Thermal conductivity of spinels and olivines from vibrational spectroscopy: Ambient conditions

Abstract

The damped harmonic oscillator model for thermal conductivity of insulators is improved, lead­ing to a formula that predicts thermal conductivity at ambient conditions (k 0 ) from various physical properties, most of which are commonly measured. Specifically, k 0 = [ρ/(3ZM)] C v [(u P + u s )/2] 2 / <Γ>, where ρ is density, Z is the number of formula units in the primitive unit cell, M is the molar weight, C v is heat capacity, u is the sound speed (P denotes compression; S denotes shear), and <Γ> is the average of the damping coefficients determined from peak widths in infrared reflectivity spec­tra, or from suitable Raman and Brillouin spectra. The classical physics and quantum-mechanical basis for this model is discussed, with emphasis on the effect of phonon-phonon interactions on mode properties. The calculated values of k 0 all lie within the experimental uncertainty of the mea­surements for all samples with the spinel or olivine structure examined by Horai (1971) with known or approximately correct chemical compositions. Other divergent measurements of k for MgAl 2 O 4 are discounted for various reasons. Early studies of Fe-bearing spinels are not generally reliable, but rough estimates from the above equation are consistent with all data, and good agreement is ob­tained for samples such as Mg 0.5 Fe 0.5 Al 2 O 4 and γ-Fe 2 SiO 4 for which the previous authors obtained chemical data, and for which IR reflectivity data exist. The theory reproduces the measured depen­dence of k 0 on composition and structure. Anisotropy in k 0 results mainly from differences in lattice constants (j): the equation for olivine is kj/k 0 =(V ⅓ /j) 0.73 which predicts the ratios within 3%. For solid solutions between Fe and Mg, the model provides a non-linear dependence of k 0 on mol% Fe, with the damping coefficient being the key factor producing non-linearity. Predicted ambient values are 11.3 ± 0.4 W/m-K for γ-Mg 2 SiO 4 , 6.5 ± 0.7 W/m-K for γ-Mg 12 Fe 0.8 SiO 4 , and 6.9 ± 0.3 W/m-K for β-Mg 2 SiO 4 . The high k 0 for ringwoodite suggests that heat in Earth’s transition zone should be conducted twice as efficiently as in the adjacent upper and lower mantles: this discontinuous depth dependence of k could impact thermal models of conduction in subducting slabs and of mantle convection