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.     

Are anharmonicity corrections needed for temperature-profile calculations of interiors of terrestrial planets?

It is shown that there is linearity between the thermal pressure P_TH and T between the Debye temperature θ and some high temperature $\text{T}{}^1$. $\text{T}{}^1$ has been measured at 1 atm and is reported for several minerals including, for example, MgO (1300 K) and forsterite (1200 K). The change in thermal pressure from room temperature for five solids, so far measured, indicate striking linearity with T at high temperatures.It is further shown that the value of $\text{T}{}^1$ increases greatly as the pressure increases. It is therefore concluded that P_TH is probably linear with T for mantle minerals under mantle conditions. The proportionality constant is derived from the measurements of thermal expansivity and bulk modulus at high temperature and zero pressure.The argument is then reversed. Assuming that the thermal pressure is in fact linear with T for the various shells in a planet, the resulting density and temperature profile of the planet is derived. The resulting density profile of the Earth compares favorably with corresponding values of recent seismic profiles.     

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.     

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.     

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.     

Effect of oxygen partial pressure on the dislocation recovery in olivine: a new constraint on creep mechanisms

The dislocation annihilation rate in experimentally deformed olivine single crystals was measured as a function of oxygen partial pressure (P_O2). It was shown that the dislocation annihilation rate decreased with increasing P_O2. This result is inconsistent with the reported P_O2 dependence of creep rate () in single olivine crystals, thus indicating that the creep in single olivine crystals is not rate-controlled by recovery, under the experimentally investigated conditions.     

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.     

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.     

High-temperature anelasticity and elasticity of mantle peridotite

An apparatus has been devised which allows precise creep and relaxation measurements to be made on minerals and rocks at temperatures up to 1600°C and at very low deviatoric stresses (1 < σ < 300 bar). This paper is concerned with measurements on mantle peridotite (lherzolite) from Balmuccia (Zone of Ivrea, Italy).The reaction of the sample to a step-like increase in stress is called its “creep function”. It is shown that the creep function contains all the necessary information to derive the spectra of the quality factor Q(ω) and of Young's modulus E(ω), within the seismic range of frequencies, provided the material behaves as a linear system. This has been proven up to a strain of 5 × 10^?5.The Q^?1-spectra at 1200 and 1300°C, obtained by Fourier inversion from the creep function, show no pronounced peak in the frequency band 0.01 < tf < 1 Hz and exhibit a general tendency to decrease slightly with frequency. The creep function: ?(t) = ?_u · [1 + 3.7 · q · {(1 + 50t)^0.27 ? }], where q is related to Q, satisfactorily describes the data at high temperatures and leads to $\text{Q}{}^{\mathrm{?1}}\text{(ω, T) = 3 × 10}{}^3\text{· ω}{}^{\mathrm{<mtext>?0.27\; ·\; </mtext><mtext>exp</mtext>}}\text{(}\text{?30}\text{RT}\text{)}$E(ω) is related to Q(ω) by the material dispersion equation. Above 1100°C the unrelaxed Young's modulus decreases rapidly with temperature according to an activation energy of about 20 kcal/mole. A lowering of short period S-wave velocity by 40% and P-wave velocity by 10% occurs below the solidus. Therefore, no partial melting is required in the asthenosphere.Steady-state creep at low axial stresses (20 < σ < 100 bar), obtained from the same rock, follows the relation $\text{?}\text{?}\text{= 3 × 10}{}^7\text{· δ}{}^{1.4}\text{·}\text{exp}\text{(}\text{?125}\text{RT}\text{)}$ indicative of grain boundary diffusion or superplasticity. At higher stresses a power law $\text{?}\text{?}\text{= 45 · δ}{}^4\text{·}\text{exp}\text{(}\text{?125}\text{RT}\text{)}$ typical of dislocation creep, is found.The frequency dependence of Q and the ratio of the activation energies of Q and are indicative of so called “high-temperature background absorption”, as the dominant mechanism, and of a diffusion-controlled dislocation mobility common to both absorption and creep. From a, b, and c, relations between the effective viscosity η_f and Q of the form: $\text{log}\text{η}{}_{\mathrm{e??}}\text{=}\text{1}\text{α}\text{·}\text{log}\text{Q ? (n ? 1) ·}\text{log}\text{ω +}\text{log}\text{D}$ are derived, where α ～ 0.25, n is the power of σ, and D is a constant.     

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.     

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.     

