Redefinition of the mode Grüneisen parameter for polyatomic substances and thermodynamic implications
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Contributed by Hokwang Mao
Abstract
Although the value of the thermal Grüneisen parameter (γ_{th}) should be obtained by averaging spectroscopic measurements of mode Grüneisen parameters [γ_{i} ≡ (K_{T}/ν_{i})∂ν_{i}/∂P, where K_{T} is isothermal bulk modulus, ν is frequency, and P is pressure], in practice, the average 〈γ_{i}〉 is up to 25% lower than γ_{th}. This discrepancy limits the accuracy of inferring physical properties from spectroscopic data and their application to geophysics. The problem arises because the above formula is physically meaningful only for monatomic or diatomic solids. We redefine γ_{i} to allow for the presence of functional groups in polyatomic crystal structures, and test the formula against spinel and olivinegroup minerals that have wellconstrained spectra at pressure, band assignments, thermodynamic properties, and elastic moduli, and represent two types of functional groups. Our revised formula [γ_{i} ≡ (K_{X}/ν_{i})∂ν_{i}/∂P] uses polyhedral bulk moduli (K_{X}) appropriate to the particular atomic motion associated with each vibrational mode, which results in equal values for 〈γ_{i}〉, γ_{th}, and γ_{LA} (the Grüneisen parameter of the longitudinal acoustic mode). Similar revisions lead to the pressure derivatives of these parameters being equal. Accounting for differential compression intrinsic to structures with functional groups improves the accuracy with which spectroscopic models predict thermodynamic properties and link to elastic properties.
The pressure dependencies of thermodynamic, elastic, and transport properties are needed to investigate the Earth's interior. Some limitations exist on direct measurements of the various physical properties. For example, pressure (P) derivatives of the bulk modulus (K_{T}) determined from volume (V) data are uncertain because the strong coupling of K_{T} with K′_{T} = ∂K_{T}/∂P through the equation of state (1). Measurement of the thermal conductivity (k) of hard materials at pressure with methods involving contact is problematic (2). To address such problems and to extrapolate the available data, pressure derivatives of C_{V}, thermal expansivity (α), K_{T}, and k are obtained from quasiharmonic models that use the measured pressure dependence of the vibrational frequencies (e.g., refs. 3–6).
These models are commonly cast in terms of mode Grüneisen parameters: 1 Usually, the pressure derivative is computed because frequency is measured against P, and bulk moduli are known at ambient conditions. This formula is considered to be a definition, as it provides a dimensionless representation of the response to compression. But, to be useful, an appropriate average 〈γ_{i}〉 must reproduce the thermal Grüneisen parameter, 2 which appears in many relationships between thermodynamic and elastic properties. However, averaging γ_{i} values obtained from Eq. 1 typically underestimates γ_{th} (e.g., refs. 7 and 8). For example, for γMg_{2}SiO_{4}, the IR modes (9) give 〈γ_{i}〉 = 0.96 which is substantially lower than γ_{th} = 1.25 calculated from thermodynamic measurements (8). Similarly, Raman modes (8) give 〈γ_{i}〉 = 1.10. The longitudinal acoustic (LA) and transverse acoustic (TA) modes also have associated Grüneisen parameters: 3 and 4 where K_{s} is the adiabatic bulk modulus and G is the shear modulus. All quantities are at ambient conditions (10). The average, γ_{ac} = (γ_{LA} + 2γ_{TA})/2 of 1.03 for ringwoodite, is low (8), as often occurs (10).
In calculating 〈γ_{i}〉, we summed only the optical modes found at the center of the Brillouin zone, accounting for their degeneracies. This approach is correct because where the wave vector q_{j} is 0, acoustic modes have ν_{i} of 0, and thus do not contribute to the zone center sum. In a rigorous determination, the sum should include both acoustic and optic modes, and be averaged over the Brillouin zone. The dependence of optic frequencies on q_{j} (dispersion) is known for only a few silicates, and the existing data obtained through inelastic neutron scattering are not complete (e.g., ref. 12). Hence, the Grüneisen parameters of the acoustic modes have been postulated to represent the thermodynamic average (e.g., ref. 13). This concept stems from the Debye model for heat capacity, wherein the acoustic modes are used to represent all vibrational energy. The mode Grüneisen parameters of longitudinal acoustic modes for many solids resemble γ_{th} whereas the Grüneisen parameters of their transverse acoustic modes are much less (10). This departure of γ_{ac} from γ_{th} was attributed to failure of the Debye model for some classes of solids (13).
