Abstract
Elastic geothermobarometry relies on the contrast between the thermal expansion and compressibility of a mineral inclusion and its surrounding host, leading to a residual pressure in the inclusion (P_{inc}) that may differ significantly from the external pressure. Quartzingarnet (QuiG) inclusionhost systems are widely used in elastic geothermobarometry to estimate the inclusion entrapment conditions and thus the rock petrogenesis. To elucidate the behaviour of QuiG at elevated pressures, we have applied in situ highpressure Raman spectroscopy on three QuiG samples having P_{inc} close to 0 GPa at room temperature. We demonstrate that upon pressure increase, the garnet host acts as a shield to the softer quartz inclusion. Consequently, the P_{inc} increases with a smaller rate compared to that of the external pressure. Up to 2.5 GPa, the evolution of P_{inc} calculated from the Raman data agrees very well with prediction from the equations of state. Furthermore, the behaviour of a quartz inclusion in a relatively thin host specimen was explored up to external pressures of 7 GPa. Our results indicate that the shielding effect of the host (even if only partial because of the insufficient distance between the inclusion and the host surface) can keep the quartz inclusions thermodynamically stable up to about 2 GPa above the equilibrium quartz–coesite phase boundary. In addition, the partial shielding leads to the development of anisotropic symmetrybreaking stresses and quartz inclusions undergo a reversible crossover to a lower symmetry state. Given that the presence of nonhydrostatic stress may influence the quartztocoesite phase boundary, especially at elevated temperatures relevant for entrapment conditions, our results emphasize the importance of elastic anisotropy of QuiG systems, especially when quartz inclusion entrapment occurs under conditions close to the coesite stability field.
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Introduction
At ambient conditions, a mineral inclusion in a mineral host can display a pressure that significantly differs from the external one due to the difference in the thermal expansion (\(\Delta \mathrm{\alpha }={\mathrm{\alpha }}_{\mathrm{inclusion}}{\mathrm{\alpha }}_{\mathrm{host}}\)) and compressibility (\(\Delta\upbeta ={\upbeta }_{\mathrm{inclusion}}{\upbeta }_{\mathrm{host}}\)) coefficients between the inclusion and the host (e.g. Rosenfeld and Chase 1961). The precise quantification of the inclusion pressure (P_{inc}) is of great interest in mineralogy and petrology, because it can provide information on the inclusion entrapment pressure (P_{trap}) and temperature (T_{trap}) and thus on its metamorphic history. This is the subject of elastic thermobarometry. This method appears particularly attractive, because it does not depend on chemical equilibrium and hence represents a good alternative, or a complementary tool, to classical thermobarometry based on thermodynamic equilibrium (e.g., Angel et al. 2015; Kohn et al. 2023). During the last few years, elastic thermobarometry has considerably advanced. The effect of hostinclusion geometry, in terms of inclusion shape, size, and position in the host, as well as the intrinsic anisotropy in the elastic properties of the crystalline inclusion and the host have been implemented in the model (e.g., Mazzucchelli et al. 2018; Campomenosi et al. 2018; Murri et al. 2018; Zhong et al. 2020; Alvaro et al. 2020; Moulas et al. 2020; Gilio et al. 2021; Gonzalez et al. 2021).
Quartzingarnet (QuiG) and zirconingarnet (ZiG) are among the most used inclusionhost systems in elastic thermobarometry, because they can be found in many metamorphic rocks (e.g. Zhong et al. 2019; Gilio et al. 2022). As QuiG systems show ∆α < ∆β, they are more sensitive to changes in pressure P and, therefore, they are good barometers. On the other hand, for ZiG systems ∆α > ∆β, making them chiefly sensitive to changes in temperature T and, consequently, good thermometers (e.g. Kohn 2014; Thomas and Spear 2018; Bonazzi et al. 2019).
