The effect of hydrostatic and uniaxial pressure on the elastic constants of cadmium

The third order elastic constants of the hep metal cadmium have been determined from measurements of the hydrostatic and uniaxial pressure dependences of the velocities of ultrasonic waves propagated through single crystals. The hydrostatic pressure dependences of the second order elastic constants have also been obtained. The compression of cadmium has been estimated by extrapolation of the data up to high pressures by using the Murnaghan equation of state. Using the generalised Grüneisen theory in the quasiharmonic approximation, the long wavelength acoustic mode Grüneisen parameters have been obtained; the mean acoustic mode parameter is compared with the thermal Grüneisen parameter. In general the anisotropy of the vibrational anharmonicity of the acoustic modes is found to be consistent with the deviation of the c/a ratio of cadmium from that expected for an ideally close-packed metal.     

Elastic stiffness constants and acoustic symmetry of LiY0.5Tb0.5F4

Measurements of ultrasound wave velocities between 4.2 and 293 K have been used to determine the temperature dependences of the seven adiabatic elastic stiffness tensor components of the laser host scheelite structure fluoride LiY_0.5Tb_0.5F₄. The sign of C₁₆ is negative, as found previously for other oxide and fluoride scheelites; therefore for LiY_0.5Tb_0.5F₄ the positions of the acoustic symmetry axes in the XY plane have the same sense as those of the other scheelites. The k and γ axes of acoustic symmetry lie at +31° and + 76° from the + X axis. Transformation of the elastic stiffness constants to a basis comprised of two acoustic symmetry axes together with the +Z axis enables a detailed comparison of the elastic behaviour of LiY_0.5Tb_0.5F₄ with those of other scheelite crystals.     

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.     

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.     

The hydrostatic pressure derivatives of the elastic constants of indium and an indium cadmium alloy

The hydrostatic pressure derivatives of the elastic stiffness constants of indium and indium-3.4 at.% cadmium alloy single crystals have been obtained from pulse echo overlap measurements of the dependence of ultrasonic wave velocities upon pressure. The softest zone centre acoustic phonon mode in indium is a shear mode propagating k along the [101] direction rather than that (k[110], e[11?0]) which drives the ferroelastic phase transition in the indium-cadmium alloys. The derivative δ((C₁₁ – C₁₂)/2)/δP is positive, accounting for the stability of the fct structure of indium under high pressure. Using the quasiharmonic, anisotropic continuum model the acoustic mode Grüneisen parameters have been calculated and are discussed in terms of mode softening. The high temperature limiting value$\text{\$}\text{?}$γ_H (= 2.56) of the mean acoustic mode Grüneisen parameter is found to be close to the thermodynamic Grüneisen parameter γ^th (=2.5).     

Second-order phase transitions and Ehrenfest equation for strained solids

The equations describing the second-order phase transitions at the hydrostatic and non-hydrostatic pressures are considered. It is shown that the proportionality coefficient between an “effective” volume and the true one V^??=?AV is inversely proportional to the compressibility of the solid at a uniaxial pressure and has a jump at the second-order phase transition. In the case of the non-hydrostatic pressure the “effective” volume of the solid is not a continuous function of temperature and has a jump at the phase transition as well. The Ehrenfest equation is generalized to the solids with an arbitrary homogeneous elastic deformation accompanied by change of the solid volume, in particular, to the solid strained by the uniaxial, biaxial or triaxial pressure. It is shown that the sum of the derivatives of the phase transition temperature with respect to uniaxial pressures applied along axes a, b, c does not coincide with the derivative of the phase transition temperature with respect to the hydrostatic pressure.     

