Thomas J. Ahrens

Equation of State

The use of plane shock waves to determine the equations of state of condensed materials to very high pressure began in 1955 with the classic papers of Walsh and Christian (1955) and Bancroft et al. (1956). Walsh and Christian described the use of in-contact explosives to determine dynamic pressure– volume relations for metals and compare these to the then available static compression data. Bancroft et al. described the first polymorphic phase change discovered in a solid, via shock waves—iron. Two years later Soviet workers (Al’tshuler et al., 1958) reported the first data for iron to pressures of several million bars (megabars) actually exceeding the pressure conditions within the center of the Earth. Since that time the equations of state of virtually hundreds of condensed materials have been studied, including elements, compounds, alloys, rocks and minerals, polymers, fluids, and porous media. These studies have employed both conventional and nuclear explosive sources, as well as impactors launched with a range of guns to speeds of approximately 10 km/s. Recently, Avrorin et al. (1986) have reported shock-compression data in lead to a record pressure of 550 Mbar.

The Melting Curve of Iron to 250 Gigapascals: A Constraint on the Temperature at Earth's Center.

The melting curve of iron, the primary constituent of Earths core, has been measured to pressures of 250 gigapascals with a combination of static and dynamic techniques. The melting temperature of iron at the pressure of the core-mantle boundary (136 gigapascals) is 4800 � 200 K. whereas at the inner core-outer core boundary (330 gigapascals), it is 7600 � 500 K. Corrected for melting point depression resulting from the presence of impurities, a melting temperature for iron-rich alloy of 6600 K at the inner core-outer core boundary and a maximum temperature of 6900 K at Earths center are inferred. This latter value is the first experimental upper bound on the temperature at Earths center, and these results imply that the temperature of the lower mantle is significantly less than that of the outer core.

Densities of Liquid Silicates at High Pressures

Densities of molten silicates at high pressures (up to ∼230 kilobars) have been measured for the first time with shock-wave techniques. For a model basaltic composition (36 mole percent anorthite and 64 mole percent diopside), a bulk modulus Ks, of ∼230 kilobars and a pressure derivative (dKs/dP) of ∼4 were derived. Some implications of these results are as follows: (i) basic to ultrabasic melts become denser than olivine-and pyroxene-rich host mantle at pressures of 60 to 100 kilobars; (ii) there is a maximum depth from which basaltic melt can rise within terrestrial planetary interiors; (iii) the slopes of silicate solidi [(dTm/dP), where Tm is the temperature] may become less steep at high pressures; and (iv) enriched mantle reservoirs may have developed by downward segregation of melt early in Earth history.

An equation of state for liquid iron and implications for the Earth's core

An equation of state is presented for liquid iron based on published ultrasonic, thermal expansion, and enthalpy data at 1 bar and on pulse-heating and shock wave compression and sound speed data up to 10 Mbar. The equation of state parameters, centered at 1 bar and 1811 K (the normal melting point of iron), are density, ρ_0 = 7019 kg/m^3, isentropic bulk modulus, K_(S0) = 109.7 GPa, and the first- and second-pressure derivatives of K_S, K′_(S0) = 4.66 and K″_(S0) = −0.043 GPa^(−1). A parameterization of the Gruneisen parameter γ as a function of density ρ and specific internal energy E is γ = γ_0 + γ′(ρ/ρ_0)^n(E - E_0) where γ_0 = 1.735, γ′ = −0.130 kg/MJ, n = −1.87, and E_0 is the internal energy of the liquid at 1 bar and 1811 K. The model gives the temperature dependence of γ at constant volume as (∂γ/∂T)_(v|1bar,1811K) = −8.4 × 10^(−5) K^(−1). The constant volume specific heat of liquid Fe at core conditions is 4.0–4.5 R. The model gives excellent agreement with measured temperatures of Fe under shock compression. Comparison with a preliminary reference Earth model indicates that the light component of the core does not significantly affect the magnitude of the isentropic bulk modulus of liquid Fe but does decrease its pressure derivative by ∼10%. Pure liquid Fe is 3–6% more dense than the inner core, supporting the presence of several percent of light elements in the inner core.

Shock melting and vaporization of lunar rocks and minerals.

The entropy associated with the thermodynamic states produced by hypervelocity meteoroid impacts at various velocities are calculated for a series of lunar rocks and minerals and compared with the entropy values required for melting and vaporization.Taking into account shock-induced phase changes in the silicates, we calculate that iron meteorites impacting at speeds varying from 4 to 6 km/s will produce shock melting in quartz, plagioclase, olivine, and pyroxene. Although calculated with less certainty, impact speeds required for incipient vaporization vary from ~ 7 to 11 km/s for the range of minerals going from quartz to periclase for aluminum (silicate-like) projectiles. The impact velocities which are required to induce melting in a soil, are calculated to be in the range of 3 to 4 km/s, provided thermal equilibrium is achieved in the shock state.

