High-pressure phase transformations and compressions of ilmenite and rutile,I. Experimental results

Natural ilmenite (Fe,Mg)TiO₃ has been found to transform to the perovskite structure and then to disproportionate into its component oxides, (Fe,Mg)O plus a cubic phase of TiO₂, at loading pressures of 140 and 250 kbar respectively, and at temperatures of 1,400 to 1,800°C. Samples were compressed in a diamond-anvil press and heated by irradiation with a YAG laser. The lattice parameters of the perovskite phase of (Fe,Mg)TiO₃ at room temperature and 1 bar are $\text{a}{}_0\text{= 4.471 ± 0.004, b}{}_0\text{= 5.753 ± 0.005,}\text{and}\text{c}{}_0\text{= 7.429 ± 0.006}\text{A}\text{?}$ with 4 molecules per cell. The zero-pressure volume change is 8.0% for the ilmenite-perovskite transition, 13.3% for the perovskite-mixed-oxides transition, and 20.2% for the ilmenite-mixed-oxides transition. The cubic phase of TiO₂ can be indexed on the basis of space group Fm3m with $\text{Z = 4}\text{and}\text{a}{}_0\text{= 4.455 ± 0.008}\text{A}\text{?}$ at room temperature and 1 bar, which corresponds to a decrease in zero-pressure volume of 29.2% for the rutile-cubic-phase transition. An isentropic bulk modulus at zero pressure of 5.75 ± 0.30 Mbar and a pressure derivative greater than 8 were calculated for the high-pressure cubic phase. The calculated bulk modulus for the mixture of (Fe,Mg)O and cubic TiO₂ is 2.48 ± 0.25 Mbar. All the phase transformations, the calculated lattice parameters, and the bulk moduli observed in this study are in good agreement with published shock-Hugoniot data for ilmenite and rutile.     

Phase transformations and elasticity in rutile-structured difluorides and dioxides

Pressure-induced phase transformations in each of the rutile-structured difluorides (NiF₂, MgF₂, CoF₂, ZnF₂, FeF₂ and MnF₂) exhibit unique behavior; however, a general trend is found in the major structural changes: rutile phase → “distorted fluorite” phase → post-“distorted fluorite” phase with volume changes of about 5–10%. For a given phase transformation sequence found commonly in two or more difluorides, the phase transformation pressure is related inversely to the unit cell volume and thus inversely to the mean cation-anion bond length. The relationship in oxides (SnO₂, TiO₂ and GeO₂) is much less systematic. It is therefore not possible to predict without uncertainty the post-stishovite phases in the lower mantle.Velocity-density systematics in the difluorides and oxides are governed, to a large extent, by cationic radius. The pressure dependence of shear elastic constant C_S = (C₁₁ ? C₁₂)/2 is negative in all of the nine difluorides and oxides. However, the C_S mode does not vanish at the initial phase transformation pressure; rather, the ratios of $\text{C}{}_{\mathrm{<mtext>S</mtext>}}\text{K}{}_{\mathrm{<mtext>S</mtext>}}$ are 0.10 and 0.04 to 0.10 for transitions of rutile → orthorhombic and of rutile → “distorted fluorite”, respectively, and are in agreement with the approach of Demarest et al.     

An experimental study of magnetite-titanomagnetite interdiffusion

Interdiffusion experiments were performed between Fe₃O₄ (single crystal) and Fe_2.8Ti_0.2O₄ (powder), under self-buffering conditions (temperature range 600–1034°C), and for various oxygen potentials at 1400°C. Profiles of Fe and Ti were obtained by electronprobe microanalysis, and the interdiffusion coefficient $\text{D}$ was calculated by the Boltzmann-Matano method. Low-temperature data at 3 mole% Ti could be described by $\text{D}\text{= (3.85}{}_{\mathrm{?1.11}}{}^{\mathrm{+1.68}}\text{) × 10}{}^{\mathrm{?3}}\text{exp}\text{(2.23 ± 0.04}\text{eV}\text{/kT)}\text{cm}{}^2\text{/}\text{s}$. An estimate is given for the time to interdiffuse 2μm at various temperatures, and the results compared with recent experiments.     

