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

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 olivine-group minerals that have well-constrained 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 γ-Mg₂SiO₄, 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 dispersion-free 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_BT, 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 γ-Mg₂SiO₄, 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 side-band 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 (Al₂O₃) or stishovite (SiO₂) 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., SiO₄ 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 single-crystal refinements at pressure (24). Values of K_X are 147 GPa for MgO₆; 151 GPa for FeO₆; 423 GPa for SiO₄; 142 GPa for MgO₄; and 255 GPa for AlO₆. These values should be appropriate for olivines, as suggested by determinations of K_X for the related wadsleyite structure as 144 ± 9 GPa for MgO₆; 160 ± 10 GPa for FeO₆; and 350 ± 60 GPa for SiO₄ (25). The lower value for Si tetrahedra in wadsleyite may be connected with the existence of the Si₂O₇ 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 SiO₄ 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 ν₃ and ν₄ in the IR is the polyhedral bulk modulus of tetrahedral Si. In contrast, the symmetric bend (ν₁) 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 (ν₂) 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 M²⁺ translations occur at similar frequencies, which promotes mixing of modes. Mode mixing occurs in other bands. The frequency of ν₃ in the Raman spectrum of γ-Mg₂SiO₄ (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, ν₄ 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.

Table 1.

Revised Grüneisen parameters for spinels

Peak	X	νi0	γi0	qi0
γ-Mg2SiO4 (Raman, ref. 8)
ν3 (T2g)	Bulk	796	1.26	1.3*
ν4(T2g)	Si	600	1.54	1.5*
T(T2g)	Mg	302	1.24	2.4
ν2(Eg)	Bulk	372	0.985	0†
ν1(Ag)	Bulk	834	0.926	0†
γ-Mg2SiO4 (IR, ref. 9)
ν3	Si	822	2.05‡	1.8
ν4	Si	541	1.09	1.1*
T(M)	Mg	422	1.41	1.35‡
T(mix)	Bulk	350	0.60	0.80
γ-Fe2SiO4 (IR, ref. 9)
ν3	Si	825	2.31	2.3*
ν4	Si	497	1.32	1.2*
T(M)	Fe	299	1.31 (8)	2.5§
T(mix)	Bulk	206	0.62	≈0†
LA	Bulk	206	1.47 (5)	≈0†
MgAl2O4 (ref. 17)
ω5(T2g)	Al oct	666	‖
ω5(T2g)	Al oct	492	1.52
T(T2g)	Mg tet	322	0.615
ω2(Eg)	Al oct	407	1.59
ω1(Ag)	Al oct	767	1.43
ω3(T1u)	Bulk	675	1.5
Al–O(T1u)	Bulk	585	1.2
ω4(T1u)	Bulk	485	2.0
Mg(T1u)	Mg tet	305	0.46

Eqs. 7 and 8 were used for γ_i0 and q_i0 using K_X from Hazen and Yang (24). K_S (48) gives K_T0 = 182.5 GPa for γ–Mg₂SiO₄. K_T0 = 207 GPa (49) for γ–Fe₂SiO₄. MgAl₂O₄ has K_T = 194 ± 6 GPa (33). tet, tetrahedra; oct, octahedra. 

^*For this internal mode, curvature was not detected and we assumed that q_i = γ_i; see text. 

^† For this external mode, curvature could not be detected due to a resonance, so that q_i is poorly constrained; we assume that q_i is roughly at its minimum value of 0; see text. 

^‡ The fit is to data from 85–370 kbar (8.5–37.0 GPa), leading to greater uncertainty. 

^§ This peak was off scale at low P. The actual value of γ_i0 may be higher. 

^‖ Assumed equal to the other ω₅ mode. 

Table 2.

Revised Grüneisen parameters for olivine

Peak	X	Forsterite, IR, ref. 7			Favalite, IR, ref. 7			Forsterite, Raman*, ref. 4
		νi0	γi0	qi0	νi0	γi0	qi0	Peak	X	νi0	γi0
T + M	Bulk	142	1.46	0.25	107	0.97	33?	T	M	183	2.43
T	M	201	2.15	1.35	230	0.77	0†	T	M	227	0.78
M	M	275	1.42	5.4	173	0.131	0†	R	Bulk	244	1.23
M	M	275	1.93	5.0	173	1.45	6.3	M	M	306	1.90
M	M	295	1.29	7.0	183	1.10	4.4	M	M	331	1.35
M	M	295	1.64	5.5	183	1.65	3.7	R	Bulk	341	1.89
M	M	320	1.40	0.9	198	1.25	3.5	R	Bulk	376	1.27
M	M	362	0.74	0†	253	1.18	0†	R + ν2	Bulk	411	1.00
M	M	421	0.58	0†	292	1.11	0†	R + ν2	Bulk	424	1.45
R	Bulk	481	0.39	0†				R + ν2	Bulk	434	1.42
ν4	Si	545	2.51	—‡	476	0.88	—‡	R + ν2	Bulk	441	1.62
ν4	Si	609	0.93	—‡	558	1.58	48?	ν4	Bulk	545	0.54
ν1	Si	837	0.48	—‡	826	0.40	—‡	ν4	Bulk	585	0.67
ν3	Si	876	1.06	—‡	876	1.21	—‡	ν4	Bulk	609	0.72
ν3	Si	925	0.78	—‡	915	1.15	—‡	ν3	Si	826	1.62
ν3	Si	968	2.29	—‡	949	1.91	—‡	ν1	Bulk	856	0.50
ν3	Si	1,005	≈1.4	—‡	975	1.53	—‡	ν3	Si	884	1.47
								ν3	Si	922	1.22
								ν3	Si	966	2.21

Eqs. 7 and 8 were used for γ_i0 and q_i0 with K_T = 129 GPa for forsterite (28) and 132 GPa for fayalite (30), and K_X from the site moduli of Hazen and Yang (24); see text. 

