Third-order elastic constants and pressure derivatives of the second-order elastic constants of β-tin

The second- and third-order elastic constants and pressure derivatives of second-order elastic constants of tetragonal β-tin have been obtained using the deformation theory. The strain energy density derived using the deformation theory is compared with the strain dependent lattice energy obtained from the elastic continuum model approximation to get the expressions for the second- and third-order elastic constants. Higher order elastic constants are a measure of the anharmonicity of a crystal lattice. The 12 non-vanishing third-order elastic constants and the six pressure derivatives of the second-order elastic constants in tetragonal β-tin are obtained in the present work and are compared with the available experimental values. The second-order elastic constant C₃₃ obtained in the present study is in reasonable agreement with the experimental values. The third-order elastic constants are generally one order of magnitude greater than the second-order elastic constants as expected of a crystalline solid. The third-order elastic constant C₃₃₃ is higher in magnitude than all other values. This shows a greater anharmonicity of β-tin along the c-axis direction of the crystal.     

Elastic constants of solids at high pressures

The isothermal and adiabatic nth-order (n ?? 2) elastic constants of a loaded crystal are defined. These constants fully determine the behavior of solids at an arbitrary load and are controlled by both an interatomic interaction and an applied load. Expressions that relate these constants (of the second, third, and fourth order) to Brugger elastic constants of the corresponding order, which are only determined by an inter-atomic interaction, are found for cubic symmetry crystals under hydrostatic pressure. These expressions are used to calculate the equation of state and the second- and third-order elastic constants of bcc tantalum at T = 0 K over a wide pressure range (0?C600 GPa) using an electron density functional method. The results of calculating the equation of state and the second-order elastic constants agree with available experimental data and the calculation results obtained in other works.     

Third-order elastic constants of ZnS and ZnSe

The complete set of non-vanishing third-order elastic constants of the semiconductors ZnS and ZnSe is obtained theoretically. The strain energy density is estimated using finite strain elasticity theory by considering the interactions up to two nearest neighbours of each atom in the unit cell of these compounds. This energy density is compared with the strain dependent lattice energy density from the continuum model approximation. The second-order parameter of the potential function φ is obtained from the measured principal axis C_ij. The third-order potential parameter is estimated by assuming a Lennard-Jones type of interatomic potential. The interlattice displacements as well as the second-order elastic constants are evaluated along with the six third-order elastic constants of ZnS and ZnSe. Using these second- and third-order elastic constants of ZnS, the pressure derivatives of second-order elastic constants are evaluated. The second- and third-order elastic constants of ZnSe are compared with the available experimental values. The third-order elastic constants show anisotropy in different directions.     

Third-order elastic constants of diamond determined from experimental data

The pressure derivatives of the second-order elastic constants (SOECs) of diamond were determined by analyzing previous sound velocity measurements under hydrostatic stress [McSkimin and Andreatch, J. Appl. Phys., vol. 43, 1972, pp. 2944] [4]. Our analysis corrects an error in the previously reported results. Using the corrected pressure derivatives, together with published data for the nonlinear elastic response of shock-compressed diamond [Lang and Gupta, Phys. Rev. Lett., vol. 106, 2011, pp. 125502] [3], a complete and corrected set of third-order elastic constants (TOECs) is presented that differs significantly from TOECs published previously.     

Third-order elastic constants of the alloy Fe72Pt28

The complete sets of second- and third-order elastic constants of the cubic Fe₇₂Pt₂₈ have been obtained using the strain energy density derived from interactions up to three nearest neighbours of each atom in the unit cell. The finite strain elasticity theory has been used to get the strain energy density of Fe₇₂Pt₂₈. The strain energy density is compared with the strain-dependent lattice energy density obtained from the continuum model approximation and the expressions for the second- and third-order elastic constants of Fe₇₂Pt₂₈ are given. The second-order potential parameter is deduced from the measured second-order elastic constants of Fe₇₂Pt₂₈ and the third-order potential parameter is estimated from the Lennard-Jones inter-atomic potential for Fe₇₂Pt₂₈. The inter-lattice displacements; the three independent second-order elastic constants and the six independent third-order elastic constants of Fe₇₂Pt₂₈ are also determined. The second-order elastic constants are compared with the experimental elastic constants of Fe₇₂Pt₂₈. We also study the effect of pressure on the second-order elastic constants of Fe₇₂Pt₂₈.     