The failure of the Debye model has further been attributed to discrepancies between the averages of γ_{i} and γ_{th} (8). This cannot be the reason because the calculation for the optic modes is unrelated to the Debye model. Instead, it is tied to Einstein's dispersionfree model of mode heat capacities (c_{ij}) through 5 (ref. 14), where the summation is over the i vibrational modes and the j wave vectors, x_{ij} = hν_{i}(q_{j})/k_{B}T, where h is Planck's constant, and k_{B} is Boltzmann's constant. The form assumes ν_{i} is independent of temperature. Perhaps this quasiharmonic approximation is at fault; however, a simple average of γ_{i0} for the IR modes gives 0.93 for γMg_{2}SiO_{4}, compared with 0.96 from Eq. 5. Thus, the errors in the weighing factors are too small to explain the discrepancy, and neither the harmonic approximation nor the dispersion relation assumed in calculating the heat capacity is the cause.
For various silicate spinels, the IR spectrum at pressure is simple, complete, and reasonably accurate, yet, γ_{th} is underestimated by 〈γ_{i}〉 by ≈25%. Discrepancies of 10–25% have been observed for complex mineral structures, regardless of whether IR data, Raman data, or both, are used (8). Even silicates for which dν/dP is constrained for all optic modes show such discrepancies (15), and thus a partial sum and lack of information on the inactive modes can be dismissed as the key factors. The final possible source of the discrepancy is if dν/dP depends on q_{i}. This seems unlikely because modes sampled inside the Brillouin zone through sideband spectroscopy (16) give dν/dP comparable to IR and Raman measurements (17) or to Brillouin data (18, 19) near or at zone center. Therefore, we examined the original papers by Grüneisen and realized that discrepancies arise because Eq. 1 does not account for differential compression in structures with functional groups. This paper tests relationships against measurements of dν/dP near zone center. Although, in principle, dispersion relations and their pressure dependence could be calculated from theoretical models, the accuracy of such models for complex solids at this time is not sufficient for such a test.
The Original Derivation
Grüneisen (20, 21) derived 6 for monatomic solids, considering explicitly the volume about the vibrating atom (V_{a}). For monatomic solids, the volume of the unit cell (or the molar volume) is proportional to the atomic volume, leading to Eq. 1. For diatomic solids, e.g., MgO, only one interatomic distance exists and the volume about the cation (or the anion) is proportional to the molar volume, so that Eq. 1 still holds for the optic modes. Structures such as corundum (Al_{2}O_{3}) or stishovite (SiO_{2}) are reasonably approximated by one interatomic distance (5) and may also described by Eq. 1. However, minerals such as quartz have more than one relevant interatomic distance. Such cases, as well as polyatomic structures with three or more different kinds of atoms, have functional groups, e.g., SiO_{4} tetrahedra. Grüneisen (20) stated that structures with functional groups differ in important ways from monatomic substances, but limited his discussion to qualitative aspects.
Generalization of the Definition of the Mode Grüneisen Parameter
We propose that for each vibration in a solid, γ_{i} should be computed by using the volume that changes during the particular atomic motion correlated with each given frequency. This approach follows the original proposal of Grüneisen (20, 21) to the letter. Instead of Eq. 1, we use Eq. 6, slightly modified: 7 where K_{X} is the bulk modulus associated with the volume vibrating (V_{a}).
Optic Modes.
For monatomic solids and many structures with two types of atoms, K_{X} = K_{T}, as discussed above, and Eq. 7 is identical to Eq. 1. For many of the optic modes in polyatomic structures, K_{X} will be a polyhedral bulk modulus = −V_{a}/[dV_{a}/dP] (22), but for optical modes involving complex motions of all atoms, K_{T} pertains. Thus, application of Eq. 7 requires band assignments.