The temperature of shock‐compressed water

Temperatures from 3300–5200 K were measured in liquid H2O shocked to 50–80 GPa (500–800 kbar). A six‐channel, time‐resolved optical pyrometer was used to perform the measurements. Good agreement with the data is obtained by calculating the temperature with a volume‐dependent Gruneisen parameter derived from double‐shock data and a heat capacity at constant volume of 8.7 R per mol of H2O.

Planetary cratering mechanics

The objective of this study was to obtain a quantitative understanding of the cratering process over a broad range of conditions. Our approach was to numerically compute the evolution of impact induced flow fields and calculate the time histories of the key measures of crater geometry (e.g. depth, diameter, lip height) for variations in planetary gravity (0 to 10^9 cm/s^2), material strength (0 to 2400 kbar), and impactor radius (0.05 to 5000 km). These results were used to establish the values of the open parameters in the scaling laws of Holsapple and Schmidt (1987). We describe the impact process in terms of four regimes: (1) penetration, (2) inertial, (3) terminal and (4) relaxation. During the penetration regime, the depth of impactor penetration grows linearly for dimensionless times τ = (Ut/a) 5.1, the crater grows at a slower rate until it is arrested by either strength or gravitational forces. In this regime, the increase of crater depth, d, and diameter, D, normalized by projectile radius is given by d/a = 1.3 (Ut/a)^(0.36) and D/a = 2.0(Ut/a)^(0.36). For strength-dominated craters, growth stops at the end of the inertial regime, which occurs at τ = 0.33 (Y_(eff)/ρU^2)^(−0.78), where Y_(eff) is the effective planetary crustal strength. The effective strength can be reduced from the ambient strength by fracturing and shear band melting (e.g. formation of pseudo-tachylites). In gravity-dominated craters, growth stops when the gravitational forces dominate over the inertial forces, which occurs at τ = 0.92 (ga/U^2)^(−0.61). In the strength and gravity regimes, the maximum depth of penetration is d_p/a = 0.84 (Y/ρ U^2)^(−0.28) and d_p/a = 1.2 (ga/U^2)^(−0.22), respectively. The transition from simple bowl-shaped craters to complex-shaped craters occurs when gravity starts to dominate over strength in the cratering process. The diameter for this transition to occur is given by D_t = 9.0 Y/ρg, and thus scales as g^(−1) for planetary surfaces when strength is not strain-rate dependent. This scaling result agrees with crater-shape data for the terrestrial planets [Chapman and McKinnon, 1986]. We have related some of the calculable, but nonobservable parameters which are of interest (e.g. maximum depth of penetration, depth of excavation, and maximum crater lip height) to the crater diameter. For example, the maximum depth of penetration relative to the maximum crater diameter is 0.6, for strength dominated craters, and 0.28 for gravity dominated craters. These values imply that impactors associated with the large basin impacts penetrated relatively deeply into the planets surface. This significantly contrasts to earlier hypotheses in which it had been erroneously inferred from structural data that the relative transient crater depth of penetration decreased with increasing diameter. Similarly, the ratio of the maximum depth of excavation relative to the final crater diameter is a constant ≃0.05, for gravity dominated craters, and ≃ 0.09 for strength dominated craters. This result implies that for impact velocities less than 25 km/s, where significant vaporization begins to take place, the excavated material comes from a maximum depth which is less than 0.1 times the crater diameter. In the gravity dominated regime, we find that the apparent final crater diameter is approximately twice the transient crater diameter and that the inner ring diameter is less than the transient crater diameter.

The equation of state of a molten komatiite: 2. Application to komatiite petrogenesis and the Hadean Mantle

