The thermodynamic properties of 154 mineral end-members, 13 silicate liquid end-members and 22 aqueous fluid species are presented in a revised and updated data set. The use of a temperature-dependent thermal expansion and bulk modulus, and the use of high-pressure equations of state for solids and fluids, allows calculation of mineral–fluid equilibria to 100 kbar pressure or higher. A pressure-dependent Landau model for order–disorder permits extension of disordering transitions to high pressures, and, in particular, allows the alpha–beta quartz transition to be handled more satisfactorily. Several melt end-members have been included to enable calculation of simple phase equilibria and as a first stage in developing melt mixing models in NCKFMASH. The simple aqueous species density model has been extended to enable speciation calculations and mineral solubility determination involving minerals and aqueous species at high temperatures and pressures. The data set has also been improved by incorporation of many new phase equilibrium constraints, calorimetric studies and new measurements of molar volume, thermal expansion and compressibility. This has led to a significant improvement in the level of agreement with the available experimental phase equilibria, and to greater flexibility in calculation of complex mineral equilibria. It is also shown that there is very good agreement between the data set and the most recent available calorimetric data.

An internally consistent thermodynamic data set for phases of petrological interest

QUANTUM gravitational effects are usually ignored in calculations of the formation and evolution of black holes. The justification for this is that the radius of curvature of space-time outside the event horizon is very large compared to the Planck length (Għ/c3)1/2 ≈ 10−33 cm, the length scale on which quantum fluctuations of the metric are expected to be of order unity. This means that the energy density of particles created by the gravitational field is small compared to the space-time curvature. Even though quantum effects may be small locally, they may still, however, add up to produce a significant effect over the lifetime of the Universe ≈ 1017 s which is very long compared to the Planck time ≈ 10−43 s. The purpose of this letter is to show that this indeed may be the case: it seems that any black hole will create and emit particles such as neutrinos or photons at just the rate that one would expect if the black hole was a body with a temperature of (κ/2π) (ħ/2k) ≈ 10−6 (M/M)K where κ is the surface gravity of the black hole1. As a black hole emits this thermal radiation one would expect it to lose mass. This in turn would increase the surface gravity and so increase the rate of emission. The black hole would therefore have a finite life of the order of 1071 (M/M)−3 s. For a black hole of solar mass this is much longer than the age of the Universe. There might, however, be much smaller black holes which were formed by fluctuations in the early Universe2. Any such black hole of mass less than 1015 g would have evaporated by now. Near the end of its life the rate of emission would be very high and about 1030 erg would be released in the last 0.1 s. This is a fairly small explosion by astronomical standards but it is equivalent to about 1 million 1 Mton hydrogen bombs. It is often said that nothing can escape from a black hole. But in 1974, Stephen Hawking realized that, owing to quantum effects, black holes should emit particles with a thermal distribution of energies — as if the black hole had a temperature inversely proportional to its mass. In addition to putting black-hole thermodynamics on a firmer footing, this discovery led Hawking to postulate 'black hole explosions', as primordial black holes end their lives in an accelerating release of energy.

Black Hole Explosions

: The unpolarized absorption and circular dichroism spectra of the fundamental vibrational transitions of the chiral molecule, 4-methyl-2-oxetanone, are calculated ab initio. Harmonic force fields are obtained using Density Functional Theory (DFT), MP2, and SCF methodologies and a 5S4P2D/3S2P (TZ2P) basis set. DFT calculations use the Local Spin Density Approximation (LSDA), BLYP, and Becke3LYP (B3LYP) density functionals. Mid-IR spectra predicted using LSDA, BLYP, and B3LYP force fields are of significantly different quality, the B3LYP force field yielding spectra in clearly superior, and overall excellent, agreement with experiment. The MP2 force field yields spectra in slightly worse agreement with experiment than the B3LYP force field. The SCF force field yields spectra in poor agreement with experiment.The basis set dependence of B3LYP force fields is also explored: the 6-31G* and TZ2P basis sets give very similar results while the 3-21G basis set yields spectra in substantially worse agreements with experiment. jg

Ab initio calculation of vibrational absorption and circular dichroism spectra using density functional force fields

