Host-inclusion Systems Research Articles

Clustering and interfacial segregation of radiogenic Pb in a mineral host-inclusion system: Tracing two-stage Pb and trace element mobility in monazite inclusions in rutile

Abstract Accessory minerals like zircon, rutile and monazite are routinely studied to inform about the timing and nature of geological processes. These studies are underpinned by our understanding of the transfer processes of trace elements and the assumption that the isotopic systems remain undisturbed. However, the presence of microstructures or Pb-bearing phases in minerals can lead to the alteration of the Pb isotopic composition. To gain insight into the relationship between Pb isotopic alterations from inclusions and microstructures, this study focused on inclusions from an ultra-high-temperature metamorphic rutile. The studied inclusions are submicrometer monazites, a common mineral rich in Pb but normally not present in rutile. The sample is sourced from Mt. Hardy, Napier Complex, East Antarctica, an ultra-high-temperature (UHT) metamorphic terrane. By applying correlative analytical techniques, including electron backscatter diffraction mapping, transmission electron microscopy (TEM), and atom probe tomography, it is shown that monazite inclusions are often in contact with low-angle boundaries and yield no preferred orientation. TEM analysis shows the monazite core has a mottled texture due to the presence of radiation damage and nanoclusters associated with the radiation damage defects that are rich in U, Pb, and Ca. Some monazites exhibit a core-rim structure. The rim yields clusters composed of Ca- and Li-phosphate that enclose Pb nanoclusters that are only present in small amounts compared to the core, with Pb likely diffused into the rutile-monazite interface. These textures are the result of two stages of Pb mobility. Initial Pb segregation was driven by volume diffusion during UHT metamorphism (2500 Ma). The second stage is a stress-induced recrystallization during exhumation, leading to recrystallization of the monazite rim and trace element transport. The isotopic signature of Pb trapped within the rutile-monazite interface constrains the timing of Pb mobility to ca. 550 Ma.

First evaluation of stiff-in-soft host–inclusion systems: experimental synthesis of zircon inclusions in quartz crystals

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 (Pinc) 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.

High-temperature behavior of quartz-in-garnet system revealed by in situ Raman spectroscopy

Quartz is one of the most abundant minerals in the Earth crust and therefore quartz inclusions in garnet are of great interest for elastic geobarometry, an approach that exploits the elastic properties of the mineral pair to back-calculate the conditions of inclusion entrapment. However, the high-temperature behavior of quartz inclusions close to the alpha–beta transition boundary has not been studied experimentally. We have therefore performed in situ high-temperature Raman spectroscopy on a quartz-in-garnet system up to 1000 K, and have also collected an improved reference data set for the temperature dependence of the Raman scattering of free quartz. Our results show that the alpha-to-beta phase transition is hindered by the stress imposed by the host on the quartz inclusion, resulting in a thermosalient effect of the whole host-inclusion system or a mechanical cracking of the host mineral.

Large-strain Elastic and Elasto-Plastic Formulations for Host-Inclusion Systems and Their Applications in Thermobarometry and Geodynamics

Mineral inclusions are trapped in a variety of geological environments and physical conditions. If brought to conditions different than their entrapment, mineral inclusions will generally experience different stress conditions than their hosts due to differences in their thermo-elastic properties and the associated deformation. These stress differences develop both in prograde and retrograde metamorphic conditions. The currently available analytical solutions consider isotropic materials and employ either fully linear-elastic behavior or they account for the non-linear-elastic volumetric deformation of minerals. Here we show that, by taking into account the finite volumetric deformation, we are able to explain the systematic differences amongst the available linear and non-linear elastic solutions. Furthermore, we employ a newly derived analytical solution for fully non-linear elastic materials (generalized Varga materials) to the host-inclusion problem. This solution considers both the geometric non-linearity and the material non-linearity by employing a Murnaghan equation of state. Our results show that the complete non-linear, hyperelastic behavior is not needed to explain the pressure differences that develop in common, unreacting, host-inclusion systems. The effects of plastic yielding are also investigated for the case of large finite deformations that can be relevant for the cases of phase transitions and mineral reactions that induce significant volume changes. Our results show that in the case of very large volumetric deformations the incorporation of finite strain effects may become important. Moreover, depending on the yield stress of the materials, the effects of plasticity may be dominant. In the latter case, significant pressure gradients will be developed as a consequence of stress balance. These results are general and they can also be used for elastic-barometry/volcanology applications and for benchmarking compressible Navier-Stokes geodynamic models. Accurate stress predictions in mechanical problems with large volumetric deformation can be significant in modeling the effects of mineral reactions that are generally non-isochoric.

