Empirical geothermometers and geothermobarometers for spinel peridotite phase assemblages

Experimental synthesis of spinel peridotite phase assemblages for a range of compositions that mimic natural samples is used to derive a set of empirical geothermometers and geothermobarometers represented by multiple linear regression best-fit surfaces that link the variables of temperature, pressure, and composition. The calibrated geothermometers use reactions that govern the solubility of Al and Cr in both pyroxenes and the Mg–Fe exchange between silicates and spinel. Geothermobarometers map the Mg–Fe exchange between coexisting olivine and clinopyroxene and pyroxenes and Ca–Mg exchange between coexisting pyroxenes. Application of the geothermometers and geothermobarometers to suites of naturally occurring samples indicates that while reactions governing the Cr and Al solubility and solvus of orthopyroxene give useful estimates of ‘original’ mantle temperatures and pressures, respectively, comparable reactions for clinopyroxene yield estimates that are variably dependent on the transport phase of the sample suites. Temperature and pressure estimates from reactions governing Mg and Fe exchange between silicates and spinel and coexisting silicates are all sensitive to the later transport stage of the samples.


Introduction
The search for geothermometers and geothermobarometers that may be used to interpret the conditions of formation of rocks of ultramafic composition represents an example of the successful interplay among experimental, theoretical, empirical, and field approaches. Earlier work, which focused on defining the boundaries among the low-, intermediate-, and high-pressure phase assemblages of plagioclase, spinel, and garnet peridotites, respectively, yielded broad estimates of the pressure and temperature fields within which each assemblage is stable (Gasparik 1984). In order to improve the resolution, later work examined the temperature and pressure dependency of exchange reactions that governed the compositions of coexisting phases. Solid solutions of particular interest have been those controlling the solubility of Al in the pyroxenes, the mutual Ca solubility between coexisting pyroxenes, and the Fe and Mg exchange between coexisting silicates and silicates and spinel.
The use of geothermometers and geothermobarometers has been effectively applied to peridotite suites in the garnet peridotite stability field (MacGregor 1975;Carswell and Gibb 1987;Ionov et al. 2010;Ashchepkov et al. 2012). In garnet peridotites, while the solubility of Al in pyroxene is pressure dependent, the exchange of Ca between coexisting pyroxenes is sufficiently pressure independent that the couplet has been used to make estimates of the pressures and temperatures of formation. In contrast, reactions controlling the Al solid solution in pyroxenes in the spinel peridotite field are essentially independent of pressure and may be used as geothermometers (Gasparik and Newton 1984). Since, within the error limits, experiments on the mutual solubility of Ca for coexisting pyroxenes in the spinel field have shown that the reaction is essentially pressure independent, this presents the predicament that there is no corresponding geothermobarometer for spinel peridotites.
An additional set of geothermometers for both garnetand spinel-bearing assemblages have used the reactions that govern Mg and Fe exchange between coexisting silicates and silicates and spinel. Examples relevant to this article include theoretical discussion of governing reactions (Irvine 1965(Irvine , 1967Wood and Banno 1973;Wells 1977), experimental studies (Engi and Evans 1980;Engi 1983;Jamieson and Roeder 1984;Perkins and Vielzeuf 1992b;Von Seckendorff and O'Neill 1993;Kawasaki and Ito 1994), and applications (Engi and Evans 1980).
A parallel evolution has been the coupling of theoretical and empirical models to help expand the application of experimentally determined reactions from simple to more complex natural systems (Wells 1977;Sachtleben and Seck 1981;Bertrand and Mercier 1985;Brey and Köhler 1990;Witt-Eickschen and Seck 1991). The interactive process has been most successfully applied to garnet-bearing assemblages where experimental results on simple and multi-component systems are available and theoretical models expand the use to natural compositions. For spinel-bearing assemblages where experimental data are restricted to simple systems there has yet to be an experimental demonstration of the complex interactions among elements in multi-component systems.
In the present study, experiments have been designed to synthesize spinel peridotite phase assemblages (olivine + orthopyroxene + clinopyroxene + spinel) that cover much of the temperature, pressure, and compositional range of naturally occurring ultramafic rocks. The results allow an empirical calculation of the variables that govern the reactions controlling the mutual solubility of Al, Cr, and Ca in coexisting pyroxenes and the Mg and Fe exchange between coexisting silicates and silicates and spinel.

Experimental design Reactions and starting compositions
Using reactants of handpicked olivine and garnet from garnet lherzolite xenoliths from South African kimberlite pipes, the reaction which defines the boundary between garnet-and spinelbearing peridotites, was used to synthesize spinel peridotite phase assemblages. In the spinel peridotite field, the solubility of Al and Cr in pyroxenes is further governed by the coupled reaction (Obata 1976;Mori 1977) where ((n i ) j ) k describes the element 'n', with the number of 'i' atoms in the structural site 'j' of the mineral phase 'k'.
Handpicked olivine and garnets with three different compositions were selected to synthesize three different reactant compositions that cover much of the compositional range of natural spinel peridotites. The starting compositions (Table 1) were made using olivine mixed successively with garnets of different composition weighed to match the stoichiometric 1:1 molecular ratio of the phases defined for the reactants (Equation (1)). The mineral mixtures (reactants) were ground under alcohol with an alumina mortar and pestle for 1 hour to yield a powder whose grain size varied from 1 to 5 µm.

Experimental procedures
The experiments were conducted in a ½-inch diameter solid media piston and cylinder device (Boyd and England 1960) using a furnace assembly described by Boyd and England (1963). Temperatures were measured using a Pt 10 Rh 90 /Pt 90 Rh 10 thermocouple held constant with a Crisel temperature controller. The controller maintained the temperature to within ±5°C. No corrections were made for any pressure effects on the thermocouple, or for longer-term effects resulting from thermocouple contamination. In all experiments the pressure was raised to a gauge pressure of 0.3 GPa in excess of that which is nominal for the experiment, and the piston retracted to the desired pressure after the experiment had stabilized at the desired temperature. Boyd et al. (1966) suggest that for this procedure the gauge pressure is within 5% of the 'true' pressure and that temperature gradients from 5 to 10°C can be expected within the capsule.
In the synthesis experiments the reactants were held at a range of temperatures and pressures in the spinel stability field. The experimental conditions (bulk composition, temperature, pressure, time, and capsule) for each run are given in Table 2. The products, which are the set of minerals that constitute the spinel peridotite phase assemblage, are also listed.
Reactants were contained in graphite, graphite sealed in Pt, and sealed Pt capsules ( For experiments conducted at temperatures in excess of 1200°C, reactants were dried for 45 minutes at 1000°C in a dry nitrogen furnace; in experiments performed between 1200 and 1100°C reactants were not dried. At temperatures of 1000°C and lower, reactants were either not dried (graphite capsules) or 5 wt.% of oxalic acid (C 2 H 2 O 4 ·2H 2 O) was added to the reactants (Pt or graphite capsules sealed in Pt capsules; Table 2).

