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Sound velocity-density systematics has long been a fruitful way to take shock wave measurements on elements, alloys, oxides, rocks, and other materials, and allow reasonable extrapolation to densities found deep in the Earth. Recent detection of super-Earths has expanded interest in terrestrial planetary interiors to an even greater range of materials and pressures. Recent published diamond anvil cell (DAC) experimental measurements of sound velocities in iron and iron alloys, relevant to planetary cores, are inconsistent with each other with regard to the validity of Birch’s Law, a linear relation between sound velocity and density. We examine the range of validity of Birch’s Law for several shocked metallic elements, including iron, and shocked ionic solids and make comparisons to the recent DAC data.

We performed compressional sound velocity and density measurements on liquid iron at pressures up to 800 GPa with a newly refined shock-compression method using a high-power laser. We found that sound velocity as a function of density can be fitted to a linear relation following Birch's law for hot dense liquid as well as for the solid phase of iron, with a slope ratio between the solid and liquid of approximately 1.5. A comparison of Birch's law for solid and liquid metals indicates that the sound velocity in the liquid phase is about 10% lower than that in the solid phase at melting point density, which is about 1.5 times larger than the initial density. We suggest that these relations between Birch's law coefficients for solid and liquid phases along the Hugoniot are universal for metals.

We measured the sound velocity of Fe3S at room temperature up to 85 GPa employing inelastic X-ray scattering to better constrain the constitution of the inner core. The density of Fe3S was also determined by X-ray diffraction under the same conditions. The relation of the P-wave velocity ( v P) and density (ρ) of Fe3S follows Birch’s law, v P(m/s) = 1.14(5) × ρ(kg/m3) − 2580(410). Based on Birch’s law determined here for Fe3S and that for ɛ-Fe reported previously, we found that sulfur decreases both density and compressional velocity of hcp-Fe at the core pressure and 300 K.

This chapter looks at the ways in which mineral physics can help in our understanding of the structure, composition, and evolution of Earth's core. The only direct observations of Earth's core are from seismology, so it is the job of mineral physics to find compositional models that exactly match the seismological data. However, the conditions of pressure (~ 130–360 GPa) and temperature (~ 3500 to > 6000 K) necessarily make this a daunting task. Nevertheless, advances in experimental techniques and theoretical methods have enabled mineral physicists to make a significant contribution to our understanding of Earth's core through measurements and simulations of the phase diagrams, and the thermoelastic and rheological behavior of iron and iron alloys at core conditions. Results from recent studies on iron and iron alloys are presented and it is shown how these lead to some very interesting interpretations of the seismological evidence available for Earth's core.

The pressure dependence of the sound velocities, single-crystal elastic constants, and shear and adiabatic bulk moduli of a natural gahnite (ZnAl2O4) spinel have been determined to ~9 GPa by gigahertz ultrasonic interferometry in a diamond anvil cell. The elastic constants of gahnite are (in GPa): C11 = 290(3), C12 = 169(4), and C44 = 146(2). The elastic constants C11 and C12 have similar pressure derivatives of 4.48(10) and 5.0(8), while the pressure derivative of C44 is 1.47(3). In contrast to magnetite, gahnite does not exhibit C44 mode softening over the experimental pressure range. The adiabatic bulk modulus KS0 is 209(5) GPa, with pressure derivative KS' = 4.8(3), and the shear modulus G0 = 104(3) GPa, with G' = 0.5(2). Gahnite, along with chromite (FeCr2O4) and hercynite (FeAl2O4) are the least compressible of the naturally occurring oxide spinels. Evaluation of Birch’s Law for isostructural minerals indicates that spinels containing transition metals on both the [4] A and [6] B sites

Birch's law, prescribing a linear dependence of elastic wave speed on the density in a given material, is an important tool in understanding the composition and thermodynamic conditions of the planetary interior. However, data from direct measurements of elastic wave speed usually have too much of scatter to permit a precise test of this law. Here, we use data from experimental measurements as well as ab initio density-functional-theory based calculations existing in the literature, supplemented by our own data of the latter type for elemental solids, for such a test. Using many such datasets, we show that, although Birch's law is satisfied fairly well in all the cases, the product of elastic wave speed and one-third power of density satisfies linear dependence on density consistently and more accurately than the speed alone. This opens the possibility of more reliable extrapolation of low density velocity data to the higher densities—the primary application of Birch's law.

