Читать книгу Mantle Convection and Surface Expressions - Группа авторов - Страница 66

The finite‐strain formalism outlined in Section 3.2 describes a smooth variation of elastic properties and pressure with finite strain and temperature. Phase transitions interrupt these smooth trends as, at the phase transition, a new phase with different elastic properties becomes stable. The series of phase transitions in (Mg,Fe)₂SiO₄ compounds from olivine (α) to wadsleyite (β) to ringwoodite (γ), for example, results in abrupt changes of elastic properties and density when going from one polymorph to another. Such abrupt changes in elastic properties typically result from first‐order phase transitions that involve a reorganization of the atomic structure of a compound. The elastic properties of the compound are then best described by constructing a finite‐strain model for each polymorph. If the phase transition consists of a gradual distortion of the crystal structure, for example by changing the lengths or angles of chemical bonds, the elastic properties and density of the compound may vary continuously across the phase transition. In this case, it is often possible to describe the effect of the phase transition by adding an excess energy contribution to the energy of the undistorted phase. The changes in elastic properties that result from the phase transition can then be calculated from the excess energy and added to the finite‐strain contribution of the undistorted phase.

Ferroelastic phase transitions are a common type of continuous phase transitions that can lead to substantial anomalies in elastic properties (Carpenter & Salje, 1998; Wadhawan, 1982). Upon cooling or compression across the transition point, a high‐symmetry phase spontaneously distorts into a phase of lower crystal symmetry. Along with the reduction in symmetry, the crystallographic unit cell changes shape, giving rise to spontaneous strains that describe the distortion of the low‐symmetry phase with respect to the high‐symmetry phase. Many minerals undergo ferroelastic distortions including the high‐pressure phases stishovite (Andrault et al., 1998; Carpenter et al., 2000; Karki et al., 1997b; Lakshtanov et al., 2007) and calcium silicate perovskite (Gréaux et al., 2019; Shim et al., 2002; Stixrude et al., 2007; Thomson et al., 2019). The excess energy associated with ferroelastic phase transitions can be described using a Landau expansion for the excess Gibbs free energy (Carpenter, 2006; Carpenter & Salje, 1998):

The first part of this expansion gives the energy contribution that arises from structural rearrangements which drive the phase transition. These rearrangements may be related to changes in the ordering of cations over crystallographic sites, in the vibrational structure, or in other properties of the atomic structure. The progress or extent of these rearrangements is captured by the order parameter Q. The last term gives the elastic energy associated with distorting the high‐symmetry phase with the elastic stiffness tensor into the low‐symmetry phase according to the spontaneous strains e_ij. Coupling between the order parameter Q and the spontaneous strains e_ij is taken into account by the central term with coupling coefficients λ_ij,m,n. The exact form and order of the coupling terms follow strict symmetry rules (Carpenter et al., 1998; Carpenter & Salje, 1998).

In the high‐symmetry phase, Q = 0 and e_ij = 0 so that the excess energy vanishes. At the transition point, the high‐symmetry phase spontaneously distorts to the low‐symmetry phase, and both the order parameter and the spontaneous strains evolve away from zero. By following a sequence of thermodynamic arguments, expressions for the variation of the order parameter Q, the spontaneous strains e_ij, and individual components of the elastic stiffness tensor c_ijkl with pressure and temperature can be derived (Carpenter & Salje, 1998). These expressions can be used to analyze experimentally observed or computed spontaneous strains or elastic properties to constrain the parameters A, B, ⋯ and the coupling coefficients λ_ij,m,n. When all parameters are known, the excess elastic properties can be calculated. For example, elastic anomalies that arise from the ferroelastic phase transitions from stishovite to CaCl₂‐type SiO₂ and from cubic to tetragonal calcium silicate perovskite have been assessed in this way (Buchen et al., 2018a; Carpenter et al., 2000; Stixrude et al., 2007; Zhang et al., 2021).

