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

Abstract

The bulk modulus KT can be related to structural parameters and to the sum of the squares of the vibrational frequencies. ∑vi2, 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 KT and sums involving vi2 and the first or second pressurderivative of vi, 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 KT(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 ∑vi2. 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 KT for the garnets is 1.367 time experimental KT, 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 KT 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 AXBY.

Redefinition of the mode Grüneisen parameter for polyatomic substances and thermodynamic implications

“…Computation of bulk moduli at pressure from IR spectroscopic measurements reproduced elastic and volumetric data for NaCl and other alkali halides (44) even though one of the assumptions is violated (second nearest neighbors are important in the interatomic potentials). This result is puzzling in that KЈ in not accurately calculated from ␥ i for silicates (5), for which nearest-neighbor interactions seem to dominate. Reliable calculation of KЈ from IR data for polyatomic structures requires that the structure scale on compression, which is difficult to meet for minerals with Si in tetrahedral coordination.…”

“…To address such problems and to extrapolate the available data, pressure derivatives of C V , thermal expansivity (␣), K T , and k are obtained from quasiharmonic models that use the measured pressure dependence of the vibrational frequencies (e.g., refs. [3][4][5][6].…”

Vibrational spectroscopy of aluminate spinels at 1 atm and of MgAl2O4 to over 200 kbar

“…[Kpoly of 7--Ni2SiO~ is >2.5 Mbar for the Si tetrahedra and 1.7 Mbar for the Ni octahedra, where as for MgA1204, K is 1.2 Mbar for the Mg tetrahedron but 2.6 for the A1 octahedron (Finger et al 1979(Finger et al , 1986). ] The relative sizes of the mode gammas are to be expected because the bulk modulus is directly connected to the vibrational frequencies (Brout 1959) and K' is proportional to their mode Grfineisen parameters (Hofmeister 1991). These relationships suggest that Kpoly should be connected with vibrational Note that the 647 cm-i sideband mode (Fig.…”

“…Thus, complete and accurate characterization of a solid provides insight into microscopic as well as macroscopic behavior and offers a means to quantitatively compare aluminate to silicate spinels. Second, a model is now available to calculate bulk modulus K and its pressure derivatives K' and K" from vibrational frequencies and their pressure dependences (Hofmeister 1991). Third, sideband fluorescence data (e.g.…”

Thermodynamic properties of MgSiO3 ilmenite from vibrational spectra

“…One possibility is that the 799 cm -1 band is an unresolved doublet. However, based on compositional trends of Raman peaks of all ilmenites measured so far (ZnTiO3, CdTiO3, MgGeOa, ZnSiO3 and ZnGeO3: Botto and Baran 1979; Ross and Navrotsky 1988;Leinenweber et al 1989) the 683 cm-1 band in MgSiO3 is the highest frequency Eg mode and the 620 cm-1 band is an unresolved doublet (Hofmeister 1991). With this correlation, Raman bands are predicted with an average difference of 7.2% from experiment, compared to a larger difference of 11.0% of IR bands.…”