Acoustic Modes.
The LA mode expands and contracts the unit cell; hence, K_{X} = K_{T}, and Eq. 3 is valid. In contrast, the pure shear motion for a TA mode does not change the volume (23); therefore, γ_{TA} is undefined. This behavior differs from that of bending modes or rotations–librations because these optic modes are likely to be impure or mixed. Because the acoustic modes are sampled at the edge of the Brillouin zone, whereas optic modes exist at zone center, the sums are considered separately (the Debye model discussed above), suggesting that γ_{LA} should equal γ_{th}. This equivalence can be rigorous only for monatomic primitive unit cells (many metals), which lack optic modes. For minerals, the Debye model is inexact, but should be a close enough representation to be useful.
Comparison of 〈γ_{i}〉, γ_{LA}, and γ_{th} for Minerals in the Spinel and Olivine Groups
Polyhedral bulk moduli of spinels were established through singlecrystal refinements at pressure (24). Values of K_{X} are 147 GPa for MgO_{6}; 151 GPa for FeO_{6}; 423 GPa for SiO_{4}; 142 GPa for MgO_{4}; and 255 GPa for AlO_{6}. These values should be appropriate for olivines, as suggested by determinations of K_{X} for the related wadsleyite structure as 144 ± 9 GPa for MgO_{6}; 160 ± 10 GPa for FeO_{6}; and 350 ± 60 GPa for SiO_{4} (25). The lower value for Si tetrahedra in wadsleyite may be connected with the existence of the Si_{2}O_{7} dimer, and hence K_{X} from spinels are used in the present calculations. However, the above differences suggest that highly accurate determinations of γ_{i0} require that K_{X} be determined for each structure type.
Nesosilicates.
The relevant polyhedron is deduced from the band assignments. The lowest frequency modes are assigned to translations of Mg (or Fe) against O, or of the SiO_{4} tetrahedron moving as a unit; the latter motion must be against the divalent cation. Because all of these motions involve the divalent cation against O atoms, K_{X} is the polyhedral bulk modulus of the divalent cation in the octahedra. For brevity, X is denoted by the central cation.
The IR modes internal to the tetrahedron are the asymmetric Si–O stretch and O–Si–O bend. These motions of Si against O need not involve the adjacent Mg ion; hence, K_{X} of ν_{3} and ν_{4} in the IR is the polyhedral bulk modulus of tetrahedral Si. In contrast, the symmetric bend (ν_{1}) active in Raman spectra, which is a simple expansion of the tetrahedron, must involve compression of the adjacent Mg octahedra; therefore, K_{X} = K_{T}.
The symmetric bend (ν_{2}) and rotation–librations (R) of the tetrahedron involve no change in the volume of the tetrahedron. These must be related to K_{T}, otherwise there would be no change in their frequency with pressure. The divalent cations could be involved by “dragging” during these motions, and also because M^{2+} translations occur at similar frequencies, which promotes mixing of modes. Mode mixing occurs in other bands. The frequency of ν_{3} in the Raman spectrum of γMg_{2}SiO_{4} (8) depends nonlinearly on P, and has a lower ∂ν_{i}/∂P than that of other stretching modes, suggesting mixing with octahdedral motions and thus K_{X} = K_{T}. Similarly, ν_{4} in the Raman spectrum of olivine is assigned to K_{T}.
The above information was used to compute γ_{i} in Tables 1 and 2. A simple average was used for 〈γ_{i}〉 in Table 3. The weighing factor (Eq. 5) would increase the reported averages by no more than 3%, which is roughly the uncertainty.
For γMg_{2}SiO_{4}, the separate averages of the Raman (1.25) and the IR (1.29) modes equal γ_{th}, given the respective uncertainties. Recent data (11) provide γ_{LA} as 1.4, which is slightly higher. For γFe_{2}SiO_{4}, 〈γ_{i}〉 = 1.39 for the IR modes. Raman data are unavailable. The LA mode is active in the IR, through a resonance, and γ_{LA} = 1.46 (Table 1). These agree with the computation of γ_{th} of 1.48 ± 7, using α = (22 ± 1) × 10^{−6}/K (27). Slightly lower α and thus γ_{th} are possible (9).