Physical constraints for the analysis of the geomagnetic secular variation

Slow changes in the magnetic field are believed to originate in the core of the Earth. Interpretation of these changes requires knowledge both of the vertical component of the field and of its rate of change at the core-mantle boundary (CMB). While various spherical harmonic models show some agreement for the field at the CMB, those for secular variation (SV) do not. SV models depend heavily on annual means at relatively few and poorly distributed magnetic observatories. In this paper, the SV at the CMB is modelled by fitting 15-year differences in the annual means of the X, Y and Z components (from 1959 to 1974). The model is made unique by imposing the constraint that $\text{?}{}_{\mathrm{CMB}}\text{B}\text{?}{}_{\mathrm{r}}{}^2\text{d}\text{S}$ be a minimum, using the method of Shure et al. (1982). If SV is attributed to motions of core fluid, then this model will yield, in some sense, the slowest core motions. The null space is determined by the distribution of observations, and therefore, to be consistent, only those observatories have been retained which recorded almost continuously throughout the interval 1959–1974.The method allows misfit between the model and the observations. The best value for the misfit can be derived from estimates of errors in the data, or alternatively, because larger misfit leads to smoother models (i.e., smaller $\text{?}\text{B}\text{?}{}_{\mathrm{r}}{}^2\text{d}\text{S}$), the best value can be estimated subjectively from the final appearance of the model. Both procedures have their counterparts in the conventional spherical harmonic expansion approach, when smoothing is achieved by lowering the truncation level. The new proposal made in this paper is to use objective criteria for determining the misfit, based on the assumption that diffusion is negligible, in which event all integrals $\text{∫}\text{B}\text{?}{}_{\mathrm{r}}{}^2\text{d}\text{S}$ will vanish when S_i is a region on the CMB bounded by a contour of zero vertical component of field. For the 1965 definitive model which is adopted here, and for most other contemporary models, there are six such areas, giving five independent integrals (the integrals over the six regions must sum to zero if ? · B = 0). Tabulating these integrals for various choices of the misfit gives minimum values near 2 nT y^?1. It is impossible to achieve this good a fit to the data using a reasonable model derived by truncating the spherical harmonic expansion. The value 2 nT y^?1 corresponds to errors of ～ 20 nT in individual annual means, which is rather larger than expected from the scatter in the data.     

Solidus and liquidus temperatures and mineralogies for anhydrous garnet-lherzolite to 15 GPa

A review of experimental data for systems, pertaining to anhydrous fertile garnet-lherzolite shows strong convergence in the liquidus and solidus temperatures for the range 6.5–15 GPa. These can converge either to a common temperature or to temperatures which differ by only ～ 100°C. The major-element composition of magmas generated by even minor degrees of partial melting may be similar to the primordial bulk silicate Earth composition in an upper-mantle stratigraphic column extending over 160 km in depth.The convergence of the solidus and liquidus temperatures is a consequence of the highly variable $\text{d}\text{T}\text{d}\text{P}$ of the fusion curves for minerals which crystallize in peridotite systems. In particular, $\text{d}\text{T}\text{d}\text{P}$ for the forsterite fusion curve is much less than that for diopside and garnet. Whether or not the solidus and liquidus intersect, the liquidus mineralogy for undepleted garnet-lherzolite compositions changes from olivine at low pressures to pyroxene, garnet, or a complex pyroxene-garnet solid solution at pressures in excess of 10–15 GPa. Geochemical data for the earliest Archean komatiites are consistent with an upper-mantle phase diagram having garnet as a liquidus phase for garnet-lherzolite compositions at high pressures. All estimates of the anhydrous solidus and liquidus for the range 10–15 GPa are consistent with silicate liquid compressibility data, which indicate that olivine may be neutrally buoyant in ultramafic magmas at these pressures.     

Evidences for a low-velocity core-mantle transition zone

Using the spectral ratios $\text{P}\text{PcP}\text{,}\text{ScS}{}_{\mathrm{n+1}}\text{ScS}{}_{\mathrm{n}}\text{,}\text{sScS}{}_{\mathrm{n+1}}\text{sScS}{}_{\mathrm{n}}\text{and}\text{SKS}\text{ScS}$, models for the core-mantle boundary are found. The models have close similarity with each other, implying an irregular surface with lateral variation in the core-mantle properties. The models are characterized by two to four low-velocity, high-density layers imbedded between the mantle and the core half space. The velocities of the imbedded layers decrease towards the core boundary with a lower bound of 9.3 km/sec for the compressional wave and 3.5 km/sec for the shear wave. The models fitted to the empirical data support the hypothesis of a finite rigid outer core with a higher bound for the shear velocity of 1.4 km/sec. Based on this finite rigidity in the outer core and a layered core-mantle transition zone, the value of Q for the whole mantle is 2,000. For the outer core Q ranges from 100–1,000 , which may indicate that it is chemically zoned.     