^*The fits were made to different slopes at high pressure and therefore q_i could not be determined. 

^† For this external mode, curvature could not be detected due to a resonance, so that q_i is poorly constrained; we assume that q_i is roughly at its minimum value of 0; see text. 

^‡ For this internal mode, curvature was not detected and we assumed that q_i = γ_i; see text. 

Table 3.

Comparison of results from using the revised mode Grüneisen parameters to experiment

Phase	Elasticity measurements					Thermal data				Spectroscopy			Free volume
	KS, GPa	G, GPa	G′	K′S	KSK″S	γLA	〈qLA〉	γtha	qtha	〈γ〉	〈q〉	K′T	K′T	KTK″T
Forsterite	131b	79.4b	1.0b	3.8b	0b	1.22	1.5 ± 1.7	1.29	2.8  ± 1.0	1.30	2.0–2.3	2.61	4.25	−6.2
Fo90	129.4c	78c	1.71c	4.29c	0c	1.59	−0.3
Fo90	129.5d	77.6d	1.9d	4.6d	−19.4d	1.79d	5.4d
Fayalite	134e	50.7e	≈1.1e	≈4e		≈1.6e	5.3f	1.21	4.0  ± 1.5	1.15	1.8–3.4	2.90	4.09	−7.2
γ-Mg2SiO4	188g	118g	1.3g	4.1g		1.40g	2.9f	1.25h	3.1  ± 1.1g	1.29	1.3–2.4	3.61	4.17	−6.2
γ-Fe2SiO4	208i	88j	0.72j	4i		1.47j	5.0f	1.45j	4  ± 2	1.39	1.5–2.5	3.65	4.44	−8.6
MgAl2O4	197k	108k	0.50k	4.89k	−94k	1.41	2.4f	1.3l	4  ± 2	1.30	0?	4.25	4.27	−6.2
	198m	109m	0.36m	5.7m	−129m	1.58

The average Grüneisen parameters are for IR spectra only. Raman averages are indistinguishable. The average q is also listed for the low-frequency IR modes. K′ is calculated both from the spectroscopic sum of Eq. 12 and from the free volume formula, Eq. 13. For spinel, K_T and α depend on Mg–Al ordering, leading to the differing values from physical measurements on various samples. See Tables 1 and 2 for K_T. 

^a From the compilation of Anderson and Isaak (29) unless indicated otherwise. 

^b Zha et al. (28) for Fo90 using Brillouin scattering. The same value was observed for K′_T. For fits involving KK" = −9.6, GG" = −5.2, the derivatives are slightly higher (K′ = 4.2, G′ = 1.6, K_S = 130 GPa, G = 78.2 GPa) giving γ_LA = 1.56, but the fit was not improved. 

^c Abramson et al. (46) for Fo90 using picosecond transient grating spectroscopy. GG" = −2.1. 

^d Webb (40) for Fo91 using ultrasonic methods. 

^e Ambient values from Isaak et al. (30) suggest the derivatives given by comparison of K₀ to that of Graham et al. (31). 

^f Assuming KK" = −7 and GG" = −4, as occurred for forsterite. For fayalite, we used K′ = 4, G′ = 0.63. 

^g Sinogeikin et al. (11) for Fo91 composition using Brillouin scattering. 

^h Calculated from thermal properties by Chopelas et al. (8). 

ⁱ From volumetric data of Hazen (47). 

^j Hofmeister and Mao (9) calculated shear moduli (and other properties) from a resonance. 

^k Chang and Barsch (34) using ultrasonic. If K′_S = K′_T = 4 (32) and G′ = 0.85, then γ_LA = 1.3. 

^l Chopelas and Hofmeister (17). 

^m Yoneda (35) using ultrasonic techniques. 

For γ-Mg₂SiO₄, 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 γ-Fe₂SiO₄, 〈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. MgAl₂O₄ does not actually have the normal modes of isolated AlO₆, but such motions were assigned to the vibrational bands as a first approximation (17). Allocation of K_X is equivocal for several of the high-frequency modes. For the two intense-frequency 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 lowest-frequency 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. AlO₆. In either case, K_X is the polyhedral bulk modulus of MgO₄. 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 high-frequency Raman modes are analyzed by using the polyhedral bulk modulus of AlO₆.

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 well-supported 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 γ-Mg₂SiO₄, have G that is more than K_S/2 and G′ that is about K′/3. Fayalite and γ-Fe₂SiO₄, both have G that is between K_S/2 and K_S/3. For γ-Fe₂SiO₄, 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 γ-Fe₂SiO₄. 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_i0γ_i0)∂²ν_i/∂P²|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 ∂²ν_i/∂P² 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 γ-Mg₂SiO₄ 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 second-order 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 short-range repulsive forces between unlike atoms are central and pair-wise 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 free-volume 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 MgAl₂O₄, 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 higher-order free-volume 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 K₀K"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 high-precision 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 γ-Fe₂SiO₄ 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 nearest-neighbor 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 free-volume 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 well-constrained 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.

Abbreviations

LA

longitudinal acoustic

TA

transverse acoustic

IR

infrared