Elastic constants,acoustic mode vibrational anharmonicity and Grüneisen parameters of hexahalometallate crystals

The room-temperature elastic constants of a number of hexahalometallate A₂MX₆ single crystals [K₂SnCl₆, K₂ReCl₆, (NH₄)₂SnCl₆, (NH₄)₂SnBr₆, (NH₄)₂SiF₆, Rb₂SnBr₆, K₂SeBr₆, (NH₄)₂TeBr₆, K₂PtBr₆ and (NH₄)2PtBr₆] have been measured either by Brillouin scattering or by the ultrasonic pulse echo overlap technique. Refractive indices have also been determined. These antifluorite structure compounds contain large MX^2?₆ ions and the interionic spacings are much greater than those of the alkaline-earth fluorite structure halides: their elastic stiffnesses are correspondingly smaller. Hydrostatic pressure derivatives of the elastic stiffness constants have been measured for K₂SnCl₆, (NH₄)₂SnBr₆ and (NH₄)₂SnCl₆ and are found to be positive; there is no marked softening of the long-wavelength acoustic-phonon modes at room temperature. The vibrational anharmonicities of these long-wavelength modes are discussed in terms of the acousticmode Grüneisen parameters, which are compared with the thermal Grüneisen parameters. For K₂SnCl₆ a mean of optic- and acoustic-mode Grüneisen parameters is shown to correlate well with the thermal Grüneisen parameter.     

Lattice-dynamical calculations of the thermodynamic properties of CsBr and CsI

Results of a quasi-harmonic calculation of the normal mode frequencies, mode-Grüneisen parameters, the coefficient of thermal expansion α, the heat capacity C_p, and the macroscopic Grüneisen function γ(itT) for CsBr and CsI are presented. Very good agreement with the experimental results for α and C_p is obtained. A self-consistent procedure is used to parametrize the models so that some often neglected vibrational and configuration-dependent effects can be accounted for. Predictions are made for the values that the potential energy derivatives and the elastic constants would have in the absence of vibrational motion.     

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 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.     

Four constant potential function of amorphous selenium and evaluation of its thermodynamic and acoustic properties

Litov and Anderson after various considerations suggested a four constant potential function for a-Se as well as a-As₂S₃. Hence we also used a four constant potential function with the sole purpose of applying this potential function to obtain several acoustic, thermodynamic and other properties. We calculated several acoustic properties of a-Se like second order elastic constants (SOECs), their pressure derivatives, the longitudinal and transverse Grüneisen constant by two different methods, phonon frequencies, absorption band position through the use of Nath-Smith-Delaunay’s equation, and the thermodynamic properties like heat capacity, bulk modulus, thermal Grüneisen constant, the pressure derivative of the bulk modulus (dK _T/dP=C ₁), the pressure derivative ofC ₁ which is related to Anderson-Grüneisen parameter, pressure derivative of Grüneisen constant namelyγ′ _g which is related to second Grüneisen constant, characteristics of phonon frequencies, potential energy function through the use of fitted parameters and third order elastic constants. Finally we calculatedK _T at the reduced density ofρ/ρ ₀=1.1.K _T is obtained from the potential function with the fitted parameters. In all the above cases the calculated values are found to be in good agreement with experiment wherever available. In this connection it is important to point out that we eliminated ‘C’ a constant in the potential function using the equilibrium condition as was done by Litovet al in a-Se and Gerlichet al in the case of a-As₂S₃ as all amorphous substances are isotropic as mentioned by several authors. We contemplate to calculate several other properties for a-Se and a-As₂S₃ and present them at a later stage.     

Generalized Grüneisen parameters and low temperature limit of lattice thermal expansion of cadmium and zirconium

The generalized Grüneisen parameters (γ′ _j ) and (γ″ _j ) for cadmium and zirconium were calculated from the second- and third-order elastic constants to determine the low temperature limit of the volume thermal expansion of these metals of hexagonal symmetry. The low temperature limit of cadmium and zirconium was calculated to be positive values indicating a positive volume expansion down to 0 K even though many Grüneisen gammas were found to be negative.     

Grüneisen parameter for HCP iron at extreme compression

The Thomas–Fermi approximation gives the Grüneisen parameter γ=γ_∞=1/2 for all materials at extreme compression (P→∞ or V→0). After re-analyzing the existing experimental data of volume dependence of Grüneisen parameter γ of hexagonal close-packed (HCP) iron, we find that γ=1/2+a(V/V ₀)^1/3+b(V/V ₀) ⁿ , where a, b and n are constants. Based on this new form of γ, the second Grüneisen parameter q, the Debye temperature θ_D and the shear sound velocity v _s of HCP iron are discussed in the present work. It is found that the zero pressure second Grüneisen parameter q ₀=0.654, which is consistent with the previously determined value of HCP iron for Earth's core physics from Dubrovinsky et al. The calculations for the Debye temperature and the shear sound velocity are also found to be in good agreement with the experimental data.     

