Amphibole thermodynamics are approximated with the symmetric formalism (regular solution model for within-site non-ideality and a reciprocal solution model for cross-site-terms) in order to formulate improved thermometers for amphibole-plagioclase assemblages. This approximation provides a convenient framework with which to account for composition-dependence of the ideal (mixing-on-sites) equilibrium constants for the equilibria:
 For A and B all possible within-site and cross-site interactions among the species □−K−Na−Ca−Mg−Fe2+−Fe3+−Al−Si on the A, M4, M1, M3, M2 and T1 amphibole crystallographic sites were examined. Of the 36 possible interaction energy terms, application of the symmetric formalism results in a dramatic simplification to eight independent parameters. Plagioclase nonideality is modelled using Darken's quadratic formalism. We have supplemented an experimental data set of 92 amphibole-plagioclase pairs with 215 natural pairs from igneous and metamorphic rocks in which the pressure and temperature of equilibration are well constrained. Regression of the combined dataset yields values for the eight interaction parameters as well as for apparent enthalpy, entropy and volume changes for each reaction. These parameters are used to formulate two new thermometers, which perform well (±40°C) in the range 400–1000°C and 1–15 kbar over a broad range of bulk compositions, including tschermakitic amphiboles from garnet amphibolites which caused problems for the simple thermometer of Blundy and Holland (1990). For silica-saturated rocks both thermometers may be applied: in silica-undersaturated rocks or magmas thermometer B alone can be applied. An improved procedure for estimation of ferric iron in calcic amphiboles is presented in the appendix.

Non-ideal interactions in calcic amphiboles and their bearing on amphibole-plagioclase thermometry

This is a review of Collected Works of John Tate. Parts I, II, edited by Barry Mazur and Jean-Pierre Serre. American Mathematical Society, Providence, Rhode Island, 2016. For several decades it has been clear to the friends and colleagues of John Tate that a “Collected Works” was merited. The award of the Abel Prize to Tate in 2010 added impetus, and finally, in Tate’s ninety-second year we have these two magnificent volumes, edited by Barry Mazur and Jean-Pierre Serre. Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate. For an overview of Tate’s work, the editors refer the reader to [4]. Before discussing the volumes, I describe some of Tate’s work. 1. Hecke L-series and Tate’s thesis Like many budding number theorists, Tate’s favorite theorem when young was Gauss’s law of quadratic reciprocity. When he arrived at Princeton as a graduate student in 1946, he was fortunate to find there the person, Emil Artin, who had discovered the most general reciprocity law, so solving Hilbert’s ninth problem. By 1920, the German school of algebraic number theorists (Hilbert, Weber, . . .) together with its brilliant student Takagi had succeeded in classifying the abelian extensions of a number field K: to each group I of ideal classes in K, there is attached an extension L of K (the class field of I); the group I determines the arithmetic of the extension L/K, and the Galois group of L/K is isomorphic to I. Artin’s contribution was to prove (in 1927) that there is a natural isomorphism from I to the Galois group of L/K. When the base field contains an appropriate root of 1, Artin’s isomorphism gives a reciprocity law, and all possible reciprocity laws arise this way. In the 1930s, Chevalley reworked abelian class field theory. In particular, he replaced “ideals” with his “idèles” which greatly clarified the relation between the local and global aspects of the theory. For his thesis, Artin suggested that Tate do the same for Hecke L-series. When Hecke proved that the abelian L-functions of number fields (generalizations of Dirichlet’s L-functions) have an analytic continuation throughout the plane with a functional equation of the expected type, he saw that his methods applied even to a new kind of L-function, now named after him. Once Tate had developed his harmonic analysis of local fields and of the idèle group, he was able prove analytic continuation and functional equations for all the relevant L-series without Hecke’s complicated theta-formulas. Received by the editors September 5, 2016. 2010 Mathematics Subject Classification. Primary 01A75, 11-06, 14-06. c ©2017 American Mathematical Society

The collected works

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

Review of global 2.1-1.8 Ga orogens: implications for a pre-Rodinia supercontinent

The eastern Tonale fault zone: a ‘natural laboratory’ for crystal plastic deformation of quartz over a temperature range from 250 to 700 °C

