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All available volume and elasticity data for the garnet end-members grossular, pyrope, almandine and spessartine have been re-evaluated for both internal consistency and for consistency with experimentally measured heat capacities. The consistent data were then used to determine the parameters of third-order Birch–Murnaghan EoS to describe the isothermal compression at 298 K and a Mie–Grüneisen–Debye thermal-pressure EoS to describe the PVT behaviour. In a full Mie–Grüneisen–Debye EoS, the variation of the thermal Grüneisen parameter with volume is defined as γ = γ 0 V V 0 q . For grossular and pyrope garnets, there is sufficient data to refine q which has a value of q  = 0.8(2) for both garnets. For other garnets, the data do not constrain the value of q and we therefore refined a q- compromise version of the Mie–Grüneisen–Debye EoS in which both γ / V and the Debye temperature θ D are held constant at all P and T , leading to ∂ C V ∂ P T = 0 , parallel isochors and constant isothermal bulk modulus along an isochor. Final refined parameters for the q- compromise Mie–Grüneisen–Debye EoS are: Pyrope Almandine Spessartine Grossular V 0 (cm 3 /mol) a 113.13 115.25 117.92 125.35 K 0T (GPa) 169.3 (3) 174.6 (4) 177.57 (6) 167.0 (2) K 0 T ′ 4.55 (5) 5.41 (13) 4.6 (3) 5.07 (8) θ D0 771 (28) 862 (22) 860 (35) 750 (13) γ 0 1.185 (12) 1.16 (fixed) 1.18 (3) 1.156 (6) for pyrope and grossular, the two versions of the Mie–Grüneisen–Debye EoS predict indistinguishable properties over the metamorphic pressure and temperature range, and the same properties as the EoS based on experimental heat capacities. The biggest change from previously published EoS is for almandine for which the new EoS predicts geologically reasonable entrapment conditions for zircon inclusions in almandine-rich garnets.

Elastic geothermobarometry is a method of determining metamorphic conditions from the excess pressures exhibited by mineral inclusions trapped inside host minerals. An exact solution to the problem of combining non-linear Equations of State (EoS) with the elastic relaxation problem for elastically isotropic spherical host-inclusion systems without any approximations of linear elasticity is presented. The solution is encoded into a Windows GUI program EosFit-Pinc. The program performs host-inclusion calculations for spherical inclusions in elastically isotropic systems with full P-V-T EoS for both phases, with a wide variety of EoS types. The EoS values of any minerals can be loaded into the program for calculations. EosFit-Pinc calculates the isomeke of possible entrapment conditions from the pressure of an inclusion measured when the host is at any external pressure and temperature (including room conditions), and it can calculate final inclusion pressures from known entrapment conditions. It also calculates isomekes and isochors of the two phases.

Elastic geobarometry for host-inclusion systems can provide new constraints to assess the pressure and temperature conditions attained during metamorphism. Current experimental approaches and theory are developed only for crystals immersed in a hydrostatic stress field, whereas inclusions experience deviatoric stress. We have developed a method to determine the strains in quartz inclusions from Raman spectroscopy using the concept of the phonon-mode Gruneisen tensor. We used ab initio Hartree-Fock/Density Functional Theory to calculate the wavenumbers of the Raman-active modes as a function of different strain conditions. Least-squares fits of the phonon-wavenumber shifts against strains have been used to obtain the components of the mode Gruneisen tensor of quartz (γ1m and γ3m) that can be used to calculate the strains in inclusions directly from the measured Raman shifts. The concept is demonstrated with the example of a natural quartz inclusion in eclogitic garnet from Mir kimberlite and has been validated against direct X-ray diffraction measurement of the strains in the same inclusion.

Raman spectroscopy provides information on the residual strain state of host-inclusion systems that, coupled with the elastic geobarometry theory, can be used to retrieve the P-T conditions of inclusion entrapment. In situ Raman measurements of zircon and coesite inclusions in garnet from the ultrahigh-pressure Dora Maira Massif show that rounded inclusions exhibit constant Raman shifts throughout their entire volume. In contrast, we demonstrate that Raman shifts can vary from the center to the edges and corners of faceted inclusions. Step-by-step polishing of the garnet host shows that the strain in both rounded and prismatic inclusions is gradually released as the inclusion approaches the free surface of the host. More importantly, our experimental results coupled with selected numerical simulations demonstrate that the magnitude and the rate of the strain release also depend on the contrast in elastic properties between the host and the inclusion and on the inclusion crystallographic orientation with respect to the external surface. These results allowed us to give new methodological guidelines for determining the residual strain in host inclusion systems.

