Equations of State of Ca-Silicates and Phase Diagram of the CaSiO3 System under Upper Mantle Conditions

The equations of state of different phases in the CaSiO3 system (wollastonite, pseudowollastonite, breyite (walstromite), larnite (Ca2SiO4), titanite-structured CaSi2O5 and CaSiO3-perovskite) are constructed using a thermodynamic model based on the Helmholtz free energy. We used known experimental measurements of heat capacity, enthalpy, and thermal expansion at zero pressure and high temperatures, and volume measurements at different pressures and temperatures for calculation of parameters of equations of state of studied Ca-silicates. The used thermodynamic model has allowed us to calculate a full set of thermodynamic properties (entropy, heat capacity, bulk moduli, thermal expansion, Gibbs energy, etc.) of Ca-silicates in a wide range of pressures and temperatures. The phase diagram of the CaSiO3 system is constructed at pressures up to 20 GPa and temperatures up to 2000 K and clarifies the phase boundaries of Ca-silicates under upper mantle conditions. The calculated wollastonite–breyite equilibrium line corresponds to equation P(GPa) = −4.7 × T(K) + 3.14. The calculated density and adiabatic bulk modulus of CaSiO3-perovskite is compared with the PREM model. The calcium content in the perovskite composition will increase the density of mineral and it good agree with the density according to the PREM model at the lower mantle region.

Influence of the principle of minimum potential energy on the distribution of density and gravitational energy of the Earth for the reference model PREM

From the point of view of modeling the internal structure of the Earth, its figure and evolution play an important role, which to one degree or another are associated with gravitational energy and the principle of its minimization. The realization of the minimum principle of potential for models of the distribution of the Earth’s density is the key in studies on the detection of inhomogeneous mass distribution. Achieving the minimum gravitational energy of the Earth is equivalent to the approximation of the internal structure to the hydrostatic state, and this condition is achieved due to variations in density. Therefore, for correct geophysical interpretation of gravimetric data, it is necessary to align the PREM (Preliminary Reference Earth Model) with harmonic coefficients of geopotential and minimum functional condition that determines gravitational energy, and only on this basis to estimate variations in density and tectonosphere. An algorithm for representing a piecewise continuous density distribution function in a spherical PREM model by Legendre polynomials is proposed in the paper to calculate the density, potential and energy distribution in an ellipsoidal planet using an additional condition — the minimum of gravitational energy. The use of such an algorithm made it possible to transform the spherically symmetric PREM model to a hydrostatically balanced state. It turned out that in the model obtained, the excess potential energy is concentrated in the inner and outer core of the Earth, and also insignificantly in the planet’s crust. The total energy E for the PREM reference model, which is subdivided into ellipsoidal layers, is 2.3364∙1024 erg, and in the modified PREM model after its correction for the hydrostatic component, it is 2.2828∙1024 erg. Estimation of density redistribution and identification of areas of their greatest change provide a mechanism for explaining the dynamic processes in the middle of the Earth.

A point dislocation in a layered, transversely isotropic and self-gravitating Earth – Part III: internal deformation

SUMMARY In this paper, we derive analytical solutions for the dislocation Love numbers (DLNs) and the corresponding Green's functions (GFs) within a layered, spherical, transversely isotropic and self-gravitating Earth. These solutions are based on the spherical system of vector functions (or the vector spherical harmonics) and the dual variable and position matrix method. The GFs for displacements, strains, potential and its derivatives are formulated in terms of the DLNs and the vector spherical harmonics. The vertical displacement due to a vertical strike-slip dislocation and the potential change (nΦ) due to a vertical dip-slip dislocation are found to be special, with an order O(1/n) on the source level and O(n) elsewhere. Numerical results are presented to illustrate how the internal fields depend on the particular type of dislocation. It is further shown that the effect of Earth anisotropy on the strain field can be significant, about 10 per cent in a layered PREM model and 30 per cent in a homogeneous earth model.

Computation for Equation of States of (Mg0.92, Fe0.08)SiO3 Perovskite Based on Mie-Grüneisen-Debye Model

In this paper, the thermal equation of state (EOS) of (Mg0.92, Fe0.08)SiO3is computed by Birch-Murnaghan and Mie-Grüneisen-Debye equations and the related parameters are also analyzed. The value of and has little effect on EOS of (Mg0.92, Fe0.08)SiO3perovskite. The effect of EOS of (Mg0.92, Fe0.08)SiO3perovskite is mainly from the temperature under high pressure. The temperature is higher; the deviation of EOS relative to the PREM model is bigger. The thermal EOS complies with PREM model at T=2000K. The thermal pressure of (Mg0.92, Fe0.08)SiO3perovskite a constant only related to temperature at the lower mantle conditions. At the same time, the EOS of (Mg0.92, Fe0.08)SiO3perovskite is insensitive to the data of and at T=2000K, but when and the thermal EOS is more agreement with PREM model. That is to say, when the value of the and is in the range of 253~273 GPa and 3.69~4.23, (Mg0.92, Fe0.08)SiO3is the perovskite phase, and (Mg0.92, Fe0.08)SiO3perovskite structure remains stable at the mantle conditions.