The thermodynamic properties of 254 end-members, including 210 mineral end-members, 18 silicate liquid end-members and 26 aqueous fluid species are presented in a revised and updated internally consistent thermodynamic data set. The PVT properties of the data set phases are now based on a modified Tait equation of state (EOS) for the solids and the Pitzer & Sterner (1995) equation for gaseous components. Thermal expansion and compressibility are linked within the modified Tait EOS (TEOS) by a thermal pressure formulation using an Einstein temperature to model the temperature dependence of both the thermal expansion and bulk modulus in a consistent way. The new EOS has led to improved fitting of the phase equilibrium experiments. Many new end-members have been added, including several deep mantle phases and, for the first time, sulphur-bearing minerals. Silicate liquid end-members are in good agreement with both phase equilibrium experiments and measured heat of melting. The new dataset considerably enhances the capabilities for thermodynamic calculation on rocks, melts and aqueous fluids under crustal to deep mantle conditions. Implementations are already available in thermocalc to take advantage of the new data set and its methodologies, as illustrated by example calculations on sapphirine-bearing equilibria, sulphur-bearing equilibria and calculations to 300 kbar and 2000 °C to extend to lower mantle conditions.

An improved and extended internally consistent thermodynamic dataset for phases of petrological interest, involving a new equation of state for solids

A series of basaltic compositions and compositions within the simple system CaO-MgO-FeO-Al2O3-SiO2 have been crystallized to garnetclinopyroxene bearing mineral assemblages in the range 24–30 kb pressure, 750°–1,300° C temperature. Microprobe analyses of coexisting garnet and clinopyroxene show that K
D(Fe2+/MgG+/Fe2+/MgCpx) for the Fe-Mg exchange reaction between coexisting garnet and clinopyroxene is obviously dependent upon the Ca-content and apparently independent of the Mg/(Mg+Fe) content of the clinopyroxene and garnet. The Ca-effect is believed to be due to a combination of non-ideal Ca-Mg substitutions in the garnet and clinopyroxene. Our data and interpretation reconciles previous inconsistencies in the temperature dependence of K
D
− values determined in experimental studies of simple systems, complex basalt, grospydite and garnet peridotite compositions. Previous differences between the effect of pressure upon K
Das predicted from simple system theory (Banno, 1970), and that observed in experiments on multicomponent natural rock compositions (Raheim and Green, 1974a) can now be resolved. We have determined K
Das a function of P, T, and X

 Gt
 Ca
 (grossular) and derived the empirical relation 
$$T\left( {^\circ {\text{K}}} \right) = \frac{{3104X_{{\text{Ca}}}^{{\text{Gt}}} + 3030 + 10.86P\left( {{\text{kb}}} \right)}}{{\ln K_{\text{D}} + 1.9034}}$$

 . This empirical relationship has been applied to garnet-clinopyroxene bearing rocks from a wide range of geological environments. The geothermometer yields similar estimates for garnet-clinopyroxene equilibration for neighbouring rocks of different composition and different K
Dvalues. In addition, temperature estimates using the above relationship are more consistent with independent temperature estimates based on other geothermometers than previous estimates which did not correct for the Ca-effect. An alternative approach to the above empirical geothermometer was attempted using regular solution models to derive Margules parameters for various solid solutions in garnets and clinopyroxenes. The derived Margules parameters are broadly consistent with those determined from binary solution studies, but caution must be exercised in interpreting them in terms of actual thermodynamic properties of the relevant crystalline solid solutions because of the assumptions which necessarily have to be made in this approach.

An experimental study of the effect of Ca upon garnet-clinopyroxene Fe-Mg exchange equilibria

Mixing properties of aluminosilicate garnets: constraints from natural and experimental data, and applications to geothermo-barometry

We have experimentally determined the displacement of the equilibrium Grossular + 2 Kyanite + Quartz ⇆ 3 Anorthite (GASP) as a function of garnet composition in the systems Mg-Ca-Mn, Fe-Mg-Ca and Fe-Mg-Ca-Mn at 1000°C. The results were treated along with selected experimental and observational data available in the literature as well as binary parameters from other workers to obtain a set of mutually compatible binary mixing parameters of the quaternary (Fe,Mg,Ca,Mn)- aluminosilicate garnet solid solution. Attempts to determine equilibrium garnet composition in the GASP equilibrium in the Ca-Mg binary were unsuccessful due to the formation of pyroxene. Calculations of binary and ternary miscibility gaps show that the P,T,X combination required for unmixing of garnet solid solution is not realized by natural samples. The solution model was applied to account for compositional effects on Fe-Mg exchange between garnet and ortho- or clino-pyroxene. Applications of the revised thermometric formulations to selected natural assemblages yield P-T conditions which are much less sensitive to compositional effects compared to the other available formulations, and are consistent with independent constraints.

https://www.geo.arizona.edu/%7Eganguly/GtSol_CMP.pdf

Thermodynamics of aluminosilicate garnet solid solution: new experimental data, an optimized model, and thermometric applications