Raman spectroscopy is a powerful method to obtain P_{inc} at atmospheric P and room T. Moreover, it provides the opportunity to study hostinclusion systems at nonambient conditions. For instance, since ZiG systems are sensitive to temperature changes, we recently demonstrated that in situ Raman spectroscopy at high temperatures and ambient pressure can be used to reach the conditions of mechanical equilibrium between zircon and garnet (Campomenosi et al. 2023). In addition, we have shown that at higher temperatures, zircon inclusions experience a tensile strain imposed by the host, leading to a negative P_{inc}, which triggers ZiG systems to instantaneous plastic relaxation. Consequently, the P_{inc} of zircon inclusions measured after a heating–cooling cycle are representative of the conditions explored via the experiments rather than the conditions of original entrapment. The possibility of instantaneous elastic resetting of the inclusion when P_{inc} is lower than the external pressure has important implications in petrology, because it shows that ZiG systems can be very sensitive to different exhumation P–T paths (Campomenosi et al. 2023). However, we also demonstrated that when the ZiG system behaves elastically, the elastic models used within the framework of the elastic thermobarometry correctly reproduce the P_{inc} measured in natural samples also at nonambient conditions.
By the same logic, since QuiG systems are good barometers, here we show that in situ Raman spectroscopy at high pressure and room temperature, using the diamondanvilcell (DAC) technique, can provide valuable information about the elastic interaction between the two minerals under nonambient conditions. Our results validate the current elastic models and EoS used to predict entrapment conditions of QuiG systems in the investigated pressure range (up to 2.5 GPa). In addition, we have performed experiments up to 7 GPa on a QuiG system in which the inclusion was only partially shielded from the external pressure to explore the hostinclusion geometry effects.
Theoretical background: a recap on residual strain and P_{inc} in hostinclusion systems
In this study, we are concerned only with the elastic deformation of the inclusion developed from its entrapment to its final state after the rock exhumation (Angel et al. 2014a, b; 2015; Mazzucchelli et al. 2019), without considering possible viscoplastic deformation (e.g. Zhong et al. 2020).
The volume equationofstate (EoS), V(P), provides a tool to calculate the volume strain ε_{V}. To take into account the elastic anisotropy of the system, we should use the strain tensor ε_{ij}, which can be calculated from the axial EoSs, l_{i}(P), where l_{i} represents the corresponding unitcell parameter. However, the EoSs do not include the elastic relaxation process to restore the mechanical equilibrium between the host and the inclusion at the final state. Therefore, the calculation of the true residual strain is a twostep procedure (Fig. 1), which for QuiG systems is (Mazzucchelli et al. 2019).

1)
the unrelaxed strain \({\upvarepsilon }_{\mathrm{ij}}^{\mathrm{unrel}}\) is calculated as
$$\varepsilon _{{\text{ij}}({\text{qz}})}^{{\text{unrel}}} = \left\{ {\begin{array}{*{20}{c}} {{{\left( {\frac{{{\text{V}}_{{\text{gr}}}^{{\text{end}}}}}{{{\text{V}}_{{\text{gr}}}^{{\text{trap}}}}}} \right)}^{1/3}}\frac{{l_{{\text{i}},{\text{qz}}}^{{\text{trap}}}}}{{l_{{\text{j}},{\text{qz}}}^{{\text{end}}}}}  1,}&{{\text{i}} = {\text{j}}} \\ {0,}&{{\text{i}} \ne {\text{j}}} \end{array}} \right.$$(1)
where V_{gr} is the volume of garnet at the entrapment (trap) and final (end) conditions, while l_{i,qz} is the corresponding unitcell parameter of quartz, derived from EoSs.

2)
the relaxed strain \({\upvarepsilon }_{\mathrm{kl}}^{\mathrm{rel}}\) is calculated aswhere R_{klij} is the fourthrank relaxation tensor, specific for a given hostinclusion system (Mazzucchelli et al. 2019), because it depends on the elastic properties of both the host and inclusion as well as their relative crystallographic orientation.
$${\varepsilon }_{\mathrm{kl}(\mathrm{qz})}^{\mathrm{rel}}={\mathrm{R}}_{\mathrm{klij}}{\varepsilon }_{\mathrm{ij}(\mathrm{qz})}^{\mathrm{unrel}},$$(2)
Knowing the residual strain tensor, we can then calculate the stress tensor σ_{mn} as
where \({\mathrm{c}}_{\mathrm{mnkl}(\mathrm{qz})}\) is the fourthrank elastic stiffness tensor of quartz, and finally, the inclusion residual pressure P_{inc} is
Note that the Einstein summation convention is used in Eqs. 2 and 3.
