Elastic Constants of Crystals from X-ray Coherent Scattering Data. Dedicated to Professor Hermann NEELS on the occasion of his 80th birthday

The X-ray diffraction method for determining the elastic constants C_ij and their temperature dependence using the Bragg reflection intensity is discussed.     

A general treatment of X-ray (residual) macro-stress determination in textured cubic materials: General expressions,cubic invariancy and application to X-ray strain pole figures

An expression in terms of general hkl and orientation distribution functions for the relationship between the elastic lattice strain measured by means of X-ray diffraction methods and the (residual) elastic stress is given for cubic textured materials. In practice the socalled sin² Ψ-method is often used to determine these macro stresses. Then, for the case of textured specimens the quasi-isotropic X-ray elastic constants (depending on hkl only) become complicated functions of the crystallographic texture (J. Appl. Crystallogr. 16 , 325 (1983)). Results are given for both orthorhombic and monoclinic specimen symmetry in terms of general (b.c.c. or f.c.c.) hkl and allow every permutation possible for a given hkl combination. From this cubic invariancy some new cubic invariant spherical surface harmonics are derived which are represented on a basis consisting of the 4-fold symmetry axes of the cubic crystals. Special textures and special stress-states are discussed. For a given texture of a cold-rolled production steel sheet values of the predicted elastic strains are given in terms of “X-ray strain pole figures”.     

Lattice dynamics and related properties of AIBIII and AIIBIV compounds,I. Elastic constants

The present state of knowledge of the elastic properties of the ternary chalcopyrite compounds and some related materials is reviewed. Results of experimental elastic constant determinations are critically evaluated by comparing with other experimentally available material parameters that directly depend on the elastic constants. Possible reasons for existing discrepancies and obvious inconsistencies in the experimental data are discussed. The results of theoretical calculations of the elastic constants of these compounds based on different theoretical models and approaches are compared with experimental data and are assessed concerning achievable accuracy and reliability. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

一种利用实验测量和数值计算确定3m点群晶体弹性系数的方法

将实验测量与数值计算相结合确定低对称性晶体弹性系数的方法可以弥补常规方法测量时造成的弹性系数非对角元误差大的不足.本文将该方法由正交系晶体推广到了三方系3m点群晶体,在数值计算中选取了目前公认的无约束最优化方法中最稳定的两种算法-单纯形法和BFGS法,并增加了弹性系数的约束条件,提高了方法的适应性和正确性.在此基础上,本文利用3m点群晶体LiNbO3和LiTaO3的弹性劲度系数[cij]和顺服系数[sij]的实验数据对该方法进行了讨论及验证.结果表明这种方法对3m点群晶体弹性系数非对角元的计算是可行的.另外,本文还对数值计算时,初值的选取和检验结果合理性等关键性问题进行了详细的讨论.     

Negative Poisson’s ratio behavior in the planar model of asymmetric trimers at zero temperature

Elastic constants of two-dimensional soft cyclic trimers, exhibiting asymmetry and atomic size dispersion, are studied at zero temperature. The molecules interact through a short-range, site–site, purely repulsive, n-inverse-power potential. The essential influence of the molecular asymmetry on the elastic properties is shown. Negative values of the Poisson’s ratio are found for strongly asymmetric molecules. It is also shown that, at fixed number density in the system, by increasing the size dispersion of particles one can significantly increase the elastic constants and the Poisson’s ratio.     

Electrooptical coefficients and temperature and pressure derivatives of the elastic constants of tetragonal Li2B4O7

New values for the following properties of tetragonal Li₂B₄O₇ are reported: dielectric, electrostrictive, electrooptic, elastic, thermoelastic and piezoelastic constants, coefficients of thermal expansion and indices of refraction. These values are considered to be of superior precision compared to earlier published data. Most elastic constants possess positive temperature derivatives. This anomalous behaviour is not reflected in the pressure derivatives. The electrooptic effects are not large enough to compete with other materials used for technical applications.     

A computational study of elastic properties of disordered systems with voids

We present a computational study of elastic properties of disordered systems with voids. The influence of hydrogen and voids on the elastic properties has been investigated by means of atomistic simulations using empirical potentials for hydrogenated amorphous silicon. The elastic constants have been obtained from the fluctuations of the simulation cell in Monte Carlo simulations at constant pressure and temperature. Our results indicate that the softening of the elastic constants of a-Si:H observed experimentally upon increasing hydrogen content cannot be explained by a weakening of the network induced by the reduced coordination but should be attributed instead to the formation of voids. A simple relation between Young’s modulus and the density is presented.     

