, Young's modulus, Poisson's ratio, average sound velocity and Debye temperature for ideal polycrystalline M₂SbP aggregates were performed in the framework of the Voigt–Reuss–Hill approximation. The calculated band structures show that all studied materials are electrical conductors. Analysis of the atomic site projected densities showed that the bonding is of covalent–ionic nature with the presence of metallic character. The density of states at the Fermi level is dictated by the transition metal d–d bands; the Sb element has little effect. Thermal effects on some macroscopic properties of M₂SbP were predicted using the quasi-harmonic Debye model, in which the lattice vibrations are taken into account. The variations of the volume expansion coefficient, heat capacity and Debye temperature with pressure and temperature in the ranges 0–50 GPa and 0–2000 K were obtained successfully.     

Ab initio study of structural,elastic, and high-pressure properties of rubidium halides RbX (X = F,Cl, Br and I)

We present first-principle calculations on the structural, elastic, and high-pressure properties of rubidium halides compounds, using the pseudo-potential plane-waves approach based on density functional theory, within the generalized gradient approximation. Results are given for lattice constant, bulk modulus and its pressure derivative. The pressure transition at which these compounds undergo structural phase transition from NaCl-type to CsCl-type structure are calculated and compared with previous calculations and available experimental data. The elastic constants and their pressure dependence are calculated using the static finite strain technique. We derived the bulk and shear moduli, Young's modulus and Poisson's ratio for ideal polycrystalline RbF, RbCl, RbBr, and RbI aggregates. We estimated the Debye temperature of these compounds from the average sound velocity.     

Punktdefekte im kubischen elastischen Kontinuum

The determination of the displacement and strain fields of a point defect in a cubic crystal requires even in the framework of continuum elasticity theory numerical calculation. These fields of elastic dipoles are expanded in suitable vector and tensor fields. The coefficients of this expansion are calculated up to polynomials of 5th and 4th order in the direction cosines using the ratios of elastic constants as parameters. With this expansion the interaction of elastic dipoles in a cubic medium can be calculated. The results have been applied to the interaction of F-centres and of O₂ ^?-centres in alkali halides.     

Insight into the structural,electronic and elastic properties of AInQ2 (A: K,Rb and Q: S,Se, Te) layered structures from first-principles calculations

First-principles calculations were performed to explore the structural, elastic and electronic properties of the ternary indium chalcogenides AInQ₂ (A: K, Rb and Q: S, Se, Te) in both monoclinic and triclinic phases. This study is carried out by using the first-principles pseudopotential plane-wave (PP–PW) method as implemented in CASTEP code. Both the generalized gradient approximation of Perdew–Burke–Ernzerhof scheme (GGA–PBE) and the Heyd–Scuseria–Ernzerhof (HSE06) hybrid functional were used to treat the exchange-correlation interactions. In order to confirm the previous reports and to understand the effect of symmetry in determining the physical properties of these layered materials we have calculated the structural and the electronic properties at the equilibrium lattice constant for both the systems. The single-crystal elastic constants C_{i j} are calculated using the stress-strain approach. The elastic moduli of the polycrystalline aggregates and their related properties are obtained in the framework of Voigt–Reuss–Hill approximations. Electronic band structure indicates the semiconducting behaviour with a direct band gap at Γ–Γ. The results obtained from the (HSE06) hybrid functional are in excellent agreement with the available experimental data and computed results for the monoclinic and triclinic structures.     

Elastic properties and electronic structures of antiperovskite-type InNCo3 and InNNi3

We have performed first-principles calculations to study the elasticity, electronic structure, and magnetism of InNCo₃ and InNNi₃. The independent elastic constants are derived from the second derivative of total energy as a function of strain, and the elastic moduli are predicted according to the Voigt–Reuss–Hill approximation. Our calculations show that the bulk modulus of InNCo₃ is slightly larger than that of InNNi₃ due to a smaller lattice constant for InNCo₃. For InNCo₃ the ferromagnetic state is energetically preferable to the non-magnetic state, while the ground state of InNNi₃ is non-magnetic. This is due to the different strength of 2p–3d hybridization for the N–Co atoms in InNCo₃ and the N–Ni atoms in InNNi₃.     