This study assesses temperatures of formation of common granulites by combining experimental constraints on the P±T stability of granulite-facies mineral associations with a garnet± orthopyroxene (Grt±Opx) thermobarometry scheme based on Al-solubility in Opx, corrected for late Fe±Mg exchange. We applied this scheme to 414 granulites of mafic, intermediate and aluminous bulk compositions. Our findings suggest that granulites are much hotter than traditionally assumed and that the P±T conditions of the amphibolite±granulite transition portrayed in current petrology textbooks are significant underestimates by over 100 C. For aluminous and intermediate granulites, mean corrected temperatures based on our method are 890 17 and 841 11 C, respectively (uncertainties reported as 95% confidence limits on the mean), consistent with minimum temperatures for orthopyroxene production by fluid-absent partial melting in these bulk compositions. In contrast, mean temperatures based on Grt±Opx Fe±Mg exchange equilibria, using the same thermodynamic data, are 732 22 and 723 11 C, respectively, well below the minimum temperatures for Opx stability. For mafic granulites, the mean corrected temperature using our method is 816 12 C, similar to the mean temperature of 793 13 C from Fe±Mg exchange. Reasons for the differences between the mafic granulites and aluminous±intermediate granulites are unclear but may be due to the lower Al concentrations in Opx in the mafic rocks and possible deficiencies in the thermodynamic modelling of these low concentrations. We discuss a number of well-known granulite terrains in the context of our findings, including the Adirondacks, the Acadian granulites of New England, the incipient charnockites of southern India and Sri Lanka, and the Kerala Khondalite Belt. Our findings carry implications for thermotectonic models of granulite formation. A computer program to perform our thermobarometry calculations, RCLC, is available from the Journal of Petrology website at http://www.petrology.oupjournals.org or from the authors at http://www.geo.ucalgary.ca/ pattison/drm_pattison-rclc.htm.

Temperatures of Granulite-facies Metamorphism: Constraints from Experimental Phase Equilibria and Thermobarometry Corrected for Retrograde Exchange

Phase equilibria and thermobarometry of metapelites

The temperature-sensitive Fe,Mg exchange equilibrium, 

$$\begin{gathered} = {1 \mathord{\left/ {\vphantom {1 3}} \right \kern-\nulldelimiterspace} 3}Mg_3 A1_2 Si_3 O_{12} + CaFeSi_2 O_6 \hfill \\ {\text{ pyrope hedenbergite}} \hfill \\ \hfill \\ {\text{ = }}{{\text{1}} \mathord{\left/ {\vphantom {{\text{1}} 3}} \right \kern-\nulldelimiterspace} 3}Fe_3 A1_2 Si_3 O_{12} + CaMgSi_2 O_6 \hfill \\ {\text{ almandine diopside}} \hfill \\ \end{gathered}$$

has been investigated experimentally at 15 and 29 kbar and 700°–1200° C, with 59 reversals and 6 half reversals of the equilibrium clinopyroxene compositions from synthetic crystalline solid solutions The effect of Ca in garnet was investigated with grossular mole fractions of 02, 025, 03, 04 and 05 Garnet compositions in garnet-clinopyroxene reversal mixes were maintained constant during the runs by overwhelming abundance of garnet (90–95%) Reequilibration of rims of high-Fe and low-Fe clinopyroxene seeds gave tight reversal brackets Ca and Al contents of the clinopyroxenes showed systematic variations with T and bulk compositions but were not reversed The results calibrate a geothermometer for garnet amphibolites, granulites and eclogites with clinopyroxene low in Na The garnet-clinopyroxene distribution curves are asymmetric, in contrast to those of the previous experimental studies of Raheim and Green (1974) and Ellis and Green (1979) There is an increase of indicated temperature of about 100° C per 10 mole percent increase of grossular in garnet and a pressure effect of about 55° C/kbar Overall, the present calibration generally gives temperatures 0–60° C lower than Raheim and Green, 60–150° C lower than Ellis and Green, and 100–300° C lower than other semi-empirical and theoretical scales When applied to many samples from granulite facies terrains, the general result is a spread of calculated temperatures, many in the range 550–650° C, with some in the range 700–800° C This appears to indicate that retrograde resetting to lower temperatures is common and that the most reliable indication of peak or near-peak temperatures for a given terrain is the upper part of the calculated temperature range The Fe,Mg distribution curves of this study require asymmetric non-ideal mixing in diopside-hedenbergite The magnitude of nonideality is not as large as found by modelling Fe,Mg distributions with ilmenite (Bishop 1980) or orthopyroxene (Saxena et al 1986)