Melting temperature distribution and fractionation in the lower mantle

The melting curve of perovskite MgSiO₃ and the liquidus and solidus curves of the lower mantle were estimated from thermodynamic data and the results of experiments on phase changes and melting in silicates.The initial slope of the melting curve of perovskite MgSiO₃ was obtained as $\text{d}\text{T}{}_{\mathrm{<mtext>m</mtext>}}\text{/}\text{d}\text{P?77}\text{K}\text{GPa}{}^{\mathrm{?1}}$ at 23 GPa. The melting curve of perovskite was expressed by the Kraut-Kennedy equation as $\text{T}{}_{\mathrm{<mtext>m</mtext>}}\text{(}\text{K}\text{)=917(}\text{1+29.6ΔV}\text{V}{}_0\text{)}$, where T_m?2900 K and P?23 GPa; and by the Simon equation, $\text{P(}\text{GPa}\text{)?23=21.2[}\text{(T}{}_{\mathrm{<mtext>m</mtext>}}\text{(}\text{K}\text{)}\text{2900)}{}^{1.75}\text{?1}\text{]}$.The liquidus curve of the lower mantle was estimated as $\text{T}{}_{\mathrm{<mtext>liq</mtext>}}\text{? 0.9 T}{}_{\mathrm{<mtext>m</mtext>}}$ (perovskite) and this gives the liquidus temperature T_liq=7000 ±500 K at the mantle-core boundary. The solidus curve of the lower mantle was also estimated by extrapolating the solidus curve of dry peridotite using the slope of the solidus curve of magnesiowüstite at high pressures. The solidus temperature is ～ 5000 K at the base of the lower mantle. If the temperature distribution of the mantle was 1.5 times higher than that given by the present geotherm in the early stage of the Earth's history, partial melting would have proceeded into the deep interior of the lower mantle.Estimation of the density of melts in the MgOFeOSiO₂ system for lower mantle conditions indicates that the initial melt formed by partial fusion of the lower mantle would be denser than the residual solid because of high concentration of iron into the melt. Thus, the melt generated in the lower mantle would tend to move downward toward the mantle-core boundary. This downward transportation of the melt in the lower mantle might have affected the chemistry of the lower mantle, such as in the D″ layer, and the distribution of the radioactive elements between mantle and core.     

Orthogonality of spherical harmonic coefficients

An essentially arbitrary function V(θ, λ) defined on the surface of a sphere can be expressed in terms of spherical harmonics $\text{V(θ, Λ) = a}\text{∑}\text{n=1}\text{∞}\text{∑}\text{m=0}\text{n}\text{p}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{(cos θ) (g}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{cos mΛ + h}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{sin mΛ)}$ where the P_n^m are the seminormalized associated Legendre polynomials used in geomagnetism, normalized so that $\text{〈[P}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{(cos θ) cos mΛ]〉}{}^2\text{=1/}\text{(2n+1)}$ The angular brackets denote an average over the sphere. The class of functions V(θ, λ) under consideration is that normally of interest in physics and engineering. If we consider an ensemble of all possible orientations of our coordinate system relative to the sphere, then the coefficients g_n^m and h_n^m will be functions of the particular coordinate system orientation, but $\text{〈:(g}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{〉) = 〈(h}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{=}\text{S}{}_{\mathrm{n}}\text{/}\text{(2n=1)}$ where $\text{S}{}_{\mathrm{n}}\text{=}\text{∑}\text{m=0}\text{n}\text{[(g}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{+ (h}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{]}$ for any orientation of the coordinate system (S_n is invariant under rotation of the coordinate system). The averages are over all orientations of the system relative to the sphere. It is also shown that 〈g^m_ng^l_p〉 = 〈h^m_nh^l_p〉 = 0 for l ≠ m or p ≠ n and 〈g^m_nh^l_p〉 = 0 fro all n, m, p, l.     

On the mechanism of the olivine-spinel phase transformation

Recent experiments on the olivine-spinel phase transformation in Ni₂SiO₄ have provided the following fundamental results: (1) the interphase structural orientation relationships with $\text{(100)}{}_{\mathrm{<mtext>ol</mtext>}}\text{|(111)}{}_{\mathrm{<mtext>sp</mtext>}}\text{and}\text{[001]}{}_{\mathrm{<mtext>ol</mtext>}}\text{|[}\text{1}\text{10]}{}_{\mathrm{<mtext>sp</mtext>}}$ between the olivine and spinel phases upon transformation; and (2) the growth of 300 μm spinel single crystal from single crystals of olivine. In the light of these observations, transformation mechanisms proposed have been compared. We conclude that neither a nucleation and growth model nor a martensite model is verified by the experimental results. The former model lacks consistency in terms of the interphase relationships, and the latter cannot offer a dislocation-multiplication mechanism for the growth process.     

