can be used to analyze experimentally observed or computed spontaneous strains or elastic properties to constrain the parameters A, B, ⋯ and the coupling coefficients λij,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):