Thermal expansion of cadmium fluoride

The thermal expansion of CdF₂ single crystal has been measured over the temperature range 80–300°K. From the latter data, the thermal Grüneisen constant as a function of temperature has been evaluated, and compared with the elastic Grüneisen constant. The correlation between the two sets of data, and with possible lattice interactions is discussed.     

Elastic constants and softening of acoustic modes in A2MX6-crystals observed by Brillouin scattering

Room-temperature Brillouin-scattering measurements of the longitudinal and transverse sound velocities for the cubic crystals K₂SnCl₆, K₂ReCl₆, (NH₄)₂SnCl₆ and (NH₄)₂SiF₆ are presented. The elastic constantsc ₁₁,c ₁₂ andc ₄₄ are determined from the angular variation of the Brillouin line shifts. Furthermore, for K₂SnCl₆ the sound velocities in the [100]- and [111]-direction are investigated as a function of temperature, 256K≦T≦330K. A strong softening of thec ₁₁–c ₁₂ mode is observed in the para-distortive phase as the temperature approaches the phase transition temperatureT _c1=262K from above, whilec ₄₄ shows no anomaly within the experimental error. These acoustic anomalies are interpreted theoretically by a mechanism based on the coupling of the elastic strain field to the fluctuations of the soft-mode coordinates.     

The Debye temperature and Grüneisen parameter of the CaLa<Subscript>2</Subscript>S<Subscript>4</Subscript>-La<Subscript>2</Subscript>S<Subscript>3</Subscript> system

The thermal expansion coefficient of solid solutions in the CaLa₂S₄-La₂S₃ system at a temperature of 300 K is investigated experimentally. The Debye temperature, the Grüneisen parameter, and the isothermal compressibility coefficient of solid solutions in the system under investigation are determined from the experimental thermal expansion coefficient. It is demonstrated that, upon substitution of calcium ions for cation vacancies in La₂S₃, the Debye temperature decreases, the isothermal compressibility coefficient increases, and the Grüneisen parameter remains constant for all compositions in the CaLa₂S₄-La₂S₃ system. A correlation between the ionic radii of Ca²⁺ and La³⁺, the concentration of cation vacancies, and the rigidity of the lattice, on the one hand, and the Debye temperature, the Grüneisen parameter, and the isothermal compressibility coefficient, on the other, is revealed for the studied samples.     

Elastic properties of Se and As2Se3 glasses under pressure and temperature

The sound velocities and their pressure and temperature variations of Se and As₂Se₃ glasses have been determined by means of a pulse superpositions method, and other elastic constants and their pressure and temperature derivatives were calculated from these data. The bulk modulus was found to be 94·6 kbar for Se glass and 143·7 kbar As₂Se₃ glass, both of which are higher than the values calculated from the previous compression data. No anomaly was observed in any of the pressure and temperature dependence of elastic behavior of these glassses. Furthermore, the comparison of the pressure and temperature derivatives of the bulk modulus indicates that the thermodynamic self-consistency is satisfied on these materials. The bulk moduli of these glasses and crystalline As and Se were used to obtain an empirical bulk modulus-volume relationship for compounds in the As?Se system. The acoustic Grüneisen parameter was calculated and compared with the thermal Grüneisen parameter.     

Thermal expansivity in semiconducting NbO2

Measurements have been made of sample lengths L_a and L_c parallel to the a- and c-crystallographic directions, respectively, in semiconducting, distorted-rutile (C⁶_4h) structure NbO₂ between 133 K and 360 K using a fused silica LVDT dilatometer system. L_a changes very slightly and goes through a minimum as a function of temperature, T, while L_c increases monotonically with T in a normal manner. This behavior is compared with that of rutile-structure compounds. The present data do not cause significant changes in the elastic stiffness moduli which were deduced previously from ultrasonic transit times using lengths extra-polated from lattice parameter data above room T. Thus the discrepancy between the elastic and calorimetric Debye temperatures reported in the literature remains. Thermal expansion coefficients and Grüneisen anharmonicity parameters are deduced. The expansion coefficients are analyzed into a quasicubic component and a term depending on the anisotropy of the Grüneisen function.     