Experimental products: Petrography and mineral chemistry
Microscopic examination showed that products were microcrystalline aggregates with variable and patchy grain sizes generally in the range 1-5 µm (Figures 1(a) and 1(b)). However, in most samples it was possible to find irregular patches of coarser-grained material (10-50 µm) suitable for chemical analysis.
Experimental products were made into polished sections, examined with a reflecting petrographic microscope, and, whenever possible, single product phases chemically analysed with an ARL-EMXSM microprobe. The Bence and Albee (1968) matrix correction algorithm was used to process the raw probe data. The chemistry of the probe standards used as a reference for the analyses is given in Supplementary Because of the small grain size and associated ambiguities arising from overlapping and/or superimposed grains, it was very tedious and, for some samples, impossible to collect mineral analyses for all four reactant phases. Mineral phases were identified by their chemical stoichiometry. In addition, it was not possible to test whether there was any chemical zoning and it is assumed that the analysed grains are chemically homogeneous. The composition of product phases is given in Tables 3 (Ol), 4 (Opx), 5 (Cpx), and 6 (Sp).

Analytical statistics
The number of analyses and associated errors for each element in each phase are provided in Supplementary Tables 2a (Ol), 2b (Opx), 2c (Cpx), and 2d (Sp). A summary of the number of analyses and average errors are given in Supplementary Tables 2e and 2f. Because of the small number of analyses (1-5) for each phase, the analytical errors are not well documented but give an indication of uncertainties arising from the sum of microprobe errors and ambiguity  from assuming that there was no analytical interference from adjacent mineral grains.

Comments
A number of experimental parameters are not fully constrained. These include the question of whether the synthesis experiments yield equilibrium product assemblages, the consequence of not buffering the oxygen fugacity, and the lack of knowledge of the Na 2 O content of orhopyroxenes and clinopyroxenes.
A basic assumption in this study is that experiments synthesized spinel peridotite phase assemblages, which represent equilibrium products for reactions (1) and (2). All experiments were conducted well within the spinel peridotite stability field and, in most cases, there was complete conversion of the reactants to the products. However, in some experiments the reaction was incomplete and some reactant olivine and garnet still remained (Figure 1(a), Table 2). For the latter cases, when it was possible to make satisfactory analyses of the product pyroxene and spinel, the mineral analyses were comparable to experiments at equivalent temperatures and pressures where complete reaction was observed. The results from the incompletely reacted experiments have been included in the calculation of the equilibrium constants for the different geothermometers and geothermobaromenters.
In order to test the assumption of equilibrium, three experiments (G0-R1, G1-R1, and G1-R2) were conducted where the starting compositions were held for two to four days at a lower temperature before being raised by 100-200°C for another two to three days (Table 2). Target compositions for the low-and hightemperature endmember product phases were calculated from the average compositions at the same temperatures and pressure derived from the synthesis experiments. The goal was to test the assumption that during the first lower temperature stage of the experiments, coupled reactions (Equations (1) and (2)) yielded compositions close to the respective lower temperature averages, while any subsequent reaction (Equation (2)) during the higher-temperature stage would change mineral compositions towards those represented by the higher-temperature target averages. Chemical analyses for the phases from the 'reversal' experiments are given in Supplementary Tables 3a, 3b, and 3c, and summarized in Table 3d. The difference between the measured and calculated changes, shown as the percentage of change, gives an indication of the degree to which phases are changing composition during the second higher-temperature stage of the experiment. Supplementary Tables 3a, 3b, 3c, and 3d distinguish between reversed changes that are within and those that exceed analytical errors, and situations where either no change could be distinguished or changes were opposite to those expected. The results for the different oxides are summarized as follows: • Al 2 O 3 : Except for one experiment (G0-R1), Al 2 O 3 increases as expected for the pyroxenes but not for spinel. • Cr 2 O 3 : For the pyroxenes, the absolute change between the low-and high-temperature products are approximate to or less than the analytical errors and no firm conclusions can be made. Confirming Although fully reversed compositions are not observed for all oxides, the directions of most observed changes are those expected to adjust to the second stage, higher-temperature experimental conditions. The limited experiments provide qualitative support for the conclusion that the phase chemistry adjusts to the higher temperature conditions and supports the assumption that the synthesis experiments can yield equilibrium products. The interpretation agrees with Wood's (1974) reversal experiments for the four-component system, CaO-MgO-Al 2 O 3 -SiO 2. His theoretical calculations show that the experiments approach equilibrium at times varying from 1½ hours at 1200°C to approximately 54 hours at 900°C. Since the experiments in the present study were held for periods significantly greater than the limits reported by Wood (1974), it is reasonable to expect that the experimental products record equilibrium conditions. The oxygen fugacity (fO 2 ) was not buffered in any experiments. However, for a few experiments where calculation of the structural formula indicates the presence of Fe 3+ in spinel, it was possible to use an olivineorthopyroxene-spinel oxygen barometer (Ballhaus et al. 1991) to calculate the experimental oxygen fugacity relative to the quartz-fayalite-magnetite (QFM) buffer. Where estimates are possible, values for fO 2 range from 2 to −3 log units above and below the QFM buffer (Supplementary Figure 1). In most experiments in graphite capsules where calculation of structural formulae indicated that there was no Fe 3+ in the spinel, it was not possible to calculate the fO 2 and more reduced values, less than that which may be expected for the C-CO-CO 2 buffer (French and Eugster 1965;Vielzeuf 1992a, 1992b). The range of observed experimental fO 2 is comparable to that calculated for natural spinel peridotite assemblages (i.e. between 3 log units higher than QFM and the ironwustite (IW) buffers (Ballhaus et al. 1991;O'Neill and Wall 1987)). It is also noted that the range of Fe 3+ M1 in orthopyroxenes and, clinopyroxenes and Fe 3+ in spinels calculated for experimental phases (Supplementary   Tables 6b, 6c, and 6d) is similar to that for natural spinel peridotite assemblages. Another shortcoming is that the Na 2 O content of the pyroxenes is not known. Because of the very low Na 2 O content of the reactant compositions (between 0.03 and 0.07 Na 2 O wt.%; Table 1) and the difficulty in analysing Na, the Na content of pyroxenes was not determined. Although the Na 2 O wt.% of bulk reactant compositions is low, it is comparable to values measured in natural peridotite samples (e.g. the Na 2 O content for bulk rock mantle peridotites ranges between 0.02 and 0.3 Na 2 O wt.%; Boyd et al. 1997;Janney et al. 2010). Comparably, the average Na 2 O wt.% for Irvine (1965), Fabries (1979) 9a Irvine (1965), Fabries (1979) 10a Irvine (1965), Fabries (1979) 11  Perkins and Vielzeuf (1992b), Kawasaki and Ito (1994) ( Wood and Banno (1973), Wells (1977) 14 a sample of 590 orthopyroxenes and 759 clinopyroxenes from spinel peridotites from a wide range of tectonic environments yield average values of 0.06 wt. % (SD = 0.3) and 1.07 wt.% (SD = 0.55), respectively. Since the experimental bulk compositions lie at the low range of natural peridotites, it is likely that the Na 2 O content of orthopyroxenes and clinopyroxenes from the experiments is comparably lower than that for natural samples. Though unconstrained, it is assumed that ignoring the Na 2 O content does not introduce significant errors into the calculated geothermometers and geothermobarometers.
In order to apply the experimentally derived geothermobarometers to natural samples, the calculated mole fractions (X) for X Ca M2 and X R 3+M1 used in the calculation of the equilibrium constants for each geothermobarometer (Supplementary Table 8) may be adjusted to Na-free equivalents by assigning Na M2 proportionately to Al, Cr, and Fe 3+ , as jadeite, kosmochlor, and acmite components. This procedure is similar to that recommended by Wood (1974) and comparable to the approach used by Bertrand and Mercier (1985) and Brey and Köhler (1990), who found that by subtracting Na M2 from Ca M2 the simplifying adjustment yields temperature estimates within analytical errors for CFMS (CaO-FeO-MgO-SiO 2 ) and natural systems.
The analysis of reactant olivine was especially problematic. Since the chemistry of the reactant olivine (Table 3) is distinctly different from the starting compositions (Table 1), the reactant olivine is readily identified. Only a relative few olivine analyses were obtained. Since the chemistry of all coexisting phases is needed for the calculation of equilibrium constants, it was necessary to supplement estimated olivine compositions for experiments where direct analysis was not possible. Correspondingly, statistical fits were made to the reactant olivine chemical analyses (Supplementary Table 4) to provide estimated compositions for experiments where analyses were not available. Since the data for the trace elements (Al, Cr, and Ca) are not statistically significant, estimates were only made for SiO 2 , MgO, and FeO. The calculated chemistries for olivine are listed and identified in Table 3.