Resetting of zircon inclusions in garnet: Implications for elastic thermobarometry

Abstract Elastic thermobarometry of host-inclusion systems for back-calculating pressure (P) and temperature (T) conditions of inclusion entrapment relies on the assumption that the hostinclusion rheology is purely elastic. In this study, we have explored both the elastic and non-elastic behavior of zircon-in-garnet (ZiG) systems by in situ Raman spectroscopy at high T and ambient P. We show that upon heating, plastic relaxation takes place immediately after the zircon inclusions experience tensile stress conditions with respect to a free crystal at the same T. On subsequent cooling, the inclusions develop a new stress state, and thus the inclusion pressures have been reset from those corresponding to their original entrapment. Resetting of inclusion pressures therefore strongly depends on the exhumation P-T path. This explains why elastic thermobarometry using ZiG systems is reliable when applied to low-P high-T rocks where the cooling path after inclusion entrapment passes quickly into the compression domain of the inclusion. On the other hand, high-P rocks exhumed along quasi-isothermal paths take zircon inclusions into the tensile domain where they are reset until significant cooling commences at low P. ZiG systems in ultrahigh-P rocks therefore commonly indicate pressures on clockwise exhumation paths instead of the conditions of original entrapment.

The role of symmetry-breaking strains on quartz inclusions in anisotropic hosts: Implications for Raman elastic geobarometry

Raman elastic geobarometry for mineral host-inclusion systems is used to determine the strains acting on an inclusion still entrapped in its host by measuring its Raman wavenumber shifts which are interpreted through the phonon-mode Grüneisen tensors of the inclusion phase. The calculated inclusion strains can then be used in an elastic model to calculate the pressure and temperature conditions of entrapment. This method is applied frequently to host inclusion systems where the host is almost elastically isotropic (e.g. garnet) and the inclusion is elastically anisotropic (e.g. quartz and zircon). In this case, when the entrapment occurs under hydrostatic conditions the host will impose isotropic strains on the inclusion which in turn will develop non-hydrostatic stress. In this scenario the symmetry of the inclusion mineral is preserved and the strains in the inclusion can be measured via Raman spectroscopy using the phonon-mode Grüneisen tensor approach. However, a more complex situation arises when the host-inclusion system is fully anisotropic, such as when a quartz inclusion is entrapped within a zircon host, because the symmetry of the inclusion can be broken due to the external anisotropic strain field imposed on the inclusion by the host, which in turn will modify the phonon modes. We therefore calculated the strain states of quartz inclusions entrapped in zircon hosts in multiple orientations and at various geologically relevant pressure and temperature conditions. We then performed ab initio Hartree-Fock/Density Functional Theory (HF/DFT) simulations on α-quartz in these strain states. These HF/DFT simulations show that the changes in the positions of the Raman modes produced by strains that are expected for symmetry broken quartz inclusions in zircon are generally similar to those that would be seen if the quartz inclusions remained truly trigonal in symmetry. Therefore, the use of the trigonal phonon-mode Grüneisen tensor to determine the inclusion strains does not lead to geologically significant errors in calculated quartz inclusion entrapment pressures in zircon. • Mineral inclusions in anisotropic hosts can be subject to symmetry-breaking strains. • Symmetry-breaking strains modify the Raman spectrum of quartz. • Applicability of the trigonal Grüneisen tensor to symmetry-broken quartz is tested. • Entrapment pressures from the isotropic elastic model are in error by <0.185 GPa. • Entrapment pressures from the anisotropic elastic model are in error by <0.034 GPa.

