Bulk Sound Speed Research Articles

Thermal conductivity of minerals at high pressure: The effect of phase transitions

Theory predicts and measurements confirm that anharmonic properties such as the lattice thermal resistivity, the Grüneisen parameter, and the coefficient of thermal expansion generally decrease as a function of increasing density upon compression of solids. The Debye‐Grüneisen continuum model further predicts a discontinuous decrease of anharmonic properties during a phase transition with a density increase, in contradiction with available experimental data on alkali halides. We present results calculated from a model based on interatomic potentials. This model correctly predicts an increase in anharmonic properties for simple compounds transforming from a six‐ to an eight‐coordinated structure. These results stress the importance of interatomic spacing as well as density in determining changes in thermodynamic properties due to polymorphism. The change in crystal structure across a phase transition also affects the thermal conductivity via Brillouin zone summations over the interacting phonon modes. We have used in the calculations either the bulk sound speed, or the Debye average velocity calculated from the longitudinal and transverse velocities (in both cases, theoretically derived from the interatomic potentials). We have found that the pressure dependence of thermal conductivity determined from the bulk sound speed agrees much better with experimental data for each phase than that determined from the average velocity. Apparently, shear modes contribute less to the change of thermal conductivity with compression than has been thought.

Shock-wave compression of lithium niobate from 2.4 to 44 GPa

Shock compression of lithium niobate above the Hugoniot elastic limit (about 2.5 GPa) reveals a succession of unusual features. Just above the Hugoniot elastic limit, the shock velocity is observed to be well below the bulk sound speed, indicative of a drastic reduction of shear strength. The shock velocity is observed to increase with particle velocity at an unusually large rate due to the reduction of strength in a very stiff material and an anomalously large pressure derivative of the bulk modulus. This later behavior may be due to the effects of localized shock heating resulting from heterogeneous shear deformation in ferroelectrics like lithium niobate and lithium tantalate in which increases in temperature are shown to have a strong effect on bulk modulus. A shock-induced polymorphic phase transition occurs at 13.9 GPa. Above the transition point the slope of the Hugoniot curve relating shock velocity and particle velocity is unusually low, indicative of a broad mixed phase region of undetermined extent. Limited work is reported on the isomorphous crystal, lithium tantalate, which exhibits features similar to lithium niobate with a Hugoniot elastic limit of 4 GPa and a phase transition in the vicinity of 19 GPa.

Shock compression data for liquids. I. Six hydrocarbon compounds

Dynamic high pressure equation-of-state data are reported for 1,3- and 1,4-cyclohexadiene, cyclohexene, cyclohexane, toluene, and n-hexane initially in the liquid state. Plane shock waves generated by high explosives produced pressures of 0.5 to 43.0 GPa and densities to twice normal density. Toluene and n-hexane exhibit transformations at 12.6 and 19.0 GPa, respectively, but cyclohexadiene, cyclohexene, and cyclohexane do not. Decomposition of the molecule is the probable cause for the transition. Results indicate that reducing the number of double bonds by adding hydrogen pairs to the benzene molecule causes the transition to disappear. The shock velocity−particle velocity (us −Up) data for toluene, toluene, cyclohexadiene, and n-hexane are best represented by one or more line segments while cyclohexene and cyclohexane are fit by a quadratic in particle velocity. An extrapolation of the Us−Up curves to zero particle velocity result in a Us value for each liquid that is approximateley 40% higher than the bulk sound speed. The specific volume at a particular pressure for these liquids is ordered such that the hydrocarbon with the larger hydrogen-to-carbon ratio has a larger specific volume. This behavior is similar to that observed for other hydrocarbons.

Equations of state of iron sulfide and constraints on the sulfur content of the Earth