The important difference between P_{inc} calculated from unrelaxed and relaxed residual strains can be understood by introducing the concept of the entrapment isomeke (Adams et al. 1975): a line in P–T space along which the inclusion volume perfectly matches the volume cavity in the host. Along the entrapment isomeke, the pressure in the inclusion is equal to that of the host P_{host}, which in turn is identical to the external pressure P_{ext} (see Fig. 1). The pressure on the entrapment isomeke at T_{end} is called P_{foot} and it can be either positive or negative depending on the entrapment conditions. For P_{foot} ≠ 0, \(\left{{P}}_{\mathrm{foot}}\right>\left{{P}}_{\mathrm{inc}}^{\mathrm{unrel}}\right>\left{{P}}_{\mathrm{inc}}^{\mathrm{rel}}\right\)(Fig. 1). The last, \({{P}}_{\mathrm{inc}}^{\mathrm{rel}}\), is the relaxed inclusion pressure (hereafter referred to as P_{inc}) that we can measure experimentally. Null or negative P_{inc} at ambient conditions may develop in natural quartz inclusions that were trapped (or reset) toward or within the granulite facies (e.g., Kouketsu et al. 2014).
The roomtemperature P_{inc} can be derived from the Ramanpeak positions via two methods: (i) the hydrostatic approach, using experimentally established ω(P) trends on reference free crystals (e.g. Enami et al. 2007; Kohn 2014; Kouketsu et al. 2014; Zhong et al. 2019), and (ii) the mode Grüneisen tensor approach, based on the relation (e.g., Angel et al. 2019 and references therein)
where \({{\varvec{\gamma}}}^{\mathbf{m}}\) is the Grüneisen tensor of phonon mode m, \(\frac{\Delta {\omega }^{\mathrm{m}}}{{\omega }_{0}^{\mathrm{m}}}\) is the relative change in the wavenumber ω of a phonon mode m measured in the inclusion with respect to a free crystal at a given temperature and pressure, and “:” represents a doublescalar product. For uniaxial crystals like quartz, the nonzero independent straintensor components are ε_{11} and ε_{33}, and therefore, Eq. 5 becomes
The Grüneisen tensor components can be calculated from densityfunctionaltheory (DFT) simulations (e.g., Murri et al. 2018; Stangarone et al. 2019), and once we measure ∆ω, we can calculate the relaxed strain tensor and, therefore, the relaxed P_{inc} by applying Eqs. 3 and 4.
The hydrostatic approach is easytohandle and requires only one Raman peak to be measured, but it assumes negligible deviatoric stress in the hostinclusion system. The Grüneisen tensor approach considers the elastic anisotropy, but the minimum number of Raman peaks to be measured should be equal to the nonzero independent straintensor components; that is, for quartz inclusions at least two Raman peaks should be used.
Materials and methods
Sample description
Three natural garnet samples (S3, S4, and S5), each containing a single natural quartz inclusion, were manually extracted from a doublepolished section of a garnetbearing paragneiss coming from the Fjortoft island (62°42′43.4ʺN 6°24′54.7ʺE, WesternGneissRegion, Norway). The chemical composition of the host garnet is (Fe_{1.7}Mg_{0.9}Mn_{0.02}Ca_{0.2})(Al_{1.95})(SiO_{4})_{3} thus consisting of ~ 56 mol% almandine, ~ 35 mol% pyrope, ~ 8 mol% grossular, and ~ 1 mol% spessartine as major endmembers.
The QuiG specimens (Fig. 2) were loaded in a BoehlerAlmax DAC equipped with diamond culets of 600 µm in diameter. Stainless steel gaskets with a thickness of 250 µm were drilled using an AlmaxeasyLab spark eroder with a tungsten carbide tip of 300 µm. Gasket preindentation, sample dimensions, and initial P_{inc} are shown in Table 1. It should be noticed that according to previous studies (Mazzucchelli et al. 2018; Campomenosi et al. 2018; Zhong et al. 2020), the inclusion in sample S5 is too close to the external surfaces of the host for it to be elastically isolated. However, the thinner host of S5 allowed us to study this QuiG specimen to higher external pressures and to explore the effects of reduced shielding of the inclusion.