International audience; We performed sound velocity and density measurements on polycrystalline hexagonal close-packed (hcp) iron at simultaneous high pressure and high temperature, up to 93 GPa and 1100 K, by inelastic x-ray scattering and x-ray diffraction. Our experimental results indicate that high-temperature anharmonic corrections are negligible at least up to 1100 K and that the aggregate compressional velocity VP scales linearly with density over the pressure and temperature range of the investigation (Birch's law). The new results are compared with literature studies and we discuss the extrapolation schemes commonly used in experimental mineral physics, with specific regard to extrapolations to the Earth's core conditions

This chapter looks at the ways in which mineral physics can help in our understanding of the structure, composition, and evolution of the Earth’s core. The only direct observations of the Earth’s core are from seismology, so it is the job of mineral physics to find compositional models which exactly match the seismological data. However, the conditions of pressure (∼130–360 GPa) and temperature (∼3500–5500 K) necessarily make this a daunting task. Nevertheless, advances in experimental techniques and theoretical methods have enabled mineral physicists to make a significant contribution to our understanding of the Earth’s core through measurements and simulations of the thermoelastic and rheological behavior of iron and iron alloys at core conditions. Results from recent studies on iron and iron alloys are presented and it is shown how these lead to some very interesting interpretations of the seismological evidence available for the Earth’s core.

Sound velocities of hexagonal close-packed iron (hcp-Fe) were measured at pressures up to 73 gigapascals and at temperatures up to 1700 kelvin with nuclear inelastic x-ray scattering in a laser-heated diamond anvil cell. The compressional-wave velocities ( V P ) and shear-wave velocities ( V S ) of hcp-Fe decreased significantly with increasing temperature under moderately high pressures. V P and V S under high pressures and temperatures thus cannot be fitted to a linear relation, Birch's law, which has been used to extrapolate measured sound velocities to densities of iron in Earth's interior. This result means that there are more light elements in Earth's core than have been inferred from linear extrapolation at room temperature.

The compressional wave velocities of polycrystalline NaCl and KCl have been measured to over 17 gigapascals, with the use of Brillouin scattering and the diamond anvil cell. This pressure corresponds to 40% compression for NaCl and 60% compression for KCl (including the volume change across the B1-B2 transition). The data obey Birch's Law, which predicts that the velocity of each material is linear with density, except across the B1-B2 phase transition in KCl. This deviation from Birch's Law can be rationalized in terms of an interatomic potential model wherein the vibrational frequencies of the nearest neighbor bonds decrease when going to the eight-coordinated B2 structure from the six-coordinated B1 structure.

The systematics of the sound velocity-density-mass patterns found in minerals and rocks have been almost universally agreed upon, although graphical representations of the data take several forms. A growing body of literature now correlates the data by means of power law expressions rather than by straight-line plots, as were first presented by Birch. A few discrepancies still remain between some mineral data and the best graphical representations of Birch's law. This paper explains the discrepancies evidenced by the high-pressure polymorphs of silica in terms of well-recognized thermodynamic formulas. It is pointed out that the simpler forms of the power law representations assume implicitly the Murnaghau equation of state. By allowing the exponent on density in the power law to be a function of density according to other equations of state, Birch's law comes much closer to predicting the sound velocities of the high-pressure phases of silica.

Birch's law implies a unique relationship between elastic wave velocity υ and density ρ for materials of common mean atomic weight M¯ irrespective of whether density changes are due to crystallographic structure, pressure, or temperature. Velocity data are presented for materials undergoing the olivine-spinel, olivine-beta phase, quartz-rutile, pyroxene-garnet, pyroxene-ilmenite, and ilmenite-perovskite phase transformations. These data indicate that υ-ρ relationships across such polymorphic transformations are not always equivalent to υ-ρ changes caused by varying pressure, temperature, or composition at constant mean atomic weight. For the earth these data imply that previous discussions of the composition (specifically the iron content) of the upper and lower mantles based on υ-ρ systematics may have to be reevaluated.

The sound wave velocity of lime-rich minerals and rocks has been a subject of inquiry, because it does not fit into Birch's law [Birch, 1961]: ‘the compressional wave velocity is expressible in terms of density, ρ, and mean atomic weight, m, by υp = a + bρ + cm, where a, b, and c are constants.’ In spite of the high value of m, lime-rich rocks exhibited higher velocity values than expected from Birch's empirical relationship. To correct this discrepancy, Simmons [1964a] introduced into Birch's equation the weight fraction of CaO as a variable in addition to ρ and m. Simmons' relationship is ((1)) Obviously, to solve this problem the actual data of sound wave velocity of CaO are needed. To accomplish this data collection, several polycrystalline CaO specimens were fabricated with the vacuum hot-press of the Mechanics Research Department of Bell Telephone Laboratories. CaO powder of Fisher-certified chemical reagent grade was used.

Birch9s law arises in the physics of solids as a linear approximation, in a certain range of density, of a power law. For a change of chemical composition within the same crystal structure, the velocity-density relation is constant with a slope of nearly -0.5 in the first-order approximation.