The elastic behaviour of the ternary zincblende structure semiconductor CuGe2P3

A single crystal has been grown of CuGe₂P₃, a ternary semiconductor with a zincblende structure in which the copper and germanium atoms are randomly distributed on the cation sites. The second order elastic constants measured by the ultrasonic pulse echo overlap technique (C₁₁ = 13.66, C₁₂ = 6.17, C₄₄ = 6.66 and bulk modulus B = 8.67 in units of 10¹⁰Nm^?2 at 291 K) are closely similar to those of GaP and conform well to Keyes' correlation for zincblende structure crystals. The hydrostatic pressure derivatives of the second order elastic constants ($\text{?C}{}_{11}\text{?P}\text{= 4.40}$, $\text{?C}{}_{12}\text{?P}\text{= 3.9}$, $\text{?C}{}_{44}\text{?P}\text{= 0.93}$ and $\text{?B}\text{?P}\text{= 4.07}$) and the third order elastic constants (C₁₁₁ = ? 8.45, C₁₁₂ = ? 3.49, C₁₂₃ = ? 1.13, C₁₄₄ = ? 2.82, C₁₅₅ = ? 1.44 and C₄₅₆ = ? 0.69 in units of 10¹¹Nm^?2) also resemble those of GaP: even the anharmonicity of the long wavelength acoustic modes of this ternary semiconductor resembles that of its binary “parent” compound. The positive signs of the hydrostatic pressure derivatives and the negative signs of the third order elastic constants show that the acoustic modes at the long wavelength limit stiffen under pressure, an effect which is quantified here by computation of the mode Grüneisen parameters, which are all positive. The harmonic and anharmonic force constants, obtained in the valence force field model, fit well into the pattern shown by related zincblende structure compounds: the ratio ($\text{β}\text{α}$) for bond bending (β) to stretching (α) force constants is greater than 4:1; the dominating anharmonic force constant is γ: most of the anharmonicity is associated with nearest neighbour atoms. Analysis of the temperature dependence of the elastic constants on the basis of a simple anharmonic model again emphasises the similarity between the elastic behaviour of CuGe₂P₃ and GaP. The thermal expansion of CuGe₂P₃ varies almost linearly with temperature between 291 K and 1000 K, the linear coefficient of thermal expansion α being 8.21 ± 0.02 × 10^?6 °C^?1. The similar lattice parameters and elastic behaviour of CuGe₂P₃ and GaP indicate that semiconducting devices made of epitaxial deposits of CuGe₂P₃ on a GaP substrate should prove to be almost strain-free.     

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.

     

高压下Gd3Ga5O12晶体的弹性性能

 利用超声“脉冲回波重合法”技术,测定了立方晶体镱镓石榴石（GGG）各对称方向的超声波声速,及它们在流体静压力和单轴应力条件下,随压力的变化,计算了GGG的二阶和三阶弹性常数。它们分别是：c₁₁=3.04,c₁₂=1.31,c₄₄=0.94和c₁₁₁=-17.97,c₁₁₂=-12.86,c₁₂₃=-1.06,c₁₄₄=-2.11,c₁₆₆=-3.29,c₄₅₆=-0.94,单位是10¹¹ N/m²。根据这些数据,我们还计算了GGG晶体的德拜温度以及沿它的[001]→[111]→[110]→[100]传播的三个声学波分支（一个纵波,两个横波）的Gruneisen参数γ(p,N)的分布。     

Pressure dependence of elastic behaviour and force constants of GaP

The six third order elastic constants of GaP have been obtained from measurements of the effect of hydrostatic and uniaxial pressures on ultrasonic wave velocities. The valence force field model has been used to obtain the bond-bending and stretching force constants. Confirmatory evidence is added to the hypotheses that zinc blende structure crystals undergo the densification transition under pressure when (1) a modified Born stability criterion that $\text{1}\text{2}\text{(C}{}_{11}\text{? C}{}_{12}\text{)}\text{B}$ equals about 0.2 is reached at the phase transition pressure P_t, and (2) the ratio $\text{β}{}_{\mathrm{t}}\text{α}{}_{\mathrm{t}}$ of the second order bond-bending β_t to bond-stretching α_t, force constants is about 0.1. These findings suggest that a macroscopic shear takes place at the transition     

Third order elastic constants of NaCl

A brief review is given of the meaning of the third order elastic constants and of the general significance of the anharmonic parameters for mechanical and thermodynamic properties. The measurements of the third order elastic constants of NaCl and KCl by Chang and of NaCl by Swartz are described. In each case the changes in ultrasonic velocities were observed as a function of applied uniaxial stress. More recently a program in beam mixing has been initiated by Dunham to study the third order constants of NaCl through the conversion efficiency of the T+T →L interaction.     

