Bulk Sound Velocity Research Articles

Erratum: Universality relationships in condensed matter: Bulk modulus and sound velocity

New forms for the bulk modulus and sound velocity of solids under compression, based on the universal equation of state of Vinet, Ferrante, Smith, and Rose (1987) are presented. These expressions are compared with a number of bulk modulus formulas previously utilized in high-pressure studies. It is demonstrated that this form yields a superior fit to experimental data to very high compressions, for a very wide range of solids. These solids cover the entire range of values of the pressure derivative of the bulk modulus which has been observed in high-pressure measurements.

Universality relationships in condensed matter: Bulk modulus and sound velocity.

New forms for the bulk modulus and sound velocity of solids under compression, based on the universal equation of state of Vinet, Ferrante, Smith, and Rose (1987) are presented. These expressions are compared with a number of bulk modulus formulas previously utilized in high-pressure studies. It is demonstrated that this form yields a superior fit to experimental data to very high compressions, for a very wide range of solids. These solids cover the entire range of values of the pressure derivative of the bulk modulus which has been observed in high-pressure measurements.

A study of some physical properties of sediments of the backwaters and the adjoining continental shelf off Cochin, India

The study area forms part of the Arabian Sea and comprises a seaward portion of Vembanadu Lake, i.e. the backwaters, the barmouth and the adjoining continental shelf off Cochin, India. Sediment samples collected from this area were analysed for their physical properties such as bulk density, grain density, porosity, water content and sound velocity. Grain-size analysis was also carried out as well as estimation of carbonate content. The sediment distribution pattern indicated that the finer terrigenous sediments — silty clays and clayey silts — are restricted to the backwaters, barmouth and shelf regions (approximately up to the 20-m isobath), while medium to fine relict sands are confined to the rest of this area. The silty clays have sound velocities in the range 1465–1520 m s −1, lower than that of sea water, with porosities between 0.76 and 0.84. The bulk densities vary from 1.18 to 1.31 g cc −1. An agreement between values of sound velocities in these sediments with those obtained using Wood's emulsion equation suggests that these sediments lack rigidity and behave like suspensions in confirmation with the process of sedimentation in estuaries, bays and lagoons. The clayey silts occupying the barmouth and inner shelf have bulk densities in the range 1.32-1.45 g cc −1 and sound speeds around 1540 m s t-1. The phi-mean values are between 6 and 8. The deposition of these sediments is attributed to the formation of a salt wedge. On the other hand, medium to fine sands occupying the middle shelf have an average velocity value of 1790 m s −1 and an average porosity of 0.43. The carbonate content varies from 12 to 43% with an average of 32%. These sediments are believed to be relict, belonging to a preglacial period. They are compact due to aging, unlike the terrigenous muds, and have a higher carbonate content. The deviation of Wood's equation from the observed velocity values for these sands confirms that they behave like solids with a specific value for rigidity modulus. From this it is likely that the physical properties of sediments are influenced by the environment, i.e. physical processes responsible for the deposition of the sediments, their age, and the carbonate and mineral constituents forming the sediments. The cross-correlation matrix indicated that the sound velocity is better correlated with porosity, bulk density and phi-mean than with other physical properties. Regression relations among a few physical properties and grain-size parameters derived from the present study are also presented.

Hugoniot overtake sound-velocity measurements on CsI

Shock-wave overtake experiments have been used to confirm that CsI melts along the Hugoniot curve at a pressure of roughly 30 GPa. The experimental results give the pressure dependence of the bulk sound velocity in the fluid along the Hugoniot curve to 150 GPa, from which the pressure dependences of both the adiabatic bulk modulus and the Grueneisen parameter ..gamma.. can be calculated. These quantities show normal behavior, with no obvious indication of the band closure and metalization which are predicted for this region of the Hugoniot curve. The results of these experiments are combined with the shock-wave optical pyrometry data of Radousky et al. (Phys. Rev. B 31, 1457 (1985)) to obtain results for the variation along the Hugoniot curve of both the constant-volume heat capacity (which increases by 30% from the melting line to 90 GPa) and the isothermal bulk modulus. The large values of C/sub V/, which are roughly twice the classical value for a solid, are the only indications of unusual behavior for CsI. Thermodynamic properties along the Hugoniot curve are consistent only with the diamond-anvil isotherm which used solid xenon as a pressure-transmitting medium.