Effective dielectric and photoelastic tensor components in thin layered superlattices

The effective photoelastic constants of a superlattice composed of thin alternating layers of orthorhombic symmetry are calculated as functions of dielectric, elastic and photoelastic constants of the constituents. In a preliminary step to this calculation, we also obtain the effective dielectric tensor for any symmetry of the layers. It is shown that appropriate combinations of these effective constants are arithmetic averages of the corresponding quantities in the constituents, each layer having a weight equal to its relative thickness.     

An ab initio study of filled skutterudites ROs4P12 (R = Sm,Eu and Gd)

The elastic, electronic, magnetic and optical properties of filled skutterudite ROs₄P₁₂ (R = Sm, Eu and Gd) have been studied by first principles calculation. The full-potential linearized augmented plane wave method based on density functional theory was employed. For the exchange-correlation potential, local spin density approximation + Coulomb repulsion (LSDA + U) is used to treat the f-electrons more effectively. The numerical values of the elastic parameters are estimated in the framework of the Voigt–Reuss–Hill approximation. OsP-based filled skutterudite with localized 4f and 5d-electrons shows dense energy bands near Fermi energy originating from rare earth and Os atoms. The dense density of states near E_F reveals that these compounds are suitable for thermoelectric application. Optical constants including dielectric function, optical reflectivity and refractive index are calculated for photon energy radiation up to 12 eV. The exchange-splitting of R-4f states were analyzed to explain the ferromagnetic behavior of ROs₄P₁₂.     

Anharmonic properties of rocksalt structure solids

An effort has been made for obtaining the anharmonic properties of rocksalt structure solids starting from primary physical parameters viz. nearest-neighbor distance and hardness parameter assuming long- and short-range potentials at elevated temperatures. The elastic energy density for a deformed crystal can be expanded as power series of strains for obtaining coefficients of quadratic, cubic and quartic terms which are known as the second-, third- and fourth-order elastic constants, respectively. When the values of the higher-order elastic constants are known for a crystal, many of the anharmonic properties of the crystal can be treated within the limit of the continuum approximation in a quantitative manner. In this study, higher-order elastic constants are computed up to their melting temperature for rocksalt structure solids, which are alkali cyanides, sodium and potassium halides. The first order pressure derivatives of second- and third-order elastic constants, the second-order pressure derivatives of second-order elastic constants and partial contractions are also evaluated at different temperatures for these substances. The results thus obtained are compared with experimental data and found in well agreement with present values.     

The Born repulsive parameters and dielectric behaviour of alkali halides

The dependence of the Born repulsive parameters of alkali halides on elastic and dielectric data has been discussed. The values of hardness parameter in alkali halides have been recalculated using the revised values of van der Waals energies. It is observed that the two sets of hardness parameter corresponding to elastic and dielectric data differ from each other but become compatible if an effective charge parameter for the ions is introduced. Its usefulness has been demonstrated by calculating the strain derivative of static dielectric constant of alkali halides.     

Temperature derivatives at constant volume of the elastic constants of alkali halide crystals

A new method has been developed for calculating the temperature derivatives at constant volume of the elastic constants of the alkali halides using the concept of phonon pressure. Calculated values of temperature derivatives of elastic constants are compared with the corresponding values recently derived from ultrasonic experimental data.     

Photoelastic constants in silver and thallium halides

In the present communication the expressions for the photoelastic constants of the solids crystallizing in cesium chloride structure have been derived on the basis of lattice theory. These expressions have been utilized to predict the magnitudes of photoelastic constants in thallium halides. On the basis of these constants the validity of the Clausius-Mossotti model of electronic polarization in silver and thallium halides has been discussed.