For forsterite, 〈γ_{i}〉 = 1.30 for each of the IR and Raman data sets. Pressure data are not available for several IR peaks observed at moderate frequency (7), it is only the more intense Raman and IR modes that are sampled (4), and the various symmetries are unevenly represented in the unpolarized spectra. The most recent Brillouin scattering data (28) give γ_{LA} = 1.27. These values compare closely with γ_{th} = 1.29 (2). Earlier measurements of the elastic properties (Table 3) provided large pressure derivatives of the elastic moduli and larger values for γ_{LA}. Possible problems in older K′ and G′ values are discussed below.
For fayalite, the pressure dependence of 16 IR modes (7) gives 〈γ_{i}〉 = 1.15. Peaks at 92, 307, 361, and 437 cm^{−1} could not be traced with pressure. This probably explains the average being slightly than γ_{th} = 1.21 ± 0.03 (see Table 2). Raman data have not been measured at pressure. Isaak et al. (30) used several methods to determine K_{S} as 134 GPa. To reconcile this value with the pressure study of Graham et al. (31) requires K′ near 4, not 5.2 as reported. Data on V(P) (32) also support K′ = 4. If the error in K′ is caused by pressure calibration, then G′ should decrease proportionately to K′, implying that γ_{LA} = 1.65, which is outside the errors in 〈γ_{i}〉 and in γ_{th}.
Spinel.
This mineral has a different type of functional group. MgAl_{2}O_{4} does not actually have the normal modes of isolated AlO_{6}, but such motions were assigned to the vibrational bands as a first approximation (17). Allocation of K_{X} is equivocal for several of the highfrequency modes. For the two intensefrequency IR modes some curvature is seen, suggesting that K_{X} = K_{T}. For the weak shoulder at high ν, the same assignment is used. The lowestfrequency IR mode was assigned to a translation of Mg, but its frequency is linear with P, up to 32 GPa, suggesting that it is more likely a translation of Mg vs. AlO_{6}. In either case, K_{X} is the polyhedral bulk modulus of MgO_{4}. However, the later assignment is compatible with the higher frequency vibrations involving the unit cell, not just Al–O motions. In contrast, the lowest Raman mode of T_{2g} symmetry seems to be a pure translation of tetrahedral Mg because its frequency depends nonlinearly on P at low pressures; hence the two higher frequency modes in T_{2g} could be pure Al–O motions. The other polarizations do not involve motions of Mg (17), and could be pure Al–O stretching and bending. As a first approximation, the highfrequency Raman modes are analyzed by using the polyhedral bulk modulus of AlO_{6}.
For both the Raman and IR categories 〈γ_{i}〉 = 1.3 (Table 1). The averages thus duplicate γ_{th} = 1.3 obtained by using K_{T} of the same highly ordered sample (33). Anderson and Isaak (29) derive a much larger γ_{th} of 1.5, because of a higher bulk modulus measured for synthetic samples with Mg–Al disorder. Large γ_{LA} = 1.41 is computed, because of the K′_{S} being high, ≈5 (34, 35). If K′_{T} is 4, as indicated by volumetric studies (33), γ_{LA} equals the other values, within its uncertainty.
Assessment of the Revised Definitions of the Mode Grüneisen Parameters
Optic Modes.
The average 〈γ_{i}〉 calculated by using the revised definition, Eq. 7, for both mode types (Raman or IR) for olivines, silicate, and aluminate spinels equals γ_{th} within the experimental uncertainties. This comparison suggests that the revised definition should be applied to all polyatomic structures, regardless of the nature of the functional group, and probably regardless of whether a functional group exists, but with due consideration of the particular atomic motion associated with each vibrational mode. No other polyatomic minerals have wellsupported band assignments in addition to the IR, Raman, bulk moduli, and thermodynamic data at pressure needed to test Eq. 7.
Why do the separate averages of mode Grüneisen parameters for the two types of optic modes individually reproduce γ_{th}? The explanation lies in is the density of states. In a glass, the IR spectrum equals the density of states (36). The IR spectrum of a glass closely resembles that of its crystalline counterpart, and hence the IR spectrum of a solid can be used to represent the density of states. Because the average of all optic modes is expected to give γ_{th} and the IR alone provide this average, then by difference, so must the Raman modes. This result should come as no surprise because C_{V} and S are calculated on the basis of IR or Raman data alone (e.g., refs. 4, 8, 9).
