The equations of state for solid (with bcc, fcc, and hcp structures) and liquid phases of Fe were defined via simultaneous optimization of the heat capacity, bulk moduli, thermal expansion, and volume at room and higher temperatures. The calculated triple points at the phase diagram have the following parameters: bcc–fcc–hcp is located at 7.3 GPa and 820 K, bcc–fcc–liquid at 5.2 GPa and 1998 K, and fcc–hcp–liquid at 106.5 GPa and 3787 K. At conditions near the fcc–hcp–liquid triple point, the Clapeyron slope of the fcc–liquid curve is dT/dP = 12.8 K/GPa while the slope of the hcp–liquid curve is higher (dT/dP = 13.7 K/GPa). Therefore, the hcp–liquid curve overlaps the metastable fcc–liquid curve at pressures of about 160 GPa. At high-pressure conditions, the metastable bcc–hcp curve is located inside the fcc-Fe or liquid stability field. The density, adiabatic bulk modulus and P-wave velocity of liquid Fe calculated up to 328.9 GPa at adiabatic temperature conditions started from 5882 K (outer/inner core boundary) were compared to the PREM seismological model. We determined the density deficit of hcp-Fe at the inner core boundary (T = 5882 K and P = 328.9 GPa) to be 4.4%.
[1] Resent experimental and theoretical studies suggested preferential stability of Fe 3 C over Fe 7 C 3 at the condition of the Earth's inner core. Previous studies showed that Fe 3 C remains in an orthorhombic structure with the space group Pnma to 250 GPa, but it undergoes ferromagnetic (FM) to paramagnetic (PM) and PM to nonmagnetic (NM) phase transitions at 6-8 and 55-60 GPa, respectively. These transitions cause uncertainties in the calculation of the thermoelastic and thermodynamic parameters of Fe 3 C at core conditions. In this work we determined P-V-T equation of state of Fe 3 C using the multianvil technique and synchrotron radiation at pressures up to 31 GPa and temperatures up to 1473 K. A fit of our P-V-T data to a Mie-Gruneisen-Debye equation of state produce the following thermoelastic parameters for the PM-phase of , m = 4.3, and g = 0.66 with fixed parameters m E1 = 3n = 12, γ ∞ = 0, β = 0.3, and a 0 = 0. This formulation allows for calculations of any thermodynamic functions of Fe 3 C versus T and V or versus T and P. Assuming carbon as the sole light element in the inner core, extrapolation of our equation of state of the NM phase of Fe 3 C suggests that 3.3 ± 0.9 wt % С at 5000 К and 2.3 ± 0.8 wt % С at 7000 К matches the density at the inner core boundary.
Based on the modified formalism of Dorogokupets and Oganov (2007), we calculated the equation of state for diamond, MgO, Ag, Al, Au, Cu, Mo, Nb, Pt, Ta, and W by simultaneous optimization of the data of shock-wave experiments and ultrasonic, X-ray diffraction, dilatometric, and thermochemical measurements in the temperature range from ~ 100 K to the melting points and pressures of up to several Mbar, depending on the material. The obtained room-temperature isotherms were adjusted with a shift of the R1 luminescence line of ruby, which was measured simultaneously with the unit cell parameters of metals in the helium and argon pressure media. The new ruby scale is expressed as P(GPa) = 1870⋅Δλ / λ0(1 + 6⋅Δλ / λ0). It can be used for correction of room-pressure isotherms of metals, diamond, and periclase. New simultaneous measurements of the volumes of Au, Pt, MgO, and B2-NaCl were used for interrelated test of obtained equations of state and calculation of the room-pressure isotherm for B2-NaCl. Therefore, the constructed equations of state for nine metals, diamond, periclase, and B2-NaCl can be considered self-consistent and consistent with the ruby scale and are close to a thermodynamic equilibrium. The calculated PVT relations can be used as self-consistent pressure scales in the study of the PVT properties of minerals using diamond anvil cell in a wide range of temperatures and pressures.
