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.