For both Raman and IR modes, 〈γ_{i}〉 using Eq. 1 is less than the 〈γ_{i}〉 using Eq. 7 because K_{T} tends to fall near the smaller of the polyhedral bulk moduli, rather than near their average. Because IR activity requires a change in the dipole moments, the motions tend to involve individual polyhedra, and thus polyhedral bulk moduli are generally appropriate. Raman modes tend to be symmetric and commonly involve the unit cell, and thus K_{T} is often appropriate. Because the Raman spectra have a larger number of modes with K_{X} = K_{T}, previous calculations of 〈γ_{i}〉 from Raman data by using the older formula (Eq. 1) were uniformly closer to γ_{th} than previous estimates of 〈γ_{i}〉 based on IR modes and Eq. 1.
Acoustic Modes: Implications for G′.
Generally γ_{LA} is larger but within ≈10% of γ_{th} or 〈γ_{i}〉, and is more uncertain. For all of the substances tested, except fayalite, γ_{LA} equals γ_{th} within the experimental uncertainties (Table 3). A likely source of the discrepancy is G′. Both forsterite and γMg_{2}SiO_{4}, have G that is more than K_{S}/2 and G′ that is about K′/3. Fayalite and γFe_{2}SiO_{4}, both have G that is between K_{S}/2 and K_{S}/3. For γFe_{2}SiO_{4}, G′ is 0.72 (10). Substitution of Fe for Mg in both the olivine and spinel polymorphs increases K_{S} whereas G decreases and K′ remains near 4 (refs. 9 and 30; Table 3). We suggest that G′ decreases in olivine as deduced for the spinel phases (figure 12 in ref. 9). Rearranging Eq. 3, and substituting γ_{th} for γ_{LA} gives G′ = 0.63 for fayalite, which is comparable to G′ for γFe_{2}SiO_{4}. Similarly, if we accept the volumetric determination of K′ = 4 for spinel, and require that γ_{LA} equals γ_{th}, then G′ = 0.85, which is in between G′ values obtained for the silicate spinels (Table 3).
The trends in pressure derivatives of elastic moduli with time are toward smaller values (Table 3); also compare MgO (18, 19). This change may be the result of technical improvements such as in pressure calibration.
Pressure Derivatives of the Grüneisen Parameters
The second mode Grüneisen parameter is redefined as q_{i} = (V_{a}/γ_{i})∂γ_{i}/∂V_{a}, leading to 8 where K′_{X} = ∂K_{X}/∂P. Data on the pressure derivatives of polyhedral bulk moduli appear to be nonexistent. We approximate K′_{X} as being 4 for Mg and Fe in the octahedral sites in the olivines and silicate spinels because the polyhedral moduli of these species are similar to the bulk moduli (i.e., most of the compression is taken up by the octahedral sites) and because for these phases, K′ is close to 4.
For Si in the tetrahedral site, the bulk modulus is extremely high, 423 GPa (24). This unit is nearly incompressible, with a bulk modulus like that of diamond, suggesting that K′_{Si} = 2, like diamond (37).
Values for q_{i0} are constrained for frequencies that depend nonlinearly on pressure (e.g., ref. 7). Most of these modes occur at low frequencies and are associated with motions involving the Mg or Fe octahedra. For frequencies that appear to be linear in P, Eq. 8 reduces to q_{i} = γ_{i} − K′_{X0} which gives q_{i} near −3 for the modes involving Mg or Fe or the whole lattice. This result is not realistic, as the vibrational modes should have some slight curvature (concave downwards) if measurements were extended to sufficiently high pressures (38). Similar modes showing curvature (Tables 1 and 2) have q_{i} from 1 to 6, with 2 being typical. Hence the curvature term, −(K/ν_{i0}γ_{i0})∂^{2}ν_{i}/∂P^{2}_{0}, is typically 5, and the minimum size resolved in the spectroscopy is 4. Roughly, the curvature term for the modes that appear linear could then be 2 to 3, suggesting that the actual q_{i} value is −1 to 0 for modes where curvature is not seen. Therefore, in obtaining the average, q_{i} is set to a plausible value of 0 for any of the external modes (X = M, T, or bulk) with frequencies that depend linearly on P.