Critical behaviour of elastically coupled systems with spatially anisotropic critical fluctuations

The spatial anisotropy of critical fluctuations has strong influence on the temperature dependence of the elastic constants of an elastic medium coupled magneto- or electrostrictive to the order parameter. Under pinned boundary conditions we find for uniaxial dipolar systems of hexagonal and trigonal symmetry a second order phase transition, where only c₁₁ and c₁₂ show critical behaviour. For other symmetries a first order transition is expected. At an uniaxial Lifshitz point only c₃₃ becomes critical and the phase transition remains of second order for any symmetry.     

Mechanical and nonlinear elastic characteristics of polycrystalline AMg6 aluminum alloy and <Emphasis Type="Italic">n</Emphasis>-AMg6/C<Subscript>60</Subscript> nanocomposite

The influence of nanostructuring on the mechanical and nonlinear elastic characteristics of polycrystalline AMg6 aluminum alloy and n-AMg6/C₆₀ nanocomposite has been experimentally studied. The independent second- and third-order elastic coefficients are measured via the ultrasonic method. The thir-dorder elastic coefficients have been evaluated via the Tearstone–Bragger approach from the experimentally established velocities of shear and longitudinal bulk acoustic waves as the functions of the uniaxial compression in the studied samples. The nonlinear elastic properties are examined via the spectral acoustic technique in AMg6 aluminum alloy and n-AMg6/C₆₀ nanocomposite, and the nonlinear acoustic parameters are determined.     

Specific propagation directions of acoustic waves for piezoelectric and nonpiezoelectric media

I reanalyze the problem of the existence of longitudinal normals inside the symmetry planes of piezoelectric crystals belonging to the symmetry class mm2. The equations determining components of longitudinal normals situated outside the symmetry planes for media of this symmetry are discussed. It is proven that nonpiezoelectric media of rhombic symmetry could have 4 or 8 distinct acoustic axes. Examples of nonpiezoelectric elastic media of monoclinic symmetry without acoustic axes are given. The method of determination of the components of acoustic axes for piezoelectric media of arbitrary symmetry is presented. With the help of this method, I discuss the problem of acoustic axes for piezoelectric media of the symmetry class mm2. The text was submitted by the author in English.     

General compliance transformation relation and applications for anisotropic hexagonal metals

In anisotropic crystals, the compliance (s_ij) and the stiffness (c_ij) matrices are usually specified in the orthogonal coordinate systems (X₁,Y₁,Z₁), which do not coincide with the crystal axes (X,Y,Z) used commonly, excepting cubic and orthorhombic crystal systems, and must be transformed to an arbitrary orthogonal coordinate system chosen to be convenient for the question. Such a transformation has been done in this paper for hexagonal crystals and a general compliance transformation relation is given. Accordingly, the useful expressions of the Young’s modulus E(hkl), Poisson’s ratio υ(hkl) and X-ray elastic constants (XREC) and are also given in terms of the Miller indices of the lattice plane (hkl) in the crystal axes (X,Y,Z) used commonly.     

Z′-boson decays as a test of the four-color symmetry of quarks and leptons

The quark and leptonic widths of the Z′ boson that arises from four-color quark-lepton symmetry are calculated and compared with the predictions of the E ₆ and the left-right (LR) model. It is shown that this four-color symmetry leads to a specific relation of the type v′ _q +a′ _q between the vector and axial coupling constants for Z′-boson interaction with quarks and leptons. Calculations with allowance for the four-color symmetry in question yield leptonic widths of the Z′ boson that are considerably greater than those predicted within the E ₆ and the LR model and result in a relativelys mall hadronic-to-leptonic width ratio. Since these features are associated with the four-color symmetry, their observation would suggest its manifestation in Z′-boson decays.     

