Garnet and Spinel Oxybarometers : New Internally Consistent Multi-equilibria Models with Applications to the Oxidation State of the Lithospheric Mantle

New thermodynamic data for skiagite garnet (Fe3Fe2 Si3O12) are derived from experimental phase-equilibrium data that extend to 10 GPa and are applied to oxybarometry of mantle peridotites using a revised six-component garnet mixing model. Skiagite is more stable by 12 kJ mol than in a previous calibration of the equilibrium 2 skiagite1⁄4 4 fayaliteþ ferrosiliteþO2, and this leads to calculated oxygen fugacities that are higher (more oxidized) by around 1–1 5 log fO2 units. A new calculation method and computer program incorporates four independent oxybarometers (including 2 pyropeþ 2 andraditeþ2 ferrosilite1⁄42 grossularþ 4 fayaliteþ3 enstatiteþO2) for use on natural peridotite samples to yield optimum log fO2 estimates by the method of least squares. These estimates should be more robust than those based on any single barometer and allow assessment of possible disequilibrium in assemblages. A new set of independent oxybarometers for spinel-bearing peridotites is also presented here, including a new reaction 2 magnetiteþ 3 enstatite1⁄4 3 fayaliteþ 3 forsteriteþO2. These recalibrations combined with internally consistent PT determinations for published analyses of mantle peridotites with analysed Fe2O3 data for garnets, from both cratonic (Kaapvaal, Siberia and Slave) and circumcratonic (Baikal Rift) regions, provide revised estimates of oxidation state in the lithospheric mantle. Estimates of log fO2 for spinel assemblages are more reduced than those based on earlier calibrations, whereas garnet-bearing assemblages are more oxidized. Importantly, this lessens considerably the difference between garnet and spinel oxybarometry that was observed with previous published calibrations.


INTRODUCTION
The redox state of the Earth's mantle is of fundamental importance in understanding how it melts, how the abundance and disposition of carbon-bearing minerals and fluids vary at greater depths, where diamonds form, and the depths at which they entrap a variety of silicate and oxide inclusions (Harte & Cayser, 2007;Frost & McCammon, 2008;Dasgupta & Hirschmann, 2010;Foley, 2010;Harte, 2010;Stagno et al., 2013).Most attempts to determine mantle log f O2 are based on the pioneering calibration of Gudmundsson & Wood (1995), who used an oxygen barometer based on the reaction 2ski where ski is skiagite (Fe 3 Fe 2 3þ Si 3 O 12 ), fa is fayalite (Fe 2 SiO 4 ) and fs is ferrosilite (Fe 2 Si 2 O 6 ).Accounting for the activities of ski, fa and fs in garnet, olivine and orthopyroxene allows determination of the oxygen activity, usually expressed in terms of log f O2 .More recently this calibration has been questioned by Stagno  (2013), who proposed an alternative oxybarometer based on the equilibrium of Luth et al. (1990): where py is pyrope (Mg 3 Al 2 Si 3 O 12 ), andr is andradite (Ca 3 Fe 2 Si 3 O 12 ), gr is grossular (Ca 3 Al 2 Si 3 O 12 ), and en is enstatite (Mg 2 Si 2 O 6 ).On the basis of their own oxygen sensor measurements, Stagno et al. (2013) suggested that reaction (2), calibrated using the thermodynamic data of Holland & Powell (2011), provides an oxybarometer that satisfies the available data better than reaction (1).Stagno et al. (2013) showed that calculations based on reaction (1) deviated from their experiments by 0Á5-2 log units in which log f O2 was measured by an Ir-Fe sensor, with the largest deviations occurring at the highest pressures (6-7 GPa).Here we show that there was an error in the original thermodynamic calibration of reaction (1) and proceed to derive new thermodynamic data for skiagite that allow reconciliation between the two barometers and pave the way to using a multi-reaction approach involving optimization of log f O2 by least squares.This should help determine whether minerals in an assemblage have equilibrated effectively and hence should provide more robust estimates for log f O2 .
The problem with the original calibration of reaction (1) lies in an error in the Gibbs free energy adopted for that reaction by Gudmundsson & Wood (1995).In their analysis the Gibbs energy change for reaction (1) was estimated as 133Á3 kJ at 1 bar and 1100 C (log K 1 ¼ -5Á07), this value being based on the calculations of Woodland & O'Neill (1993) on the reaction involving hercynite (hc, FeAl 2 O 4 ): for which they derived a Gibbs energy of -69Á3 kJ.This led them, through a sequence of calculations based on thermodynamic data from Holmes et al. (1986) and Holland & Powell (1990), to deduce the free energy of reaction (4) involving quartz (qz, SiO 2 ) and iron metal (Fe) as 981Á0 kJ at 1 bar and 1100 C.This value was used by Gudmundsson & Wood (1995) in their derivation of the free energy for reaction (1).A recalculation of the free energy of reaction (4) (see Appendix), with the data sources used by Woodland & O'Neill (1993), leads to a significantly smaller value (970Á4 kJ), which in turn causes the oxybarometer to be systematically in error.