As the Landau excess energy is typically defined in terms of the Gibbs free energy, the excess elastic properties will be functions of pressure and temperature. In finite‐strain theory, however, the variation of elastic properties is formulated in terms of finite strain and temperature. One way to couple Landau theory to finite‐strain theory consists in replacing the pressure P in excess terms by an EOS of the form P(V, T) (Buchen et al., 2018a). Alternatively, the excess energy can be defined in terms of the Helmholtz free energy with finite strain and temperature as variables (Tröster et al., 2014, 2002). Both approaches have been used to analyze pressure‐induced ferroelastic phase transitions (Buchen et al., 2018a; Tröster et al., 2017).

Compression‐induced changes in the electronic configuration of ferrous and ferric iron give rise to another class of continuous phase transitions, often referred to as spin transitions. Spin transitions are associated with substantial elastic softening, mainly of the bulk modulus, and have been the subject of numerous experimental and computational studies (see Lin et al., 2013 and Badro, 2014 for reviews). Most iron-bearing phases that are relevant to Earth’s lower mantle have been found to undergo spin transitions, including ferropericlase (Badro et al., 2003; Lin et al., 2005), bridgmanite (Badro et al., 2004; Jackson et al., 2005; Li et al., 2004), and the hexagonal aluminum‐rich (NAL) and calcium ferrite‐type aluminous (CF) phases (Wu et al., 2017, 2016). In addition to numerous studies on the EOS of these phases, the variation of sound wave velocities across spin transitions has been directly probed by experiments for ferropericlase (Antonangeli et al., 2011; Crowhurst et al., 2008; Lin et al., 2006; Marquardt et al., 2009b, 2009c; Yang et al., 2015) and recently for bridgmanite (Fu et al., 2018). In parallel to experimental efforts, the effects of spin transitions on elastic properties and potential seismic signatures have been evaluated by DFT computations, e.g., Fe²⁺ in ferropericlase (Lin and Tsuchiya, 2008; Muir and Brodholt, 2015b; Wentzcovitch et al., 2009; Wu et al., 2013, 2009; Wu and Wentzcovitch, 2014) and Fe³⁺ in bridgmanite (Muir and Brodholt, 2015a; Shukla et al., 2016; Zhang et al., 2016). Despite substantial progress in understanding the effect of spin transitions on elastic properties, discrepancies remain between computational and experimental studies at ambient temperature (Fu et al., 2018; Shukla et al., 2016; Wu et al., 2013) and in particular between computational (Holmström & Stixrude, 2015; Shukla et al., 2016; Tsuchiya et al., 2006; Wu et al., 2009) and experimental (Lin et al., 2007; Mao et al., 2011) studies that address the broadening of spin transitions at high temperatures.

Common to most descriptions of elastic properties across spin transitions is the approach to treat phases with iron cations in exclusively high‐spin and exclusively low‐spin configurations separately and to mix their elastic properties across the pressure–temperature interval where both electronic configurations coexist (Chen et al., 2012; Speziale et al., 2007; Wu et al., 2013, 2009). We will see below that the electronic structure of transition metal cations in crystal structures is more complex than this simple two‐level picture and how measurements at room temperature can be exploited to construct more detailed models. Most approaches to spin transitions are based on the Gibbs free energy and hence describe elastic properties as functions of pressure and temperature (Speziale et al., 2007; Tsuchiya et al., 2006; Wu et al., 2013, 2009). Coupling a thermodynamic description of spin transitions to finite‐strain theory, however, requires to express the changes in energy that result from the redistribution of electrons in terms of volume strain. Building on the ideas of Sturhahn et al. (2005), a formulation for compression‐induced changes in the electronic configurations of transition metal cations can be proposed that accounts for energy changes in terms of an excess contribution to the Helmholtz free energy. Crystal‐field theory proves to provide the right balance between complexity and flexibility for a semi‐empirical and strain‐dependent model for the electronic structure of transition metal cations in crystal structures.

