Measured Pressure Derivatives Research Articles

Single-crystal elasticity of iron-bearing majorite to 26 GPa: Implications for seismic velocity structure of the mantle transition zone

Single-crystal elastic moduli of synthetic iron-bearing majoritic garnet were measured using Brillouin spectroscopy up to a pressure of 26 GPa. We found that the iron-bearing majoritic garnet (Mg 0.79Fe 0.08Al 0.30Si 0.84O 3) remains elastically isotropic throughout the pressure range of this study. The shear and adiabatic bulk moduli and their pressure derivatives at zero pressure were determined to be µ 0 = 88.7(7) GPa, K S0 = 166(2) GPa, (∂ µ / ∂ P) 0 = 1.28(8), (∂ K / ∂ P) 0 = 4.2(2). These values are almost indistinguishable, within the experimental uncertainties, from those obtained in previous studies of pure MgSiO 3 majorite and other solid solutions of the majorite–pyrope join. Our direct acoustic velocity measurements of iron-bearing majorite indicate that the incorporation of iron into majoritic garnet does not drastically affect its elastic moduli and their pressure derivatives. The measured pressure derivatives of iron-bearing majorite are about 3.6 times lower than required to account for the high seismic velocity gradients observed in the mantle transition zone.

Elasticity of fused silica spheres under pressure using resonant ultrasound spectroscopy

Resonant ultrasound spectroscopy (RUS) is a powerful tool for determining the elastic properties of solids at high temperature. Before RUS can be used to measure the pressure derivatives of elasticity, however, effects of boundary conditions between the pressurizing gas and specimen must be understood. Data are presented that show effects of different pressurizing gases on the apparent or measured values of (∂G/∂P)T of fused silica spheres, where G is the shear modulus, P is pressure, and T is temperature. The value of (∂G/∂P)T is found to depend on the molecular mass, M, of the pressurizing gas via −3.425(0.032)−5.9(1.6)×10−3M, when M is in grams. Extrapolating to zero gas mass gives (∂G/∂P)T=−3.42(0.03), a value bracketed by results from previous plane-wave, ultrasonic measurements. An alternative analysis in which effects of the pressurizing gas are removed by theoretical concerns is also presented and suggests that (∂G/∂P)T is −3.32(0.03). Our results quantify the effect of pressurizing gas on measured pressure derivatives of G when using RUS and indicate that systematic problems in transducer-specimen bonding from previous acoustic plane-wave measurements on fused silica at elevated pressure are not yet measurable.

Elasticity of stishovite at high pressure

The elastic-wave velocities of stishovite, the rutile-structured polymorph of SiO 2, were measured to 3 GPa at room temperature in a piston cylinder apparatus using ultrasonic interferometry on polycrystalline samples. These polycrystalline samples (2–3 mm in length and diameter) were hot-pressed at 14 GPa and 1050°C in a 2000 ton uniaxial split-sphere apparatus (USSA-2000) using fused silica rods as starting material. They were characterized as low porosity (less than 1%), single phase, fine grained, free of cracks and preferred orientation, and acoustically isotropic by using density measurement, X-ray diffraction, scanning electron microscopy, and bench-top velocity measurements. On the basis of subsequent in situ X-ray diffraction study at high P and T on peak broadening on similar specimens, it is evident that the single crystal grains within these polycrystalline aggregates are well equilibrated and that these specimens are free of residual strain. P- and S-wave velocities measured at 1 atm are within 1.5% of the Hashin-Shtrikman bounds calculated from single-crystal elastic moduli. Measured pressure derivatives of the bulk and shear moduli, K′ 0 = 5.3 ± 0.1 and G′ 0 = 1.8 ± 0.1, are not unusual compared with values measured for other transition zone phases such as silicate spinel and majorite garnet. Isothermal compression curves calculated with the measured values of K 0 and K′ 0 agree well with experimental P-V data to 16 GPa. The experimental value of dG dP is in excellent agreement with predictions based on elasticity systematics. Theoretical models are not yet able to replicate the measured values of K′ 0 and G′ 0.