General relationships among sound speeds: II. Theory and discussion

The dependence of bulk sound speed V_φ upon mean atomic weight $\text{m}$ and density ρ can be expressed in a single equation:

$\text{V}{}_{\mathrm{\phi }}\text{=Bρ}{}^{\mathrm{\lambda }}\text{(}\text{m}{}_0\text{m}{}^{\mathrm{<mtext>[</mtext><mtext>1</mtext><mtext>2</mtext><mtext>+\lambda (1?c)]</mtext>}}\text{(}\text{km/sec}\text{)}$

Here B is an empirically determined “universal” parameter equal to 1.42, $\text{m}{}_0\text{= 20.2}$, a reference mean atomic weight for which well-determined elastic properties exist, and λ = 1.25 is a semi empirical parameter equal to $\text{γ ?}\text{1}\text{3}$ where γ is a Grüneisen parameter. The constant c = (? ln V_M/? ln $\text{m}\text{)}{}_{\mathrm{X}}$, 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 Fe²⁺Mg²⁺ 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.     

Spatial power spectra of the crustal geomagnetic field and core geomagnetic field

Lowes (1966, 1974) has introduced the function R_n defined by $\text{R}{}^{\mathrm{n}}\text{=(n + 1)}\text{∑}\text{m=0}\text{∞}\text{[(g}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{+ (h}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{]}\text{where}\text{g}{}_{\mathrm{n}}{}^{\mathrm{m}}\text{and}\text{h}{}_{\mathrm{n}}{}^{\mathrm{m}}$ are the coefficients of a spherical harmonic expansion of the scalar potential of the geomagnetic field at the Earth's surface. The mean squared value of the magnetic field B = ??V on a sphere of radius r > α is given by $\text{〈}\text{B}\text{·〉 =}\text{∑}\text{n=1}\text{∞}\text{R}{}^{\mathrm{n}}\text{(}\text{a/}\text{r}\text{)}{}^{\mathrm{2n=4}}\text{where}\text{a}$ is the Earth's radius. We refer to R_n as the spherical harmonic spatial power spectrum of the geomagnetic field.In this paper it is shown that Rⁿ = R^M_n = R^C_n where the components R_n^M due to the main (or core) field and R_n^C due to the crustal field are given approximately by $\text{R}{}^{\mathrm{M}}{}_{\mathrm{n}}\text{= [}\text{(n =1)/}\text{(n + 2)}\text{](1.142 × 10}{}^9\text{)(0.288}{}^{\mathrm{n}}\text{Λ}{}^2\text{R}{}^{\mathrm{C}}{}_{\mathrm{n}}\text{= [}\text{(n =1){[1 — exp(-n/290)]/}\text{(n/290)} 0.52 Λ}{}^2\text{where}\text{Iγ = 1 nT}$. The two components are approximately equal for n = 15.Lowes has given equations for the core and crustal field spectra. His equation for the crustal field spectrum is significantly different from the one given here. The equation given in this paper is in better agreement with data obtained on the POGO spacecraft and with data for the crustal field given by Alldredge et al. (1963).The equations for the main and crustal geomagnetic field spectra are consistent with data for the core field given by Peddie and Fabiano (1976) and data for the crustal field given by Alldredge et al. The equations are based on a statistical model that makes use of the principle of equipartition of energy and predicts the shape of both the crustal and core spectra. The model also predicts the core radius accurately. The numerical values given by the equations are not strongly dependent on the model.Equations relating average great circle power spectra of the geomagnetic field components to R_n are derived. The three field components are in the radial direction, along the great circle track, and perpendicular to the first two. These equations can, in principle, be inverted to compute the R_n for celestial bodies from average great circle power spectra of the magnetic field components.     

High-pressure recovery of olivine: implications for creep mechanisms and creep activation volume

High-temperature and high-pressure recovery experiments were made on experimentally deformed olivines at temperatures of 1613–1788 K and pressures of 0.1 MPa to 2.0 GPa. In the high-pressure experiments, a piston cylinder apparatus was used with BN and NaCl powder as the pressure medium, and the hydrostatic condition of the pressure was checked by test runs with low dislocation density samples. No dislocation multiplication was observed. The kinetics of the dislocation annihilation process were examined by different initial dislocation density runs and shown to be of second order, i.e.

