Finite-Strain Formulation of High-Pressure Elastic Properties

We follow the self-consistent finite-strain formalism as developed by Stixrude and Lithgow-Bertelloni (2005) but restrict the discussion to those relations relevant for the inversion procedure. Accordingly, we use the Eulerian definition of the finite-strain tensor E_{i j} and assume the finite strain that results from hydrostatic compression to be isotropic, i. e. E_{i j} =

−f_Eδi j with f_E=

(V₀/V)^2/3−1

/2 and the unit cell volumes V in the strained state andV₀ at ambient conditions. For a truncation of the Helmholtz free energy expansion after the fourth-order term in finite strain (Stixrude and Lithgow-Bertelloni, 2005), the components of the adiabatic elastic stiffness tensor vary with finite strain and temperature as (Stixrude and Lithgow-Bertelloni, 2005):

functions of finite strain (Stixrude and Lithgow-Bertelloni, 2005). The thermal part of the internal energyU_QHand the isochoric heat capacityC_Vare computed from a quasi-harmonic Debye model (Ita and Stixrude, 1992) with a finite-strain-dependent Debye temperature (Ita and Stixrude, 1992; Stixrude and Lithgow-Bertelloni, 2005). At a given finite strain,∆U_QH and∆(C_VT)are the changes in the thermal part of the internal energyU_QHand in the prod-uct of isochoric heat capacity C_V and temperatureT,C_VT, respectively, with respect to the reference temperature.

Expressions (5.18) to (5.20) give the second-order (5.18), third-order (5.19), and fourth-order (5.20) contributions of finite strain to the cold part of the respective compo-nent of the elastic stiffness tensor. As we assumed the finite strain to be isotropic, the bulk modulus in expressions (5.18) to (5.20) is calculated from the components of the isother-mal elastic stiffness tensor according to the Voigt bound (Watt et al., 1976; Stixrude and Lithgow-Bertelloni, 2005):

K₀=c_{i jkl0}δi jδkl/9 (5.23)

Analogous expressions can be written down for the pressure derivativesK₀⁰andK₀⁰⁰by replac-ing the components of the elastic stiffness tensor with their respective pressure derivatives

c⁰_{i jkl0} and c_{i jkl0}⁰⁰ . Expression (5.21) gives the thermal contribution and expression (5.22)

the isothermal-to-adiabatic conversion (Wehner and Klein, 1971; Davies, 1974). In the following, we will mainly focus on the cold part of the c_{i jkl}. More details on the thermal contributions can be found in Stixrude and Lithgow-Bertelloni (2005).

Based on expressions (5.18) to (5.22), the components of the adiabatic elastic stiffness tensor at any finite strain and temperature can be combined to the adiabatic bulk modulus K_S and the shear modulus G. For the Voigt bound, this yields the equations for the bulk and shear modulus as a function of finite strain and temperature given by Stixrude and Lithgow-Bertelloni (2005):

5.2 Outline of the Inversion Procedure

with the respective (isothermal) moduli K₀ and G₀ at ambient conditions and their first (primed) and second (double-primed) pressure derivatives. The isotropic Grüneisen param-eter is related to the Grüneisen tensor by γ =γi jδi j/3 . The derivatives of the Grüneisen parameter with respect to compressional and shear strain, ηB and ηS, are computed from the componentsηi jkl in the same way as the bulk and shear moduli, respectively, from the componentsc_{i jkl} of the elastic stiffness tensor (Watt et al., 1976).

The pressurePcan be linked to finite strain and temperature by the corresponding equa-tion of state (EOS) (Stixrude and Lithgow-Bertelloni, 2005):

P=1

with up to third-order finite-strain contributions in expression (5.33), fourth-order contri-butions in (5.34), and the thermal contribution in (5.35). Note that the EOS (5.33)–(5.34) uses the isothermal bulk modulus K₀ = K_T0 at ambient conditions. At given temperature and finite strain, adiabatic and isothermal bulk moduli are related by (Wehner and Klein, 1971; Davies, 1974):

K_S−K_T =γ²C_VT

V (5.36)

i. e. by subtracting the term (5.28) from the adiabatic bulk modulus. Since the differences between the pressure derivatives of the components of the isothermal and adiabatic elastic stiffness tensors and between the pressure derivatives of the isothermal and adiabatic bulk moduli are typically smaller than experimental uncertainties, we will neglect these differ-ences. As a consequence, adiabatic elastic properties can be analyzed along an isotherm using only the (isothermal) cold parts of the c_{i jkl} and the bulk modulus K_S, i. e. with re-lations (5.18)–(5.20) and (5.24)–(5.26). For pressure calcure-lations, however, the obtained adiabatic bulk modulus at ambient conditions (K₀ in expressions (5.24)–(5.26)) has to be converted to the isothermal modulus before being used in the EOS (5.33)–(5.34).

Expressions (5.18) to (5.22) describe the variation of the components of the elastic stiff-ness tensorc_{i jkl} with finite strain and temperature and can be inserted into equation (5.10) to compute anisotropic sound wave velocities at a given combination of finite strain and

ments of the unit cell volume at the same set of conditions where sound wave velocities are determined. Alternatively, unit cell volumes (or densities) can be indirectly constrained by pressure measurements via the EOS (5.33)–(5.35). In this case, the unit cell volumes (or densities) at all pressure-temperature combinations are treated as adjustable parameters that simultaneously have to satisfy equations (5.10) and (5.33)–(5.35).

The evolution of the elastic stiffness tensor with finite strain and temperature can also be analyzed with expressions (5.18) to (5.22) without linking them to sound wave veloci-ties. Note that individual components of the elastic stiffness tensor are linked to each other through the bulk modulus at ambient conditions (eqn. (5.23)). To maintain self-consistency, the variation of individual c_{i jkl} with pressure and temperature should always be analyzed taking into account expressions (5.18)–(5.22) for all other c_{i jkl} at the same time. For ex-ample, individually adjustingc_{i jkl0} andc_{i jkl0}⁰ for each c_{i jkl} and calculating the bulk modulus and its pressure derivative at ambient conditions with equation (5.23) using the results for

allc_{i jkl} may result in values for the bulk modulus and its pressure derivative that differ from

those initially inserted into expressions (5.18)–(5.22) during the refinement.

Although we focus here on the analysis of single-crystal sound wave velocities, relations (5.24)–(5.28) and (5.29)–(5.32) can be coupled to aggregate sound wave velocities in an analogous way with:

v_P= v u

tK_S+⁴₃G

ρ (5.37)

v_S= v tG

ρ (5.38)

Instead of inverting single-crystal sound wave velocities to single-crystal finite-strain pa-rameters (c_{i jkl0}, c_{i jkl0}⁰ , c⁰⁰_{i jkl0}, etc.), aggregate sound wave velocities can be self-consistently inverted to aggregate finite-strain parameters, i. e. K₀,K₀⁰,G₀,G₀⁰, etc..