Using the elastic properties of zircon-garnet host-inclusion pairs for thermobarometry of the ultrahigh-pressure Dora-Maira whiteschists: problems and perspectives

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.

A Grüneisen tensor for rutile and its application to host-inclusion systems

Abstract Rutile is often found as inclusions in garnet, quartz, and several other rock-forming minerals, and it is also a common accessory phase in high-pressure metamorphic rocks. Its relatively simple structure, chemistry, broad P-T stability field, and its wide occurrence in nature makes it a candidate for the application of elastic geobarometry. However, thermodynamic studies coupled with observations on natural samples predict that rutile inclusions in garnets should exhibit zero residual pressure. This implies that the rutile inclusions are detached from the inclusion walls in the host garnet after entrapment. We determined the elastic and vibrational properties of rutile via ab initio hybrid Hartree-Fock/Density Functional Theory simulations under different strain states. Our results confirmed the thermodynamic behavior of rutile in garnet and allowed us to determine for the first time the components of the phonon-mode Grüneisen tensors of rutile. We demonstrated that pure rutile inclusions in garnets from metamorphic rocks exhibit no residual strain or stress, consistent with thermodynamic modeling. Nevertheless, there are rutile inclusions in garnet surrounded by optical birefringence haloes, which are indicative of residual inclusion pressures. Careful examination of these show that they contain significant amounts of amphibole, which reduce the elastic moduli of the composite inclusion to less than that of the garnet hosts. A calculation method for the residual pressures of multi-phase inclusions is described.

Calculating pressure with elastic geobarometry: A comparison of different elastic solutions with application to a calc-silicate gneiss from the Rhodope Metamorphic Province

Raman elastic geobarometry has increasingly been used complementary to metamorphic phase equilibria to estimate the conditions of recrystallization in metamorphic rocks. The procedure of applying Raman elastic barometry to host-inclusion mineral systems requires several steps that involve various assumptions. One of the most essential assumptions is that the mineral host-inclusion system behaves in an elastic and reversible manner. We discuss the discrepant results obtained by different authors employing different analytical solutions for elasticity and explore the assumptions lying behind each method. Furthermore, we evaluate numerically linear and non-linear elastic solutions and show their discrepancies. Both formulations are tested against recently published experiments on quartz inclusions in garnet (QuiG) at pressures up to 3 GPa, and we find a very good agreement between calculated and experimental pressure values (within 10% relative error). We subsequently apply our new elastic geobarometer to a calc-silicate gneiss from the Rhodope Metamorphic Province (N. Greece). The results of Raman elastic barometry combined with garnet-clinopyroxene geothermometry yield eclogite-facies conditions (~720 ± 40 °C, ~1.5 ± 0.2 GPa). These results are comparable to a high-temperature metamorphic overprint deduced from phase equilibria modeling in surrounding lithologies (730 ± 40 °C, ~1.2 ± 0.1 GPa). Our findings indicate that the estimated pressure from Raman elastic barometry is consistent with a significant viscous relaxation at high temperatures. We conclude that although Raman elastic barometry is a powerful tool for pressure estimation in metamorphic rocks, its pressure estimates do not necessarily correspond to entrapment conditions. Our results are consequential for the estimates of reaction overstepping in high-grade metamorphic rocks.