With the aim to interpret the optical absorption spectra of octahedrally coordinated first‐row transition metal cations, Tanabe and Sugano (1954a) described the energies of multi‐electron states resulting from different 3d electron configurations in terms of the crystal‐field splitting Δ = 10Dq and the Racah parameters B and C. The electric field generated by the negative charge of an octahedral coordination environment causes the 3d orbitals of the central transition metal cation to split into the e and t₂ levels, separated by an energy equal to the crystal‐field splitting Δ. The Racah parameters B and C account for the interelectronic repulsion between d electrons that results from a given distribution of electrons over the e and t₂ orbitals. In the strong‐field limit, as appropriate for the treatment of compression‐induced changes in the electronic configuration, the energy of a multi‐electron state can be approximated by a sum of the form (Tanabe and Sugano, 1954a):

Each multi‐electron state is characterized by the symmetry of the electron distribution as expressed by the symbol of the corresponding irreducible representation Γ and by the spin multiplicity M = 2S + 1 with the sum S of unpaired electrons. The coefficients z₁, z₂, and z₃ depend on the number of d electrons and have been calculated and tabulated for each state (Tanabe & Sugano, 1954a). Figure 3.4 shows how the 3d electron configurations d⁵ (Fe³⁺) and d⁶ (Fe²⁺) give rise to multi‐electron states and how their energies vary as a function of the ratio Δ/B (Tanabe and Sugano, 1954b). We will see below that the ratio Δ/B increases with compression.

To incorporate the changes in state energies that result from compression, I assume the crystal‐field parameters to scale with the volume ratio V₀/V as:

An electrostatic point charge model for an octahedrally coordinated transition metal cation suggests that δ = 5 (Burns, 1993). High‐pressure spectroscopic measurements, however, have shown that real materials may deviate from this prediction (Burns, 1985; Drickamer & Frank, 1973). Enhanced covalent bonding might also lead to a reduction of the Racah B parameter with increasing compression (Abu‐Eid & Burns, 1976; Keppler et al., 2007; Stephens & Drickamer, 1961a, 1961b) and would imply b < 0.

Most multi‐electron states are degenerate and allow a number m > 1 of different electron configurations. The total degeneracy of a given state ^MΓ is then given by the product mM and contributes a configurational entropy equal to kln(mM), where k is the Boltzmann constant. The Helmholtz free energy of a given state can then be expressed as (Badro et al., 2005; Sturhahn et al., 2005):

Figure 3.4 Energy diagrams for multi‐electron states that arise from the 3d electron configurations d⁵ (Fe³⁺) (a) and d⁶ (Fe²⁺) (b) in the limit of strong octahedral crystal fields. Note the change in electronic ground state for each electron configuration when the ratio Δ/B reaches the value marked by a vertical line. The distributions of d electrons over the crystal‐field orbitals t₂ and e are illustrated for the three states that have the lowest energies at the change in electronic ground state.

At T > 0, and in particular at temperatures of Earth’s mantle, the d electron configuration can be thermally excited into states of higher energies. The thermal population of higher energy states becomes most important when the energies of other states approach the energy of the ground state, for example, as a result of compression. A spin transition results from a compression‐induced change of the electronic ground state from a high‐spin to a low‐spin state (Figure 3.4). At finite temperatures, the broadening of the transition reflects the incipient population of the low‐spin state before and the decaying population of the high‐spin state after their energies cross over as long as the energies of both states remain close enough to allow for thermally induced transitions between both states (Holmström & Stixrude, 2015; Sturhahn et al., 2005; Tsuchiya et al., 2006). However, states other than the high‐spin or low‐spin state may get close enough in energy to the ground state to become thermally populated by a significant fraction of d electrons. The thermal distribution of electrons over accessible multi‐electron states contributes to the overall configurational entropy. The excess contribution to the Helmholtz free energy per transition metal cation that results from compression‐ and temperature‐induced changes in the occupation of electronic states can then be identified as:

where Δϕ = ϕ(V, T) − ϕ₀(V₀, T₀) are the changes in the fractions ϕ of d electrons that occupy each electronic state at a given volume and temperature with respect to the fractions ϕ₀ at ambient conditions. Note that the configurational entropy term is multiplied by the number d of d electrons per transition metal cation. The fraction of d electrons that occupy a given state can be found by applying the equilibrium condition (∂F_EL/∂n_i)_V,T,N = 0 to a micro‐canonical ensemble with the absolute numbers n_i = ϕ_iN of electrons in each state and the total number N of electrons (Sturhahn et al., 2005):

The denominator sums over all multi‐electron states being considered.

The excess contributions to pressure and bulk modulus then follow from the definitions:

and

All excess quantities need to be multiplied by the number and fraction of crystallographic sites that are occupied by the transition metal cation.

The parametrization of spin transitions presented above builds on the formulation proposed by Sturhahn et al. (2005) and resembles earlier attempts to predict the effect of spin transitions in Fe²⁺ on thermodynamic properties of mantle minerals (Badro et al., 2005; Gaffney, 1972; Gaffney & Anderson, 1973; Ohnishi, 1978). Unlike these forward modeling approaches, I will use the formulation here in a semi‐empirical way with adjustable parameters that will be constrained by experimental observations. In contrast to formulations based on the Gibbs free energy with pressure as variable and elastic compliances as parameters that have proven useful in describing the results of DFT computations (Shukla et al., 2016; Wentzcovitch et al., 2009; Wu et al., 2013), a formulation of excess contributions in terms of volume and temperature is more consistent with finite‐strain theory based on the assumption of homogeneous and isotropic finite strain and can, in principle, be generalized to evaluate the effect on individual components of the elastic stiffness tensor and hence on the shear modulus. The impact of spin transitions on the shear modulus, however, appears to be minor (Fu et al., 2018; Marquardt et al., 2009b; Shukla et al., 2016; Wu et al., 2013). Note that by accounting for changes in the electronic configurations in terms of excess properties it is no longer necessary to derive sets of finite‐strain parameters for each individual electronic configuration as required by earlier formulations based on high‐spin and low‐spin states only (Chen et al., 2012; Speziale et al., 2007; Wu et al., 2013, 2009), since the excess contributions are simply added to the (cold) elastic and thermal contributions. Although I assume here that transition metal cations with different electronic configurations mix ideally among each other and with other cations, it is in principle possible to add further terms to account for nonideal mixing behavior and interactions between transition metal cations (Holmström and Stixrude, 2015; Ohnishi and Sugano, 1981; Sturhahn et al., 2005). To some extent and for low concentrations of transition metal cations, deviations from ideal mixing will be captured by the semi‐empirical parameters when fit to experimental data.