Synthesis of a new high-pressure ohase of tin dioxide and some geophysical implications

Tin dioxide (SnO₂) in the rutile structure as starting material has been found to transform to the orthorhombic α-PbO₂ structure (S.G. Pbcn) at about 155 kbar and 1000–1400°C when compressed in a diamond-anvil cell and heated by irradiation with a YAG laser. The lattice parameters at room temperature and 1 bar are $\text{a}{}_{\mathrm{<mtext>o</mtext>}}\text{= 4.719 ± 0.002, b}{}_{\mathrm{<mtext>o</mtext>}}\text{= 5.714 ± 0.002,}\text{and}\text{c}{}_{\mathrm{<mtext>o</mtext>}}\text{= 5.228 ± 0.002}\text{A}\text{?}\text{with}\text{Z = 4}$ for the orthorhombic form of SnO₂, which is 1.5% more dense than the rutile form. Crystal-chemical arguments suggest that stishovite (SiO₂) may also transform to the α-PbO₂ structure at elevated pressure and temperature with an increase in zero-pressure density of about 2–3%. Mineral assemblages containing the orthorhombic SiO₂ are unstable relative to those containing the perovskite MgSiO₃ under lower-mantle conditions.     

Rupture process of the Miyagi-Oki,Japan, earthquake of June 12, 1978

The faulting mechanism and multiple rupture process of the M = 7.4 Miyagi-Oki earthquake are studied using surface and body wave data from local and worldwide stations. The main results are as follows. (1) P-wave first motion data and radiation patterns of long-period surface waves indicate a predominantly thrust mechanism with strike N10° E, dip 20°W, and slip angle 76°. The seismic moment is 3.1 × 10²⁷ dyne-cm. (2) Farfield SH waveforms and local seismograms suggest that the rupture occurred in two stages, being concordant with the two zones of aftershock activity revealed by the microearthquake network of Tohoku University. The upper and lower zones, located along the westward-dipping plate interface, are separated by a gap at a depth of 35 km and have dimensions of 37 × 34 and 24 × 34 km², respectively. Rupture initiated at the southern end of the upper aftershock zone and propagated at N20°W subparallel to the trench axis. About 11 s later, the second shock, which was located 30 km landward (westward) of the first, initiated at the upper corner of the lower aftershock zone and propagated down-dip N80°W. Using Haskell modelling for this rupture process, synthetic seismograms were computed for teleseismic SH waves and nearfield body waves. Other parameters determined are: seismic moment $\text{M}{}_0\text{= 1.7 × 10}{}^{27}\text{dyne-cm, slip dislocation}\text{u}\text{= 1.9}\text{m}\text{, Δσ = 95}\text{bar, rupture velocity}\text{ν = 3.2}\text{km s}{}^{\mathrm{?1}}\text{,}\text{rise time}\text{τ = 2}\text{s}$, for the first event; $\text{M}{}_0\text{= 1.4 × 10}{}^{27}\text{dyne-cm}\text{,}\text{u}\text{= 2.4}\text{m}\text{, Δσ = 145}\text{bar}$, for the second event; and time separation between the two shocks ΔT = 11 s. The above two-segment model does not explain well the sharp onsets of the body waves at near-source stations. An initial break of a small subsegment on the upper zone, which propagated down-dip, was hypothesized to explain the observed near-source seismograms. (3) The multiple rupture of the event and the absence of aftershocks between the two fault zones suggests that the frictional and/or sliding characteristics along the plate interface are not uniform. The rupture of the first event was arrested, presumably by a region of high fracture strength between the two zones. The fracture energy of the barrier was estimated to be 10¹⁰ erg cm^?2. (4) The possible occurrence of a large earthquake has been noted for the region adjacent to and seaward of the area that ruptured during the 1978 event. The 1978 event does not appear to reduce the likelihood of occurrence of this expected earthquake.     

Electrical conductivity of ultramafic rocks to 1820 kelvin

Electrical conductivity, σ, of six ultramafic rocks (garnet-bearing peridotites and an eclogite) has been investigated in the temperature range 670–1820 K under known environment. Between 670 and ～1320 K σ increases with 1/T monotonically, first slowly, by 1–1.5 orders of magnitude. Above 1320 K σ increases sharply; an increase of 3–4 orders of magnitude is observed between 1320 and 1670 K. In the same temperature range the σ values for all the six rocks fall within 1.5 orders of magnitude, the lowest conductivity being for a spinel lherzolite. The range of σ values is narrowed at higher temperatures. The differences in σ values above 1470 K may be explained on the basis of varying degrees of partial melting in the rocks. Over the entire range of temperature, the σ values for the ultramafic rocks are lower than those reported for basalts but higher (by 1–2 orders of magnitude) than those for single-crystal olivines.