A numerical application of the Eshelby theory for geobarometry of non-ideal host-inclusion systems

In the complex geodynamic processes occurring at convergent plate margins, rocks can be subducted at depth into the Earth experiencing metamorphism. A mineral inhomogeneity entrapped into another mineral, after exhumation to the Earth surface, will exhibit stress and strain fields different from those of the host because of the different thermoelastic properties. In the present paper, we propose a finite-element-based approach to determine the Eshelby and the relaxations tensors for any morphology of the inhomogeneity and for any crystallographic symmetry of the host. The proposed procedure can be directly applied in the framework of elastic geobarometry to estimate, on the basis of the Eshelby theory, the entrapment conditions (pressure and temperature) from the residual strain field measured in the inhomogeneity. This aspect represents a step forward to currently available models for geobarometry allowing the investigation of complex morphologies of the inhomogeneity in systems with general material anisotropy. We validate the proposed approach versus Eshelby analytical solutions available for spherical and ellipsoidal inclusions and we show the application to a real geological case of high pressure metamorphic rocks.

Assessment of the reliability of elastic geobarometry with quartz inclusions

Elastic geobarometry makes use of the contrast in elastic proprieties between host and inclusion crystals to determine the entrapment conditions of the inclusions from the residual stress and strain measured in the inclusion when its host is at ambient conditions. The theoretical basis has been developed extensively in the past few years, but an experimental validation of the method is still required. We performed two syntheses experiments with quartz inclusions in pure almandine garnet at eclogitic conditions. Experiment labelled Alm-1 with synthesis performed at P = 3.0 GPa and T = 775 °C and Alm-2 at P = 2.5 GPa and T = 800 °C. All the experiments have been carried out in a piston-cylinder press. Isolated, fully-enclosed quartz inclusions in the recovered garnets have been then measured using micro-Raman spectroscopy. All fully-buried inclusions exhibit Raman peaks at higher frequencies and wavenumbers than those obtained from quartz crystals at ambient pressure. If these peak shifts are interpreted as a remnant pressures by use of hydrostatic calibrations of the Raman shifts of quartz with pressure, the remnant pressures show a large spread in values and lead to significant errors in back-calculated entrapment pressures, of up to 1.4 GPa for inclusions synthesised at 3.0 GPa. These results confirm that quartz inclusions trapped inside garnet are not subject to hydrostatic pressure.We therefore used the phonon-mode Grüneisen tensors of quartz to calculate the full strain state of each inclusion, from which the full anisotropic stress state can be calculated by using the elastic properties of quartz. The mean residual remnant stress of the inclusions determined in this way show a much smaller spread in values. Entrapment pressures calculated from this mean stress with the isotropic model for host-inclusion systems differ from the known experimental values by <0.2 GPa, which is of the order of the combined experimental uncertainties. These results show that the most significant effect of the elastic anisotropy of quartz is on the Raman shifts of the inclusion, and not on the subsequent calculation of entrapment conditions.

The effects of non-hydrostatic stress on the structure and properties of alpha-quartz

The study of the effects of non-hydrostatic stresses on rock-forming minerals is fundamental for understanding how minerals respond to tectonic stresses in the Earth’s interior. Larger deviatoric stresses of the order of GPa can arise from physical interaction between minerals (e.g. host–inclusion systems, compositional gradients) at the micrometric scale, even if at the large-scale large stresses are annealed by plastic flows. Thus, experimental data under hydrostatic pressure do not provide us access to all the possible modifications and structural changes experienced by minerals subjected to deviatoric stresses. Therefore, we carried out ab initio hybrid Hartree–Fock/Density Functional Theory simulations to determine the properties of alpha-quartz, since it is one of the most abundant minerals in the Earth’s crust with a very simple chemistry and structure. We calculated its structure, elastic parameters and Raman-active vibrational modes as a function of different applied strains, which allow us to show how phonon frequencies and structural parameters (bond lengths and angles) are affected by deviatoric stress conditions rather than hydrostatic pressure.