New shock and release wave data for pyrrhotite (Fe0.9S) obtained over the pressure range of 3–158 GPa (0.03–1.58 Mbar) suggest that this mineral undergoes a major shock‐induced phase change(s) with an onset in the range 2.7–3.8 GPa. Free‐surface velocities of Fe0.9S released from states between 8.9 and 24.7 GPa indicate a maximum postshock density of ∼5.54 g/cm3. A pressure of ∼25 GPa appears to be required to drive the phase transition(s) to completion. A density for the high‐pressure phase (hpp) consistent with present static high‐pressure X ray data, ∼5.34 g/cm3, was used to calculate a zero‐pressure, adiabatic bulk modulus of 126–128 GPa for the hpp. Release adiabat measurements centered at 152 and 158 GPa are consistent with the assumption that the Hugoniot curve in the 25‐ to 158‐GPa range reflects the properties of a denser polymorph, possibly with eight‐fold coordination, which may be similar to the local bonding of sulfur in the liquid core of the earth. Similar, but less well constrained, reductions are presented for pyrite, FeS2, based on the three data points of Simakov et al. (1974) in the pressure range 88–320 GPa. These are inferred to represent the behavior of an unknown hpp (approximate zero‐pressure density of 5.3 g/cm3) and indicate that this phase forms at a pressure above ∼29 GPa. Reduction of these data yields a zero‐pressure bulk modulus for the hpp in the range 205–244 GPa. The raw Hugoniot data for Fe0.9S, FeS2, and Fe when constrained to the seismologically obtained density‐pressure profiles of the outer core of the earth indicate a systematic decrease of apparent sulfur content from 10 to 6.5% with depth. When the shock data are reduced to isotherms, a nearly constant sulfur content in the range 9–12% is inferred. Using these bounds on the sulfur content of the core, and depending on whether an olivine or pyroxene mantle stoichiometry is assumed, the earth can be modeled as being depleted in S by a factor ranging from 1.7 to 3.2 with respect to the abundances of Si and from 2.8 to 7.5 relative to the abundance of Fe, in CCl carbonaceous chondrites. It is concluded that although the shock wave data permit the major light element in the core to be sulfur, the earth can be modeled as being depleted in sulfur, along with other volatile elements. A systematic relation, C0(km/s) = 7.15 ‐ 0.47 , was also discovered upon comparison of the inferred densities and bulk sound speeds (C0) of the hpp's of the iron sulfides with other measurements for 12 sulfides and elemental sulfur. Here is the volume (in cubic centimeters) per mole of atoms.

Shock-compression and release behavior near melt states in aluminum

The response of porous aluminum (ρ0=1.6 g/cm3) when shock-compressed to states of partial melting and then released has been investigated. There are no detectable changes in any of the Hugoniot properties in the region where melting should occur. However, release wave speeds in the completely compacted state of the porous aluminum specimens were found to depend critically upon the magnitude of the initial shock loading. For impact stresses below about 7 GPa, the measured release-wave velocity is well described by the assumption that the initial release is elastic. For initial impact stresses above 7 GPa, the speed of the release wave is observed to be about 20% lower than the elastic wave velocity and approaches the bulk sound speed predicted by equilibrium thermodynamics. All calculations of wave speeds were made with a complete equation of state which was obtained by constructing semiempirical free-energy functions for both solid and liquid pure phases. These were developed by first assuming self-consistent forms for the second derivatives of the Helmholtz free energy and then adjusting coefficients in the second derivatives using known quasistatic thermodynamic properties for each phase until agreement with all available data was achieved. Calculation of the intersection of the Hugoniot with the melt boundary (7.5 GPa, 1340 K) using this equation of state when compared with the observed change in sound speed gives strong support to the assumption that melting occurred in these experiments. The release-wave technique shows promise for measuring phase boundaries in regions of pressure and temperature not accessible by ordinary quasistatic techniques.

Mach waves and reflected rarefactions in aluminum

A rarefaction wave in aluminum shocked to 27.7 GPa (277 kbar) is examined with radiographic techniques, and the bulk sound speed is determined. A Mach reflection in aluminum is also examined radiographically and the results are compared with the simple three-shock theory of Mach waves. Then a modified three-shock model is developed which is consistent with the experimental results. The modified model leaves some quantities undetermined but closely bounded.

Acoustic slow waveguide antenna

The acoustic analog of the dielectric rod antenna for electromagnetic radiation is investigated both theoretically and experimentally. In acoustics, a cylinder of silicone rubber (whose bulk sound speed is about two-thirds that of water) is analogous to the dielectric rod. An acoustic wave in the direction parallel to the axis of the waveguide is concentrated within the guide by progressive refraction. A hydrophone (whose free-field sensitivity is omnidirectional) is attached near one end of the waveguide and detects this concentrated radiation as a gain in signal for this direction relative to any other direction. Similarly, if the slow waveguide is attached to an acoustic source, it will enhance the source level in the direction of the waveguide. The antenna is mathematically modeled by coupling the surface Helmholtz integral equation appropriate to the region within the silicone waveguide to a similar equation for the water region exterior to the cylinder (which includes the point hydrophone) by requiring continuity of acoustic pressure and normal velocity on the common surface. The coupled integral equations are solved numerically. Predicted beam patterns and directivity gains are in excellent agreement with experiment. Broad-band directivity gains of over 9 dB are both predicted and observed for two test antennas.