Ab initio investigations of the elastic properties of chlorates and perchlorates

Elastic properties of NaClO₃, KClO₃, LiClO₄, NaClO₄, and KClO₄ have been investigated from first principles by the method of linear combination of atomic orbitals in the gradient approximation of the density functional theory using CRYSTAL software. The elastic constants and moduli, hardness, Poisson’s ratio, and the anisotropy parameters have been calculated. The velocities of sound, the Debye temperature, the thermal conductivity, and the Grüneisen parameter have been estimated. It has been found that these compounds are mechanically stable, anisotropic, and ductile materials. The dependences of their elastic parameters on the atomic number of the cation have been calculated. The obtained results are in good agreement with the available experimental data.     

Grüneisen parameter and thermal expansion of V3Si and V3Ge

The Grüneisen parameter and lattice thermal expansion of the A-15 compounds V₃Si and V₃Ge at room temperature are evaluated on the basis of the method due to Brugger and Fritz [1] from the third order elastic constants reported earlier [2]. The calculated values are compared with available experimental values and are found to fit satisfactorily.     

Thermal expansion of the C15 Laves-phase materials TaV2 and TaV2H x

The thermal expansion of the C15 Laves-phase material TaV₂H _x was measured over the temperature range of 130–350?K for x?=?0, 0.34 and 0.53. A primary objective of the study was to see if the anomalous temperature dependence of elastic constants found in an earlier study is due to, or associated with, unusual thermal expansion. The expansion was typical of transition metals, and quite close to theoretical results for related Laves-phase materials. Grüneisen parameters were determined from the measurements, and these were also typical of transition metals, and close to calculated values for related materials. Both the thermal expansion and the Grüneisen parameters of TaV₂ decreased somewhat with the addition of hydrogen. No anomalous behavior of the expansion was found that might account for the highly anomalous temperature dependence of elastic moduli reported earlier. This absence is consistent with the attribution of the elastic constant behavior to an electronic structure effect.     

Lattice dynamical calculation of the second Grüneisen constant of CsBr

The second order Grüneisen constant constitutes an additional measure of the anharmonicity in a solid (the Grüneisen constant itself being the first measure). This quantity and the corresponding second mode-Grüneisen parameters are calculated for CsBr using a modified rigid ion model of lattice dynamics. The model parameters are deduced from the three elastic constants and the long wavelength optical mode frequencies. Using the first and second order pressure dependences of these physical observables, the individual second mode-Grüneisen parameters are obtained as a function of wave vector. The temperature variation of the second Grüneisen constant is then deduced in a quasi-harmonic approximation. The second Grüneisen constant is also calculated from available thermodynamic data and is in reasonable agreement with the present theoretical prediction.     

Pressure derivatives of the elastic moduli of strontium,barium and lead nitrate

The pressure derivatives of the elastic moduli of Sr(NO₃)₂, Ba(NO₃)₂ and Pb(NO₃)₂ single crystal, have been measured by the ultrasonic pulse superposition method. The results for Sr(NO₃)₂ and Pb(NO₃)₂ are similar in magnitude and character, while the ones for Ba(NO₃)₂ differ strongly, $\text{?c}{}_{11}\text{?P}$ and $\text{?c}{}_{12}\text{?P}$ even From the measured pressure derivatives, the mode Grüneisen gammas and the Grüneisen constant as a function of temperature have been determined, and the latter is correlated with the experimental values, deduced from thermal expansion. The explicit temperature dependence of the elastic moduli is calculated, and found to be always negative, increasing in absolute value from Sr(NO₃)₂ to Pb(NO₃)₂.     

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).     

First-principles lattice dynamics,thermodynamics, and elasticity of Cr2O3

We present the calculation of the lattice dynamics of chromia (Cr₂O₃), a typical Mott–Hubbard insulator, employing the first-principles density functional theory plus U approach. We first report the phonon dispersions at the theoretical equilibrium volume. Then the phonon density-of-states is calculated as a function of volume. Finally, the atomic volume, heat capacity, linear thermal expansion coefficient, bulk modulus, Grüneisen constant, and elastic constants are calculated as functions of temperature.     

Mode-Grüneisen parameters in disordered molecular crystals: Raman spectra of NH4Br and NH4I under hydrostatic pressure

The Raman spectrum of NH₄I has been measured under hydrostatic pressure. The high pressure phase of NH₄I was identified with the disordered CsCl-type phase by comparing the spectrum with that of NH₄ Br. The mode-Grünesen parameters of NH₄Br and NH₄I were determined from the frequency shift of phonon bands under hydrostatic pressure. These values were compared with the thermodynamic value and that obtained from the Slater relation. Contributions to the thermodynamic Grüneisen constant are discussed from free energy terms of NH⁺₄-orientation and -interactions with lattice modes.