$\text{d}\text{ρ}\text{d}\text{t}\text{= ?p}{}^2\text{K}{}_0\text{exp}\text{[?(}\text{E}{}^1\text{+PV}{}^1\text{RT}\text{]}$

where ρ is the dislocation density, k₀ is a constant, $\text{E}{}^1\text{and}\text{V}{}^1$ are the activation energy and volume respectively, and P, R and T are pressure, gas constant and temperature, respectively. Activation energy and volume were estimated from the temperature and pressure dependence of the dislocation annihilation rate as $\text{E}{}^1\text{=389±59}\text{kJ mol}{}^{\mathrm{?1}}$ and $\text{V}{}^1\text{=14±2}\text{cm}{}^3\text{mol}{}^{\mathrm{?1}}$, respectively.The diffusion constants relevant to the dislocation annihilation process were estimated from a theoretical relation k=αD where $\text{k=k}{}_0\text{exp}\text{[?(E}{}^1\text{+ PV}{}^1\text{)/RT], D}$ is the diffusion constant and α is a non-dimensional constant of ca. 300. The results agree well with the self-diffusion constant of oxygen in olivine. This suggests that the dislocation annihilation is rate-controlled by the (oxygen) diffusion-controlled dislocation climb.The mechanisms of creep in olivine and dry dunite are examined by using the experimental data of static recovery. It is suggested that the creep of dry dunite is rate-controlled by recovery at cell walls or at grain boundaries which is rate-controlled by oxygen diffusion. Creep activation volume is estimated to be 16±3 cm³ mol^?1.     

Elasticity of ilmenites

Ultrasonic data for the velocities of the ilmenites MgTiO₃ and CoTiO₃ 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 ($\text{M}$) diagram, however, all of the ilmenite consistent with a single $\text{υ}{}_{\mathrm{\phi }}\text{M}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{=}\text{constant trend}$. Elasticity systematics for isostructural sequences are used to e the bulk modulus (2.09 Mbar) and bulk sound velocity (7.4 km/sec) of MgSiO₃-elmenite.     

Applications of liquid state physics to the Earth's core

By use of the modern theory of liquids and some guidance from the hard-sphere model of liquid structure, the following new results have been derived for application to the Earth's outer core. (1) dK/$\text{d}\text{P ? 5 ? 5. 6P}$/K, where K is the incompressibility and P the pressure. This is valid for a high-pressure liquid near its melting point, provided that the pressure is derived primarily from a strongly repulsive pair potential φ. This result is consistent with seismic data, except possibly in the lowermost region of the outer core, and demonstrates the approximate universality of dK/dP proposed by Birch (1939) and Bullen (1949). (2) dlnT_M/dlnρ = (γC_V ? 1)/($\text{C}{}_{\mathrm{<mtext>V</mtext>}}\text{?}\text{3}\text{2}\text{),}\text{where}\text{T}{}_{\mathrm{<mtext>M</mtext>}}$ is the melting point, ρ the density, γ the atomic thermodynamic Grüneisen parameter and C_V the atomic contribution to the specific heat in units of Boltzmann's constant per atom. This reduces to Lindemann's law for C_V = 3 and provides further support for the approximate validity of this law. (3) It follows that the “core paradox” of Higgins and Kennedy can only occur if $\text{γ <}\text{2}\text{3}$. However, it is shown that $\text{γ <}\text{2}\text{3}\text{? ∫}{}^{\mathrm{\infty }}{}_0\text{(?g/?T)}{}_{\mathrm{\rho }}\text{r(}\text{d}\text{/}\text{d}\text{r)(r}{}^2\text{φ)}\text{d}\text{r > 0}$, which cannot be achieved for any strongly repulsive pair potential φ and the corresponding pair distribution function g. It is concluded that $\text{γ >}\text{2}\text{3}$ and that the core paradox is almost certainly impossible for any conceivable core composition. Approximate calculations suggest that γ ～ 1.3–1.5 in the core. Further work on the thermodynamics of the liquid core must await development of a physically realistic pair potential, since existing pair potentials may be unsatisfactory.     