The ultrahigh-pressure (UHP) whiteschists of the Brossasco-Isasca unit (Dora-Maira Massif, Western Alps) provide a natural laboratory in which to compare results from classical pressure (P)–temperature (T) determinations through thermodynamic modelling with the emerging field of elastic thermobarometry. Phase equilibria and chemical composition of three garnet megablasts coupled with Zr-in-rutile thermometry of inclusions constrain garnet growth within a narrow P–T range at 3–3.5 GPa and 675–720 °C. On the other hand, the zircon-in-garnet host-inclusion system combined with Zr-in-rutile thermometry would suggest inclusion entrapment conditions below 1.5 GPa and 650 °C that are inconsistent with the thermodynamic modelling and the occurrence of coesite as inclusion in the garnet rims. The observed distribution of inclusion pressures cannot be explained by either zircon metamictization, or by the presence of fluids in the inclusions. Comparison of the measured inclusion strains with numerical simulations shows that post-entrapment plastic relaxation of garnet from metamorphic peak conditions down to 0.5 GPa and 600–650 °C, on the retrograde path, best explains the measured inclusion pressures and their disagreement with the results of phase equilibria modelling. This study suggests that the zircon-garnet couple is more reliable at relatively low temperatures (< 600 °C), where entrapment conditions are well preserved but chemical equilibration might be sluggish. On the other hand, thermodynamic modelling appears to be better suited for higher temperatures where rock-scale equilibrium can be achieved more easily but the local plasticity of the host-inclusion system might prevent the preservation of the signal of peak metamorphic conditions in the stress state of inclusions. Currently, we cannot define a precise threshold temperature for resetting of inclusion pressures. However, the application of both chemical and elastic thermobarometry allows a more detailed interpretation of metamorphic P–T paths.

Quartz crystals with zircon inclusions were synthesized using a piston-cylinder apparatus to experimentally evaluate the use of inclusions in “soft” host minerals for elastic thermobarometry. Synthesized zircon inclusion strains and, therefore, pressures ( P inc ) were measured using Raman spectroscopy and then compared with the expected inclusion strains and pressures calculated from elastic models. Measured inclusion strains and inclusion pressures are systematically more tensile than the expected values and, thus, re-calculated entrapment pressures are overestimated. These discrepancies are not caused by analytical biases or assumptions in the elastic models and strain calculations. Analysis shows that inclusion strain discrepancies progressively decrease with decreasing experimental temperature in the α-quartz field. This behavior is consistent with inelastic deformation of the host–inclusion pairs induced by the development of large differential stresses during experimental cooling. Therefore, inclusion strains are more reliable for inclusions trapped at lower temperature conditions in the α-quartz field where there is less inelastic deformation of the host–inclusion systems. On the other hand, entrapment isomekes of zircon inclusions entrapped in the β-quartz stability field plot along the α–β quartz phase boundary, suggesting that the inclusion strains were mechanically reset at the phase boundary during experimental cooling and decompression. Therefore, inclusions contained in soft host minerals can be used for elastic thermobarometry and inclusions contained in β-quartz may provide constraints on the P – T at which the host–inclusion system crossed the phase boundary during exhumation.

Diamonds and their inclusions are unique fragments of deep Earth, which provide rare samples from inaccessible portions of our planet. Inclusion-free diamonds cannot provide information on depth of formation, which could be crucial to understand how the carbon cycle operated in the past. Inclusions in diamonds, which remain uncorrupted over geological times, may instead provide direct records of deep Earth’s evolution. Here, we applied elastic geothermobarometry to a diamond-magnesiochromite (mchr) host-inclusion pair from the Udachnaya kimberlite (Siberia, Russia), one of the most important sources of natural diamonds. By combining X-ray diffraction and Fourier-transform infrared spectroscopy data with a new elastic model, we obtained entrapment conditions, P trap  = 6.5(2) GPa and T trap  = 1125(32)–1140(33) °C, for the mchr inclusion. These conditions fall on a ca. 35 mW/m 2 geotherm and are colder than the great majority of mantle xenoliths from similar depth in the same kimberlite. Our results indicate that cold cratonic conditions persisted for billions of years to at least 200 km in the local lithosphere. The composition of the mchr also indicates that at this depth the lithosphere was, at least locally, ultra-depleted at the time of diamond formation, as opposed to the melt-metasomatized, enriched composition of most xenoliths.

