40 years of mineral elasticity: a critical review and a new parameterisation of equations of state for mantle olivines and diamond inclusions

Angel, R. J.; Alvaro, Matteo; Nestola, Fabrizio

doi:10.1007/s00269-017-0900-7

Cited by 55 publications

(47 citation statements)

References 68 publications

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“…There are several formalisms linking the variation of volume to that of pressure (isothermal EoS), each with its own assumption and mathematical development (e.g., Angel, ; Angel et al, , ; Duffy & Wang, ; Kroll et al, ). Here we employed a Birch‐Murnaghan EoS, that is a finite strain EoS , in which the strain energy of a solid undergoing compression is expressed as a Taylor series in the finite Eulerian strain f E , where

f_{E} = \frac{1}{2} ([], {(\frac{V_{0}}{V})}^{\frac{2}{3}} - 1)

(Angel, ).…”

Section: Methodsmentioning

confidence: 99%

“…γ and θ D only depend on the volume and are constant along isochors. The vibrational energy, together with the Grüneisen parameter and the volume, defines the thermal pressure, with the Ec and Tc being energy and temperature at reference conditions:

P_{th} = \frac{γ ((), V)}{V} ([], E ((),, T, θ_{D}) - E_{c} ((),, T_{c}, θ_{D})) .

In summary, in the MGD model, parameters that control the thermal pressure and hence allow the description of the thermal state of the system are only three: the Debye temperature, the Grüneisen parameter, and q (see Angel et al, , and references therein).…”

Section: Methodsmentioning

confidence: 99%

“…The second approach is the TP model. In this model the slope of an isochor in the pressure‐temperature space is written as

{(\frac{\partial P}{\partial T})}_{V} = α_{V} K_{0},

(Anderson ; Angel et al, ; Poirier, ) with α V being the thermal expansion. Consequently, during an isochoric heating, thermal pressure can be written as

P_{th} = \int_{T_{0}}^{T} α_{V} K_{0} \partial T .

The product α V K 0 is usually assumed to be constant along an isochor, as isochors are linear for T above the Debye temperature.…”

Section: Methodsmentioning

confidence: 99%

See 2 more Smart Citations

Equation of State of SiC at Extreme Conditions: New Insight Into the Interior of Carbon‐Rich Exoplanets

et al. 2018

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There is a direct relation between the composition of a host star and that of the planets orbiting around it. As such, the recent discovery of stars with unusual chemical composition, notably enriched in carbon instead of oxygen, supports the existence of exoplanets with a chemistry dominated by carbides instead of oxides. Accordingly, several studies have been recently conducted on the Si–C binary system at high pressure and temperature. Nonetheless, the properties of carbides at the pressure‐temperature conditions of exoplanets interiors are still inadequately constrained, effectively hampering reliable planetary modeling. Here we present an in situ X‐ray diffraction study of the Si–C binary system up to 200 GPa and 3,500 K, significantly enlarging the pressure range explored by previous experimental studies. The large amount of collected data allows us to properly investigate the phase diagram and to refine the Clapeyron slope of the transition line from the zinc blende to the rock salt structure. Furthermore, the pressure‐volume‐temperature equation of state is provided for the high‐pressure phase, characterized by low compressibility and thermal expansion. Our results are used to model idealized C‐rich exoplanets of end‐members composition. In particular, we derived mass‐radius relations and performed numerical simulations defining rheological parameters and initial conditions which lead to onset of convection in such SiC planets. We demonstrate that if restrained to silicate‐rich mantle compositions, the interpretation of mass‐radius relations may underestimate the interior diversity of exoplanets.

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f_{E} = \frac{1}{2} ([], {(\frac{V_{0}}{V})}^{\frac{2}{3}} - 1)

(Angel, ).…”

Section: Methodsmentioning

confidence: 99%

P_{th} = \frac{γ ((), V)}{V} ([], E ((),, T, θ_{D}) - E_{c} ((),, T_{c}, θ_{D})) .

Section: Methodsmentioning

confidence: 99%

“…The second approach is the TP model. In this model the slope of an isochor in the pressure‐temperature space is written as

{(\frac{\partial P}{\partial T})}_{V} = α_{V} K_{0},

(Anderson ; Angel et al, ; Poirier, ) with α V being the thermal expansion. Consequently, during an isochoric heating, thermal pressure can be written as

P_{th} = \int_{T_{0}}^{T} α_{V} K_{0} \partial T .

