Mantle Convection and Surface Expressions. Группа авторов

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СКАЧАТЬ metal oxides, such as FeO, despite of their partially filled d orbitals.

      By far the most common strategy to account for the repulsion between localized d electrons is to add an energy term E_U that depends on the Hubbard parameter U and on the occupation numbers of d orbitals (Anisimov et al., 1997, 1991; Cococcioni, 2010; Cococcioni and de Gironcoli, 2005). This approach, referred to as LDA+U or GGA+U, can reduce discrepancies between predicted and experimentally observed elastic properties (Stackhouse et al., 2010) and led to substantial improvements in treating spin transitions of ferrous and ferric iron with DFT calculations (Hsu et al., 2011, 2010a, 2010b; Persson et al., 2006; Tsuchiya et al., 2006). Despite important progress in modeling the electronic properties of iron cations in oxides and silicates, elastic properties extracted from DFT computations still seem to deviate significantly from experimental observations in particular across spin transitions in major mantle minerals (Fu et al., 2018; Shukla et al., 2016; Wu et al., 2013), highlighting persistent challenges in the treatment of localized d electrons.

      While most first‐principle calculations assume the atomic nuclei to be static, thermal motions of atoms at finite temperatures can be addressed by coupling DFT to molecular dynamics (MD) (Car & Parrinello, 1985) or by computing vibrational properties using density functional perturbation theory (DFPT) (Baroni et al., 2001, 1987b; Giannozzi et al., 1991). Sometimes referred to as ab initio molecular dynamics, DFT‐MD allows computing elastic properties at pressures and temperatures that span those in Earth’s mantle (Oganov et al., 2001; Stackhouse et al., 2005b). Within the limitations imposed by the finite sizes of systems that can be simulated, DFT‐MD includes anharmonic effects that go beyond the approximation of atomic vibrations as harmonic oscillations and become discernible at high temperatures (Oganov et al., 2001; Oganov and Dorogokupets, 2004, 2003). Alternatively, the frequencies of lattice vibrations can be derived from DFPT for a given volume (Baroni et al., 2001, 1987b) and then used in the quasi‐harmonic approximation (QHA) to compute temperature‐dependent elastic properties (Karki et al., 1999, 2000; Wentzcovitch et al., 2004, 2006, 2010a). The pressure‐temperature space for which the QHA remains valid for a given material can be estimated from the inflexion points (∂²α/∂T²)_P = 0 in computed curves of the thermal expansivity α(P, T) as the QHA appears to overestimate thermal expansivities at higher temperatures (Carrier et al., 2007; Karki et al., 2001b; Wentzcovitch et al., 2010b). This criterion suggests that the QHA should remain valid throughout most of Earth’s mantle for some materials while others are expected to deviate from purely harmonic behavior (Wentzcovitch et al., 2010b; Wu & Wentzcovitch, 2011). The results of QHA‐DFT computations can be corrected for anharmonic contributions by adding a semi‐empirical correction term to match experimental observations (Wu and Wentzcovitch, 2009). Anharmonic effects can also be addressed in DFPT computations (Baroni et al., 2001; Oganov & Dorogokupets, 2004). For example, a recent study on MgO combined DFPT calculations with infrared spectroscopy and IXS to relate experimentally observed indications of anharmonicity, such as phonon line widths, to multi‐phonon interactions (Giura et al., 2019). Such efforts demonstrate the possibility to assess anharmonicity in first‐principle calculations. Among other advancements, all these developments facilitate the routine application of DFT computations to study the elastic properties of structurally and chemically complex minerals at high pressures and high temperatures (Kawai and Tsuchiya, 2015; Shukla et al., 2015; Wu et al., 2013; Zhang et al., 2016).

