Temperature dependence of the second-order elastic constants of cubic crystals

A simplified phenomenological theory of the temperature dependence of the second-order elastic constants of crystals is considered. The temperature dependences of the second-order elastic constants are calculated for a series of cubic crystals with various types of predominant chemical bonding. Satisfactory agreement between the calculation results and experimental data is obtained. Fiz. Tverd. Tela (St. Petersburg) 41, 235–240 (February 1999)     

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.     

Third order elastic constants of rubidium chloride

The third order elastic constants of RbCl, determined by measurements of the static stress dependence of ultrasonic waves, are found to be (in units of 10¹²$\text{dynes}\text{cm}{}^2$)

$\text{C}{}_{111}\text{= ?6.71 ± 0.1 C}{}_{123}\text{= 0.05 ± 0.07}$

$\text{C}{}_{112}\text{=?0.18 ± 0.04 C}{}_{144}\text{= 0.11 ± 0.02}$

$\text{C}{}_{166}\text{=?0.17 ± 0.01 C}{}_{456}\text{= 0.4 ± 0.01}$

. The calculation of third order constants using a rigid ion Born model is briefly discussed, and results are compared to the measurements. The comparison qualitatively supports the model, but no quantitative evaluation of the repulsive interaction is possible.     

Temperature derivatives at constant volume of the elastic constants of the alkali halides

Values of dC/dT)_V have been computed from experimental dC/dT)_P and dC/dP)_T for the three elastic constants (expressed as $\text{B}{}_{\mathrm{T}}\text{, C}{}_{44}\text{and}\text{(C}{}_{11}\text{?C}{}_{12}\text{)}\text{2)}$ of each of the 16 Li, Na, K, and Rb halides. The dC/dT)_V measure the explicit dependence of C on T, the effect of thermal expansion having been removed. The dC/dT)_V are all small (compared with dC/dT)_P), are all negative, and vary quite systematically and smoothly with anion, with cation and with the three elastic constants. Negative dB_T/dT)_V is accounted for both thermodynamically and by available lattice dynamical calculations. dC/dT)_V for shear constant is expressed in terms of two vibrational mode parameters, whose values can then be estimated from the observed value, and the trend of dC/dT)_V with Debye temperature.     

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.     

The elastic constants and their temperature and pressure derivatives of AgBr-AgCl mixed crystals

The three independent second-order elastic constants and their temperature and pressure derivatives have been measured for four AgBr-AgCl mixed crystals, with 19.5, 39.1, 56.6 and 78.7 mole % AgCl, using the ultrasonic pulse-echo technique at room temperature. The explicit temperature dependence of the elastic constants is calculated and is found to be much larger than that of other NaCl structure crystals. The violation of the Cauchy relation C₁₂ = C₄₄ is found to be significant and increases between AgBr and AgCl. The high temperature limit of the Gruneisen parameter is calculated from the elastic data. A comparison is made between the elastic properties of the silver halides and the alkali halides.     

Crystal equilibrium and the electronic contribution to the elastic constants of potassium and rubidium

Electronic contribution to the elastic constants of K and Rb are calculated by applying the crystal equilibrium condition to an eight parameter lattice dynamical model. For comparison, these quantities have also been calculated according to Fuch's criterion. The two results are found to be in fairly good agreement. Also, the calculated phonon frequencies agree well with the experimental data.     

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).     

Temperature and pressure derivatives of elastic constants in single crystal Au-47.5 at.% Cd near martensitic phase transformation

The behavior of the elastic properties and their pressure derivatives near the martensitic transformation β→β′ was determined from sound velocity measurements at a frequency of 10 MHz, on a single crystal of Au-47.5 at.% Cd.The transformation of the β phase to the martensitic β' phase was obtained from the anisotropic elastic constants c₁₁, c₄₄ and c′, and their pressure derivatives. The behaviour of c₁₁ and c₄₄ is normal as a function of temperature, while the variation of c′ is positive with temperature. Before the β→β′ phase change the value of c′ is very small and exhibits high elastic anisotropy. The small value of c′ and the anisotropy were explained according to the crystallographic mechanism of the martensitic transformation and the Zener theory of instability of the b.c.c. structure. The pressure derivatives of c′ are negative and the variation of $\text{d}\text{c}{}_{44}\text{d}\text{P}$ with temperature is anomalous. This behavior can be explained by the influence of pressure on the transformation temperature, and by the instability of the β phase. It was found that the elastic constants c′ determine the mechanism of the transformation. From the pressure derivatives of c₁₁, c₄₄ and c′ the Grüneisen parameters for different modes of vibration were calculated. For the c′ mode the values are negative, for the c₄₄ mode the variation of the parameter with temperature is negative. This anomalous behavior can be explained as due to the anisotropic softening of the atom bonding on the (110) planes of the cubic structure, as a consequence of the phase transformation mechanism.     

Analysis of the pressure derivatives of the second order elastic constants in alkaline earth oxides

The three body force shell model (TSM) has been applied to evaluate the magnitude of the pressure derivatives of second order elastic constants of alkaline earth oxides and a good agreement with the experimental values has been obtained. It is also pointed out that the calculated values of dC₄₄?dp show a smooth variation with ionicities. The theory has been modified by taking into account next nearest neighbour interaction.     

Temperature dependence of the elastic constants of aluminum

The single crystal elastic constants of aluminum have been measured using a piezoelectric composite oscillator from room temperature to just 20 K below the melting point. The elastic moduli differ markedly from previous high temperature results, but match in well with previous cryogenic results. Over the temperature range investigated the isothermal bulk modulus and the two shear moduli have a simple exponential dependence on isobaric volume, and the cryogenic data indicate this dependence may be preserved down to absolute zero. As has been found previously for a wide range of materials, the isothermal bulk modulus and the shear modulus $\text{(c}{}_{11}\text{– c}{}_{12}\text{)}\text{2}$ appear to be continuous functions of volume through the melting expansion, and melting seems to find its origin in the mechanical instability associated with this shear modulus vanishing at the volume of the melt at the freezing point. Grüneisen's parameter divided by the molar volume is very nearly independent of isobaric volume.     

Temperature dependence of the elastic constants of aluminium

Published results for the temperature variation of the elastic constants of single crystal aluminium are discussed with special reference to the possibility that error propagation in the conversion of compliances to stiffness may contribute to the discrepancies between the different sets of data for the stiffnesses.     

Temperature and volume dependence of the elastic constants of palladium

A separation between lattice and electronic contributions to the elastic constants of Pd is obtained. The temperature dependence at constant volume is measured by applying hydrostatic pressure. The apparatus and results for Pd are presented.     

立方结构Fe基磁性材料弹性系数第一性原理计算

通过赝势平面波法(CASTEP)及全电势线性缀加平面波法(FLAPW)，以bcc-Fe为对象，研究第一性原理计算立方结构Fe基磁性材料弹性系数的方法，分析影响计算立方结构Fe基磁性材料弹性系数准确度的各项因素. 结果表明，在第一性原理弹性系数计算中，晶格常数是决定弹性系数计算准确度的关键因素；势函数的选择也会影响计算准确度. 使用全电势基矢的FLAPW法可以得到更为精准的弹性系数计算结果. 计算得到bcc-Fe的弹性系数C₁₁，C₁₂，C₄₄分别为246 GPa，121 GPa，113 GPa，与实验值基本一致. 利用本方法，计算了新型Fe-Ga磁致伸缩材料的弹性系数C₁₁，C₁₂，C₄₄分别为207 GPa，166 GPa及108 GPa. 关键词： 弹性系数 磁致伸缩材料 赝势平面波法 全电势线性缀加平面波法