Pressure derivatives of elastic wave velocities from ultrasonic interferometric measurements on jacketed polycrystals

A buffer-rod technique is described for the measurement by ultrasonic pulse interferometry of elastic wave velocities in jacketed polycrystalline specimens as functions of pressure to 3 GPa. The technique is tested on a suite of synthetic polycrystalline specimens (alumina and forsteritic olivine) for which bounds on the pressure-dependent wave velocities may be calculated from single-crystal elasticity data. Measurements conducted at pressures below ∼1 GPa with an unbonded buffer-rod configuration are compromised by the behavior of the interface between the buffer rod and the specimen. It appears that complex reflection and transmission coefficients associated with the contact between microscopically rough surfaces give rise to significant travel-time perturbations that decrease markedly with increasing pressure. These complications can be minimized but not eliminated by insertion between the buffer rod and the specimen of a disk of gold foil that amounts to a ‘‘bond’’ with more clearly defined acoustic properties. The range of pressures within which the interface effect is negligible may be significantly increased by the use of this bonded configuration. Measured elastic wave velocities are influenced by the presence of both microcracks and pores. Very large pressure derivatives measured between 0 and 0.5 GPa on one specimen are attributed to the closure of 0.1% porosity associated with cracks with aspect ratios less than 10−3. Otherwise, pressure derivatives are normal with velocities approximately 1%, 3%, and 4%–6% below the Hashin–Shtrikman bounds throughout the 0- to 3-GPa interval for specimens of 0.3%, 2.1%, and 4.8% porosity, respectively. These modest reductions in velocity and their survival to 3 GPa indicate that essentially all of the porosity in these specimens is contributed by relatively equidimensional pores of aspect ratio ∼0.1—consistent with their origin as sintered aggregates. The measured pressure derivatives are compared with those of the Hashin–Shtrikman bounds calculated from the pressure-dependent single-crystal elastic moduli. With the sole exception of the microcracked specimen at pressure below 0.5 GPa, the measured derivatives of wave velocities reproduce within ±10% the slopes of the Hashin–Shtrikman bounds demonstrating the viability of the technique for measurement of pressure derivatives of elastic wave velocities on fine-grained polycrystals.

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

The pressure derivatives of the elastic moduli of Sr(NO 3) 2, Ba(NO 3) 2 and Pb(NO 3) 2 single crystal, have been measured by the ultrasonic pulse superposition method. The results for Sr(NO 3) 2 and Pb(NO 3) 2 are similar in magnitude and character, while the ones for Ba(NO 3) 2 differ strongly, ∂c 11 ∂P and ∂c 12 ∂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(NO 3) 2 to Pb(NO 3) 2.

Lattice dynamics, third order elastic constants and thermal expansion of rhenium

A systematic investigation of the lattice dynamics, second and third order elastic constants and thermal expansion has been carried out in rhenium using Keating's approach. The ten third-order elastic constants of rhenium are calculated using four anharmonic parameters. The present model reproduces the measured pressure derivatives of the second order elastic constants well. The low and high temperature limits γ̄ L and γ̄ H of the lattice thermal expansion are calculated. The agreement between the calculated γ̄ H and that obtained by Gschneidner from the thermal expansion and specific heat data of rhenium is satisfactory.

Cauchy discrepancy in the second order elastic constants, their pressure derivatives on the three body force shell model

Expressions for the Cauchy discrepancies in the second order elastic constants and their pressure derivatives have been derived on the basis of the simplified three body force shell model of Verma and co-workers for NaCl and CsCl structures. It is shown that the three body force shell model qualitatively explains the Cauchy discrepancy in the measured pressure derivatives of the second order elastic constants for the NaCl structures.

On the Calculation of Mode Grüneisen Parameters in Rubidium Halides

AbstractRecently both authors have calculated the mode Grüneisen parameters of RbI using different methods. A critical comparison is made and the consequences of different approximations for the third order coupling constants are discussed. It is shown that the pressure dependence of at least five shell model parameters has to be taken into account in order to reproduce the measured pressure derivatives of the elastic and dielectric constants. In the most stringent approximation, however, where only the third order terms of the rigid part of the shell model are taken into account in the central potential approximation, qualitative agreement with the measured mode Grüneisen dispersion curves is still maintained.

Mode Gruneisen parameters and thermal expansion of thallous bromide

Some anharmonic properties of TlBr are calculated using a variable-charge shell model. The Gruneisen parameters of the long-wavelength acoustic modes calculated from the measured pressure derivatives of the second-order elastic constants disagree with the theoretical values computed by Cowley and Okazaki (see abstr. A37556 of 1967). Expressions are derived for the third-order elastic constants on the variable-charge shell model and the anharmonic parameters are obtained from the measured pressure derivatives. The dispersion of the mode Gruneisen parameters has been calculated along the three symmetry directions. There is a need to remeasure the thermal expansion of TlBr at low temperatures.

