In search of the mixed derivative ∂² M/∂P∂T (M = G, K): joint analysis of ultrasonic data for polycrystalline pyrope from gas- and solid-medium apparatus

Abstract

Elastic wave velocities for dense (99.8% of theoretical density) isotropic polycrystalline specimens of synthetic pyrope (Mg₃Al₂Si₃O₁₂) were measured to 1,000 K at 300 MPa by the phase comparison method of ultrasonic interferometry in an internally heated gas-medium apparatus. The temperature derivatives of the elastic moduli [(∂Ks/∂T) _P  = −19.3(4); (∂G/∂T) _P  = −10.4(2) MPa K⁻¹] measured in this study are consistent with previous acoustic measurements on both synthetic polycrystalline pyrope in a DIA-type cubic anvil apparatus (Gwanmesia et al. in Phys Earth Planet Inter 155:179–190, 2006) and on a natural single crystal by the rectangular parallelepiped resonance (RPR; Suzuki and Anderson in J Phys Earth 31:125–138, 1983) method but |(∂Ks/∂T) _P | is significantly larger than from a Brillouin spectroscopy study of single-crystal pyrope (Sinogeikin and Bass in Phys Earth Planet Inter 203:549–555, 2002). Alternative approaches to the retrieval of mixed derivatives of the elastic moduli from joint analysis of data from this study and from the solid-medium data of Gwanmesia et al. in Phys Earth Planet Inter 155:179–190 (2006) yield ∂² G/∂P∂T = [0.07(12), 0.20(14)] × 10⁻³ K⁻¹ and ∂² K _S /∂P∂T = [−0.20(24), 0.22(26)] × 10⁻³ K⁻¹, both of order 10⁻⁴ K⁻¹ and not significantly different from zero. More robust inference of the mixed derivatives will require solid-medium acoustic measurements of precision significantly better than 1%.