The volume–pressure data relations have been determined by a least-square fitting of room-T data (Table 1) to the third-order Birch–Murnaghan (BM) equation of state whose general form is as follows:

where K₀, K’₀, and V₀ are the isothermal bulk modulus, its pressure derivative, and the zero-pressure unit-cell volume, respectively. Fitting of all parameters simultaneously using 300-K data collected at the end of each heating cycle, however, yielded large uncertainties (K₀ = 166.2 ± 2.4 GPa, K’₀ = 3.6 ± 0.4, and V₀ = 1735.7 ± 0.3 ų), suggesting that all parameters cannot be resolved simultaneously due to the limited pressure range in this study. Therefore, we chose to fix the pressure derivative K’₀ at 4 here, which is compatible with the slightly positive slope of the normalized pressure against the Eulerian strain (inset of Fig. 3) that indicates a value of K’₀ ≥ 4 (Angel, 2000). Fitting of equation (1) with K’₀ fixed at 4 yielded the bulk modulus K₀ = 164.2 ± 0.7 GPa, which is in excellent agreement with K₀ = 164 ± 1 GPa (K’₀ = 4) proposed by Fan et al. (2015) on the basis of angle-dispersive XRD in the diamond-anvil cell. Another fitting of equation (1) with K’₀ fixed at 4.5 gave K₀ = 161.6 ± 0.9 GPa (V₀ = 1736.1 ± 0.3 ų), which is also in good agreement with K₀ = 162 ± 2 GPa, K’₀ = 4.5 ± 0.3 derived by Fan et al. (2015) when they refined all parameters simultaneously (Table 2). The agreement on the results of elastic parameters and the small offset between data sets (Fig. 3), despite using two different techniques, suggests limited experimental uncertainties in our study and that of Fan et al. (2015).

When we fixed K’₀ at 4.7, we obtained a smaller K₀ = 160.6 ± 1.0 GPa, which, within the uncertainties, is also in agreement with the values of K₀ = 162 and K’₀ = 4.7 proposed by Léger et al. (1990). Our results show that it is difficult to determine a unique K₀/K’₀ tradeoff for uvarovite garnet. This can be explained by the lack of constraints on the first pressure derivative K’₀, which is partly due to the limited pressure range addressed in all studies, respectively. One alternative way to determine the most appropriate set of moduli and derivatives would be to fit all available data simultaneously. We noted, however, that although below 10 GPa, the original P–V data of Léger et al. (1990) agree well with our data (Fig. 3), volumes at P higher than 10 GPa are larger, which is likely caused by anisotropic stress above 10 GPa due to the solidification of their methanol/ethanol pressure-transmitting medium (Léger et al., 1990), and therefore we excluded those data from the global fitting. The resulting elastic parameters K₀ = 162.1 ± 1.9 GPa, K’₀ = 4.3 ± 0.3 and V₀ = 1736.5 ± 0.4 ų agree well, within uncertainties, with our data and those of Fan et al. (2015).

It is worth noting that none of the parameters fitted here can explain the correlation between bulk modulus and its pressure derivatives for natural uvarovite samples (Diella et al., 2004; Wang and Ji, 2001), suggesting the Cr³⁺/Al³⁺ ratio would have a substantial effect on the compressibility of those silicate garnets, which will be discussed a few sections below. Moreover, we cannot explain the low bulk modulus K₀ = 143 ± 4 GPa for K’₀ = 4.7 proposed by Milman et al. (2001) based on an ab initio calculation by the density functional theory within the generalized gradient approximation. It is, however, difficult to draw any conclusion, because their calculations did not take into account the possibility of a Jahn–Teller effect that is expected to affect the compressibility of minerals containing octahedrally coordinated transition metals (Bass and Weidner, 1984; Pacalo et al., 1992; Zhang et al., 1998).

Fitting of the pressure–volume–temperature data was carried out by a modified third-order Birch–Murnagham equation of state with the following form:

In this equation, the thermal dependences of the zero-pressure volume V_0,T and bulk modulus K_0,T are expressed using the following equations:

where V_0,300 is the volume at room-P,T, and α_0,T, the thermal expansion, which is empirically assumed to be α_0,T = a₀ + b₀T, Eq. (3). K_0,T is expressed as a linear function of temperature, its first temperature derivative (∂K_0,T/∂T)_P and the zero-pressure bulk modulus K_0,300, Eq. (4). Here we assumed that K’₀ is constant with temperature.