Lattice dynamics, third order elastic constants and thermal expansion of gadolinium

A systematic investigation of the lattice dynamics, second and third order elastic constants and the temperature variation of the effective Grüneisen functions has been carried out in gadolinium using Keating's approach. The ten third order elastic constants are calculated using five anharmonic parameters. The present model reproduces the measured pressure derivatives of the second order elastic constants of gadolinium well. The low and high temperature limits gg L and gg H of the lattice thermal expansion are evaluated. The agreement between the calculated gg H and that obtained from the thermal expansion and specific heat data of gadolinium is good.

Dispersion relations and lattice thermal expansion of dysprosium

The lattice dynamics, second and third order elastic constants and the lattice thermal expansion of dysprosium have been calculated using Keating's approach. The ten third order elastic constants are calculated using four anharmonic parameters. The present model reproduces the measured pressure derivatives of the second order elastic constants of dysprosium well. The low and high temperature limits γ̄ I and γ̄ H of the lattice thermal expansion are evaluated and the agreement between the calculated γ̄ H and that obtained from the thermal expansion and specific heat data is satisfactory.

Phonon dispersion relations and thermal expansion of erbium

The lattice dynamics, second- and third-o rder elastic constants, and the temperature variation of the lattice thermal expansion of erbium are worked out on the basis of Keating's method. The ten third-order elastic constants are calculated using four third-order anharmonic parameters. The model used here reproduces the measured pressure derivatives of the second-order elastic constants of erbium well. The low- and high-temperature limits γ̄L and γ̄H of the lattice thermal expansion are evaluated and the calculated γ̄H is compared with that obtained from the thermal expansion and specific heat data. The dispersion relations in erbium are found to be similar to those in holmium.

Lattice dynamics, third order elastic constants and thermal expansion of Cadmium

The lattice dynamics, second and third order elastic constants, pressure derivatives of the second order elastic constants and the temperature variation of the lattice thermal expansion of Cadmium have been worked out utilising Keating's approach. The ten TOE constants are calculated using four anharmonic parameters. The model used here reproduces the measured pressure derivatives of the SOE constants of Cadmium extremely well. The low temperature limit of the volume Gruneisen function γ L is found to be nearer to the value of Andres. The high temperature limit γ v ( T) is in good agreement with the value calculated by Gschneidner Jr. from the Cv data for Cadmium. The lattice dynamics of Cadmium is found to be essentially similar to that of Zinc.

Anharmonic properties of the hexagonal metals, magnesium, zinc and beryllium—I. Lattice dynamics and third order elastic constants

The lattice dynamics and the anharmonic properties of the hexagonal metals magnesium, zinc and beryllium have been worked out utilising the approach of Keating. In Magnesium, the dispersion curves are fitted using ten second order parameters and the six second order elastic constants are evaluated. The ten third order elastic constants are calculated using three third order parameters. The experimental measurements on the pressure derivatives of the second order elastic constants in magnesium are in good agreement with the calculated values. The dispersion relations and the second order elastic constants in zinc are fitted using twelve second order parameters. The third order elastic constants of zinc evaluated with the use of three anharmonic parameters are in good agreement with experiment. The dispersion curves and the second order elastic constants of beryllium are obtained using twelve second order parameters. The three third order parameters are obtained from the measured pressure derivatives of three of the second order elastic constants and the third order elastic constants of this metal are computed. The anisotropic thermal expansion of these metals on the present model will be discussed in Section 2 of this paper.

The Pressure Dependence of the Elastic Constants of Cubic Crystals in the NaCl and Spinel Structures from a Lattice Model

S-Y The method of homogeneous static deformation is used to compute the pressure dependence of the elastic constants of spinel under the assumption of a Born-Mayer two-body potential. The method, originally used by Fuchs (1935), is first applied to a number of alkali halides and periclase which have the sodium chloride structure, and reasonable agreement with the measured pressure derivatives is obtained. When applied to the spinel structure, the theory predicts that dc,,/dP is positive at low pressure but becomes negative at pressures of a few hundred kilobars. The geophysical implications of a negative d~~~/dP for the spinel region of the mantle are discussed.

Pressure Derivatives of the Sound Velocities of Polycrystalline Alumina

The sound velocities and the pressure derivatives of the sound velocities were measured on a small specimen of polycrystalline alumina by the method of “pulse superposition”. The pressure derivatives found from experiments up to 4 kbars at 25°C are: image These values lead to the variation of Poisson's ratio with pressure of dσ/dP = 1.02 × 10−4/ kbar and to the variation of (isothermal) bulk modulus with pressure of dBT/dP= 3.99. Using the above values the compression is computed up to 400 kbars and compares well with shock wave measurements. Consequently, the measured pressure derivatives may hold up to several hundred kilobars.