PAC analysis of defect motion by Blume's stochastic model forI=5/2 electric quadrupole interactions

Using Blume's stochastic model and the approach of Winkler and Gerdau, we have computed time-dependent effects on perturbed angular correlation (PAC) spectra due to defect motion in solids in the case ofI=5/2 electric quadrupole interactions. We report detailed analysis for a family of simple models: “XYZ+Z” models, in which the symmetry axis of an axial EFG is allowed to fluctuate among orientations alongx, y, andz axes, and a static axial EFG oriented along thez axis is added to the fluctuating EFGs. When the static EFG is zero, this model is termed the “XYZ” model. Approximate forms are given forG ₂(t) in the slow and rapid fluctuation regimes, i.e. suitable for the low and high temperature regions, respectively. Where they adequately reflect the underlying physical processes, these expressions allow one to fit PAC data for a wide range of temperatures and dopant concentrations to a single model, thus increasing the uniqueness of the interpretation of the defect properties. Application of the models is illustrated with data from a PAC study of tetragonal zirconia.     

Pressure and temperature dependences of the elastic properties of rutile (TiO2)

Using ultrasonic techniques we have measured the pressure and temperature dependences of the six elastic constants of single-crystalline rutile (TiO₂). The pressure measurements were made in the range 0–20 kbar at room temperature, and the temperature measurements were made in the range 80–300°K at atmospheric pressure. Our results are in good agreement with the results recently reported by Manghnani et al., although our pressure derivatives are generally a few per cent smaller than theirs and our temperature derivatives a few per cent more negative (at 298°K). Nonlinear pressure dependences were observed for some of the elastic constants. A relation between the pressure-induced softening of two lattice modes (one optic and one acoustic) and a possible pressure-induced transition to the CaCl₂ structure is discussed. An approximate separation of the isobaric temperature derivatives of the elastic constants into “pure” volume and temperature contributions is considered. In particular, an attempt is made to estimate the magnitude of the error made in effecting this separation using equations applicable to cubic rather than tetragonal symmetry. It is shown that, in principle, these errors could be significant.     

Magnetoelastic effects in La1−xCexB6

The elastic constants of La_1−xCe_xB₆ (x = 0.00, 0.01, 0.03) single crystals were determined by ultrasonic measurements. The temperature dependence of the elastic symmetry modes c₄₄ and (c₁₁ − c₁₂)/2 is explained on the basis of the crystalline-electric-field level scheme of CeB₆ by taking the magnetoelastic interaction into account. An influence of the Kondo effect on the elastic constants could not be detected.     

Electronic, elastic, optical properties of rutile TiO2 under pressure: A DFT study

The electronic, elastic constants and optical properties of rutile TiO₂ have been investigated using first principle pseudopotential method within generalized gradient approximation (GGA) proposed by Perdew-Burke-Ernzerhof (PBE). The calculated volume, bulk modulus and pressure derivative of bulk modulus are in good agreement with previous experimental and computational results. An underestimated band gap (1.970 eV) along with the higher density of states and expanded energy bands around the fermi level is obtained. Calculated elastic constants satisfying the Born stability criteria suggest that rutile TiO₂ is mechanically stable under higher hydrostatic pressure. The acoustic wave speeds in [1 0 0], [0 1 0], [0 0 1], [1 1 0] and [45° to [1 0 0] and [0 0 1]] directions are predicted using the investigated elastic constants. The dielectric constant is identified with respect to electronic band structure and is utilized to derive the other optical properties like refractive index, energy loss function, reflectivity and absorption. The effect of hydrostatic pressure (0-70 GPa) is described for listed properties. Our investigated results are in good accord with the existing theoretical and experimental results.