Derivation of geothermometers and geothermobarometers
Introduction For all reactions that define chemical exchange between coexisting phases, the associated equilibrium constants used to calculate the geothermometers and geothermobarometers assume ideal mixing models. This follows the simplifying assumptions used by previous authors in the derivation of equilibrium constants referenced in this article. The simplification follows the empirical approach used by Witt-Eickschen and Seck (1991), who found that the set of thermodynamic data needed to model reactions governing elemental exchange between coexisting phases are currently inadequate to derive equilibrium constants accurate enough for satisfactory geothermobarometers. This is especially the case for multi-component chemistries of natural spinel peridotites. Correspondingly, they advocated an empirical approach to extend the work of Sachtleben and Seck (1981). Witt-Eickschen and Seck (1991) used a suite of natural peridotites as the norm for the derivation of empirical geothermometers. In the present study, experimentally derived mineral chemistries for compositions that cover much of the range of natural samples are used as the empirical norm. For the experimental case, governing bulk chemistries, temperatures, and pressures are known and do not have to be estimated. For convenient reference, Table 7 lists the reactions and equilibrium constants proposed as geothermometers and geothermobarometers.
The variables examined in this study include temperature, pressure, and 'bulk' composition. 'Bulk' compositions are calculated as the sum of the atomic ratios of the mineral products defined independently for each governing reaction (Supplementary Table 8). This artifice allows for a more general application to estimate temperatures for natural phase assemblages where, although the chemistry of the products (minerals) is often known, the bulk composition (rock) has seldom been determined. The temperature dependencies of the different variables with associated errors are given in the Appendix and Supplementary Tables 9a and 9b.
The proposed geothermometers and geothermobarometers are calculated as an empirical set of best-fit multiple linear regression surfaces relating equilibrium constants to the respective set of experimental variables of temperature, pressure, and 'bulk' composition. For convenient reference, Tables 8-10 list the equations, variables, and coefficients for the best-fit multiple linear regression surfaces for the proposed geothermometers and geothermobarometers. Tables 11 and 12 are statistical measures (residual sum of squares (r ss ) and coefficients of determination (r 2 )) to evaluate the calculated best-fit multiple linear regression surfaces. An additional statistical reference is the ability of the proposed geothermometers and geothermobarometers to recover experimental temperatures and pressures. The latter, listed as the average of the differences between experimental and calculated temperatures and pressures (Δ (°C, GPa) (Exp-Calc) %), provide a further means to evaluate the proposed geothermobarometers. Since the latter sum both experimental and analytical errors, they better describe the uncertainties inherent in the estimates of temperature and/or pressure when applied to natural spinel peridotites.

Included in
Theoretical background: Trivalent ion solubility in pyroxenes Mori (1977), following Obata (1976), has shown that the trivalent ion solubility in orthopyroxene (Opx) or clinopyroxene (Cpx) coexisting with olivine and spinel is governed by reactions (3) and (4): and Mori (1977) includes consideration of the coupled occupancy of Al-for Mg-and Ca-Tschermak components in M 1 Octohedral and Z Tetrahedral sites when deriving the equilibrium constants so that and Using reactions (5) and (6), Mori's model may also be used to define the Cr solubility in the pyroxenes: for orthopyroxene and clinopyroxene, respectively. Cr occupies the pyroxene M1, and Al both the M1 (octahedral) and tetrahedral sites. Following Mori's model for ideal mixing, the equilibrium constants may be defined in terms of mole fractions as follows:  Previous experimental work on three-, four-, and five-component systems (Anastasiou and Seifert 1972;Fujii 1976;Fujii and Takahashi 1976;Herzberg and Chapman 1976;Presnall 1976;Mori 1977;Danckwerth and Newton 1978;Gasparik 1984;Gasparik and Newton 1984;Sen 1985;Walter and Presnall 1994;Gudfinnsson and Presnall 2000) has shown that equilibrium constants (3a and 4a) are essentially independent of pressure so that the Al solubility in pyroxenes may be used effectively as a geothermometer. The experimental results that confirm the pressure independence of Al solubility in pyroxenes uphold the theoretical considerations proposed by Wood (1974), Obata (1976), Mori (1977), and Danckwerth and Newton (1978).
This study replicates experiments that examine the Al solubility in pyroxenes but adds new data on Cr solubility. The coefficients calculated for the best-fit surfaces that relate the equilibrium constants (geothermometers) to temperature, pressure, and composition are listed in Table 8, and the data plotted in Figures 2 (Alss Opx), 3 (Alss Cpx), 4 (Crss Opx), and 5 (Crss Cpx), which relate the data to subsets of the family of curves that map out the temperature, pressure and compositional space for the reactions that govern the trivalent solubility in pyroxenes. Calculated surfaces for the geothermometers give acceptable statistical fits to the data (r ss range from 0.69 to 1.14; Table 11) and   Table 11. Statistical estimates (r ss ) for multiple least squares best-fit surfaces for different geothermometers and geothermobaromometers, and differences between experimental and calculated temperatures and pressures (Δ (°C, GPa) (Exp-Calc)).    recover experimental temperatures to within averages ranging from −0.099% to 0.003% (SD < 3.4%). Sensitivity tests (Appendix; Supplementary Table 9a) indicate that while the trivalent ion solubilities of Al in orthopyroxene and Cr in both pyroxenes are, within the range of error, essentially independent of pressure, the Al solubility in clinopyroxene is slightly dependent on pressure and 'bulk' Al (−44°C (±25°C) per one GPa; 24°C (±18°C) per 0.1 'Bulk' Al (Supplementary  Table 9a). Cr solubility in both pyroxenes is dependent on 'bulk' Al. With some caution for clinopyroxene, the trivalent ion solubility in pyroxenes may be used as geothermometers.