Equilibrium Compositions of Coexisting Garnet and Orthopyroxene: Experimental Determinations in the System FeO-MgO-Al2O3-SiO2, and Applications

We have determined the quartz-coesite transition by reversed experiments in a pistoncylinder apparatus in the range 500-1200 0c. The differencebetween the sample pressure and apparent pressure was calibrated by (1) studying the friction decay in the hysteresis loop defined by the relationship between apparent ("nominal") pressure and piston position in the compression and decompression cycles, and (2) determining the melting temperature of LiCl by DTA in pressure cells similar to those used in the reversal experiments and comparing the results with those determined in the gas apparatus. The equilibrium transition boundary can be expressed as P (kbar) = 21.945 (:to.1855) + 0.006901 (:to.0003)T (K). It is subparallel to, but -1.5 kbar higher than, the transition boundary determined by Bohlen and Boettcher (1982). We have also retrieved the entropy [39.56 :t 0.2 J/(mol. K)] and enthalpy off ormation (-907.25 :t 0.007 kJ/mol) from elements of coesite at 1 bar, 298 K., from our phase-equilibrium data and selected thermochemical data from the literature. From the characteristics of the hysteresis loop we conclude that the often-used practice of maintaining a constant nominal pressure by repeated pressure adjustment during an experiment leads to variation of pressure on the sample.

/pdf/quartz-coesite-transition-revisited-reversed-experimental-2lx5iu37hl.pdf

Quartz-coesite transition revisited; reversed experimental determination at 500-1200 degrees C and retrieved thermochemical properties

We present new experimental data on diffusion of divalent cations in almandine-spessartine diffusion couples in graphite capsules in the P-T range of 14–35 kb, 1100–1200° C. The tracer diffusion coefficients of the major divalent cations, viz. Fe, Mg and Mn, retrieved from the multicomponent diffusion profiles, have been combined with earlier data from our laboratory at 29–43 kb, 1300–1480° C (Loomis et al. 1985) to derive expressions of the P-T dependence of the diffusion coefficients at fO2 approximately corresponding to that defined by equilibrium in the system graphite-O2. We review the conditions, discussed earlier by Cooper, under which the flux of a component in a multicomponent system becomes proportional to its concentration gradient (Fickian diffusion), as if the entire solvent matrix behaves as a single component, and also suggest a method of incorporating the thermodynamic effect on diffusion in the same spirit. Regardless of the magnitude or sign of the off-diagonal terms of the D matrix, it is always possible to define an effective binary diffusion coefficient (EBDC) of a component in a semi-infinite multicomponent diffusion-couple experiment such that it has the property of the Fickian diffusion coefficient, provided that there is no inflection on the diffusion profiles. It is shown that the success of Elphick et al. in fitting the experimental diffusion profiles of all components over a limited concentration range by a single diffusion coefficient is due to fortuitous similarity of the EBDCs of the components (Fe, Mg, Mn and Ca) in their diffusion couple experiments. In common metapelitic garnets showing compositional zoning, the EBDCs of the divalent cations do not differ from each other by more than a factor of 2.5. However, the EBDC of a component changes from core to rim by a factor of 3 to 12, depending on the composition. We suggest a method of volume averaging of the EBDC which should prove useful in approximate calculations of diffusion flux during relaxation of compositional zoning. The EBDC of Mn is found to reduce essentially to DMnMn, the main diagonal term of the D matrix, and consequently can be calculated quite easily. Evaluation of EBDC of Fe, Mg and Mn in garnets from a prograde Barrovian sequence did not reveal any significant dependence on the extent of relaxation of garnet. The diffusion data have been applied to calculate the cooling rate of natural biotite-garnet diffusion couple from eastern Finland and diffusional modification of growth zoning in garnet in early Proterozoic Wopmay orogen, Canada. The results are in good agreement with geochronological and other independent constraints.

Cation diffusion in aluminosilicate garnets: experimental determination in spessartine-almandine diffusion couples, evaluation of effective binary diffusion coefficients, and applications

Jibamitra Ganguly

Papers

Mixing properties of aluminosilicate garnets: constraints from natural and experimental data, and applications to geothermo-barometry

Thermodynamics of aluminosilicate garnet solid solution: new experimental data, an optimized model, and thermometric applications

Equilibrium Compositions of Coexisting Garnet and Orthopyroxene: Experimental Determinations in the System FeO-MgO-Al2O3-SiO2, and Applications

Quartz-coesite transition revisited; reversed experimental determination at 500-1200 degrees C and retrieved thermochemical properties

Cation diffusion in aluminosilicate garnets: experimental determination in spessartine-almandine diffusion couples, evaluation of effective binary diffusion coefficients, and applications