Modes that involve internal motions of the Si tetrahedron are almost all linear, or nearly so. Taking K′ = 2 gives q_{i} near −1. Because some slight amount of curvature would exist for this site, too, the q_{i} values are underestimated. The curvature term for internal vibrations can be large because K_{Si} is squared in Eq. 8. If ∂^{2}ν_{i}/∂P^{2} is [1/10] the size of the minimum observed curvature for IR modes in olivines and silicate spinels (7, 9), then q_{i} for the internal modes would be 1–2. The Si–O stretching mode of γMg_{2}SiO_{4} has curvature close to this minimum. Because its q_{i} value is close to its γ_{i} value (Table 1), we approximate q_{i} as γ_{i} for the internal modes of the Si tetrahedron that appear to have frequencies that are linearly dependent on pressure.
From thermodynamic identities, the second thermal Grüneisen parameter is 9 where all quantities are measured at ambient conditions. The term Tαγ_{th} is much smaller than unity, and can be ignored. The term (2Tγ_{th}/K_{T}) ∂K_{T}/∂T_{P} is about [1/40] of the last term, and about [1/20] of value of K′, and thus neglecting this term scarcely affects the uncertainty. The term (Tγ_{th}/α) ∂α/∂T is about 0.2, and thus ignoring it contributes a small amount to the uncertainty, leading to q_{th} being slightly overestimated. To a good approximation, 10 where δ_{T} = −(1/αK_{T}) ∂K_{T}/∂T_{P} is the Anderson–Grüneisen parameter. Note that the uncertainty in q_{th} is the sum of the uncertainties in K′_{T} and δ_{T}, which roughly are ±10% each, and thus q_{th} is uncertain by about ±1, which is large compared with the neglected terms.
Because dγ_{LA}/dP equals dγ_{th}/dP, then 11 where γ represents 〈γ_{i}〉, γ_{LA}, or γ_{th}. The utility of Eq. 11 is limited by the uncertainties in K" and G", and is approximate, as it is a corollary of the Debye model.
Calculations for Spinels and Olivine.
The only phase in Table 3 with sufficient data for comparison of the three types of q is forsterite, which has q_{th} = 2.8 ± 1.0 (29). The spectroscopic results are compatible. For the IR modes of forsterite showing curvature, the q_{i} values average about 2.6. These are, however, the minority of the IR modes. Raman modes do not show curvature, but instead have breaks in slopes, and thus cannot be used to constrain q. If q_{i} is taken as 0 for the external modes with constant dν/dP (Table 2) and as γ_{i} for the internal modes, as discussed above, then 〈q_{i}〉 for the IR is 2.0. The acoustic q value depends on the second pressure derivatives of the bulk modulus. For the linear fits of the elastic moduli with pressure (28), q_{LA} is slightly negative; it increases proportionately with K" and G" (Table 3), up to a value of 3.2, corresponding to KK" = −9.6. Given that the fit to the elastic moduli is not obviously improved by the secondorder terms, we suggest that a reliable value for q_{LA} is 1.45, which is the average of the two fits of Zha et al. (28). Averaging the results for the three elasticity data sets in Table 3 gives q_{LA} = 2.1 ± 0.1.
For the remaining phases, agreement is good, considering that various parameters were estimated (Table 3). For all phases, q is ≈2–3, and γ is close to 1.3.
Calculation of the First Pressure Derivative of the Bulk Modulus
Hofmeister (5) related pressure derivatives of K_{T} to vibrational frequencies and mode Grüneisen parameters, assuming that forces between like atoms are electrostatic, that shortrange repulsive forces between unlike atoms are central and pairwise additive, that ions are rigid, and that the structure scales on compression. The relationship at ambient pressure 12 was derived for cubic structures or substances with one dominant interatomic distance. If all of the pressure derivatives are the same, Eq. 12 reduces to the freevolume equation (39): 13 Given the approximations needed for Eq. 12 and that mode Grüneisen parameters are not available for all of the optic modes, Eq. 13 could be just as valid as Eq. 12.