By taking into account the three electronic states with lowest energies for ratios Δ/B close to the compression‐induced change in electronic ground state for ferrous iron, Fe²⁺ (d⁶), and ferric iron, Fe³⁺ (d⁵), in octahedral coordination, i.e., ⁵T₂, ¹A₁, and ³T₁ for Fe²⁺ and ⁶A₁, ²T₂, and ⁴T₁ for Fe³⁺ (Tanabe & Sugano, 1954a, 1954b), I analyzed recent experimental result on the elastic properties of ferropericlase (Yang et al., 2015), Fe³⁺‐bearing bridgmanite (Chantel et al., 2012; Fu et al., 2018), and Fe³⁺‐bearing CF phase (Wu et al., 2017) across their respective spin transitions. When fitting experimental data, I assumed a constant ratio C/B = 4.73 (Krebs & Maisch, 1971; Lehmann & Harder, 1970; Tanabe & Sugano, 1954b) and consequently set b = c. The crystal‐field splitting of Fe²⁺ and Fe³⁺ in octahedral coordination in periclase and corundum, respectively, has been derived from optical spectroscopy (Burns, 1993; Krebs and Maisch, 1971; Lehmann and Harder, 1970) and from early quantum‐mechanical computations (Sherman, 1991, 1985). Hence, I assumed Δ₀ = 10800 cm^–1 for Fe²⁺ in ferropericlase (Burns, 1993; Sherman, 1991) and adopted Δ₀ = 14750 cm^–1 for Fe³⁺ in corundum as approximation for Fe³⁺ in bridgmanite and in the CF phase (Krebs & Maisch, 1971; Lehmann & Harder, 1970; Sherman, 1985). Similarly, the Racah B₀ parameter at ambient conditions has been determined for octahedrally coordinated Fe³⁺ in corundum from optical spectra (Krebs & Maisch, 1971; Lehmann & Harder, 1970) and was set to B₀ = 655 cm^–1 for Fe³⁺ in the CF phase. The volume exponent δ was initially set to δ = 5 as suggested by the point charge model. The exponents c = b and, when required, δ and B₀ were treated as adjustable parameters in addition to the finite‐strain parameters that describe the compression‐induced changes of elastic moduli without excess contributions.

The fitting results are shown in Figures 3.5a–c and demonstrate that the semi‐empirical model captures the softening of the bulk modulus of ferropericlase (Yang et al., 2015), the dip in P‐wave velocities of Fe³⁺‐bearing bridgmanite (Fu et al., 2018), and the segment of enhanced volume reduction in the compression curve of the CF phase (Wu et al., 2017) that have been interpreted to result from compression‐induced changes in the electronic structures of ferrous and ferric iron. The Racah B₀ parameter found for Fe²⁺ in ferropericlase is compatible with values derived from optical spectroscopy (Burns, 1993; Tanabe & Sugano, 1954b). For all three data sets, −3 < b = c < −2 indicating a decrease of the Racah B parameter with compression as suggested by results from high‐pressure optical spectroscopy (Abu‐Eid & Burns, 1976; Keppler et al., 2007; Stephens & Drickamer, 1961a, 1961b). It is important to note, however, that the exponents b = c are positively correlated with the exponent δ that was fixed at δ = 5 for ferropericlase and the CF phase. A combination of slightly higher values for δ with less negative values of b and c can explain the observations equally well, suggesting that the difference δ − b might be more meaningful than the individual parameters. The P‐wave velocity data for bridgmanite required significantly higher values for the exponent δ and the Racah B₀ parameter than suggested by the point charge model or optical spectroscopy. The very large exponent δ for bridgmanite might reflect the different compression behaviors of A and B sites in the perovskite crystal structure that might not be related in a simple way to the compression mechanism of the crystal structure as a whole and to the ratio V₀/V of unit cell volumes (Boffa Ballaran et al., 2012; Glazyrin et al., 2014). Distortions of the coordination environment away from an ideal octahedron will also result in crystal‐field parameters that deviate from their values for more regular and symmetric arrangements of coordinating anions. However, the general consistency between crystal‐field parameters from optical spectroscopy when used in the semi‐empirical model for electronic excess properties and high‐pressure experimental data on elastic properties, in particular for close‐packed oxide structures, may motivate further testing and development of the model.