The product α V K 0 is usually assumed to be constant along an isochor, as isochors are linear for T above the Debye temperature.…”

Section: Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Equation of State of SiC at Extreme Conditions: New Insight Into the Interior of Carbon‐Rich Exoplanets

et al. 2018

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“…The presence of a relatively compressible fluid film around the inclusion (cf. Nimis et al, ) or plastic deformation in the diamond might account for some pressure release, but their possible effects on the final P inc are still unknown (Angel et al, ). The proximity of this garnet to the external surface of the diamond, which would affect the P inc , cannot be invoked as the garnet was located deep within the diamond host.…”

Section: Resultsmentioning

confidence: 99%

“…The procedure to calculate the depth of formation using elastic geobarometry is described in detail in a number of publications; the first approaches were detailed in Rosenfeld and Chase () and Adams et al (, ), followed by more recent publications by, for example, Zhang (), Angel, Gonzalez‐Platas, & Alvaro (), Angel, Alvaro, et al (), Angel, Nimis, et al (), and the introduction of software EosFit7c and EosFit‐P inc by Angel, Nimis, et al () and Angel, Alvaro, & Nestola (). All these works are based on the concept of the isomeke.…”

Section: Resultsmentioning

confidence: 99%

Toward a Robust Elastic Geobarometry of Kyanite Inclusions in Eclogitic Diamonds

et al. 2018

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Here we report the first results from elastic geobarometry applied to a kyanite inclusion entrapped within an eclogitic diamond (from Voorspoed mine, South Africa) using micro‐Raman and Fourier transform infrared spectroscopy, electron microprobe analysis, ab initio calculations, and finite element modeling. Application of elastic geobarometry to very elastically anisotropic kyanite inclusions is challenging, as current models do not allow for elastic anisotropy. In order to minimize the effects of anisotropy, we have explored the effects of deviatoric stress on Raman modes via ab initio density functional theory. The results allowed us to select the Raman mode (at ca. 638 cm−1) that is the least sensitive to deviatoric stress. The shift of this band in the inclusion while still trapped within the diamond relative to the inclusion in air (once liberated) was used under hydrostatic approximation to determine a residual pressure on the inclusion of 0.184 ± 0.045 GPa and an entrapment pressure of 5.2 ± 0.3 GPa (~160 km depth) for an FTIR N‐aggregation residence temperature of 1119 ± 50 °C. This is the first geothermobarometric determination for a diamond from the Voorspoed kimberlite. It overlaps with P–T estimates obtained by traditional chemical geobarometry for diamonds from other kimberlites from the Kaapvaal craton, suggesting that the hydrostatic approximation does not introduce significant errors in the geobarometric evaluation. Our protocol of Raman peak selection can be used for geobarometry of further kyanite‐bearing diamonds and may provide a guide for more robust geobarometry of other types of mineral inclusions in diamonds, both eclogitic and peridotitic.

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Garnet EoS: a critical review and synthesis

Angel¹,

Gilio

Mazzucchelli

et al. 2022

Contrib Mineral Petrol

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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 $$\gamma = {\gamma }_{0}{\left(\frac{V}{{V}_{0}}\right)}^{q}$$ 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 $$\left( {{\raise0.7ex\hbox{${\partial C_{{\text{V}}} }$} \!\mathord{\left/ {\vphantom {{\partial C_{{\text{V}}} } {\partial P}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial P}$}}} \right)_{{\text{T}}} = 0$$ ∂ 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 V0 (cm3/mol)a 113.13 115.25 117.92 125.35 K0T (GPa) 169.3 (3) 174.6 (4) 177.57 (6) 167.0 (2) $$K^{\prime}_{{0{\text{T}}}}$$ 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) γ0 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.

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40 years of mineral elasticity: a critical review and a new parameterisation of equations of state for mantle olivines and diamond inclusions

Cited by 55 publications

References 68 publications

Equation of State of SiC at Extreme Conditions: New Insight Into the Interior of Carbon‐Rich Exoplanets

Equation of State of SiC at Extreme Conditions: New Insight Into the Interior of Carbon‐Rich Exoplanets

Toward a Robust Elastic Geobarometry of Kyanite Inclusions in Eclogitic Diamonds

Garnet EoS: a critical review and synthesis

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