A first-principles study on the structural and elastic properties and the equation of state of wurtzite-type cadmium selenide under pressure

A first-principles investigation on the crystal structural and elastic properties and the equation of state of wurtzite-type cadmium selenide (w-CdSe) has been conducted using the plane-wave pseudo-potential density functional theory and the quasi-harmonic Debye model. The elastic constants, the aggregate elastic moduli, the elastic anisotropy, and Poisson's ratio under pressure have been investigated. Our calculated equilibrium lattice constants, the elastic constants, and the aggregate elastic moduli at zero pressure are in good agreement with the experimental data and other theoretical results. The variations in the compressional and shear elastic wave velocities with pressure at zero temperature up to pressure 2.7 GPa have been studied; the computed Debye temperature at zero pressure and zero temperature is in reasonable agreement with the result of Bonello et al., In addition, the equation of state of w-CdSe in the pressure range of 0–2.7 GPa and up to a temperature of 900 K has also been obtained.     

On the determination of the volume dependence of Grüneisen parameters in cubic and non-cubic solids

For a material of orthorhombic symmetry under hydrostatic stress, three generalized Grüneisen parameters relative respectively to the three principal directions of the deformation can be defined. Finite strain expressions for these parameters are derived in terms of the Lagrangian strain tensor and the frame-indifferent analogue of the Eulerian strain tensor. The expressions require for their evaluation the thermal expansion coefficients, the elastic moduli and their pressure and temperature derivatives, and the specific heat of the material, so that there are no arbitrary constants or “curve fitting.” In the case of cubic materials, it is possible to determine the unique Grüneisen parameter as a function of volume directly. For non-cubic materials, the volume dependence of the Grüneisen parameters is calculated using a quasi-harmonic finite strain model of equation of state. Numerical applications are given for some cubic and hexagonal metals and the results are discussed.     

Elastic behaviour of lead fluoride under pressure

The variation of the second-order elastic constants (SOECs) and the longitudinal and shear modulii with hydrostatic pressure for the lead fluoride (PbF₂) has been investigated for the first time by means of a three-body force potential (TBP) model. The significance of three-body interactions (TBI) has been clearly demonstrated in reproducing the elastic constant variations and the pressure derivatives of SOECs of PbF₂. The equation of state for this crystal has also been reported.     

金刚石压砧内单轴应力场对物质状态方程测量的影响

金刚石压砧的几何结构使得在高压下封垫内的样品通常处于单轴应力场中：压砧轴向加载应力最大,径向应力最小.由于金刚石压砧内非静水压单轴应力场的影响,用传统的高压原位X射线衍射方法测得的物质压缩曲线一般位于理想静水压压缩曲线之上.利用金刚石压砧径向X射线衍射技术以及晶格应变理论,结合最近的钨、金刚石和硼六氧样品的高压原位同步辐射径向X射线衍射实验结果,从宏观差应力、样品强度、标压物质和待测物质强度的关系三个方面分析讨论了金刚石压砧内单轴应力场对物质状态方程测量的影响及解决方案. 关键词： 金刚石压砧 单轴应力场 高压原位X射线衍射 状态方程     

High pressure X-ray diffraction study of copper hydride at room temperature

Abstract

High pressure X-ray studies on CuH up to 23 GPa have been performed at room temperature using a gasketed diamond anvil cell. The experimental data on the molar volume of CuH as a function of pressure have been fitted to Murnaghan's equation of state giving a bulk modulus: B₀ = 72.5±2 GPa and B₀ = 2.7 ± 0.3. By comparison with the equation of state for pure copper the effective additive volume of hydrogen has been evaluated as a function of pressure. It decreases from 3.2 cm³/mol H, at ambient pressure reaching a flattening value of 1.7cm³hol H at about 60 GPa. This suggests a continuous transition of CuH from ionic or covalent character at normal pressure to metallic hydride behavior at high pressure     