The eclogite to garnetite transition — Experimental and thermodynamic constraints

We have found that the forms of the experimentally determined phase diagrams for pyroxene‐garnet transformations require the volume changes for these reactions to be much smaller at 130 kbar (400 km) than at 1 atm. Using these volume data we calculated the width of the eclogite‐garnetite transition in natural quartz tholeiite and alkali olivine basalt compositions taking account of mixing in multicomponent pyroxene and garnet phases. We find that the eclogite‐garnetite transition does not exhibit any discontinuity in bulk sound velocity in the pressure range 110 to 230 kbar. In addition, this velocity, when plotted as a function of pressure, possesses a curvature which is in the opposite sense to that given by the Preliminary Reference Earth Model (PREM). This phase transition, therefore, does not appear to be a satisfactory model for the 400 km seismic discontinuity.

Subsurface imaging of glass fibres in a polycarbonate composite by acoustic microscopy

Using reflection acoustic microscopy at 800 MHz, we have imaged 10 μm diameter glass fibres embedded in an optically opaque Makrolon™ (polycarbonate) matrix. Maximum depth of acoustic waves contributing to image formation was 105 μm, with imaging resolution of several microns at shallower depths. The areal fibre distribution, lengths, and orientations were readily determined non-destructively from the acoustic images. Additionally, marked differences were observed between known normally bonded samples with high strengths, and poorly bonded samples from parts which exhibited fracturing and premature failures. In addition to acoustic microscopy, bulk sound velocity measurements were made and used to compute focus aberration for subsurface imaging. Results were correlated with destructive SEM techniques.

Shock compression of zirconia ZrO2 and zircon ZrSiO4 in the pressure range up to 150 GPa

The shock compression state of zirconia ZrO2 and zircon ZrSiO4 in the pressure range up to 150 GPa (1.5 Mbar) are studied on the basis of the measurements of shock velocities, particle-velocity histories, free surface motions, and electrical conductivities. Zircon transforms, and zirconia probably does, to high pressure phases up to 90 GPa. The shock velocity (U s ) — particle velocity (U p ) Hugoniots can be described as U s =4.38+1.37 U p km/s above 90 GPa for ZrO2, and U s =6.50+0.49 U p km/s (mixed phase region), and U s =1.54+2.30 U p km/s (high pressure phase region) for ZrSiO4. The corrected isothermal densities of the high pressure phase ZrSiO4 are roughly consistent with the isothermal ones of mixtures of ZrO2 and SiO2. Bulk sound velocities in the high-pressure phase region of these oxides are discussed in comparison with other dioxides. Electrical conductivities of these oxides increase from lower than 10−12 S/m to greater than 100 S/m in the shock-stress range up to 70 GPa, and remain as constant values up to higher than 100 GPa.

Phase transition in SrO

Static‐compression experiments to 59 GPa (590 kbar), employing X ray diffraction through a diamond cell, demonstrate that strontium oxide (SrO) transforms from its initial B1 (NaCl‐type) to the B2 (CsCl‐type) structure at 36 (±4) GPa on the ruby fluorescence scale with a volume change at the transition ΔV/V1 ∼ −13%. These results confirm earlier predictions based upon systematics for oxides but are in lesser agreement with ab initio (modified electron gas) calculations. The new data for the B1 phase are consistent with previous ultrasonic and static‐compression measurements, which indicate that K0′ > 4. Data spanning the 25‐GPa range in pressure constrain the equation of state of the B2 phase. The change in bulk sound velocity at the transition is shown to be essentially zero, which is a significant deviation from the behavior predicted by velocity‐density systematics.