The pressuretransmitting medium used was a mixture of methanol, ethanol, and distilled water with proportion 16:3:1, respectively, providing hydrostatic pressure up to 10 GPa (Angel et al. 2007). The external P applied on the QuiG was determined via the ruby photoluminescence R1 line (Munro et al. 1985) with a precision of ~ 0.1 GPa.
Raman spectroscopy
Raman spectra were collected using a Horiba Jobin–Yvon T64000 triplegrating monochromator spectrometer (1800 gr/mm) equipped with a Symphony LN2cooled CCD detector and an Olympus BH41 microscope with a 50 × longworkingdistance objective (N.A. 0.5). Raman scattering was excited with the 488.0nm line of a Coherent Innova 90C FreD Ar + laser. The resulting spectral resolution was ~ 2 cm^{−1} and the instrumental precision in determining the peak position was 0.35 cm^{−1}. The acquisition time for each spectrum was between 60 s (sample S3 and S4) and 90 s (sample S5) averaged over 10 accumulations (i.e., total acquisition time of 600 and 900 s respectfully) to improve the signal–background ratio. The spectral range set for quartz and garnet Raman spectra was between 50 and 1250 cm^{−1}. Instrumental drift was monitored by collecting the Raman spectra of a free quartz crystal at each pressure step and the resulting average standard deviation in the peak positions was ~ 0.2 cm^{−1}. A background spectrum was measured alongside the samples and then subtracted from the sample spectrum during data evaluation. In addition, the spectra were temperaturereduced according to the Bose–Einstein phonon population factor (e.g., Kuzmany 2009). The OriginPro^{®} 2021 software package was used for data evaluation. Peak position (ω, fullwidth at halfmaximum (FWHM) and integrated intensity were determined by fitting the Raman spectra with a pseudoVoigt function \(\mathrm{PV}=\mathrm{qL}+\left(1\mathrm{q}\right)\mathrm{G}\) where L and G are the Lorentz and Gauss peakshape functions, respectively, and q is a weight coefficient.
Estimating inclusion residual strain and pressure from Raman spectra
The residual pressure P_{inc} of quartz inclusions was calculated with respect to a free quartz crystal at atmospheric pressure, i.e., at P_{ext} ~ 0 GPa. In addition, to reveal better the effect of the garnet host, the residual pressure of the inclusion was also calculated with respect to a free quartz crystal at the same external pressure as that applied to the host, i.e., the pressure in the DAC. For clarity, we will refer to this pressure hereafter as to ∆P, which is the difference between P_{inc} and P_{ext}. Both P_{inc} and ∆P were determined using both the hydrostatic approximation and the Grüneisen tensor approach. For the hydrostatic approximation, we used the P calibration of Morana et al. (2020) for the Raman peaks generated by the E mode near 128 cm^{−1}, the A_{1} mode near 207 cm^{−1}, and A_{1} mode near 464 cm^{−1} (hereafter designated as \({\mathrm{E}}^{128}\), \({\mathrm{A}}_{1}^{207},\) and \({\mathrm{A}}_{1}^{464}\), respectively). For the second approach, we used the same phonon modes with the Grüneisen tensor components calculated via DFT simulations (Murri et al. 2018). The residual stress and, subsequently, the residual P_{inc} at ambient conditions, were calculated from the strain, using the elastic stiffness tensor c of quartz reported by Lakshtanov et al. (2007) normalized to the isothermal bulk modulus of quartz (Angel et al. 2017). The corresponding c_{ijkl} values are available on the EntraPT online platform (Mazzucchelli et al. 2021). At high pressures, the same calculations were carried out, but employed a modified c tensor components according to the dc_{ijkl}/dP values obtained from linear fits to the available experimental data points of c_{ijkl}(P) at room temperature (Wang et al. 2015).
Predicting inclusion residual strain and pressure from available EoS
The prediction of inclusion residual strain and P_{inc} according to the available EoS was performed using the EoSFit7c software (Angel et al. 2014a, b) available at www.rossangel.net. For the quartz inclusions, we used the EoS of Angel et al. (2017), while for the garnet host, we used the purealmandine EoS (Angel et al. 2022), because the exact garnet composition has negligible effects in the case of QuiG (Angel et al. 2022).