React #
Theoretical background: Fe-Mg exchange between silicate and spinel Following Irvine (1965), Fabries (1979) has provided a theoretical basis for the Mg and Fe exchange between coexisting olivine and spinel assemblages. The set of reactions governing the Mg and Fe exchange between coexisting olivine and spinel (Irvine 1965 Both sets of reactions and may be summed as follows: where (Y n j ) k is the mole fraction, 'Y', of component, 'n j ' in phase, 'k'.
For ideal solutions, Irvine (1965) summed the governing exchange reactions to derive the equilibrium constant Using Irvine's model, reactions ((7a), (7b), and (7c)) for olivine may be extended to describe Mg and Fe exchange between orthopyroxene and clinopyroxene by substituting those phases for olivine: and For the Mg and Fe exchange between olivine and spinel (reaction (8)) and trivalention exchange among spinel (reactions (7b), (7d), and (7e)), reactions for pyroxenespinel may be summed as and The equilibrium constants for (9a) and (10a) may be replicated as follows: Since Y Fe3+ Sp for spinels in natural peridotites (Irvine 1965;Fabries 1979) is generally less than 0.05, the term Y Fe3+ Sp LnK d(9e) is approximately zero and may be ignored. Correspondingly, the equilibrium constants (8a), (9b), and (10b) may be treated as equations where LnK d (7 b, 9, 10) are linearly related to X Cr Sp . A number of attempts have been made to calibrate Irvine's (1965) equilibrium constant for the Mg and Fe exchange between coexisting olivine and spinel. Evans and Frost (1975) first used olivine-spinel assemblages from naturally occurring metamorphic and igneous rocks to estimate an empirical geothermometer. Later, Fabries (1979) and Engi and Evans (1980) revaluated Evans and Frost's (1975) geothermometer adding results from experiments on synthetic systems (Fujii 1978) and natural compositions (Roeder et al. (1979). Fujii and Scarfe (1982), Ono (1983), Jamieson and Roeder (1984), and Ballhaus et al. (1991) further calibrated the compositional and temperature dependence of the exchange of Mg and Fe between coexisting olivine and spinel. O'Neil and Wall (1987) used a theoretical approach to illustrate the use of orthopyroxene-spinel pairs as an oxygen geobarometer and provided a revised geothermometer for olivine-spinel that included the Ti content of spinel as an additional variable. There have been no previous experiments on the exchange of Mg and Fe between orthopyroxe and spinel or clinopyroxene and spinel pairs. The data from this experiment for olivine-spinel pairs do not agree with previous experiments (Fujii 1978;Roeder et al. 1979;Ono 1983;Jamieson and Roeder 1984) or with attempts to use natural data to help calibrate a potential geothermometer (Evans and Frost 1975;Engi and Evans 1980). Further, there is no agreement with the geothermometers proposed by Evans and Frost (1975), Roeder et al. (1979), Engi andEvans (1980), O'Neil andWall (1987), and Ballhaus et al. (1991). Because of the uniform disagreement between results from this and other studies, no geothermometer is proposed for the Mg and Fe exchange between olivine and spinel.
Coefficients for the calculated best-fit surfaces for the equilibrium constants (9b) (Opx-Sp) and (10 b) (Cpx-Sp) that use the Mg and Fe exchange reaction between coexisting pyroxene and spinel are listed in Table 9. The data are plotted in Figures 6 (Opx-Sp) and 7 (Cpx-Sp). Best-fit surfaces for the equilibrium constants (9b) (Opx-Sp) and (10b) (Cpx-Sp), give acceptable statistical fits (r 2 = 0.81 (Opx-Sp); r 2 = 0.80 (Cpx-Sp)) to the data (Table (12)) and recover experimental temperatures to within an average of -1.64%   (SD = 4.76%) for orthopyroxene-spinel and 2.4% (SD = 7.35%) for clinopyroxene-spinel pairs. Since the reactions governing Mg and Fe exchange are pressure independent, both may be used as a geothermometer.
Theoretical background: Fe-Mg exchange between olivine and pyroxene Perkins and Vielzeuf (1992b), Von Seckendorff and O'Neill (1993), and Kawasaki and Ito (1994) for clinopyroxene. Assuming ideal mixing, the equilibrium constants may be written in terms of mole fractions (X) as follows: and or by taking structural site occupancies into consideration (Mori 1977), may be expanded as follows: for orthopyroxene and clinopyroxene, respectively. Von Seckendorff and O'Neill (1993) summarized previous work on the exchange of Mg and Fe between olivine and orthopyroxene and pointed out that in the composition ranges of ultramafic rocks the distribution coefficient is essentially insensitive to temperature. In contrast, Perkins and Vielzeuf (1992b) and Kawasaki and Ito (1994) have shown that the exchange of Mg and Fe between olivine and clinopyroxene is sufficiently dependent on temperature and composition (bulk Mg/(Mg + Fe)) that the equilibrium constant (12a) may potentially be used as a geothermometer. As part of their study on the Mg and Fe exchange between coexisting phases, Brey and Köhler (1990) confirm that for olivine-orthopyroxene pairs, the exchange is essentially independent of temperature and pressure; while for olivine-clinopyroxene pairs, the exchange is both temperature and mildly pressure dependent.
The results from this study are similar to previous work. For olivine-orthopyroxene pairs (reaction (11)) within the narrow range of experimental Mg/(Mg + Fe) ratios, LnK d (11a) is essentially independent of temperature. In contrast, for olivine-clinopyroxene pairs (reaction (12a)), LnK d (12b) varies with temperature, pressure, and composition. For coexisting olivine-orthopyroxene pairs, the data agree with Von Seckendorff and O'Neill (1993). For olivine-clinopyroxene pairs, the data agree with Brey and Köhler (1990). Absolute values for the equilibrium constants are different from those for three-component systems (Perkins and Vielzeuf 1992b;Von Seckendorff and O'Neill 1993;Kawasaki and Ito 1994) but similar to experiments using multi-component compositions (Brey and Köhler 1990). Coefficients for the calculated best-fit surfaces for the LnK d (12b) that use the Mg and Fe exchange between coexisting olivine and clinopyroxene are listed in Table 8 and the data plotted in Figure 8 (Ol-Cpx). For olivine-clinopyroxene pairs, the best-fit multiple linear regression surface relating LnK d to temperature and pressure has a residual sum of squares (r ss ) of 2.51. Average differences between experimental temperatures and pressures calculated with the geothermobarometer are 0.37% (SD = 6.34%) for temperature and 2.21% (SD = 14.66%) for pressure (Table 11). Mg and Fe exchange between olivine-clinopyroxene is pressure dependent (Supplementary Table 9a) but for the small compositional range, independent of 'bulk' Mg and may be used as a geothermobarometer.
Theoretical background/basis: Fe-Mg exchange between coexisting pyroxenes Wood and Banno (1973) and Wells (1977) have expressed the exchange of Mg and Fe between coexisting pyroxenes as follows: Assuming ideality, the equilibrium constant may be written in terms of mole fractions (X) as Following Wood and Banno's (1973) derivation of a geothermometer for coexisting pyroxenes that included Fe 2+ /(Fe 2+ + Mg) Opx as a compositional variable, Wells (1977), , and Brey and Köhler (1990) have used Wood and Banno's (1973) theoretical basis, coupled with experimental results from synthetic and natural compositions, to propose revised temperature-and composition-dependent geothermometers.
This study extends previous work by examining the temperature, pressure, and compositional dependence of reaction (13) for a set of natural compositions within the spinel peridotite stability field. Unlike previous data, the present results suggest that the exchange reaction is pressure dependent. The coefficients calculated for the best-fit surfaces that relate LnK d (13a) to temperature, pressure, and composition are listed in Table 8. Figure 9 (Opx-Cpx) relates the data to subsets of the family of curves that map out the temperature and pressure dependence. Calculated surfaces for the geothermobarometer give acceptable statistical fits (r ss = 1.55; Table 11) and recover experimental temperatures and pressures to within an average ranging from −0.17%°C (SD = 8.51%) to 2.00% GPa (SD = 11.53%), respectively. When using LnK d (13a) to estimate pressures or temperatures for natural assemblages, it must be assumed that the coexisting pyroxenes are in equilibrium and that either an equilibrium temperature or pressure may be independently determined. Sensitivity tests (Supplementary Table 9a) indicate that reaction (13) is dependent on both pressure (−184°C (± 66°C) per one GPa) and the 'bulk' Mg (−26°C (± 22°C) per 0.1 'Bulk' Mg). The reaction governing the Mg and Fe exchange in coexisting pyroxenes may be used as geothermobarometers.