How geometry and anisotropy affect residual strain in host-inclusion systems: Coupling experimental and numerical approaches

How geometry and anisotropy affect residual strain in host-inclusion systems: Coupling experimental and numerical approaches

Elastic geothermobarometry: Corrections for the geometry of the host-inclusion system

Abstract Elastic geothermobarometry on inclusions is a method to determine pressure-temperature conditions of mineral growth independent of chemical equilibrium. Because of the difference in their elastic properties, an inclusion completely entrapped inside a host mineral will develop a residual stress upon exhumation, from which one can back-calculate the entrapment pressure. Current elastic geobarometric models assume that both host and inclusion are elastically isotropic and have an ideal geometry (the inclusion is spherical and isolated at the center of an infinite host). These conditions do not commonly occur in natural rocks, and the consequences for inclusion pressures can only be quantified with numerical approaches. In this paper, we report the results of numerical simulations of inclusions with the finite element method on elastically isotropic systems. We define and determine a geometrical factor (Γ) that allows measured residual pressures to be corrected for the effects of non-ideal geometry. We provide simple guidelines as to which geometries can safely be used for elastic geobarometry without correcting for the geometry. We also show that the discrepancies between elastic and conventional geobarometry reported in literature are not due to geometrical effects, and therefore result from other factors not yet included in current models.

EosFit-Pinc: A simple GUI for host-inclusion elastic thermobarometry

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.

Crystallographic orientation relationships in host–inclusion systems: New insights from large EBSD data sets

Crystallographic orientation relationships (CORs) between mineral inclusions and their hosts could potentially deliver information about inclusion formation processes and conditions. Most previous studies are based on small numbers of analyses. This paper uses EBSD to study host–inclusion CORs in an inclusion-rich Permian metapegmatite garnet (Koralpe region, Eastern Alps, Austria), demonstrating the importance of large data sets and of EBSD in particular for the analysis of CORs. The distribution of measured orientations reflects host garnet point group symmetry for 89% of inclusions analyzed (total N = 530). Each inclusion phase (rutile, corundum, and ilmenite) shows at least three different CORs to host garnet. “Statistical” CORs are introduced to describe distributions of inclusion orientations that have one or two degrees of freedom with respect to the host, but still reflect host crystal symmetry. Two end-member characteristics of statistical CORs are distinguished: rotation and dispersion. Most statistical CORs observed show a mixture of both. Each inclusion phase shows at least one statistical COR. Multiple coexisting CORs and statistical CORs are not restricted to rutile. Re-examination of previous garnet–rutile COR studies in light of the new results indicates that COR information may have been overlooked when using small data sets. Variation in COR parameters correlates with broad differences in assumed metamorphic conditions for new and literature samples, suggesting that petrogenetic information may be available if COR formation can be understood. The favorability of the detected CORs cannot be explained by a simple model involving minimization of misfit between lattice planes, implying that other interface properties or the inclusion formation mechanism are important controls on COR development.

How large are departures from lithostatic pressure? Constraints from host–inclusion elasticity

AbstractMinerals trapped as inclusions within other host minerals will develop non‐lithostatic pressures during both prograde and retrograde metamorphism because of the differences between the thermoelastic properties of the host and inclusion phases. There is only a single possible path in P–T space, the entrapment isomeke, along which no residual pressure would be developed in a host–inclusion system; non‐lithostatic pressures are developed in inclusions as a result of the external pressure and temperature deviating from the isomeke that passes through the entrapment conditions. With modern equation of state and elasticity data for minerals now available, it is possible to perform precise calculations of the isomekes for mineral pairs. These show that isomeke lines are not straight lines in P–T space at metamorphic conditions. We show that silicate inclusions in silicate hosts tend to have flat isomekes, with small values of , because of the small range of thermal expansion coefficients of silicate minerals. As a consequence, the general behaviour under decompression is for soft silicate inclusions in stiffer hosts to develop excess pressures, whereas a stiff silicate inclusion in a softer matrix will experience lower pressures than lithostatic pressure. The opposite effects occur for compression after entrapment on the prograde path. The excess pressures in inclusions, including allowance for mutual elastic relaxation of the host and inclusion, are most easily calculated by using the isomeke as a basis. Analysis of the simplest possible model of a host–inclusion system indicates that deviations from lithostatic pressure in excess of 1 GPa can be readily produced in quartz inclusions within garnet in metamorphic rocks. For softer host minerals such as feldspar, the pressure deviations are smaller, because of greater elastic relaxation of the host. The maximum pressure deviation from lithostatic pressure in the host phase around the inclusion is one‐third of the pressure deviation in the inclusion. Routines for performing these calculations have been added to the EosFit7c software package.

