Papers on Physics and Chemistry of Minerals and Rocks Volcanology

# Calculation of bulk modulus and its pressure derivatives from vibrational frequencies and mode Grüneisen Parameters: Solids with cubic symmetry or one nearest-neighbor distance

Article first published online: 20 SEP 2012

DOI: 10.1029/91JB01381

Copyright 1991 by the American Geophysical Union.

Issue

## Journal of Geophysical Research: Solid Earth (1978–2012)

Volume 96, Issue B10, pages 16181–16203, 10 September 1991

Additional Information

#### How to Cite

1991), Calculation of bulk modulus and its pressure derivatives from vibrational frequencies and mode Grüneisen Parameters: Solids with cubic symmetry or one nearest-neighbor distance, J. Geophys. Res., 96(B10), 16181–16203, doi:10.1029/91JB01381.

(#### Publication History

- Issue published online: 20 SEP 2012
- Article first published online: 20 SEP 2012
- Manuscript Accepted: 20 MAY 1991
- Manuscript Received: 17 OCT 1989

- Abstract
- References
- Cited By

The bulk modulus *K*_{T} can be related to structural parameters and to the sum of the squares of the vibrational frequencies. ∑v_{i}^{2}, for N-atom crystals of cubic symmetry or for any symmetry that contains only one nearest-neighbor distance. Required assumptions are the existence of (1) electrostatic forces exclusively between atoms at equivalent positions in the primitive unit cell, (2) pair-wise central repulsive potentials between all other atoms, and (3) rigid ions. Including pressure in the derivation requires a more stringent version of point 2, namely, (4) that the structure scales isotropically upon compression. This leads to structurally independent formulas for the pressure derivatives of bulk modulus at 1 atm: the only variables are *K*_{T} and sums involving v_{i}^{2} and the first or second pressurderivative of v_{i}, i.e., the mode Grüneisen parameters. Implicit in all equations is the independence of the sums on wave vector. Thus, knowledge of zone center phonons (mostly infrared and Raman bands) is sufficient to calculate elastic properties. Investigating these relationships for 45 minerals with 10 different structures shows that *K*_{T}(0) is predicted within 7% of experiment for 21 solids, all of which should lack strong interactions between atoms at equivalent sites in the Bravais unit cell (i.e., rutile, corundum, ilmenite, and spinel structures). The accuracy of the model strongly depends on the accuracy of the sum ∑v_{i}^{2}. Agreement for B1, B2, fluorite, and wurtzite structures is moderate to poor, as expected, because these structures place like atoms as second nearest neighbors and can thus violate assumption 1. For rocksalt, agreement varies, depends on the size of the cation, not on the polarizability (i.e. the ionic rigidity) as inferred earlier. Calculated *K*_{T} for the garnets is 1.367 time experimental *K*_{T}, instead of unity, which is related to the structure not scaling perfectly with pressure. Agreement is variable (3–20%) for the perovskite structure with large discrepancies associated with ill-constrained sums. However, for all structures, the relative size of *K*_{T} is predicted, so that the theory can be used for systematics for all minerals with the proper symmetry, reguardless of whether the assumptions are satisified. For all of the above mentioned structures, *K*′_{o} and *K*″_{o} are predicted within the experimental uncertainties from the available spectroscopic measurements of 13 solids for *K*′_{o} and 6 solids for *K*″_{o} at pressure. Based on this, the good agreement for ilmenite, which does not meet the symmetry requirements, and the structural independence of the formulas, the relations involving *K*′_{o} and *K*″_{o} are inferred to hold for structures more complex than cubic or A_{X}B_{Y}.