The thermo-elastic behavior of synthetic single crystals of grossular garnet (Ca3Al2Si3O12) has been studied in situ as a function of pressure and temperature separately. The same data collection protocol has been adopted to collect both the pressure-volume (P-V) and temperature-volume (T-V) data sets to make the measurements consistent with one another. The consistency between the two data sets allows simultaneous fitting to a single pressure-volume-temperature Equation of State (EoS), which was performed with a new fitting utility implemented in the latest version of the program EoSFit7c. The new utility performs fully weighted simultaneous fits of the P-V-T and P-K-T data using a thermal pressure EoS combined with any P-V EoS. Simultaneous refinement of our P-V-T data combined with that of KT as a function of T allowed us to produce a single P-V-T-KT equation of state with the following coefficients: V0=1664.46(5)°A3, KT0=166.57(17) GPa and K'=4.96(7)α(300K,1bar)=2.09(2)×10-5K-1 with a refined Einstein temperature (θE) of 512 K for a Holland-Powell-type thermal pressure model and a Tait third-order EoS. Additionally, thermodynamic properties of grossular have been calculated for the first time from crystal Helmholtz and Gibbs energies, including the contribution from phonons, using density functional theory within the framework of the quasi-harmonic approximation.

Elastic thermobarometry (or piezobarometry) is the process of determining the P (pressure) and T (temperature) of entrapment of inclusions from their pressure, stress or strain measured when their host mineral is at room conditions. The methods and software used for piezobarometry are currently restricted to inclusions consisting of single phases. In this contribution we describe the theory of the elasticity of mixtures of different phases and combine it with the existing isotropic analysis of the elastic interactions between single-phase inclusions and their hosts to calculate the inclusion pressures of mixed-phase inclusions. The analysis shows that the reliability of calculated entrapment conditions for mixed-phase inclusions, including those containing fluid plus minerals, depends in a complex way upon the contrasts between the elastic properties of the host and the phases in the inclusion. The methods to calculate the entrapment conditions of mixed-phase inclusions have been incorporated into the EosFit7c program (version 7.6) that is available as freeware from http://www.rossangel.net.

The dehydration of antigorite is an important reaction in subduction zones with implications on both geochemical and geophysical processes. In this experimental study we focus on the onset of antigorite dehydration and investigate various chemical and physical parameters as possible drivers for the fluid release. We performed hydrostatic and co-axial Griggs experiments on antigorite serpentinites with variable chemical composition and microstructures at high-pressure and high-temperature conditions across the antigorite dehydration (1.5 GPa, 620–670 °C). For these conditions, our thermodynamic models predict the formation of olivine from magnetite decomposition and partial dehydration of antigorite. Detailed analyses of the run products reveal limited magnetite decomposition. Antigorite dehydration is restricted to samples that have been deformed. Nano-sized olivine and orthopyroxene formed locally in oblique dehydration bands and exhibit neither a clear crystallographic preferred orientation nor a topotactic relation with precursor antigorite. We argue that limited local dehydration in our experiments is related to strain and variations in reaction kinetics. Systematic investigation excludes mineralogical and chemical heterogeneities, and temperature gradients as reaction driving potentials. The structural relation of the dehydration bands suggests deformation-related dehydration, which is supported by numerical simulations that couple reaction kinetics with mechanical work rate and self-consistently predict dehydration bands. In this scenario, strain concentration due to applied axial stress locally increases the internal energy of antigorite to reach the activation energy of the dehydration reaction, enabling dehydration. This study highlights the importance of coupled mechanical and chemical processes and provides a mechanistic framework for deformation-induced dehydration of antigorite.