Negative Poisson’s ratio of two-dimensional hard cyclic tetramers

A two-dimensional system of hard, cyclic tetramers has been investigated by constant pressure Monte Carlo (MC) simulations with variable shape of the periodic box. The molecules (tetramers) are composed of four identical hard disks (atoms) with centers forming a square with sides equal to the diameter of the disks. The elastic constants and the Poisson’s ratio are determined by the analysis of strain fluctuations and the reference state of the system. The singular behavior of the elastic constants is in agreement with free volume theory. It has been found that the hard cyclic tetramer system is auxetic – that is, the Poisson’s ratio of the tetramer system is negative.     

Pressure dependence of elastic properties of low-silica calcium alumino-silicate glasses

The hydrostatic and uni-axial pressure dependence of elastic properties of a low-silica calcium alumino-silicate glass (LSCAS) is determined by ultrasonic pulse-echo techniques at room temperature. The experimental results are used to obtain the third-order elastic constants (TOECs) of these glasses. The pressure dependence of fractal bond connectivity of these glasses is discussed. The normal behavior of positive pressure dependence of ultrasonic velocities was observed for the glass. The pressure dependence of both shear modulus and bulk modulus are positive for these glasses.     

Elastic,Piezoelectric and Dielectric Constants of Pentaerythritol

The elastic, piezoelectric and dielectric constants of pentaerythritol, C(CH₂OH)₄, have been computed employing the method of long waves and the unscreened rigid ion potential model developed by RAMAMOORTHY and KRISHNAMURTHY. Piezoelectric contribution is taken into account in the computation of elastic constants. This is the first report on the piezoelectric and dielectric constants of pentaerythritol. In the notations followed by LANDOLT-BORNSTEIN the computed piezoelectric constants are: _g14 = 4.82, _g15 = 4.37, _g31 = 16.21 and _g36 = 5.89 (in unit of 10⁻¹Vm/N) and the static dielectric constants are: K₁ = 2.38 and K₃ = 2.47. The hydrogen bonding in pentaerythritol has no effect on its dielectric behaviour. Calculation of acoustic velocities in piezoelectric crystals and as a special case to the crystals of class 4 is explained in detail. Sound velocities calculated from the elastic, piezoelectric and dielectric constants computed in the present investigation agree with the experimental values determined by the ultrasonic method.     

Third-order elastic constants determination in soda–lime–silica glass by Brillouin scattering

Brillouin light scattering allows the measurement of sound velocity and elastic moduli in transparent materials. The ability to select a small scattering volume and to use specific scattering configurations gives important information about the gradient and anisotropy of mechanical properties. Brillouin experiments are often used to measure the second-order elastic constants. When a high accuracy in frequency measurements is achieved, Brillouin scattering may allow the determination of third-order elastic constants in pre-stressed media. Hence, Brillouin scattering provides in principle, a method for the analysis of stress fields in tempered glasses. In order to validate the technique, samples of float soda-lime–silica glass submitted to controlled stresses by four-point flexion were investigated. The results show the expected profile for the velocity of sound waves propagating in directions parallel and perpendicular to the surface of the samples, respectively. They allow the determination of several third-order elastic constants of the investigated glass. This technique was applied to several samples of tempered glass corresponding to different values of the surface stress. The main result is the observation of the expected general trend, namely, through the thickness of the sample, a parabolic variation of the sound velocity whose amplitude increases with the magnitude of the surface stress.     

Piezoelektrische und elastische Eigenschaften von Triglycinselenatkristallen im Bereich des Phasenüberganges

The piezoelectric constants d₂₁, d₂₃, d₂₅ and the elastic constants S₁₁, S₁₅, S₃₃ and S₃₅ of triglycine selenate are measured by the resonance method in the region of phase transformation. The elastic and piezoelectric constants are shown to have sharp maxima in the Curie point. The influence of a constant electric field on the temperature dependence of the piezoelectric constant d₂₃ and of the elastic constant S₃₃ is studied as well as the dependence of the mentioned constants on the intensity of a constant electric field at temperatures below and above the Curie temperature. A considerable increase of the elastic constant S₃₃ is found in weak electric fields of the ferroelectric phase. The electrostrictional coefficients Q₂₁, Q₂₃ and Q₂₅ are determined by means of the piezoelectric constants, spontane polarisation and dielectric receptivity. The found experimental dependences are explained on the base of the character of the domain structure of this crystal.     

Elastic and Thermoelastic Properties of the Twinned Perovskite LaGaO3 and the Cubic Untwinned Perovskite La0.294Sr0.706Al0.647Ta0.353O3

The elastic and thermoelastic constants of twinned crystals of LaGaO₃ and untwinned crystals of La_0.294Sr_0.706Al_0.647Ta_0.353O₃ have been measured by ultrasonic resonance methods. Both materials exhibit a quite similar elastic behaviour characteristic, as for perovskites. Employing a simple formula for ionic crystals, the elastic constants of both oxygen perovskites reveal a depression of about one third in comparison to the fluor perovskite LiBaF₃.     