Localization of submicron inclusion re-equilibration at healed fractures in host garnet

Microstructures in Permian inclusion-bearing metapegmatite garnets from the Koralpe (Eastern Alps, Austria) reveal re-equilibration by coarsening of abundant submicron-sized inclusions (1 μm–2 nm diameter) at the site of healed brittle cracks. The microstructures developed during Cretaceous eclogite-facies deformation and the related overprinting of the host–inclusion system. Trails of coarsened inclusions (1–10 μm diameter) crosscut the garnet, defining traces of former fractures with occasional en-echelon overlaps. Trails are flanked by 10- to 100-μm-wide ‘bleaching zones’ characterized by the absence of ≤1-μm-sized inclusions in optical and SE images. FEG-microprobe data show that trails and bleaching zones can form isochemically, although some trails exhibit non-isochemical coarsening. Cross-correlation-based EBSD analysis reveals garnet lattice rotation of up to 0.45°, spatially correlated with bleaching zones. The garnet lattice in the center of trails is misoriented around different axes with respect to the lattice either side of the trail. Elevated dislocation density within bleaching zones is confirmed by TEM observations. Dislocations represent a plastic wake formed by crystal plastic deformation at the crack tip. Fracture enhanced diffusion rates in the lattice adjacent to crack planes by introducing dislocations, priming these areas to behave differently to the bulk of the garnet during Cretaceous metamorphism and facilitating localized coarsening of inclusions. Diffusion within the bleaching zone was enhanced by a minimum factor of 102. The partially closed host–inclusion system records the influence of deformation mechanisms on re-equilibration and contributes to understanding of the interaction between deformation and chemical reaction during metamorphism.

Geobarometry from host-inclusion systems: The role of elastic relaxation

Minerals trapped as inclusions within other host minerals can develop residual stresses on exhumation as a result of the differences between the thermo-elastic properties of the host and inclusion phases. The determination of possible entrapment pressures and temperatures from this residual stress requires the mutual elastic relaxation of the host and inclusion to be determined. Previous estimates of this relaxation have relied on the assumption of linear elasticity theory. We present a new formulation of the problem that avoids this assumption. We show that for soft inclusions such as quartz in relatively stiff host materials such as garnet, the previous analysis yields entrapment pressures in error by the order of 0.1 GPa. The error is larger for hosts that have smaller shear moduli than garnet.

Rounded mineral inclusions in upper mantle peridotite

Mantle-derived peridotite xenoliths contain abundant olivine with rounded spinel and orthopyroxene inclusions, and orthopyroxene with rounded olivine and spinel inclusions. The shape-change of spinel, olivine and orthopyroxene inclusions from the primarily polyhedral outline to the spherical outline is governed by interfacial diffusion of oxygen in spinel (the most sluggish atom) due to reduction of total interfacial energy of the host-inclusion system. The critical radius of maximum rounded inclusions of spinel in olivine is a function of temperature and annealing time. Assuming that the activation energy for the interfacial diffusion is 40–70 kcal mol −1 and that the annealing time for the spinel lherzolite from Salt Lake Crater in Hawaii is 100 Ma, the annealing times for perioditites under the island arcs of Japan are estimated to be 1–10 Ma.