Elastic behaviour under pressure of the cluster compound (Ag6Ge4P12)Ge6

The elastic stiffness constants of the cluster compound (Ag₆Ge₄P₁₂)Ge₆ have been measured by the ultrasonic pulse-echo overlap technique and are found to conform quite closely with those of zincblende structure compounds. The hydrostatic pressure dependences of the elastic constants and also the third-order elastic constants have been measured. It is found that, unlike the zincblende structure compounds, this compound does not show any tendency towards shear mode softening under pressure, all the long wavelength acoustic mode Grüneisen gammas being positive.     

Pressure Dependences of Elastic Constants of AMg6 Aluminum–Magnesium Alloy and n-AMg6/С60 Nanocomposite Alloy

The ultrasonic study results for dependence of the elastic wave velocities and second-order elasticity coefficients of the polycrystalline aluminum alloy AMg6 and its nanocomposite n-AMg6/C₆₀ on hydrostatic pressure up to 1.6 GPa have been described. The ultrasonic research has been carried out using a highpressure ultrasonic piezometer based on the piston-cylinder device. The pressure derivatives of the secondorder elastic constants of these materials established in the present study have been compared with the results of the third-order elastic constants measurements of the test alloys using the Thurston–Brugger method. Involving available literature data, we determined the relationships between the pressure derivatives of the second-order elastic constants of the AMg6 alloy and the Mg-content and nanostructuring.

     

The elastic dielectric

Macroscopic field equations, boundary conditions and equations of state are derived for the non-linear, macroscopic elastic and dielectric response of an insulator. A centrosymmetric polynomial representation of order four is introduced for the energy density; the equations of state for the electric field and stress tensor are then deduced as polynomials of degree three in the displacement gradients and electric displacement field. The results are applied to the special case of m3m material symmetry.

A finite, point-charge model of a centrosymmetric ionic crystal is introduced and used to determine 0°K microscopic expressions for the electric field and stress tensor equation of state coefficients introduced in the macroscopic analysis. The results are used to calculate the full set of second and third-order non-linear coefficients for NaI, based on a Born-Mayer potential and the 4·2°K elastic stiffness data of Claytor and Marshall.     

MgSiO3钙钛矿高温高压特性的第一性原理研究

本文采用第一性原理方法,计算了MgSiO3钙钛矿在零温和0 ~ 150 GPa静水压范围内的晶体结构和弹性模量,并利用准简谐近似Debye模型,拟合三阶Birch-Murnaghan物态方程得到了其高温高压下的热力学性质。通过与现有的理论和实验的结果数据比较,确认在0 ~ 2000 K的温度区间内,第一性原理计算结合Debye模型能够较可靠地模拟在下地幔压力范围内MgSiO3钙钛矿的热力学性质。     

MgSiO_3钙钛矿高温高压特性的第一性原理研究

本文采用第一性原理方法,计算了MgSiO_3钙钛矿在零温和0～150 GPa静水压范围内的晶体结构和弹性模量,并利用准简谐近似Debye模型,拟合三阶Birch-Murnaghan物态方程得到了其高温高压下的热力学性质.通过与现有的理论和实验的结果数据比较,确认在0～2000 K的温度区间内,第一性原理计算结合Debye模型能够较可靠地模拟在下地幔压力范围内MgSiO_3钙钛矿的热力学性质.     