Fitting of our P–V–T data with all parameters simultaneously gave the following thermoelastic properties: K_0,300 = 163.6 ± 2.6 GPa, K’₀ = 4.1 ± 0.5, V₀ = 1735.9 ± 0.4 ų, (∂K_0,T/∂T)_P = –0.014 ± 0.004 GPa K⁻¹, a₀ = 2.32 ± 0.13 × 10⁻⁵ K⁻¹, b₀ = 2.13 ± 2.18 × 10⁻⁹ K⁻² (e.g., α_0,300 = 2.37 ± 0.20 × 10⁻⁵ K⁻¹). The bulk modulus K_0,300 and its pressure derivative determined here are consistent with our fitting of P–V data at 300 K. It is also in good agreement with K_0,300 = 162 GPa and K’₀ = 4.3, derived from Fan et al.’s (2015) P–V–T data. However, we found that their temperature derivative (∂K_0,T/∂T)_P = −0.021 GPa K⁻¹ is ∼30% smaller than our present value, while their thermal expansion α_0,300 = 2.72 × 10⁻⁵ K⁻¹ is about ∼15% larger than our α_0,300 (Table 3), which is in closer agreement with the theoretical value α₀ = 2.15 × 10⁻⁵ K⁻¹ proposed by Ottonello et al. (1996) based on lattice energy and vibrational energy calculations. The thermal gradient in our cell (see experimental methods) is likely to affect the result of our EoS fitting by ∼0.8% for K_T and ∼4% for the thermal expansion, but these variations are smaller than the differences between our EoS data and those of Fan et al. (2015), and therefore cannot explain the different thermal parameters for the two studies. Although our room temperatures are in good agreement with those of Fan et al. (2015), our study suggests that unconstrained uncertainties in the data collected at high temperature might still remain between the multi-anvil-apparatus and the diamond-anvil cell, an issue that should be addressed in future experiments.

Fitting of Eqs. (2)–(4) by fixing the first pressure derivative resulted in variations of K₀ and V₀ (Table 3), due to the strong correlation between K₀ and K’. In contrast, the thermal properties (∂K_0,T/∂T)_P and α_0,300 did not change in a significant manner, indicating that the thermal properties of uvarovite are either less affected or unaffected by the variation of the bulk modulus and of its pressure derivative. A similar conclusion was reached by other experimental studies on the elasticity of silicate garnets (Fan et al., 2015; Gréaux and Yamada, 2013; Zou et al., 2012a). Fitting of all available data from our study and that of Fan et al. (2015) gave the following parameters: K_0,300 = 161.0 ± 2.8 GPa, K’₀ = 4.5 ± 0.4, V₀ = 1736.7 ± 0.6 ų, (∂K_0,T/∂T)_P = –0.018 ± 0.003 GPa K⁻¹, α_0,300 = 2.63 ± 0.11 × 10⁻⁵ K⁻¹ (Fig. 4). The temperature derivative of the bulk modulus (∂K_0,T/∂T)_P is similar to those of grossular and andradite in previous studies (Gréaux et al., 2010; Pavese et al., 2001), suggesting that the values of (∂K_0,T/∂T)_P of ugrandite garnets are independent of the 3+ cations in the garnet octahedral site. The thermal expansion coefficient α₀ = 2.40–2.53 × 10⁻⁵ K⁻¹ is smaller than that of grossular (Gréaux et al., 2010) and andradite (Pavese et al., 2001), although the difference between uvarovite and grossular fall within their respective uncertainties. Besides, the higher value proposed for andradite could have resulted from the use of a different expression for the thermal parameters in the EoS as Pavese et al. (2001) expressed α = cste instead of α = a₀ + b₀T used in the present study and Gréaux et al. (2010).

The ugrandite is a family of isostructural garnets with the formula Ca₃Y₂Si₃O₁₂, where the Y-site is commonly occupied either by Al (grossular), Fe (andradite), Cr (uvarovite), or V (goldmanite). The bulk moduli K₀ = 162–165 GPa determined in this study showed that the compressibility of uvarovite is in between that of andradite (K₀ = 154–158 GPa) (Jiang et al., 2004; Pavese et al., 2001) and grossular (K₀ = 166–171.5 GPa) (Gréaux et al., 2010; Gwanmesia et al., 2013), although differences in K₀ depend on the value of K’ adopted in the respective studies. In their study, Fan et al. (2015) neglected this aspect when they proposed representing the bulk modulus of ugrandite garnet as a linear function of the unit-cell volume. The validity of such a relation can be tested by using the derived elastic properties to calculate the compression curve of a garnet solid solution and comparing the result to actual data.