Eq. 13 gives K′_{T0} similar to 4 for the phases considered here, in agreement with existing volumetric data and recent elasticity measurements (Table 3). Therefore, Eq. 13 serves as a useful crosscheck on K′ and γ_{th}.
For the silicates, compression is unevenly distributed among the polyhedral units. The bulk modulus of olivine (≈130 GPa) is less than that of its weakest polyhedra (≈150 GPa), suggesting that these units and some bending are responsible for compression. That is, olivine does not scale on compression, and the assumptions behind Eq. 12 are not met. The bulk modulus of silicate spinel (≈200 GPa) lies between that of the weakest polyhedra and the average of ≈285 GPa, suggesting uneven compression, but less so than the olivine structure. As a result, Eq. 12 comes closer to predicting K′ for silicate spinels, but is still low. In contrast, for MgAl_{2}O_{4}, the bulk modulus (194 GPa) equals the average of the polyhedral moduli (198 GPa), suggesting that the crystal roughly scales. For this case, Eq. 12 gives K′ of 4.25, which lies within the uncertainty of K′ from V(P) data and K′ from Eq. 13 (Table 3). We propose that this result best represents spinel.
Calculation of the Second Pressure Derivative of the Bulk Modulus
The second derivative of the bulk modulus (K"_{T0}) is related to pressure derivatives of the mode Grüneisen parameters (5). The individual q_{i} values are too poorly constrained for this approach to be useful. However, the pressure derivative of Eq. 13 leads a higherorder freevolume equation 14 Eqs. 11 and 14 are compatible. Specifically, q_{LA} determined from inserting a given KK" values in Eq. 11 returns about the same KK" value through Eq. 14. Eq. 14 is not exact, but provides a check for consistency. On this basis, Webb's (40) value of −19.4 for olivine predicts q = 7, which is too high by a factor of about 2. Chang and Barsch's (34) and Yoneda's (35) fits to K_{0}K"_{0} of ≈100 for spinel are untenable.
Discussion
The agreement of 〈γ_{i}〉 with γ_{th} points to the volume of the vibrating unit being relevant to the mode Grüneisen parameter, not the volume of the whole crystal. This conclusion is supported by rough agreement with γ_{LA}. Close correspondence is not expected because the Debye model is not rigorous. We emphasize that the connections here and previously (10, 13) between acoustic and thermal Grüneisen parameters serve as a reasonable guide, but cannot be considered as exact.
The pressure derivatives of three Grüneisen parameters are difficult to constrain. Uncertainties could be reduced by acquiring additional data on thermal expansivity, as suggested by the existence of discrepancies among independent measurements of α for various substances (41). Either highprecision spectral data or measurements to very high pressures (e.g., ref. 38) would help to constrain q_{i}. For q_{LA}, improvements are more difficult because K" and G" depend on K′ and G′ (42).
The redefinition of γ_{i} provides the pressure derivative of the shear modulus, for which relatively few measurements exist, in comparison to the bulk modulus. Eq. 3, rearranged is 15 The accuracy depends largely on that of K′_{T}, which represents an average over the pressures attained, whereas all other parameters are obtained at 1 atm. Nevertheless, Eq. 15 provides G′ = 0.63 for fayalite and = 0.67 for spinel (using K′ = 4.25), which are similar to that of γFe_{2}SiO_{4} and other spinels (Table 3 and refs. 34 and 43).
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 K′ in not accurately calculated from γ_{i} for silicates (5), for which nearestneighbor interactions seem to dominate. Reliable calculation of K′ 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. However, for polyatomic substances that meet this criterion (e.g., spinel), use of the revised definition of the mode Grüneisen parameter provides K′ consistent with volumetric data (33) and internal consistency among the elastic and thermodynamic properties (Table 3).