The equations of state of forsterite, wadsleyite, ringwoodite, MgSiO3-perovskite, akimotoite, and postperovskite are set up by joint analysis of experimentally measured isobaric heat capacity, bulk moduli, thermal expansion depending on temperature at ambient pressure, and volume at room and higher temperatures. Modified equations of state based on the Helmholtz free energy are used to construct a thermodynamic model. The derived equations of state permit calculation of all thermodynamic functions for the minerals depending on temperature and volume or temperature and pressure. A phase diagram of the system MgSiO3–MgO is constructed based on the Gibbs energy calibrated using the referred experimental points. The seismic boundaries at depths of 410 and 520 km and in the zone D’ are interpreted on the basis of the phase transitions. The global upper/lower mantle discontinuity at a depth of 660 km remains debatable; it is in poor agreement with experimental and computational data on the dissociation of ringwoodite to perovskite and periclase.
Using the modified formalism of [Dorogokupets, Oganov, 2005, 2007, equations of state are developed for diamond, Ag, Al, Au, Cu, Mo, Nb, Pt, Ta, and W by simultaneous optimization of shock-wave data, ultrasonic, X-ray, dilatometric and thermochemical measurements in the temperature range from ~100 K to the melting temperature and pressures up to several Mbar, depending on the substance. The room-temperature isotherm is given in two forms: (1) the equation from [Holzapfel, 2001[Holzapfel, , 2010 which is the interpolation between the low pressure (x≥1) and the pressure at infinite compression (x=0); it corresponds to the Thomas-Fermi model, and (2) the equation from [Vinet et al., 1987]. The volume dependence of the Grüneisen parameter is calculated according to equations from [Zharkov, Kalinin, 1971;Burakovsky, Preston, 2004] with adjustable parameters, t and δ. The room-temperature isotherm and the pressure on the Hugoniot adiabat are determined by three parameters, K', t and δ, and K 0 is calculated from ultrasonic measurements. In our study, reasonably accurate descriptions of all of the basic thermodynamic functions of metals are derived from a simple equation of state with a minimal set of adjustable parameters.The pressure calculated from room-temperature isotherms can be correlated with a shift of the ruby R1 line. Simultaneous measurements of the shift and unit cell parameters of metals are conducted in mediums containing helium [Dewaele et al., 2004b;2008;Takemura, Dewaele, 2008;Takemura, Singh, 2006], hydrogen [Chijioke et al., 2005] and argon [Tang et al., 2010]. According to [Takemura, 2001], the helium medium in diamond anvil cells provides for quasi-hydrostatic conditions; therefore, the ruby pressure scale, that is calibrated for the ten substances, can be considered close to equilibrium or almost absolute. The ruby pressure scale is given as P(GPa)=1870⋅Δλ/λ 0 ⋅(1+6⋅Δλ/λ 0 ). The room-temperature isotherms corrected with regard to the ruby scale can also be considered close to equilibrium or almost absolute. Therefore, the equations of state of the nine metals and diamond, which are developed in our study, can be viewed as almost absolute equations of state for the quasi-hydrostatic conditions. In other words, these equations agree with each other, with the ruby pressure scale, and they are close to equilibrium in terms of thermodynamics. The PVT relations derived from these equations can be used as mutually agreed pressure scales for diamond anvil cells in studies of PVT properties of minerals in a wide range of temperatures and pressures. The error of the recommended equations of the state of substances and the ruby pressure scale is about 2 or 3 per cent. Calculated PVT relations and thermodynamics data are available at http://labpet.crust.irk.ru. T e c t o n o p h y s i c s P.I. Dorogokupets et al.: Near-absolute equations of state… 130Аннотация: По единой схеме с использованием модифицированного формализма из [Dorogokupets, Oganov, 2005, 2007 построены уравнения состояния алмаза, Ag, Al, Au, ...
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