Wave velocities in shock-compressed cubic and hexagonal single crystals above the elastic limit

When solids are subjected to high-pressure shock-wave loading, multiple stress waves propagate with velocities dependent upon the elastic and inelastic compressibilities of the solid. The present paper shows that the inelastic or plastic waves in cubic and hexagonal single crystals do not necessarily propagate with the bulk sound speed as they do in isotropic elastic-plastic solids. This result is a consequence of anisotropy in the plastic deformation which depends on the slip plane orientation in the crystal and has important consequences with regard to the determination of compressibilities from shock-wave data. In particular, for wave propagation in the <110> directions of cubic crystals the departure from the bulk velocity can be significant (5–25 per cent). For wave propagation normal to the c-axis in hexagonal crystals, the plastic wave velocity also differs from the bulk sound speed (10–25 per cent). Plastic wave velocities are tabulated for a number of cubic crystals on the basis of the various slip systems common to these materials. The calculated velocities are then compared with experimental data on shock-loaded single-crystal aluminum and sodium chloride.

Acoustic slow waveguide antenna

The acoustic analog of the dielectric rod antenna for electromagnetic radiation is investigated both theoretically and experimentally. In acoustics, a cylinder of silicone rubber (whose bulk sound speed is about two-thirds that of water) is analogous to the dielectric rod. An acoustic wave in the direction parallel to the axis of the waveguide is concentrated within the guide by progressive refraction. A hydrophone (whose free-field sensitivity is omnidirectional) is attached near one end of the waveguide and detects this concentrated radiation as a gain in signal for this direction relative to any other direction. Similarly, if the slow waveguide is attached to an acoustic source, it will enhance the source level in the direction of the waveguide. The antenna is mathematically modeled by coupling the surface Helmholtz integral equation appropriate to the region within the silicone waveguide to a similar equation for the water region exterior to the cylinder (which includes the point hydrophone) by requiring continuity of acoustic pressure and normal velocity on the common surface. The coupled integral equations are solved numerically. Predicted beam patterns and directivity gains are in excellent agreement with experiment.

General relationships among sound speeds: II. Theory and discussion

The dependence of bulk sound speed V φ upon mean atomic weight m and density ρ can be expressed in a single equation: V φ=Bρ λ( m 0 m [ 1 2 +λ(1−c)] ( km/sec) Here B is an empirically determined “universal” parameter equal to 1.42, m 0 = 20.2 , a reference mean atomic weight for which well-determined elastic properties exist, and λ = 1.25 is a semi empirical parameter equal to γ − 1 3 where γ is a Grüneisen parameter. The constant c = ( ∂ ln V M/ ∂ ln m) X , where V M is molar volume, is in general different for different crystal structure series and different cation substitutions. However, it is possible to use c Fe = 0.14 for Fe 2+Mg 2+ and GeSi substitutions and c Ca ≅ 1.3 for CaMg substitutional series. With these values it is pos to deduce from the above equation Birch's law, its modifications introduced by Simmons to account for Ca-bearing minerals, variations in the seismic equation of state observed by D.L. Anderson, and the apparent proportionality of bulk modulus K to V M −4.

The existence and implications of curvature in the relation between shock and particle velocities for metals

Evidence is presented for the existence of curvature in the relation between shock velocity ( U) and particle velocity ( u) for metals. Existing U, u data is fit to an expansion of the form U = C b + b 1 u + b 2 2 u 2 where C b is the bulk sound speed at zero pressure. The slopes b 1 thus calculated are shown to agree well with the slopes ( b I) predicted on the basis of low pressure ultrasonic data. In addition, possibility of estimating the U, u relation from low pressure data is demonstrated.

On the equation of state of stishovite

Because of the discovery and subsequent synthesis of stishovite, a high-density polymorph of silica, Wackerle's shock-wave data on quartz have been re-examined to see if the high-pressure phase he reports could indeed be stishovite. Although the temperatures used for synthesis are about 1000°C higher than the temperature calculated by Wackerle for the transformation, the fact that the pressures are nearly the same (about 140 kb) is the prime evidence that the two transformations are the same, since the small change in entropy for a solid-solid phase change and the large observed volume change indicate that the slope of the phase line, dP/dT, should be quite small. The large increase in density shown by the shock-wave data above the transformation is comparable to the increase in density of quartz to stishovite; this gives additional support for identification of the transformations. On the assumptions that Wackerle's high-pressure data, for both crystalline and vitreous quartz, correspond to stishovite, a Hugoniot equation of state has been found for stishovite. This material has a Gruneisen ratioV(∂P/∂E)υ = 0.9, a heat of formation relative to quartz of 1.5×1010 ergs/g, and a bulk sound speed of about 10 km/sec.