Majorite: Vibrational and compressional properties of a high‐pressure phase

Majorite, the garnet‐structured high‐pressure phase of pyroxene, has been characterized by infrared (IR) spectroscopy, and X ray diffraction between 0 and 8 GPa (80 kbar). In this structure, three out of four (Mg, Fe) are in eightfold coordination and one out of four Si is in sixfold coordination. The IR spectrum indicates the existence of large tetrahedral distortions, probably associated with the small cation radii (significant anion‐anion interactions). For a sample corresponding approximately to (Mg0.79Fe0.21)SiO3 the lattice parameter and density are a = 1151.4 (±0.1) pm and ρ = 3.737 (±0.001))Mg/m3. Hydrostatic compression measurements yield a zero‐pressure bulk modulus and its pressure derivative of K0 = 221 (±15) GPa and K0′ ∼ 4.4 (±4.8). The composition‐dependent volume and bulk modulus in the system pyrope‐almandine‐majorite are well constrained by the present data; for majorite, these are substantially smaller and larger, respectively, than predicted by systematics for garnets. Elasticity systematics largely fail to account for the moduli of garnets, although Anderson's seismic equation of state provides a satisfactory correlation. In transforming to higher pressure phases beyond majorite (with substantially higher densities), pyroxene does not exhibit significant increases in bulk sound velocity. Transformation to majorite probably occurs within the upper mantle and further phase transitions in pyroxene may be difficult to observe seismologically.

Shock compression measurements of single‐crystal forsterite in the pressure range 15–93 GPa

Shock compression experiments have been performed on pure synthetic single‐crystal forsterite (Mg2SiO4) in the pressure range 15–93 GPa using a newly installed two‐stage light gas gun. The values of the Hugoniot elastic limit (HEL) vary with the shock propagation directions with respect to the crystallographic orientations, reflecting a notable elastic anisotropy in the olivine crystal. The largest value of HEL up to 12 GPa is observed in the shock direction parallel to [010]. In the hydrostatic regime, shock data in the shock velocity‐particle velocity (Us‐up) plane are segmented into two parts by a transient plateau starting at around Us = 8.4 km/s, indicating a phase transition. For the shock data of the low‐pressure olivine phase below about 50 GPa, a linear fit with the equation Us = C0 + sup has yielded parameters of C0 = 6.26 km/s and s = 1.12. The value of C0 is close to the bulk sound velocity of forsterite. Murnaghan‐Birch fit of shock data for the low‐pressure olivine phase incorporated with the static compression data by Olinger (1977) has resulted in K0 = 132 GPa and K0′ = 3.4, which are in reasonable agreement with the ultrasonic data. The close correspondence of the present shock data to the observation of shock residual effects in olivines is noteworthy. The appearance of melt zone or diaplectic glass observed above about 50 GPa seems to correspond to the onset of the phase transition inferred in the present measurements.

Analysis of LMTO band results for A15 compounds by pressure calculations

Self-consistent LMTO band results for 27 A15 compounds have been analysed in terms of pressure calculations. A canonical scheme which describes the band structure at compressed and expanded lattice dimensions is presented and used for some of the compounds to estimate bulk moduli and sound velocities. Partial site and l-decomposed pressures obtained from the ambient lattice constant band results are used to calculate semi-empirical sound velocities and Debye temperatures.

On the 650-km seismic discontinuity

The pressure-temperature conditions and the variations of both density and bulk sound velocity in the vicinity of the 650-km discontinuity have been compared with those calculated for the phase transitions in both the olivine and the pyroxene-garnet components of the mantle material. These studies suggest that the mantle below about 650 km is composed primarily of perovskite phase, as distinct from the olivine-rich upper mantle. Thus, the “650-km” discontinuity is not likely to be associated with any of the equilibrium phase boundaries observed in olivine, pyroxene, and garnet, and is proposed instead to be a chemical change. It is suggested that the following factors may be responsible for chemical separation: the pyroxene-garnet component transforms to much denser phases possessing the ilmenite and perovskite structures before the breakdown of the spinel phase into a mixture of perovskite plus rocksalt phases. The perovskite phase is also much denser than the rocksalt phase and the two phases may not form a gravitationally stable mixture. Thus, the denser phases may tend to sink to or stay at the deep part of the mantle, causing chemical separation. Possible separation processes are discussed and the supporting observations are presented.