Figures 3.5d–f show the predicted fractions ϕ of d electrons that occupy each of the considered multi‐electron states for Fe²⁺ in ferropericlase and Fe³⁺ in bridgmanite and in the CF phase along different adiabatic compression paths. The change in electronic ground states from ⁵T₂ (high spin) to ¹A₁ (low spin) for Fe²⁺ and from ⁶A₁ (high spin) to ²T₂ (low spin) for Fe³⁺ is gradual and broadens with increasing temperatures as suggested earlier (Holmström & Stixrude, 2015; Lin et al., 2007; Sturhahn et al., 2005; Tsuchiya et al., 2006). The crystal‐field model outlined above, however, predicts additional broadening that results from thermal population of the higher energy states ³T₁ for Fe²⁺ and ⁴T₁ for Fe³⁺. At realistic mantle temperatures, these states are predicted to host up to 25% of d electrons. Population of these states will reduce the effect of spin transitions on mineral densities and elastic properties by diluting the contrasts in properties between pure high‐spin and low‐spin states. The spin transition of Fe²⁺ in ferropericlase appears to be most susceptible to thermal broadening while spin transitions of Fe³⁺ in bridgmanite and in the CF phase remain somewhat sharper even at high temperatures.

The effect of spin transitions on P‐wave velocities is shown in Figures 3.5g–i as relative velocity reductions along typical adiabatic compression paths. In qualitative agreement with results of DFT computations for ferropericlase and Fe³⁺‐bearing bridgmanite (Shukla et al., 2016; Wentzcovitch et al., 2009; Wu et al., 2013), both the pressure interval and the pressure of maximum P‐wave velocity reduction increase with temperature. Absolute velocity reductions and their exact pressure intervals at high temperatures as predicted by DFT computations, however, seem to differ from those predicted by the semi‐empirical crystal‐field model. Figures 3.5g–i also show how ignoring the population of the higher energy states ³T₁ for Fe²⁺ and ⁴T₁ for Fe³⁺ would overestimate P‐wave velocity reductions at realistic mantle temperatures. Although I considered only one additional state for each Fe²⁺ and Fe³⁺, more high‐energy states might become populated at relevant temperatures, further depleting high‐spin and low‐spin ground states. Experiments at combined high pressures and high temperatures are needed to directly assess the thermal broadening of spin transitions and their effects on mineral elasticity. Since the volume changes that result from spin transitions can be subtle, in particular at high temperatures and for typical iron contents of mantle minerals (Komabayashi et al., 2010; Mao et al., 2011), experiments that constrain elastic properties in addition to volume might be best suited to resolve the impact of spin transitions on sound wave velocities at high temperatures.

Figure 3.5 (a–c) Reanalysis of elastic moduli of ferropericlase (a), sound wave velocities of bridgmanite (b), and compression data on the CF phase (c) across spin transitions of ferrous (a) and ferric (b,c) iron. Respective data are from Yang et al. (2015) (a), Chantel et al. (2012), and Fu et al. (2018) (b), and Wu et al. (2017) (c). Bold black curves show the results of fitting a semi‐empirical crystal‐field model to the data as explained in the text with the respective crystal‐field parameters given in each panel. The values of crystal‐field parameters that were free to vary during fitting are marked with an asterisk (*). For bridgmanite, data of Chantel et al. (2012) and Fu et al. (2018) have been analyzed together to better constrain pressure derivatives of elastic moduli. The offset between both data sets arises from slightly different estimates for densities as reported in both studies. (d–f) Fractions of d electrons in multi‐electron states across spin transitions of Fe²⁺ in ferropericlase (d), Fe³⁺ in bridgmanite (on B site) (e), and Fe³⁺ in the CF phase (f) as predicted by the semi‐empirical crystal‐field model and along typical adiabatic compression paths (see Figure 3.8). Shading indicates differences in fractions that result from starting adiabatic compression at temperatures 500 K above and below 1900 K at 25 GPa. (g–i) P‐wave velocity reductions that result from spin transitions of Fe²⁺ in ferropericlase (g), Fe³⁺ in bridgmanite (on B site) (h), and Fe³⁺ in the CF phase (i) as predicted by the semi‐empirical crystal‐field model and along adiabatic compression paths starting at 1400 K, 1900 K, and 2400 K at 25 GPa (see Figure 3.8). The dashed curves show P‐wave velocity reductions along the central compression path (1900 K at 25 GPa) when the population of a third multi‐electron state with intermediate spin multiplicity is ignored.