The output of Eosfit7c (from the EPTH command in the HOSTINC utility) is the unrelaxed residual strain, from which the relaxed strain was calculated using the relaxation tensor approach (Mazzucchelli et al. 2019). Then, the relaxed stress components and P_{inc} were calculated, using the c tensor specified in the previous subsection. We employed a relaxation tensor R for quartzinpyrope, because R for the quartzinalmandine system is not available. To verify that the garnet composition has negligible effect on the strain relaxation, we performed the same calculations with the available quartzingrossular R tensor; the difference in the relaxed volume strains was ~ 0.00015, corresponding to differences in P_{inc} of ~ 0.005 GPa. We further assumed that dR(P)/dP = 0 for calculations at high pressures. Error propagation in the resulting P_{inc} at high pressure was estimated following the approach of Mazzucchelli et al. (2021). A worked example of this calculation of strains and P_{inc} from EoSs is given in the appendix.
The entrapment conditions (P_{trap} and T_{trap}) used to predict P_{inc} were chosen to fit (within error) the inclusion volume strain and P_{inc} measured at ambient conditions: P_{trap} and T_{trap} of 1.5 GPa and 1173 K for the sample S3 and of 1.4 GPa and 1173 K for the samples S4 and S5. Note that these entrapment conditions are consistent with the metamorphic history of rocks coming from the same area (Gilio et al. 2022).
Results and discussion
Main Raman spectral features: QuiG vs free quartz
For all the analysed samples, the quartz Raman peaks generated by the \({\mathrm{E}}^{128}\), \({\mathrm{A}}_{1}^{207}\), and \({\mathrm{A}}_{1}^{464}\) modes were very well resolved in the entire pressure range investigated (Fig. 3A); thus, we will focus on these three phonon modes. Figure 3 B shows the pressure dependence ω(P) for QuiG and a free quartz. As can be seen, the pressure derivatives \({(\partial \omega /\partial {P})}_{\mathrm{T}}\) are significantly smaller for QuiG than for a free quartz crystal (Fig. 3 B). This clearly demonstrates the shielding effect of the host garnet: the wavenumber of the quartz inclusion is not a function of the external pressure (P_{ext}), but rather of the strain imposed by the host at that P_{ext}. Because garnet is stiffer than quartz, the inclusion experiences an effectively lower pressure than the actual external pressure acting on the host. The shielding effect of the host is also responsible for the significant difference in FWHM shown by the \({\mathrm{A}}_{1}^{207}\) mode measured in the inclusion with respect to that of a free crystal. At ambient pressure, this mode is involved in strong phonon–phonon interactions, leading to a large FWHM (Jayaraman et al. 1987). Upon a pressure increase, the phonon–phonon interactions weaken, and consequently, the FWHM strongly decreases (e.g., Morana et al. 2020). Hence, the systematically larger FWHM for the \({\mathrm{A}}_{1}^{207}\) mode measured in the inclusion than the free crystal implies that P_{inc} < P_{ext}.
Quantifying P_{inc}: the hydrostatic approach
Figure 4A and Table S1 show the P_{inc} calculated from the difference in wavenumber of the inclusions with respect to a free crystal at ~ 0 GPa, ∆ω_{0}, and the hydrostatic P(ω) relations (Morana et al. 2020). The P_{inc} inferred from the three different modes agree within 0.1 GPa for all the three samples. Samples S3 and S4 also show a relatively good agreement with EoS predictions across the full pressure range investigated, whereas sample S5 does not. The P_{inc} in sample S5 are shifted from the pressures predicted by the elastic model of isolated inclusions toward the external pressure (blue line in Fig. 4 A). This reflects a lower degree of elastic isolation of the inclusion in sample S5 from the external environment, due to it being closer to the external surface of the host (Table 1). Nevertheless, all three quartz inclusions show P_{inc} significantly smaller than the external pressure, thus demonstrating the shielding effect of the host garnet. Figure 4A also shows the possibility to directly determine P_{foot} by highpressure experiments as the pressure at which P_{inc} = P_{ext}, thus providing the locations of the entrapment isomeke.