3.5 PARAMETER UNCERTAINTIES

The results of a series of experiments or computations are typically inverted into sets of parameters that describe the variation of elastic properties as a function of finite strain and temperature. Different definitions of finite strain result in different functional forms of expressions for elastic properties. For equations of state, different definitions and assumptions give rise to a remarkable diversity in EOS formulations (Angel, 2000; Angel et al., 2014; Holzapfel, 2009; Stacey & Davis, 2004). The variation of components of the elastic stiffness tensor has been formulated using both the Lagrangian and Eulerian definitions of finite strain (Birch, 1947; Davies, 1974; Thomsen, 1972a). For interpolations between individual observations, i.e., experiments or computations, differences between formulations should be small since, in a semi‐empirical approach, finite‐strain parameters are chosen to best reproduce the observations. Extrapolations of elastic properties beyond observational constraints, however, can be extremely sensitive to the chosen formalism. Both different definitions of finite strain and expansions to different orders of finite strain can result in substantial deviations between different finite‐strain models when extrapolated beyond observational constraints. Owing to the prevalence of elasticity formalisms based on Eulerian finite strain (Davies & Dziewonski, 1975; Ita & Stixrude, 1992; Jackson, 1998; Sammis et al., 1970; Stixrude & Lithgow‐Bertelloni, 2005), this source of uncertainty is not commonly taken into account when computing seismic properties and tends to become less important as experiments and computations are being pushed towards the verges of the relevant pressure–temperature space. Additional uncertainties arise from extrapolating the thermal or vibrational properties of minerals beyond the limitations of underlying assumptions, such as the quasi-harmonic approximation. For example, anharmonic contributions to elastic properties at high temperatures are mostly ignored as they remain difficult to assess in experiments and computations. Cobden et al. (2008) have explored how different combinations of finite‐strain equations and thermal corrections, including anelastic contributions, affect the outcomes of mineral‐physical models.

      Whether derived from experiments or computations, elastic properties are affected by uncertainties that need to be propagated into the uncertainties on finite‐strain parameters. Inverting elasticity data on a limited number of pressure–temperature combinations to find the optimal set of finite‐strain parameters will inevitably result in correlations between finite‐strain parameters. The uncertainties on derived finite‐strain parameters are only meaningful when the uncertainties on the primary data have been assessed correctly, a requirement that can be difficult to meet in particular for first‐principle computations. Uncertainties on modeled seismic properties of rocks arise from uncertainties on individual finite‐strain parameters and from the anisotropy of the rock‐forming minerals as captured by the bounds on the elastic moduli. When constructing mineral‐physical models, uncertainties have been addressed by varying the parameters that describe the elastic properties of minerals in a randomized way within their individual uncertainties (Cammarano et al., 2003; Cammarano et al., 2005a; Cobden et al., 2008). While capturing the combined variance of the models, randomized sampling of parameters cannot disclose how individual parameters or properties affect the model. Identifying key properties might help to define future experimental and computational strategies to better constrain the related parameters in mineral‐physical models.

      To illustrate how modeled seismic properties of mantle rocks are affected by different sources of uncertainties, I first concentrate on the properties of monomineralic and isotropic aggregates of major minerals in Earth’s upper mantle, transition zone, and lower mantle, i.e., olivine, wadsleyite and ringwoodite, and bridgmanite. For these minerals, complete elastic stiffness tensors have been determined together with unit cell volumes for relevant compositions and at relevant pressures. Such data sets can be directly inverted for the parameters of the cold parts of finite‐strain expressions for the components of the elastic stiffness tensor. For olivine compositions with Mg/(Fe+Mg) = 0.9, i.e., San Carlos olivine, high‐pressure elastic stiffness tensors at room temperature (Zha et al., 1998) can be combined with recent experiments at simultaneously high pressures and high temperatures (Mao et al., 2015; Zhang & Bass, 2016) to self‐consistently constrain most anisotropic finite‐strain parameters. For wadsleyite and ringwoodite, recent experimental results on single crystals (Buchen et al., 2018b; Schulze et al., 2018) are combined with tabulated parameters for the isotropic thermal contributions (Stixrude & Lithgow‐Bertelloni, 2011). Similarly, high‐pressure elastic stiffness tensors of bridgmanite at room temperature (Kurnosov et al., 2017) are complemented with results of DFT computations (Zhang et al., 2013) and high‐pressure high‐temperature experiments on polycrystals (Murakami et al., 2012) that constrain the isotropic thermal contributions.

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