New experimental data for the equation of state of a komatiitic liquid were used to model adiabatic melting in a peridotitic mantle. If komatiites are formed by >30% partial melting of a peridotitic mantle, then komatiites generated by adiabatic melting come from source regions that began their unmelted ascent in the lower transition zone (≈500–670 km) or the lower mantle (>670 km). The great depth of incipient melting implied by this model suggests that komatiitic liquids may form in a pressure regime where they are denser than their coexisting crystals, possibly the bulk mantle. Although komatiitic magmas are thought to separate from residual crystals in the mantle at a temperature ≈200°C greater than modem mid-ocean ridge basalts (MORBs), their ultimate sources are predicted to be diapirs that, if adiabatically decompressed from initially solid mantle, were more than 700°C hotter than the sources of MORBs and were derived from great depth. We also studied the evolution of an initially molten mantle, i.e., a magma ocean. Our model considers the thermal structure of the magma ocean, density constraints on crystal segregation, and approximate phase relationships for a nominally chondritic mantle. Crystallization will begin at the core-mantle boundary. Perovskite buoyancy at >70 GPa may lead to a compositionally stratified lower mantle with iron-enriched magnesiowustite content increasing with depth. Large convective velocities in the magma ocean would prohibit crystal-liquid fractionation by settling or flotation until quiescent boundary layers form. Such boundary layers could form when the crystal content of the magma reaches a critical value (near 44 vol %) and, in the late stages of crystallization, around unmelted blocks of foundered protocrust. Matrix compaction and diapirism could also lead to fractionation effects. The upper mantle could be depleted or enriched in perovskite components relative to the bulk mantle. Olivine neutral buoyancy may lead to the formation of a dunite septum in the upper mantle, partitioning the ocean into upper and lower reservoirs, but this septum must be permeable.

B1-b2 transition in calcium oxide from shock-wave and diamond-cell experiments.

Volume and structural data obtained by shock-wave and diamond-cell techniques demonstrate that calcium oxide transforms from the B1 (sodium chloride type) to the B2 (cesium chloride type) structure at 60 to 70 gigapascals (0.6 to 0.7 megabar) with a volume decrease of 11 percent. The agreement between the shockwave and diamond-cell results independently confirms the ruby-fluorescence pressure scale to about 65 gigapascals. The shock-wave data agree closely with ultrasonic measurements on the B1 phase and also agree satisfactorily with equations of state derived from ab initio calculations. The discovery of this B1-B2 transition is significant in that it allows considerable enrichment of calcium components in the earths lower mantle, which is consistent with inhomogeneous accretion theories.

Petrologic properties of the upper 670 km of the earth's mantle; geophysical implications☆

Abstract The relative distribution of Fe2+ and Mg2+ between mantle minerals is calculated for various mantle thermal models and is used to estimate compressional and shear velocity and density profiles to depths of 670 km using ultrasonic, equation of state, and thermodynamic data for olivines, pyroxenes, garnets, and spinels. Except for uncertainties, arising from the lack of a quantitative description of the low-velocity zone, and inadequate elasticity data for Ca-bearing minerals, especially the clinopyroxenes, the calculated seismic profiles for pyrolite and lherzolite upper mantles are virtually identical. Varying the total iron content ratio, Fet = Fe/(Fe + Mg) from 0.1 to 0.2, brackets the seismic observation Important constraints to the composition and thermal profile of the mantle underlying continental regions are obtainable by fitting theoretical models to the detailed Vp-profile of Helmberger and Wiggins. This profile, although possibly inaccurate in the vicinity of the low-velocity zone, is well determined below 350 km. Values of Fet = 0.15 − 0.5 with a pyroxene-garnet (pyrope-almandine, majorite, solid-solution) content of 25 ± 10% and a relatively cool geotherm (1630–1650°C, 570–670 km) are implied from the position of the α (olivine) to α + β to β (modified spinel) transition, at a depth centered at ∼420 km, the position of the β to β + γ to γ (spinel) transition, centered at ∼500 km as well as the absolute value of the P-wave velocity. Because of the low resolution of the VS- and ρ-profiles, presently offered by seismic techniques, the concomitant VS- and ρ-profiles are useful as predictive, rather than comparitive, tools. If, as has been suggested by Anderson and co-workers, substantial enrichment of iron occurs with increasing depth in the mantle, this study demonstrates that for at least the continental regions, such an increase, if it occurs, would take place at, or below the 670-km discontinuity in the mantle. Distribution coefficients, based on thermochemical data, predict that within the upper 100 km of the mantle, successively greater relative iron contents will be present in the order: orthopyroxene, clinopyroxene, olivine, and garnet. The distribution of Fe2+ between orthopyroxene and the coexisting mantle phases is calculated (using the Mossbauer data of Virgo and Hafner for the M1, M2 intersite energy in the formulation of Grover and Orville). The relative enrichment of iron in orthopyroxene (relative to clinopyroxene) in the upper 100 km decreases rapidly with depth. Below 100 km, orthopyroxene composition becomes insensitive to the assumed intersite energy. In the upper 100 km Fe2+ is markedly concentrated in the garnet phase. For a total mantle value of Fet of 0.15 the garnet will have a composition corresponding to Fe/(Fe + Mg) = 0.30. At a depth of about 100–150 km the Fe2+ content of the garnet decreases and approaches that of the bulk mantle. Below this depth range, the relative content of iron in the garnet again increases.