An alternative way to demonstrate the shielding effect of garnet would be to calculate ∆P (Table S2), which is the difference between the pressure in the inclusion and the external pressure. One might think to calculate ∆P from the ∆ω_{ext} with respect to a free quartz crystal at P_{ext} (Fig. 4 B) using the same hydrostatic P(ω) relations (Morana et al. 2020). In this way, the three modes yield negative ∆P, emphasizing that the stiffer garnet host effectively reduces the pressure acting on the softer quartz inclusion. However, only the \({\mathrm{A}}_{1}^{464}\) mode gives results that are consistent with the EoS predictions (sample S3 and S4), while for P_{ext} > 0.5 GPa, the pressures derived from the \({\mathrm{E}}^{128}\) and \({\mathrm{A}}_{1}^{207}\) modes strongly deviate from the EoS results. This discrepancy can be understood by considering the nonlinearity of the phonon wavenumbers with respect to pressure. The \({\mathrm{A}}_{1}^{464}\) mode shows linear behaviour of the wavenumber with increasing pressure, whereas the other two modes show a nonlinear behaviour (Schmidt and Ziemann 2000; Morana et al. 2020). The latter makes the \({\mathrm{E}}^{128}\) and \({\mathrm{A}}_{1}^{207}\) phonon modes unsuitable to calculate ∆P directly from the measured ∆ω_{ext} using the hydrostatic calibration with respect to a free quartz crystal at P_{ext}. A more correct approach is to obtain dP/dω at P_{ext} from the hydrostatic calibration, and apply that to obtain ∆P = ∆ω_{ext}. dP/dω, and this yields values consistent with the P_{inc}, as shown in Fig. 4A. In practical terms, this means that one should take into account adhoc polynomial equations obtained by fitting the data collected under hydrostatic pressure starting from the external pressure at which the ∆P has to be determined rather than from ambient conditions. Consequently, each ∆P point in the diagram would require a different hydrostatic calibration.
Quantifying residual volume strain and P_{inc}: the anisotropic approach
The volume strain of the inclusion \({\varepsilon }_{V}^{inc}\) calculated with respect to a free quartz at ~ 0 GPa and, consequently, the P_{inc} determined by the Grüneisen tensor approach are also in a good agreement with the EoS predictions within the investigated pressure range (Figs. 5A and 6A, Table S3). On the other hand, when considering the difference between the volume strain of the inclusion and the volume strain of a free quartz at P_{ext} (i.e. \({\Delta \varepsilon }_{V}\)) and the corresponding ∆P, (Fig. 5B and 6B), the Grüneisen approach gives consistent results with EoS predictions up to P_{ext} ~ 1.2 GPa for samples S3 and S4. The good agreement obtained in quantifying \({\Delta \varepsilon }_{V}\) and ∆P with the anisotropic approach employing the mode Grüneisen tensor of all the three modes is because the Grüneisen tensor components γ_{ij} of quartz can be considered as constant up to ca. 2 GPa (Murri et al. 2019). Consequently, the observed disagreement between EoS and our data at P_{ext} > 1GPa may indicate that at high pressures, γ_{ij} become straindependent. In addition, Murri et al. (2018, 2019) and Morana et al. (2020) have already pointed out the limitation of the Grüneisentensor approach in predicting the correct ∆ω for the \({\mathrm{A}}_{1}^{207}\) mode above 2.5 GPa on a free quartz crystal under hydrostatic conditions due to the relatively strong phonon–phonon interactions this mode experiences. However, test calculations of \({\Delta \varepsilon }_{V}\) carried out without this mode did not improve the agreement between ∆P derived from Raman data and from EoS.
Modified metastability of quartz and symmetry breaking
Previous studies on free quartz at high pressures revealed that the change in the pressure dependence of the FWHM of the \({\mathrm{A}}_{1}^{207}\) mode reflects the metastability of quartz as it enters the thermodynamic stability field of coesite (Morana et al. 2020) at ~ 2.4 GPa at 298 K (Bose and Ganguly 1995). As described above, all samples studied here show an FWHM of this mode larger than that of a free crystal at the same P_{ext} (Fig. 3), suggesting that quartz inclusions become metastable with respect to coesite at an external pressure higher than that for a free quartz crystal because of the shielding effect of garnet. This hypothesis can be demonstrated by exploring quartz inclusions at higher pressure.