Fig. 5 shows the calculated densities of a grossular/uvarovite solid solution compared with the experimental study of Diella et al. (2004), who reported the compressibility of a natural garnet sample containing ∼62 mol% of uvarovite using a diamond- anvil cell and synchrotron X-ray diffraction. The density of the garnet solid solution was estimated from a weighted summation of the individual elastic properties of the corresponding end members (e.g., Pamato et al., 2016). In this approach we assumed that elastic properties vary linearly as a function of the endmember proportions in the solid solution. The calculated densities are in relatively good agreement with experimental data, while both experimental and theoretical datasets stand in between the densities of the uvarovite (this study) and grossular (Gréaux et al., 2010) endmembers. The experimental densities of Diella et al. (2004) are generally smaller than those of uvarovite (Fig. 3), which can be explained by the smaller unit-cell volume of grossular (∼1662–1664 ų) compared to that of uvarovite (∼1727–1738 ų). We noted, however, that when using garnet endmember proportions as given by their electron probe analyses (Fig. 5, Uv62Gr33+), the calculated densities are slightly lower than the experimental data. Instead, we found a better agreement between the weighted average and experimental data (Diella et al., 2004) when we used the endmember proportions (in mol%): 65% uvarovite, 30% grossular, 2% pyrope, 1% almandine, 1% spessartine, and 1% andradite (Fig. 5, Uv65Gr30+). The amount of uvarovite that we propose here is slightly higher than the value measured with an electron probe; this difference could be derived from unconstrained uncertainties in our respective experiments. On the other hand, we can see in Fig. 5 that density calculated using Fan et al.’s (2015) linear model fits well with Diella et al.’s (2004) data at low pressure, but fails to explain densities at pressures higher than ∼5 GPa, which is likely due to their higher K’ values, as discussed above.

Such features could be explained based on the crystal structure of silicate garnets. The cubic garnet structure consists of large dodecahedral sites, which share edges with corner-linked chains of alternating SiO₄ tetrahedra and YO₆ octahedra, and with each other (Novak, 1971). Previous theoretical studies have demonstrated that the size of the dodecahedral cation likely controls the variations in the elasticity of silicate garnets with Al and Si in the octahedral site (Milman et al., 2000; Pacalo et al., 1992). However, there is no firm explanation for the lower values observed for garnets with transition metals in the octahedral site such as andradite and uvarovite garnets. In addition, the dodecahedral cavity of ugrandite garnet is filled with a large Ca²⁺ cation, so the rotational freedom within the corner-linked chains is limited, resulting in the structure becoming more rigid compared to other garnet groups (Milman et al., 2000; Pacalo et al., 1992). Based on the similarities in site occupancies, it is therefore likely that the characteristics of the trivalent cation and the resulting compressibility of the octahedral site might play a major role in controlling the variations of elasticity between ugrandite garnets. Our results go in that direction, showing that ugrandite garnets with a small octahedral site (e.g., grossular) have higher bulk modulus than garnets with a larger octahedral site (e.g., uvarovite, andradite). It is compatible with former studies in pyroxenes, olivines, and spinels, which showed that transition metals in octahedral coordination tend to decrease the elastic moduli of those minerals (Bass and Weidner, 1984; Pacalo et al., 1992; Zhang et al., 1998). Our results also suggest that ugrandite garnet with transition metals in the octahedral site (e.g., uvarovite, andradite) might become more incompressible upon increasing pressure, hence yielding higher K’ than garnet with Al in the octahedral site. It is, however, difficult to come to a firm conclusion about the K’ value of silicate garnets, as experimental values generally fall between 4 and 5 (Gwanmesia et al., 2006; Kono et al., 2010; Zou et al., 2012b) with differences between species barely standing outside of their respective uncertainties.