Reversed experimental calibration of the garnet-clinopyroxene Fe — Mg exchange thermometer

Orthopyroxene-free garnet + clinopyroxene + plagioclase ± quartz-bearing mineral assemblages represent the paragenetic link between plagioclase-free eclogite facies metabasites and orthopyroxene-bearing granulite facies metabasites. Although these assemblages are most commonly developed under P–T conditions consistent with high pressure granulite facies, they sometimes occur at lower grade in the amphibolite facies. Thus, these assemblages are characteristic but not definitive of high pressure granulite facies. Compositional factors favouring their development at amphibolite grade include Fe-rich mineral compositions, Ca-rich garnet and plagioclase, and Ti-poor hornblende. The generalized reaction that accounts for the prograde development of garnet + clinopyroxene + plagioclase ± quartz from a hornblende + plagioclase + quartz-bearing (amphibolite) precursor is Hbl + Pl + Qtz=Grt + Cpx + liquid or vapour, depending on whether the reaction occurs above or below the solidus. There are significant discrepancies between experimental and natural constraints on the P–T conditions of orthopyroxene-free garnet + clinopyroxene + plagioclase ± quartz-bearing mineral assemblages and therefore on the P–T position of this reaction. Semi-quantitative thermodynamic modelling of this reaction is hampered by the lack of a melt model and gives results that are only moderately successful in rationalizing the natural and experimental data.

Petrogenetic significance of orthopyroxene-free garnet+clinopyroxene+plagioclase±quartz-bearing metabasites with respect to the amphibolite and granulite facies

Compositional maps of orthopyroxene and garnet of contrasting grain size and in contact with different minerals were made from two paragneiss granulites from the Minto terrane of northern Quebec. The compositional maps provide clear evidence of late exchange of Fe/(Fe + Mg) after Ca in garnet and Al in orthopyroxene had been quenched-in. The extent of late Fe-Mg exchange was controlled by neighbouring minerals, with negligible Fe-Mg gradients against plagioclase and quartz, and substantial gradients against exchangeable Fe-Mg minerals. Cores of grains in contact with exchangeable Fe-Mg neighbours are progressively more reset in Fe/(Fe + Mg) as grain size decreases, whereas cores of even small grains surrounded by only plagioclase and quartz are not significantly different in Fe/(Fe + Mg) than cores of the largest grains. Gradients of Ca in garnet and of Al in orthopyroxene in grains of uniform Fe/(Fe + Mg) preserve a high-temperature retrograde history during which intergranular exchange effected compositional uniformity of mineral rims and intragranular Fe-Mg diffusion in garnet and orthopyroxene was rapid enough to homogenize Fe/(Fe + Mg). The transition from efficient intergranular exchange at relatively high temperatures to local Fe-Mg exchange at lower temperatures may have been controlled by loss of an intergranular exchange medium in the rock, possibly an internally generated dehydration melt phase. Implications for geothermometry of granulites include the following (numerical values are particular to this study): (1) core compositions of garnet and orthopyroxene grains in contact with exchangeable neighbours may be reset in Fe/(Fe + Mg) relative to the most refractory compositions by an amount equivalent to 120d C; (2) Fe-Mg exchange thermometry using even the most refractory Fe/(Fe + Mg) compositions may not record peak granulite conditions, possibly recording instead the temperature at which an intergranular exchange medium was lost from the rock; and (3) temperature-sensitive net transfer equilibria involving Al solubility in orthopyroxene yield temperatures up to 150d C higher than maximum Fe-Mg exchange temperatures, even in grains with flat Fe/(Fe - Mg) compositional profiles, making them a better means of estimating peak granulite temperatures than Fe-Mg exchange thermometry.

David R. M. Pattison

Papers

Temperatures of Granulite-facies Metamorphism: Constraints from Experimental Phase Equilibria and Thermobarometry Corrected for Retrograde Exchange

Phase equilibria and thermobarometry of metapelites

Reversed experimental calibration of the garnet-clinopyroxene Fe — Mg exchange thermometer

Petrogenetic significance of orthopyroxene-free garnet+clinopyroxene+plagioclase±quartz-bearing metabasites with respect to the amphibolite and granulite facies

Zoning patterns in orthopyroxene and garnet in granulites: implications for geothermometry