Theoretical background: Ca-Mg exchange between coexisting pyroxenes
The exchange of Ca and Mg between coexisting pyroxenes may be expressed by the exchange reaction which yields the equilibrium constant It is also possible to define temperature-, pressure-, and composition-dependent geothermobarometers using the respective pyroxene solvi (Ca/(Ca + Mg) (Px) ) as a measure of the Ca and Mg exchange between coexisting pyroxenes. At equilibrium, reaction (15) Summarizing previous theoretical and experimental studies, Davidson (1985) and Davidson and Lindsley (1985) have provided a thermodynamic model for quadrilateral (CaO-MgO-FeO + SiO 2 ) pyroxene solutions, which allows the calculation of the temperature dependence of pyroxene solvi. Subsequent experimental results by Howells and O'Hara (1975), Mori and Green (1975), Lindsley and Dixon (1976), Nehru (1976), and Mori (1977) showed that the pyroxene solvi are dependent on both temperature and pressure. Other experimental studies where coexisting pyroxenes have been synthesized over a wide range of temperatures and pressures include Fujii (1976), Mori (1977), Mysen and Boettcher (1975), Perkins III and Newton (1980), Gasparik and Newton (1984), Sen (1985), Sen and Jones (1989), Walter and Presnall (1994), Presnall (2000, 2005), and Keshav et al. (2011). The experiments all confirm the dependence of the solvi on both temperature and pressure. However, the wide temperature, pressure, and limited compositional ranges of previous experimental studies make it difficult to rigorously apply their results to the relatively narrow temperature and pressure range and multi-component system used in this experimental study.
Although the above experimental and theoretical work has shown the pressure dependence of the ortho-and clinopyroxene solvi, it has previously been assumed that the pressure dependence is sufficiently small so that the governing exchange reactions ( (14) and (15)) may be effectively used as a geothermometer.
The experimental results for this study may be presented either as that shown in reaction (14) or as solvi (15). In the case of reaction (14), the coefficients calculated for the best-fit surfaces that relate LnK (14a) to temperature, pressure, and composition are listed in Table 8. Figure 10 (Opx-Cpx) relates the data to subsets of the family of curves that map out the temperature, pressure, and compositional space for the reaction. Calculated surfaces for the geothermobarometer give acceptable statistical fits (r ss = 1.25; Table 8) and recover experimental temperatures and pressures to within an average ranging from 0.02% (SD = 3.99%) to 1.73% (SD = 10.39%), respectively. When using LnK d (14a) to estimate pressures or temperatures for natural assemblages, it must be assumed that the coexisting pyroxenes are in equilibrium and that an equilibrium temperature or pressure may be independently determined. Sensitivity tests (Appendix; Supplementary Table 9a) indicate that reaction (14) is dependent on both pressure (−181°C (± 63°C) per one GPa) and the 'bulk' Mg (−24°C (± 20°C) per 0.1 'Bulk' Mg) and may be used as a geothermobarometer.
In the case of reaction (15), the coefficients calculated for the best-fit surfaces of the respective solvi are listed in Table 10 and plotted in Figures 11 (Opx) and 12 (Cpx). Calculated surfaces for the solvi give acceptable statistical fits (r 2 = 0.64 (Opx solvus); r 2 = 0.78 (Cpx solvus); Table 12) and recover experimental pressures within an average of 1.1% (SD = 0.09%) and −0.23% (SD = 8.56%) for ortho-and clinopyroxene, respectively. Sensitivity tests (Appendix; Supplementary Table 9b) indicate that temperature estimates are dependent on pressure (158°C (± 59°C) and 124°C (± 69°C) per one GPa for ortho-and clinopyroxene, respectively). The calculated solvi do not take into account the known dependence of the solvi on the Mg/ (Mg + Fe) ratio. Since the narrow range of pyroxene compositions for natural spinel peridotites (Opx: Average 'Mg' 0.91 (SD = 0.01); Cpx: Average 'Mg' 0.94 (SD = 0.02) is very small and replicates the experimental compositions, the geothermobarometer may be usefully applied to natural assemblages. In order to apply the solvi to natural assemblages, adjustments are made by subtracting Na from Ca (Supplementary Table 8), effectively projecting the solvi onto a Na-free surface. The pyroxene solvi may be used as geothermobarometers. Alternately, respective solvi for each pyroxene (reaction (15); Figures 11 (Opx solvus) and 12 (Cpx solvus)) may be used as geothermobarometers to estimate independent pressures for each pyroxene.