The relationship (Eq. 14) derived from the freevolume formulation suggests that KK" is near −6.5 for olivines and spinels, independent of elasticity data. Similar values have been observed for MgO, with of KK" of 0 to −7.2 ± 3.3 (18, 19). NaCl also has multiple measurements and wellconstrained values of KK" from −4 to −10 (45). Also, various equation of state formalisms converge to the range −2 < KK" < −7 for K′ near 4 (42). Independent of the value of K′ and of any assumed equation of state, Eq. 14 suggests a limited range of −1 < KK" < −9, because γ and q occupy narrow ranges. Although Eqs. 13 and 14 are not exact, they provide a crosscheck in that all thermodynamic variables, all elastic properties, and all spectroscopic parameters are tied together through the particulars of the interatomic potentials.
Acknowledgments
We thank R. M. Hazen, J. D. Bass, and S. V. Sinogeikin for helpful input and for sharing data in advance of publication. Reviews by O. L. Anderson, A. A. Maradudin, and D. A. Young led to substantial improvements. Support was provided by National Science Foundation Grants EAR9712311 (to A.M.H.), and EAR892039 and EAR0001173 (to H.k.M.).
Abbreviations
 LA,
 longitudinal acoustic;
 TA,
 transverse acoustic;
 IR,
 infrared
 Accepted November 27, 2001.
 Copyright © 2002, The National Academy of Sciences
References
 ↵
 ↵
 Ross R G,
 Andersson P,
 Sundqvist B,
 Bäckström G
 ↵
 Kieffer S W
 ↵
 Chopelas A
 ↵
 ↵
 Hofmeister A M
 ↵
 Hofmeister A M,
 Xu J,
 Mao Hk,
 Bell P M,
 Hoering T C
 ↵
 Chopelas A,
 Boehler R,
 Ko R
 ↵
 Hofmeister A M,
 Mao Hk
 ↵
 Carmichael S
 Sumino Y,
 Anderson O L
 ↵
 ↵
 ↵
 Anderson O L
 ↵
 Krishnan R S,
 Srinivasan R,
 Devanarayanan S
 ↵
KochMuller, M., Hofmeister, A. M., Fei, Y. & Zha, X. (2002) Am. Mineral.87, in press.
 ↵
 Chopelas A
 ↵
 Chopelas A,
 Hofmeister A
 ↵
 Zha CS,
 Mao Hk,
 Hemley R J
 ↵
 ↵
 Grüneisen E
 ↵
 Geiger H,
 Scheel K
 Grüneisen E
 ↵
 Hazen R M,
 Downs R T,
 Prewitt C T
 ↵
 Beyer R T,
 Letcher S V
 ↵
 ↵
 Hazen R M,
 Wienberger M B,
 Yang H,
 Prewitt C T
 ↵
 ↵
 Zha CS,
 Duffy T S,
 Downs R T,
 Mao Hk,
 Hemley R J
 ↵
 Ahrens T J
 Anderson O L,
 Isaak D G
 ↵
 ↵
 Graham E K,
 Schwab J A,
 Sopkin S M,
 Takei H
 ↵
 ↵
 ↵
 Chang Z P,
 Barsch G R
 ↵
 Yoneda A
 ↵
 Farmer V C
 White W B
 ↵
 Alexsandrov I V,
 Goncharov A F,
 Zisman A N,
 Stishov S M
 ↵
 Kingma K J,
 Cohen R E,
 Hemley R J,
 Mao Hk
 ↵
 Vashchenko V Y,
 Zubarev V N
 ↵
 Webb S L
 ↵
 Ahrens T J
 Fei Y
 ↵
 Jeanloz R
 ↵
 ↵
 ↵
 ↵
 ↵
 Hazen R M
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 Article
 Abstract
 The Original Derivation
 Generalization of the Definition of the Mode Grüneisen Parameter
 Comparison of 〈γ_{i}〉, γ_{LA}, and γ_{th} for Minerals in the Spinel and Olivine Groups
 Assessment of the Revised Definitions of the Mode Grüneisen Parameters
 Pressure Derivatives of the Grüneisen Parameters
 Calculation of the First Pressure Derivative of the Bulk Modulus
 Calculation of the Second Pressure Derivative of the Bulk Modulus
 Discussion
 Acknowledgments
 Footnotes
 Abbreviations
 References
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