Elasticity of aluminate, titanate, stannate and germanate compounds with the perovskite structure

Ultrasonic data for the velocities of a large number of perovskite-structure compounds have been determined as a a function of pressure to 6 kbar at room temperature for polycrystalline specimens hot-pressed at pressures up to 100 kbar in solid-media devices: ScAlO 3, GdAlO 3, SmAlO 3, EuAlO 3, YAlO 3, CdTiO 3, CdSnO 3, CaSnO 3 and CaGeO 3. The elasticity data for these orthorhombic and cubic perovskites define systematic patterns on bulk modulus ( K S)-volume ( V O) and bulk sound velocity ( υ φ —mean atomic weight ( M) diagrams which are insensitiv to the details of cation chemistry and crystallographic structure. These isostructural trends are used to estimate K S = 2.5 ± 0.3 Mbar and υ φ = 7.9 ± 0.4 km/s for the perovskite polymorph of MgSiO 3. On a Birch diagram of veloc vs. density, the perovskite data define linear trends which lead to erroneous estimates of velocity for MgSiO 3 unless specific account is taken of ionic radius effects in isomorphic substitutions.

Elasticity of ilmenites

Ultrasonic data for the velocities of the ilmenites MgTiO 3 and CoTiO 3 have been determined as a function of pressure to 7.5 kbar at room temperature for polycrystalline specimens hot-pressed in a piston-cylinder apparatus at pressures up to 30 kbar. Titanate and germanate ilmenites define divergent isostructural trends on a Birch diagram of bulk sound velocity ( υ φ ) vs. density (ρ). On a υ φ vs. mean atomic weight ( M ) diagram, however, all of the ilmenite consistent with a single υ φ M 1 2 = constant trend . Elasticity systematics for isostructural sequences are used to e the bulk modulus (2.09 Mbar) and bulk sound velocity (7.4 km/sec) of MgSiO 3-elmenite.

Elasticity of the ilmenite - perovskite phase transformation in CdTiO 3

Ultrasonic data for the velocities of the ilmenite and perovskite polymorphs of CdTiO 3 have been determined as a function of pressure to 7.5 kbar at room temperature for polycrystalline specimens hot-pressed at pressures up to 25 kbar. This transition is characterized by the following velocity (ν)-density (ϱ) relationships: (1) the changes in compressional (ν p) and bulk sound (ν ϕ) velocities are comparable in percentage magnitude to the density jump, while the shear (ν s) velocity jump is three times greater than that for ϱ; (2) (ν p/ν s) decreases across the transition from the low- to high-pressure phase; and (3) low slopes (linear or logarithmic) on ν-ϱ diagrams. The (ν p/ν s) behaviour for the ilmenite-perovskite transformation is unusual for the transitions studied in our laboratory. The observed relationships (1) and (2) are typical of the elasticity behaviour across phase transformations which involve increases in cation-anion co-ordination and in nearest-neighbour interatomic distances, such as those exhibited by CdTiO 3 in transforming from the ilmenite to the perovskite phase. Elasticity systematics for isostructural sequences are used to estimate the bulk moduli of the perovskite polymorphs of CaSiO 3 (2.7 Mbar) and MgSiO 3 (2.8 Mbar).

Shock compression of dolomite

Shock compression studies on a dolomite rock (ρ0 = 2.84 g/cm3) have been conducted in the stress range 150–450 kbar. The entire loading and unloading history as well as the Hugoniot properties were investigated, using continuous recording piezoresistant stress gages. The following results were obtained: (1) The Hugoniot shock velocity-particle velocity curve shows deviation from linearity in a region corresponding to stresses between 100 and 250 kbar. (2) Comparison of the experimental Hugoniot with a Murnaghan pressure-volume relation extrapolated from low-pressure ultrasonic data indicates anomalous compressibility. (3) Unloading experiments from peak stresses between 180 and 300 kbar show excessive hysteresis in the stress-volume plane. (4) Shock compression to a peak stress in excess of 400 kbar is followed by immediate stress relaxation. (5) Overtaking relief wave velocities measured at the Hugoniot state are found to be considerably higher than extrapolated bulk sound velocities. Our interpretation of the results is a rate-dependent, low- to high-density phase transformation occurring in the stress range 100–500 kbar.