Application of geothermometers and geothermobarometers to estimates of temperatures and pressures for natural examples Introduction
A fundamental predicament in applying geothermometers and geothermobarometers from a set of experimentally derived equilibrium assemblages to geological examples is that natural samples have a dynamic petrogenesis and, depending on reaction rates and emplacement history, will yield temperature and pressure estimates that invariably do not represent equilibrium. Further, reaction rates for the different geothermometers and geothermobarometers are poorly known and dependent on a number of unconstrained variables. These include external parameters such as the cooling rates that are controlled by the history of the sample, the temperature of the host or interstitial magma, ambient oxygen fugacity, pressure, and the geothermal gradient. Unconstrained variables also include those that are intrinsic to the phases, such as diffusion rates that are dependent on crystallography, stress-induced dislocations, bulk chemistry, nucleation rates, grain size, and gradients induced by differential stress or composition. Information on the geological and petrographic context of each sample will help to qualitatively inform, but not necessarily resolve, the relative importance of the different external and intrinsic variables on reaction rates. As an illustration of the efficacy of geothermometers and geothermobarometers, they are applied to a small set of examples from different tectonic environments. These include xenolith localities from Dish Hill, CA (Luffi et al. 2009), a kimberlite locality (Undachnaya, Siberia; Boyd et al. 1997; Smith D., data base) and suites of samples from Hawaii (Jackson and Wright 1970;Sen and Presnall 1986;Sen 1988;Chen et al. 1992;Keshav et al. 2007), and peridotite massifs from ocean floor abyssal peridotites (Dick et al. 2010) and the Coast Range Ophiolite, CA (Jean et al. 2010;Marlon et al. 2010). Sample suites also illustrate possible differences between xenoliths and peridotite massifs (e.g. abyssal peridotites or ophiolites). Peridotite massifs rising advectively to the surface are dominated by solid-solid reactions, or possibly reactions with interstitial partial melts, while xenoliths have a twostage history: changes responding to slow, advectively responsive solid-solid reactions in the mantle followed by a relatively rapid short-lived transport stage in host magmas migrating to the surface.

Diffusion rates
As a limiting condition, the effective rate of the exchange reactions for each geothermometer or geothermobaromenter is dependent on the slowest acting of all the variables (Lasaga 1981). Since, for any set of conditions, the diffusion rate is the slowest mass transfer variable, it will be the rate-determining process for most reactions. Experimental measurements of the diffusion rates for different elements in phases from spinel peridotites are difficult to compare. Experimental and measurement techniques, ambient conditions (e.g. fO 2 ), and sample compositions vary and lead to an overlapping range of diffusion rates that make the data difficult to apply. In general, diffusion rates vary from approximately one to two orders of magnitude per 100°C, but an added complication is that the rate of temperature dependency varies from element to element and phase to phase. Nevertheless, it is possible to use existing experimental and observational studies (Brady and McAllister 1983;Rietmeijer 1983;Sautter et al. 1988;Chakrahorty et al. 1994;Ganguly and Tazzzoli 1994;Chakraborty 1997;Béjina et al. 2003;Cherniak and Liang 2007;Ganguly et al. 2007;Holzapfel et al. 2007;Cherniak 2010;Zhang et al. 2010;Müller et al. 2013) to infer relative diffusion rates for the elements in spinel peridotite phase assemblages. Broad categories that distinguish different ranges of diffusion rates are given below.
(1) The fastest diffusion rates are for Mg and Fe exchange between coexisting phases and range from being fastest to slowest in the sequence spinel-olivine-orthopyroxene-clinopyroxene (Müller et al. 2013). Diffusion in clinopyroxene is therefore likely to control the blocking temperatures estimated from Mg and Fe exchange reactions. Müller et al. (2013) add the observation that diffusion rates of Mg and Fe between clinopyroxene and orthopyroxene or garnet are considerably slower than those between clinopyroxene and olivine, spinel, and melt so that under equivalent conditions reactions between pyroxene pairs are likely to record higher temperatures than those for clinopyroxene-olivine and clinopyroxene-spinel pairs. Estimates for Log D (m 2 /s) from experimental data at 1000°C range from −12 to −19.
(2) Diffusion rates for Ca-Mg exchange in pyroxenes range from two to three orders of magnitude slower than that for Mg and Mg and Fe exchange (Zhang et al. 2010). Using Sr and Eu 2+ as proxies for Ca 2+ , Sneeringer et al. (1984) show that diffusion of Sr (and hence Eu 2+ and Ca) in diopside is also slower than diffusion of Eu 2+ in enstatite at temperatures less than 1200°C. The analogy suggests that relative diffusion rates for Ca in orthopyroxene are slower at high but faster at lower temperatures than clinopyroxene. Although proxy studies indicate a threshold temperature of 1200°C for the relative change, there are no corresponding data for Ca diffusion. Estimates for Log D (m 2 /s) from experimental data at 1000°C range from −19 to −22.
(3) Diffusion rates for Al in pyroxenes overlap but are slightly slower than those for Ca and Mg and Fe or Ca and Mg exchange. Al diffusion rates range from being fastest to slowest in the sequence spinel-clinopyroxene-orthopyroxene (Brady and McAllister 1983;Zhang et al. 2010). Estimates for Log D (m 2 /s) from experimental data at 1000°C range from −19 to −20. (4) For equivalent temperatures and pressures, the diffusion rate for Cr is slower than that for Al in spinel (Suzuki et al. 2008) and enstatite by approximately one order of magnitude. In addition, Al and Cr diffusion rates range from fastest to slowest in the sequence spinel-clinopyroxene-orthopyroxene (Ganguly et al. 2007). Estimates for Log D (m 2 /s) from experimental data at 1000°C range from −20 to −22.