Study of phase transformation and elastic properties of ThX (X = N,P, As and Sb) under high-pressure

An improved interaction potential model (IIPM) has been formulated to theoretically predict the pressure induced phase transition, elastic properties and thermophysical properties of thorium monopnictides (ThX; X = N, P, As and Sb). The phase transition pressures and volume drop obtained from this model show a better agreement with the available experimental than theoretical results. We have achieved elastic moduli, anisotropy factor, Poisson's ratio, Kleinman parameter, shear and stiffness constants on the basis of the calculated elastic constants. To know the anharmonic properties, we have also computed the third-order elastic constants, first-order pressure derivatives of second-order elastic constants and thermophysical quantities. Our results are in reasonable agreement with available measured and others reported data which supports the validity of model.     

Anderson-grüneisen parameter δ, bulk modulus and the compression curves of rhenium

The third-order elastic (TOE) constants of rhenium obtained from a model based on Keating's approach have been used to calculate the Anderson-Grüneisen (AG) parameter δ for this metal following the procedure suggested by Ramji Rao. The temperature dependence of the bulk modulus of rhenium has been calculated using Anderson's theory. The agreement with the experimental results of Fisher and Dever is good. The AG parameter has also been used to calculate the second Grüneisen constant q for rhenium. The variation of the lattice parameters of rhenium with hydrostatic pressure upto 500 kbars has been calculated using the theoretical TOE constants and Thurston's extrapolation formula. There is very good agreement with the experimental results of Liu et al.     

Theories of surface elasticity for nanoscale objects

The emergence of nanotechnology has driven recent interest in systems having surface atoms as a significant fraction of all atoms present, in particular nano-sheets (ultra-thin slabs), nano-wires, and nano-particles. In these systems, the bulk (i.e. non-surface region or interior) is typically strained in response to the stress of the surface. This elastic strain of the bulk in turn changes the surface lattice constants. Since the bulk and the surface are coupled, the problem must be solved self-consistently. Solving this problem requires a quantitative model of the surface elastic properties which are different from the bulk. In this paper we consider various models that have been proposed for surface elasticity. Our goal is to elucidate the relationship between two contrasting approaches: (1) the Shuttleworth equation which defines a surface stress based on the strain derivative of the surface energy and (2) the Gurtin-Murdoch (GM) theory which considers the surface layer as a membrane with residual strain and with elastic constants different from the bulk. The GM theory is analogous to the 2-D Frenkel-Kontorova (FK) model and can be used to obtain quantitative parameters for the FK model. We present an embedded atom method calculation of the surface elastic constants of Cu(1 1 1) using the GM theory with the surface represented by a membrane one atomic layer thick. This quantitative approach describes the elastic properties of surfaces in a physically appealing way. Just as the bulk elastic constants provide direct information regarding the stress/strain relationship in a bulk material, the surface elastic constants provide similar information for a surface monolayer. This theory will allow elasticity analysis and atomistic calculations of properties of nano-scale objects.     

The third-order elastic moduli and pressure derivatives for AlRE (RE=Y,Pr, Nd,Tb, Dy,Ce) intermetallics with B2-structure: A first-principles study

The third-order elastic moduli and pressure derivatives of the second-order elastic constants of novel B2-type AlRE (RE=Y, Pr, Nd, Tb, Dy, Ce) intermetallics are presented from first-principles calculations. The elastic moduli are obtained from the coefficients of the polynomials from the nonlinear least-squares fitting of the energy–strain functions. The calculated second-order elastic constants of AlRE intermetallics are consistent with the previous calculations. To judge that our computational accuracy is reasonable, the calculated third-order constants of Al are compared with the available experimental data and other theoretical results and are found to have good agreements. In comparison with the theory of the linear elasticity, the third-order effects are very important with the finite strains which are lager than approximately 3.5%. Finally, the pressure derivatives have been discussed.     

Study of semiconductor lasers under simultaneous uniaxial stress and hydrostatic pressure

Abstract

We present the design of a device for the simultaneous application of uniaxial stress and hydrostatic pressure. This new apparatus will for the first time allow measurements at constant strain. Results of the simultaneous application of uniaxial stress and hydrostatic pressure to a semiconductor laser are presented and discussed.