The thinner sample S5 (see Table 1) allowed us to analyse the quartz inclusion up to 7 GPa, i.e. far beyond the pressure at which coesite becomes thermodynamically stable with respect to quartz under hydrostatic stress. The pressure dependence of the FWHM of the \({\mathrm{A}}_{1}^{207}\) mode indicates that the quartz inclusion in sample S5 becomes metastable with respect to coesite at P_{ext} of ~ 4.4 GPa, i.e. apparently 2 GPa above the quartz–coesite phase boundary (Fig. 7A). However, this external pressure corresponds to an inclusion pressure P_{inc} of ~ 2.4 GPa (Figure S1), which is the pressure of the quartz–coesite phase boundary at room temperature (Bose and Ganguly 1995). This result confirms that the difference in the observed P_{ext} at which the quartz metastability begins is almost entirely the shielding effect of the host garnet. In natural systems, this shielding effect means that quartz inclusions in a garnet core remain thermodynamically stable even on prograde paths that take the garnet into the stability field of coesite. If further growth of the garnet occurs in the coesite field then the recovered garnets have quartz inclusions in their cores together with coesite inclusions in the rim of the same garnet (e.g. Reinecke 1998; Parkinson 2000).
In addition, at P_{ext} > 4.4 GPa, the doubly degenerate E^{128} mode of sample S5 splits into two nondegenerate components, with a wavenumber difference of ~ 3 cm^{−1} (Fig. 7B). The lifting of mode degeneracies is a result of the socalled morphic effect: a symmetry reduction of the crystal point group due to an external anisotropic field (e.g., electric field, uniaxial stress field, and magnetic field) (Gregora 2006). Since we apply pressure on the QuiG system, the observed symmetry reduction of the quartz inclusion is caused by stressinduced morphic effects. The effect of symmetrybreaking stress on the Raman scattering of quartz has been investigated in several studies using both experimental and theoretical approaches (Tekippe et al. 1973; Briggs and Ramdas 1977; Murri et al. 2019, 2022). According to Tekippe et al. (1973), if quartz undergoes a stressinduced symmetry reduction from trigonal to monoclinic (D_{3} → C_{2}; Schönflies notation), the splitting magnitude of the E^{128} mode observed in our experiment would correspond to a deviatoric stress of ~ 0.7 GPa, acting perpendicularly to the trigonal caxis on the quartz inclusion. It is worth noting that the splitting of the doubly degenerate \({\mathrm{E}}^{128}\) mode and the change in the slope of the pressure dependence of FWHM of the \({\mathrm{A}}_{1}^{207}\) mode for QuiG occur at the same external pressure. This suggests that in the presence of symmetrybreaking stresses, quartz undergoes a crossover to a lowersymmetry state when it becomes thermodynamically metastable with respect to coesite.
Given that garnet under hydrostatic stress undergoes isotropic strain, a fully isolated quartz inclusion keeps the original symmetry unchanged. Thus, the main source of stressinduced symmetry breaking in our experiment is the effects of hostinclusion geometry, namely the position of the inclusion inside the host. The distance between the inclusion boundary and the surface of the host is different along the directions perpendicular and parallel to the diamond culets (see Fig. 1 and Table 1), leading to an anisotropic shielding of the inclusion by the host subjected to the external hydrostatic pressure (e.g., Mazzucchelli et al. 2018; Zhong et al. 2020). That the hostinclusion geometry of sample S5 (Table 1) does not fully meet the requirements for perfectly isolated inclusions, can also be deduced from the deviation in \({\varepsilon }_{V}^{inc}\) and P_{inc} from the corresponding EoS predictions (Figs. 5, 6A). Nevertheless, our results still represent possible natural processes, because during the garnet crystallization, the inclusion cannot be fully isolated from the surrounding environment. For instance, hostinclusion geometry effects and the consequent anisotropic stress state developed in the inclusion can be very important during quartz entrapment close to the coesite stability field at high temperature (e.g. Richter et al. 2016). Therefore, the interpretation of quartz and coesite inclusions, especially when trapped within the same garnet host but at different stages of growth (e.g. Parkinson 2000), can be quite complex and the elastic anisotropy together with the evolving geometry of the hostinclusion system should not be neglected.