Reaction rates
Using the diffusion rates as a guide, it is possible to categorize semi-quantitative sets of reaction rates that distinguish relative differences among the geothermobarometers.
(1) Reaction rates that govern Mg and Fe exchange are the fastest.
(2) The Ca and Mg exchange reaction rates in pyroxenes are significantly slower than the Mg and Fe exchange reactions. Proxy data from Eu 2+ and Sr diffusion suggest that relative reaction rates for Ca and Mg exchange change at a certain threshold temperature (approximately 1200°C; see above), below which diffusion rates for orthopyroxene become faster than those for clinopyroxene.
(3) Reaction rates governing Al solubility in pyroxenes are comparable to but slightly slower than those for Ca and Mg exchange. Reaction rates for Al solubility in orthopyroxene are distinctly slower than those for clinopyroxene. (4) Orthopyroxenes reaction rates governing Cr solubility are slower by one order of magnitude than those for Al solubility. Clinopyroxene reaction rates governing Al and Cr solubility are faster than for orthopyroxene.
Summarizing this categorization, it is expected that the reaction for Cr solubility in orthopyroxene is best suited to record 'original' source or mantle temperatures. While reactions for Al solubility in orthopyroxene are likely to record similar temperatures, the slightly faster rates for Al than Cr solubility suggest that Al solubility geothermometers may respond to later changes. Since comparable reactions for trivalent ion solubility in clinopyroxene are slightly faster than for orthopyroxene, they will be even more sensitive to later stages in any sample's history. Reaction rates for Ca and Mg exchange in pyroxenes have the potential to record 'original' temperatures and pressures, but proxy data (see above) suggest a reversal in reaction rates above and below threshold temperatures. All reactions governing Mg and Fe exchange between silicates and spinel and silicates and silicates are sufficiently rapid that they are likely to respond to changes in the thermal and pressure regimes during transport or migration to the surface. Within the latter group the rates of response to changing conditions are expected to range from slower to faster in the sequence from pyroxene to silicate-spinel and olivine-clinopyroxene pairs.
Translating the reaction rates into effective times for the reactions to reach equilibrium, one can reference lab experiments for Mg and Fe exchange (e.g. for temperatures in the range from 900-1300°C), equilibrium on the micron scale (5-10 µm) may be reached in times from hours to days (Roeder et al. 1979;Engi 1983). For larger grain sizes and comparable temperatures, the experimentally derived diffusion rates indicate that equilibrium may be reached in times ranging from weeks to years. Klügel (2001) calculates an example where temperature and pressure estimates for a 2 mm-diameter pyroxene using Al solubility in pyroxenes and Ca-Mg exchange between pyroxenes, respectively, would take approximately 250,000 years at 1200°C to reach equilibrium. Sautter et al. (1988) calculated that at 1180°C a time of 10 million years is required to homogenize the Al content of a clinopyroxene crystal of 1 mm diameter. The time estimates for the different reactions to reach equilibrium provide a qualitative reference to evaluate the significance of temperature and pressure estimates from each set of geothermometers and geothermobarometers in terms of interpreting the petrogenesis of the sample.

Temperature estimates
Figures 13(a)-(e), 14(a)-(e), and 15(a)-(e) compare temperature estimates derived using the different geothermomometers and geothermobarometers for xenoliths from Dish Hill, CA, a kimberlite pipe (Udachnaya, Siberia) and a set of Hawaiian localities, and massif-like abyssal peridotites from the Kane Fracture zone and samples from the Coast Range Ophiolites, CA. All temperature estimates are referenced to the geothermometer based on the Cr solubility in orthopyroxene.
For all five occurrences temperature estimates, geothermometers using reactions governing trivalent ion (Cr and Al) solubility in orthopyroxenes are similar; most samples agree within ±50°C. In contrast, trends for temperature estimates using Cr and Al solubility in clinopyroxene depart slightly from the 1:1 line, being lower at higher and higher at lower temperatures. The differences may be interpreted as Cr and Al solubility reactions continuing during the transport phase to the surface. The 'flattening' of the clinopyroxene trends could indicate an approach to blocking temperatures that respond to the combination of reaction rates and sample history. Estimates for clinopyroxene based on Cr solubility tend to depart less from the 1:1 line than those based on Al solubility. The observations suggest that while reaction rates governing trivalent ion solubility in clinopyroxenes are slightly faster than for orthopyroxene, reaction rates based on Cr in clinopyroxene are slightly more sluggish than those based on Al solubility. The observations conform to expectations based on the categorization of diffusion and reaction rates discussed above.
For reactions that govern Mg and Fe exchange in silicate-spinel pairs, the discordant spread of temperature estimates from those for Cr solubility in orthopyroxene (Figures 14(a)-(e)) indicate that the Mg and Fe geothermometers record frozen equilibria as the xenoliths and peridotite massifs continue to respond to the changing thermal and pressure regimes on the way to the surface. While each locality gives a different distribution, temperature estimates from orthopyroxene-spinel pairs tend to cluster over smaller temperature intervals (1000 ± 200°C ), suggestive of blocking temperatures during cooling. Clinopyroxene-spinel pairs, which plot over a larger temperature interval (400°-1400°C), suggest that this pair record temperatures over the heating cycle but are less responsive to changes during cooling stages of the transport history. The latter agrees with the findings of Müller et al. (2013), who show that diffusion rates in clinopyroxene are slower than those for orthopyoxene. In the case of Hawaii (Figure 14(b)), the extreme temperature estimates (1500-2400°C) suggest that the clinopyroxenespinel pairs are not in equilibrium.
For Mg and Fe exchange geothermobarometers using silicate-silicate pairs, temperature estimates are illustrated in Figures (15(a)-(e)) for the assumption that the samples equilibrated at surface pressures. For all localities the temperature estimates are discordant to the reference temperature and plot over narrow temperature ranges (generally < ±200°C) that are suggestive of blocking temperatures controlled by the interaction between reaction rates and changing thermal regimes during transport. In general the 'blocking' temperatures range from higher to lower in the sequence Ca and Mg, then Mg and Fe exchange between coexisting pyroxenes, then lowest for olivineclinopyroxene pairs. The observations agree with Zhang et al. (2010) that diffusion rates for Ca-Mg exchange in pyroxenes range from two to three orders of magnitude slower than that for Mg and Mg and Fe exchange, and with Müller et al. (2013), who show that diffusion rates for Mg and Fe exchange between coexisting pyroxenes are considerably slower than those between clinopyroxene and olivine. estimates based on a single solvus will yield maximum values. By independently estimating pressure using their respective solvi, it is possible to show that in most natural samples the pyroxenes are not in equilibrium (Figure 16(a)-(e)). In all figures, pressure estimates using the Cr solubility and solvus of orthopyroxene form a common reference. In the case of clinopyroxenes, only the temperature estimate that uses the slower Cr rather than the faster Al solubility reaction rates is used as a base to calculate the pressure. It is assumed that as a consequence of the slow diffusion rates for Cr solubility and Ca and Mg exchange in orthopyroxene, the solvus provides a record of 'mantle' pressures rather than changing pressures during ascent to the surface. Because of the illustrated disequilibrium between coexisting pyroxenes (Figure 16(a)-(e)), the claim that the orthopyroxene solvus records 'mantle' pressures rests on two further assumptions. First, orthopyroxenes were in equilibrium with clinopyroxene during the mantle stage; and second, because of the faster diffusion rates for trivalent ions in clinopyroxene than for orthopyroxene, clinopyroxene geothermometers yield temperature and, correspondingly, pressure estimates that record the transport rather than 'mantle' stage of the sample.
In the case of the Dish Hill xenoliths (Figure 16(a)), where it was possible to estimate pressures for the cores and rims of both pyroxenes, pressure estimates for rims are uniformly lower than those for cores. In addition, pressure estimates are uniformly higher for orthopyroxene than those for clinopyroxene. The pattern seen for Dish Hill is similar to that for the Salt Lake Crater, Oahu (Figure 16 xenoliths. The difference is consistent with diffusion rates for Ca in clinopyroxene being faster than those for orthopyroxene during the higher-temperature magma transport stage. In contrast to the xenolith suites, massifstyle peridotites from the Kane Fracture Zone (Figure 16(d)), and the Coast Range Ophiolites (Figure 16(e)), the pressure estimates for orthopyroxene are uniformly lower than those for clinopyroxene. A possible explanation for the contrasting patterns is to consider the proxy data from rare earths (Sneeringer et al. (1984), which show that at higher temperatures diffusion rates are faster in clinopyroxene than orthopyroxene but reverse at threshold temperatures of approximately 1200°C. The higher temperatures of host magmas transporting xenoliths contrasted with the lower-temperature advective path followed by peridotite massif-like bodies may explain the different responses for the two groups of samples. At higher temperatures, pressure estimates using clinopyroxene adjust to decreasing pressure, while at lower temperatures reactions are sufficiently slow that changes are not recorded. More importantly, the patterns for the two sets of data indicate that the pyroxenes are out of equilibrium and adjust independently to the changing petrological environment during transport to the surface.