Evaluation of relations among stress-wave parameters and cohesive energy of condensed materials

The shock-wave velocity U and the particle velocity u for many condensed materials are linearly related by the equation U =a + b u along one or more sections of the Hugoniot. Departures from linearity can usually be attributed to porosity, elastic-wave precursors, or phase changes. If there are no such effects to cause nonlinearity, a is approximately equal to the adiabatic, bulk, or hydrodynamic sound velocity ak. Two equations involving the cohesive energy Ec are compared for 56 metals and 13 simple compounds (12 alkali metal halides and MgO): Ec = −(1/2)(a/b)2 and |Ec|=aμ2, where aμ is the shear wave velocity. It is shown that the experimental data are such that the energy of sublimation Es ≈ (1/2)(a/b)2 for the metals and compounds considered, Es≈ aμ2 for the metals, but Es≈ 0.4 aμ2 for the compounds. It is concluded that the shock-wave parameter equation, |Ec| = Es = (1/2)(a/b)2 is preferred because it applies without coefficient adjustments to both metals and simple compounds, and it may be applied to liquids as well as solids if the energy of vaporization Ev is substituted for Es. This equation is also applied to four polymers with |Ec| equated to the initial activation energy of failure Ea, which is equal to the energy of thermal decomposition. Despite the unsatisfactory nature of some shock-wave date (i.e., a ≠ ak), it appears the Ea ≈ (1/2)(a/b)2 although the fit is not as good as for the other materials considered. Therefore, the equations U = a + b u and Ec = −(1/2)(a/b)2 help provide information about the relations between macroscopic and microscopic properties of condensed materials.

Elastic properties of polycrystalline SnO 2 and GeO 2: comparison with stishovite and rutile data

Polycrystalline aggregates of SnO 2-cassiterite and GeO 2-rutile have been hot-pressed in a piston-cylinder apparatus at P = 30 kbar and T ⩽ 1,000°C. These specimens are less than 2% porous and less than 20μ in grain size. Compressional ( V p) and shear ( V s) velocities are determined as a function of pressure to 7.5 kbar at 293 K by the ultrasonic-pulse superposition method. Extrapolation of the velocity—pressure data to P = 1 ba yields the following velocities (units: km/sec) for the compressed polycrystalline aggregate SnO 2: V p = 6.91, V s = 3.75; for GeO 2: V p = 8.56, V s = 4.83. Comparison with published data for single-crystal GeO 2 indicates that our specimen is elastically isotrop within 1%. The bulk sound velocities ( V φ ) for the rutile minerals SiO 2, GeO 2 and SnO 2 are systematically related to density, with TiO 2 being anomalous. Shankland's (1972) equation (∂ ln V φ ∂ ln M) x = − 1 2 (where M is mean atomic weight) represents these rutile data well.

Grüneisen constants of polymers

One result of recent interest in Grüneisen constants, γ = − d lnν/d lnV, of polymers is a considerable spread of reported γ's for solids like polyethylene. From elasticity data (bulk modulus or sound velocities for example) one finds γ ≈ 6 for linear polymer solids. Values of γ from thermal data are much lower than 6 because the relationship usually employed, γ = αB V/CV, is not valid for polyatomic solids. Measurements of the shifts in lattice frequencies with volume strains are the most direct way of measuring γ. However, the results for pressure-induced strains differ from those for temperature-induced strains, and both differ from the results from elasticity data. In this paper we consider vibrations in a simple anharmonic well and show how the apparent shifts in vibrational frequency with pressure and temperature can be derived from changes in force constants.

Chemistry of the Earth's lower mantle

A modified Birch equation of state has been used to relate the bulk sound velocity C and density ρ for geophysically interesting compounds. Also derived here is a sequence of equations of state relating the bulk sound velocity with the density of a mixture. Calculations have been based on available zero pressure data for the bulk modulus and its first pressure derivative obtained by ultrasonic and X ray techniques. The bulk sound velocity and density of mixtures of periclase, wustite, and stishovite are consistent with recently published models for the C-ρ relationship observed in the lower mantle. We based our calculations for the oxide mixtures on two compositions, pyroxene and olivine, since most proposed compositions for the lower mantle lie between these two extremes. If the higher silica content of the pyroxene composition is assumed for the oxide mixture, the results of this study indicate that the FeO content increases by at least 5% with depth from 800 to 2800 km. If, however, the lower silica content of the olivine composition is assumed for the oxide mixture, the results of this study indicate that the FeO content remains essentially constant with depth. This latter case is consistent with a chemically homogeneous mantle and with Birch's hypothesis of an isochemical transition zone.