Conclusions
The analysis of residual volume strain and P_{inc} of quartz inclusions within garnet (QuiG) at high external pressures reported in this study demonstrates the shielding effect of the host garnet and the applicability of Raman spectroscopy to determine the P_{inc} even when the host is not at atmospheric pressure. For geobarometers like QuiG, in situ Raman spectroscopy at high pressure can be used to directly determine P_{foot} by determining when the volume strain and residual pressure of the inclusion match those of a free quartz crystal included inside the DAC.
We further show that the apparent metastability of quartz inclusions with respect to coesite at room temperature is simply the result of the shielding effect of the garnet host; quartz metastability indicated by sharpening of the \({\mathrm{A}}_{1}^{207}\) mode actually occurs at inclusion pressures corresponding to the equilibrium phase boundary at the temperature of the experiment. Moreover, the development of nonhydrostatic stresses due to the hostinclusion geometry may lead to a reduced symmetry state of quartz inclusions to adopt to the metastability of quartz with respect to coesite at high pressure. Finally, because under geologically relevant pressure and temperature conditions, symmetrybreaking stresses may also affect the mean pressure at which quartz transforms into coesite, the partial shielding of the garnet host may have important consequences in the formation of coesite inclusions.
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Acknowledgements
This work was financially supported by the Alexander von Humboldt Foundation in the form of a fellowship to N.C., and from the ERC under the European Union’s Horizon 2020 research and innovation program under Grant Agreement 714936 (ERCSTG TRUE DEPTHS) to M.A. This manuscript was greatly improved by the comments of two anonymous reviewers and by the editorial handling of Othmar Müntener.
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Appendix
Appendix
Calculated residual strain and P _{ inc } from quartz and garnet equations of state EoSs given the entrapment conditions ( P _{ trap } and T _{ trap } ) and the final conditions ( P _{ end } and T _{ end } ): a worked example
To apply Eq. 1 and calculate the unrelaxed volume strain tensor of the inclusion at different pressure with respect to a free crystal at the same external conditions, we need to know the volume of garnet and the volume and unitcell parameters of quartz at the entrapment (trap) and final (end) conditions. Such values along with their relative change (e.g., Volume_{end}/Volume_{trap} etc.) are listed in Table 2 for entrapment conditions of 1.5 GPa and 1173 K.
The corresponding unrelaxed independent strain tensor components of quartz ε_{11} = ε_{22}, ε_{33} and the volume strain eε_{V} obtained by employing data from Table 2 into Eq. 1 are listed in Table 3
The deformation along the caxis ε_{33} was calculated, by symmetry, as ε_{33} = ε_{V}−(2ε_{11}). Note that these are the unrelaxed strains with respect to a free crystal at the same external conditions (i.e. at the same T_{end} and P_{end}).
Then, relaxed strains are calculated from the unrelaxed ones according to Eq. 2 and using the quartzinpyrope relaxation tensor R (Mazzucchelli et al. 2019)
The obtained relaxed strains are listed in Table 4.
To calculate the strains of the inclusion with respect to a free crystal kept at ~ 0 GPa and ~ 300 K we add to the relaxed strains calculated above (Table 4), the strains developed in a free quartz crystal brought from ~ 0 GPa and ~ 300 K to the P_{end} and T_{end} of interest (Table 5). The results are listed in Table 6.
Finally, residual stress and P_{inc} were calculated according to Eqs. 3, 4 by considering how the elastic stiffness tensor c changes with P (e.g. Wang et al. 2015). Note that the elastic stiffness tensor components were modified according to the expected inclusion P_{inc} independently estimated with the program EoSfitPinc (Angel et al. 2014a, b).
The residual stress tensor σ and P_{inc} of a quartz inclusion with respect to a free crystal kept at ambient conditions are listed in Table 7.
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Campomenosi, N., Angel, R.J., Alvaro, M. et al. Quartzingarnet (QuiG) under pressure: insights from in situ Raman spectroscopy. Contrib Mineral Petrol 178, 44 (2023). https://doi.org/10.1007/s00410023020268
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DOI: https://doi.org/10.1007/s00410023020268