Variation of the magnetic properties in a basaltic dyke with concentric cooling zones

The effects of the variation of magnetic grain size on the magnetic properties of rocks have been studied throughout a reversely magnetized basaltic dyke with concentric cooling zones.Except in a few tachylites in which the magnetic mineral is a Ti-rich titanomagnetite, in the bulk of the dyke the magnetization is carried by almost pure magnetite grains. Although the percentage p of these magnetic oxides varies slightly, the large changes in the various magnetic parameters observed across the dyke are essentially attributable to large variations in the grain size of the magnetic particles.From the outer scoria region, where the magnetic grains are a mixture of single-domain (SD) and superparamagnetic (SP) grains, to the tachylite zone with finely crystallized basaltic glass containing interacting elongated SD particles, one observes an increase of both the ratio of the saturation remanent magnetization and the saturation induced magnetization J_rs/J_is, the bulk coercive force H_c, the median destructive field MDF, the intensity of the remanent magnetization J_r, and the Koenigsberger ratio Q. In the tachylites these parameters reach unusually high values, for subaerial basalts:

$\text{〈}\text{J}{}_{\mathrm{rs}}\text{J}{}_{\mathrm{is}}\text{〉 = 0.3, 〈H}{}_{\mathrm{c}}\text{〉 = 460 O}{}_{\mathrm{e}}\text{, 〈MDF〉 = 620 O}{}_{\mathrm{e}}\text{r.m.s., 〈J}{}_{\mathrm{r}}\text{〉 = 2.7 · 10}{}^{\mathrm{?2}}\text{e.m.u. cm}{}^{\mathrm{?3}}\text{〈Q〉 = 24}$

These parameters decrease in the basalt toward the centre of the dyke where pseudo-single-domain (pseudo-SD) particles coexist together with multidomain (MD) grains. The susceptibility remains approximately constant from the inner basalt to the tachylite, but increases in the scoria up to values 10 times higher owing to the presence of SP particles. The magnetic viscosity increases also drastically toward the margin of the dyke due to an increase of the fraction of the SD particles just above the superparamagnetic threshold.     

Transient and steady-state creep of pure forsterite at low stress

Thirty-one single crystals of synthetic forsterite, Fo₁₀₀, were deformed in 69 compressional creep tests in a 0.1-MPa confining atmosphere of H₂ + Ar. Temperature ranged from 1753 to 2023 K and stress σ (= σ₁ - σ₃) from 1.5 to 37.8 MPa. Steady-state creep under these conditions follows an empirical law of the form: strain rate $\text{?}\text{?}\text{= A}{}_{\mathrm{zigma;}}{}^{\mathrm{n}}\text{exp}\text{(}\text{?Q}\text{RT}\text{)}$ where A, n, and Q are constants. General characteristics of Fo₁₀₀ creep — uniformity of strain, shape change as a function of orientation of σ, relative deformation resistance of different orientations — match those of natural olivine single crystals of composition Fo₉₂. Specific constants in the flow law, however, are distinctly new: for σ oriented along [111]_c (equidistant from the three principal crystallographic axes), values for Fo₁₀₀ are n = 2.9 ± 0.2 and Q = 0.67 ± 0.03 MJ/mol (160 ± 7 kcal/mol). A single law covers the range 3 < σ < 30 MPa and 1753 < T < 1953 K. Steady-state deformation is preceded by a transient period of strain softening. High strain rates at σ ? 10 MPa render the transient barely resolvable; it apparently displaces the steady-state flow law by approximately ?0.5% in strain. At σ ? 7.8 MPa, the amount of strain imparted to a sample of the [101]_c orientation is typically <0.1% after one hour.     

Interionic repulsive force and compressibility of ions

The compressibility of an individual ion is examined, in comparison with a known set of data for the alkali halides. A simple extrapolation of ionic radius to high pressure is not acceptable, because the pressure derivative of ionic radius changes for different salts. According to the classical concept of an elastic ion, the repulsive potential energy between the ions i and j is specified by the nature of each ion as:

$\text{(ρ}{}_{\mathrm{i}}\text{+ ρ}{}_{\mathrm{j}}\text{) exp[}\text{(ρ}{}_{\mathrm{i}}\text{+ ρ}{}_{\mathrm{j}}\text{? r)}\text{(ρ}{}_{\mathrm{i}}\text{+ ρ}{}_{\mathrm{j}}\text{)}\text{]}$

as a function of the interionic distance r. In this expression, q_i and ρ_i are the ionic radius and ionic compressibility, respectively, in a suitably modified meaning. Such a form of the repulsive potential fits well to the data of lattice constants and bulk moduli. The parameters q_i and ρ_i are evaluated for alkali and halogen ions, and an anion turns out to be much more compressible than a cation. The present treatment may be usefully applied to the minerals in the Earth's mantle, which contain only a few major ions.