Temperature and pressure estimates
The above discussion of diffusion-controlled reaction rates argued that the 'original' mantle setting of the samples is best derived by using the Cr solubility and solvus of orthopyroxene to estimate temperatures and pressures, respectively. This procedure makes the tacit assumption that orthopyroxene and clinopyroxene solvi were in equilibrium while recording 'mantle' pressures.
Estimates have been made for a number of suites of spinel peridotite samples from a variety of tectonic environments. These include xenolith suites from Dish Hill, CA (Figure 17(a)), Hawaiian localities (Figure 17(b)) and the Udachnaya kimberlite (Figure 17(c)), and a suite of abyssal peridotites [Kane Fracture Zone (Figure 17(d)) and the Coast Range Ophiolite (Figure 17(e))]. To provide a reference for comparison, the temperatures and pressures are plotted against a similar backdrop of melting curves, geothermal gradients, and phase changes. Each suite gives different pressure versus temperature profiles.
Within xenolith suites, distinctions are observed. For Hawaiian localities, the~130 k.y. Hualalai xenoliths (Chen et al. 1992;Vazquez et al. 2007) have a higher temperature profile than xenoliths from the Salt Lake Crater, Pali #1 and #2, and Kaau and Kalihi cinder cones from the 1.8 m.y. Honolulu volcanic series, Oahu (Jackson and Wright 1970;Bohrson and Clague 1988;Sen et al. 1993). The Salt Lake Crater pyroxenites are from veins interpreted as fracture-filled magma that cut the lherzolites (Keshav et al. 2007). For the Dish Hill xenoliths (Figure 17(a)) it was possible, by assuming that the olivine composition applied to both cores and rims, to estimate temperatures and pressures for both. Xenoliths from the Udachnaya kimberlite match the lowest thermal gradient profile. The Coast Range Ophiolites show different trends for each ophiolite. Abyssal peridotites from the Kane Fracture Zone record an isothermal distribution from about 1.5 GPa to the surface.
In most cases the estimated temperatures and pressures fall within the spinel peridotite field. Exceptions are samples from the Coast Range Ophiolite ( Figure 5 (e)) and abyssal peridotites from the Kane Fracture Zone (Figure 5(d)) that plot in the plagioclase peridotite field. The samples that fall into the plagioclase stability field may be explained by the expansion of the spinel field to lower pressures for natural Cr-rich compositions (Borghini et al. 2010). The data provide improved resolution, in temperature-pressure space, to help understand the petrogenesis of spinel peridotite suites from different tectonic settings. For example, the sequence abyssal peridotites and ophiolites may be interpreted as recording their advective transit to the surface, while xenoliths from Dish Hill, Hawaii, and kimberlite occurrences record mantle conditions from tectonic environments characterized by decreasing geothermal gradients. In addition, it may be possible to illustrate differences in temperature and pressure between core and rim (e.g. Dish Hill) and relative differences among sample suites from the same general locality (e.g. Salt Lake Crater, Oahu, Hawaii, and the Coast Range Ophiolites).

General comments
Since the purpose of this section is solely to illustrate the potential of applying the new set of geothermometers and geothermobarometers to natural occurrences, interpretations are based purely on the context of the reaction rate groups discussed above. Detailed knowledge of the petrogenesis of the sample suites is not included.
Application of the geothermometers and geothermobarometers to the few occurrences indicates that the fundamental assumption of equilibrium is seldom met. The different reaction rates that control different geothermometers and geothermobaromenters mean that phases are responding independently to changing conditions during the two main stages of the petrogenetic history of the sample (i.e. residence in the mantle and transport to the surface). Fortunately, there is a sufficiently large difference in the reaction rates for the different sets of geothermometers and geothermobarometers so that they may be used to unambiguously characterize the different stages in the samples' history. The slowest set of reaction rates that govern the Cr solubility and Ca and Mg exchange in orthopyroxene may be used to capture an 'original' range of mantle source conditions. Similar but slightly faster reaction rates involving clinopyroxenes, which record later changes of the samples' history, lead to disequilibrium between pyroxenes and complicate interpretations. The set of Mg and Fe exchange reactions that are significantly faster by two to three orders of magnitude than the reactions governing Ca and Mg exchange and trivalent ion solubility in pyroxenes clearly respond to the rapidly changing conditions during the transit stage to the surface. Quantitative insights of the transit stage are confounded by the fact that there are no independent estimates. Further ambiguities result from reactions being stalled by blocking temperatures that rely on unknown dependencies between reaction rates and thermal histories.

Acknowledgements
My gratitude goes to Paul Moreau who contributed significantly to the assembly of the ½-inch-diameter solid media piston and cylinder apparatus, and to Raymond Wittkopp who provided considerable assistance and advice in the electron microprobe analyses of the experimental products. Although the final manuscript remains entirely my responsibility, I am indebted to Dexter Perkins for a comprehensive and insightful review, which significantly improved the paper.

Disclosure statement
No potential conflict of interest was reported by the author.