Ab initio calculation of the structural and elastic properties,anisotropy, and hardness of nitrogen-doped diamond

The elastic properties of nitrogen-doped diamond are studied with the use of the density functional theory in the plane-wave basis, the electron density gradient approximation, and ultrasoft pseudopotentials. It is established that doping increases the lattice parameter and decreases the elastic constants, elasticity moduli, elastic wave velocities, and the anisotropy of elastic properties and hardness. It is shown that the hardness of the (111) face exceeds that of the (100) face in both pure and nitrogen-doped diamond.     

The Effective Elastic Constants of Textured Polycrystals in Second Order Approximation

The effective elastic constants of textured polycrystalline materials are calculated in two approximations on the basis of a minimum energy assumption which is more realistic than the VOIGT and REUSS assumptions. The formulae are specified for the case of cubic crystal symmetry and orthorhombic specimen symmetry (sheet symmetry). Numerical values calculated for a copper and a steel sheet are compared with the corresponding experimental values.     

Diffuse X-ray scattering study of the temperature variation of elastic constants in indium arsenide

The elastic constants of indium arsenide have been determined in the temperature range from 80 to 750 K on n-type samples with a carrier concentration of about 8 · 10¹⁷ cm⁻³ by Wooster's method from the measurements of the thermal X-ray diffuse scattering intensities. An average decrease in the elastic constants with increasing temperature over the above mentioned temperature range is found to be about 8%.     

Statistical theory of dynamical diffraction of Mössbauer radiation

The statistical theory of dynamical diffraction of Mössbauer radiation providing the account for the mutual influence of coherent and elastic diffusion scattering has been considered. The sources of elastic and inelastic diffuse scattering for a hematite crystal are analyzed. The theoretical results are compared with the resonance nuclear scattering data obtained in the experiments performed at the Photon Factory in Tsukuba, Japan.     

The Role of Diffusion-induced Misfit at Epitaxial Heterointerfaces

The diffusion-induced misfit at heterointerfaces and related defect-structure phenomena are derived analytically. On the basis of a simple diffusion model for epitaxial combinations the resulting distributions of misfit, stress, and misfit dislocation density along the diffusion direction can be quantified. Their diffusion-induced parameters can be expressed by relatively simple relations of material parameters (e.g. diffusion coefficients, Vegard's constants, elastic constants), which allow the quick estimation of their influence at epitaxial heterointerfaces.     

The elastic behaviour of TeO2 glass under uniaxial and hydrostatic pressure

TeO₂ glass of purity exceeding 98.5 mol.% has been made, despite earlier suggestions that some 7.5–10 mol.% of modifier is required to form vitreous telluride networks. It is argued that in view of the high purity of the glass obtained, TeO₂ may well be able to form glass by itself, given an appropriate thermal history in preparation. The hydrostatic and uniaxial pressure dependences of ultrasonic waves propagated in this glass at room temperature have been measured. The results provide the second and third order elastic constants of the glass. The bulk modulus is consistent with a ring diameter averaging about 8 atoms (Te₄O₄ rings) suggesting that the glass is a disordered version of paratellurite; however, if the TeO bending force constant were to be unusually strong, then a larger ring diameter (as in tellurite) would be indicated. Although the anion coordination number is only 4, the pressure derivatives of the second order elastic constants are positive and the third order elastic constants are negative, in marked contrast to the anomalous behaviour of silica-based glasses. These findings suggest that bond bending motions of bridging atoms between the trigonal bypyramidal groups (which are the structural units) do not play an important role in the elastic properties of TeO₂ glass. In consequence the shear and oongitudinal acoustic mode Grüneisen parameters are both positive (γ₁ = +2.14, γ_s = +1.11): the long wavelength acoustic modes stiffen under hydrostatic pressure.     

Thermoelastische und elastische Eigenschaften des tetragonalen Quecksilberdijodids

All elastic constants c_ij and thermoelastic constants T_ij = d log c_ij/dT (T temperature) of tetragonal HgI₂ have been measured by ultrasonic methods on single crystals which had been grown from the vapour phase by the method of dynamical gradient reversal. HgI₂ exhibits strong anisotropic elastic properties which are qualitatively explained by structural details. The temperature dependence of the elastic constants is smaller than expected. Only the very small shear constant c₆₆ is strongly decreasing with increasing temperature. This behaviour might be seen in connection with the known phase transition at about 126°C.