Petrological studies of minerals in kimberlite and alkali basalts that returned from the deep mantle revealed that the upper mantle is largely constituted of aluminous lherzolite. It is also known that lherzolite undergoes an important transition from spinel lherzolite to garnet lherzolite at high pressures between ∼2 GPa and ∼7 GPa, which strongly depends on the Cr/(Cr + Al) ratio of the system (Klemme, 2004). In this reaction, Cr and Ca increasingly substitute into the garnet with increasing pressure, resulting in an increase in the grossular and uvarovite components of mantle garnets. Using our elasticity data on uvarovite, combined with those of other garnet endmembers, we estimated the density variation in garnet lherzolite as a function of depth along an average mantle geotherm. Here we postulated that Cr is only present through the uvarovite endmember, assuming that the knorringite component may be minor at the pressures and temperatures of the upper mantle, although its proportions might increase with increasing pressure (Zou and Irifune, 2012). Either way, it is difficult to implement this parameter in our model, as the precise thermoelastic data of pure knorringite garnet are unavailable to this day.

Fig. 6 shows the calculated density profiles of simplified mantle garnets with the following compositions: 75% pyrope and 25% grossular (Py75Gr25), and 75% pyrope and 25% uvarovite (Py75Uva25), compared to those of pyrope, which is the major component of upper-mantle garnets. It is interesting to note that densities of Py75Gr25 are only ∼0.5% higher than those of pyrope, suggesting that mantle garnet may not be sensitive to variations of the grossular component; e.g., Cr/(Cr + Al) = 0. In contrast, we found a more substantial ∼2% density increase when incorporating the uvarovite component (Py75Uva25); e.g., Cr/(Cr + Al) = 1. Garnet compositions in lherzolite xenoliths from alkali basalts and kimberlites generally yield a Cr/(Cr + Al) < 0.2–0.35 (Grutter et al., 2006), which would correspond to a ∼1% density increase compared to the pyrope endmember. Higher Cr/(Cr + Al) ratios > 0.5 are reported in chromite-saturated harzburgite garnets, but those compositions are Ca-depleted, implying the formation of a knorringite-rich pyrope garnet, for which our model is non-applicable. It should also be mentioned that lherzolite garnets bear a substantial amount of Fe, although its exact chemical behavior is not well known. If we consider only Fe²⁺ as the main garnet species (Py75Alm25, Fig. 6), density increases drastically due to the high density of the almandine endmember. In contrast, the presence of Fe³⁺ would decrease density compared to Py75Alm25 due to the lower density of the andradite endmember, which is almost the same as that of uvarovite. However, the increase of andradite garnet along with the presence of Fe³⁺ would be incompatible with the relatively low Ca contents of garnet lherzolite (McDonough and Rudnick, 1998), suggesting that Fe²⁺ might be the main species in natural systems. Fig. 6 shows that high Fe²⁺ content is more likely to affect the bulk density of mantle garnet rather than the incorporation of Cr³⁺. However, because Cr³⁺ and Fe³⁺ are positively correlated in upper mantle garnets (e.g., Canil and O’Neill, 1996; Nimis et al., 2015), the presence of the Cr-rich component might also favor the formation of Fe³⁺, hence resulting in a decrease of the mantle garnet density. Our results emphasize that the mantle garnet density could vary substantially upon Cr/(Cr + Al) increase, which seems to occur at upper mantle depths, as Cr-rich garnets are commonly found in diamond inclusions.

The density of a natural garnet lherzolite was estimated based on the elasticity of garnet endmembers and the chemical analyses of a fertile peridotite garnet brought up by alkali basalts (McDonough and Rudnick, 1998); it yielded the following: 75% pyrope, 13% almandine, 9% grossular, 4% uvarovite and 1% andradite. Our estimates show that the density of garnet lherzolite would be ∼2.5% higher compared to the Cr- and Fe-free Py75Gr25 garnet (Fig. 6), which may have some implications for the seismic discontinuity at depths of 50–100 km. This discontinuity has been observed globally, but the depth is not the same everywhere (Hales, 1969; Yamasaki and Nakada, 1997); a difference that is attributed to the shift of the spinel-garnet phase boundary from ∼2 GPa in Al-rich fertile mantle compositions to ∼7 GPa in the deeper continental crust (Klemme, 2004; Klemme and O’Neill, 2000), where Al is depleted and Cr and Fe are more abundant, as a possible consequence of melting events (Walter et al., 2002). The higher density of Cr- and Fe-rich garnets further emphasizes the importance of the spinel-garnet transition and offers an alternative to thermal anomalies in order to explain the increase in seismic velocity at depths of ∼100 km (Hales, 1969; Yamasaki and Nakada, 1997). These conclusions, however, would need to be completed by the contribution of the knorringite garnet endmember, which is likely to change such behavior at higher pressure and high Cr/(Cr + Al) ratios.