The newer thermodynamic data of Holland & Powell (2011), as opposed to those of Holland & Powell (1990), lead to essentially the same value (969Á0 kJ) and therefore a reassessment and recalibration of reaction (1) is warranted.
Reaction (1) requires thermodynamic data for skiagite that may be combined with existing data for fayalite, ferrosilite and oxygen.The following three equilibria, involving hc (hercynite, FeAl 2 O 4 ), alm (almandine, Fe 3 Al 2 Si 3 O 12 ), frw (ferroringwoodite, Fe 2 SiO 4 ), mt (magnetite, Fe 3 O 4 ), and coe (coesite, SiO 2 ) are involved in the experimental data of Woodland & O'Neill (1993): It should be noted that reaction ( 6) is a linear combination of (3) and ( 5) but can provide a test of consistency between the experimental results of Woodland & O'Neill (1993) and the thermodynamic data of Holland & Powell (2011) that is independent of skiagite.Woodland & O'Neill (1993) used only reaction (3) above to derive thermodynamic data for skiagite, because the spinel in their lower pressure experiments (<45 kbar) was dominantly a hercynite-magnetite solid solution, whereas in the higher pressure runs increasing amounts of ferroringwoodite component were present.
Availability of thermodynamic data for ferroringwoodite in the more recent dataset of Holland & Powell (2011) allows use of reaction ( 5) as an additional check on internal consistency of the data.The lowest pressure studied by Woodland & O'Neill (1993) involved quartz rather than coesite, and so, although the results reported here involve coesite, quartz was used in place of coesite in our analysis of the 27 kbar runs.
Clearly, use of reactions (3), ( 5) and ( 6) requires a solution model for ternary hc-mt-frw spinel.In this study we assume complete disorder in spinel and use the three-site model simplification as proposed by Bryndzia & Wood (1990) and Wood et al. (1990).Several lines of evidence suggest that this is reasonable and adequate for the purposes of this study: first, a binary mt-hc spinel using W hc,mt ¼ 38 kJ allows calculation of the solvus of Turnock & Eugster (1962) almost perfectly; second, O'Neill & Navrotsky (1983) in their treatment of order-disorder in spinels also found that W hc,mt ¼ 38 kJ was required to fit the solvus using their more complex model; third, the calorimetry performed by Navrotsky (1986) on MgFe 2 O 4 -MgAl 2 O 4 demands a value for W hc,mt of 40Á4 6 4 kJ, which matches closely that found here for the analogous binary FeFe 2 O 4 -FeAl 2 O 4 ; fourth, the activity of hercynite and magnetite in a binary spinel at 1300 C has been experimentally measured by Petric et al. (1981), and the calculated activities of both hercynite and magnetite are close to 0Á5 at X mt ¼ 0Á5 as in the experimental determination; finally, the ternary spinel model when fitted to the three reactions (3), ( 5) and (6) using the experiments of Woodland & O'Neill (1993) yields thermodynamic data that are entirely consistent with the dataset of Holland & Powell (2011), as shown below.These considerations suggest that the simple spinel model that we have used is an adequate approximation for this system at these conditions.
RECALIBRATION OF THE SKIAGITE BAROMETER Woodland & O'Neill (1993) equilibrated garnet and spinel experimentally at 1100 C and analysed their results using equilibrium (3).They deduced the free energy of skiagite from their data, using only experiments at pressures below 45 kbar (where the spinels contain very little of the frw end-member).Woodland & O'Neill (1993) also assumed that, at these conditions, garnet and spinel form binary alm-ski and mt-hc solid solutions.We have extended their calibration by incorporating all of their high-pressure data using the ternary mt-hc-frw solution model for spinel outlined above.For reaction (3) the equilibrium constant is K ð3Þ ¼ a alm a mt a ski a hc : In the binary alm-ski garnet (gt), with mixing on only the Y sites, the ideal activities (id) are given as and in a ternary spinel (sp) the ideal mixing activities become Adding in non-ideality, the usual condition of equilibrium at 1 bar and 1100 C (1373 K) may be written as where DG (3)1,1373 is the Gibbs free energy of reaction (3) at 1 bar and 1373 K and the P RTln c i terms come from a regular solution model (see Appendix) in both garnet (ski, alm) and spinel (hc, mt, frw), giving for reaction (3) where DW ¼ (W frw,mt -W hc,frw ), and p i is the proportion of end-member i.The end-member proportions are as follows: Si .The data of Woodland & O'Neill (1993) allow evaluation of the activity terms, and DV (3) is taken as approximately -0Á205 kJ kbar -1 (linearizing the data in Table 1 at the pressures and temperatures of the experiments).We can take W alm,ski ¼ 2Á0 kJ, under the assumption that Al-Fe 3þ mixing is the same as in grossularandradite (Holland & Powell, 2011; see Appendix), and W hc,mt ¼ 38 kJ as described above.Plotting against p frw yields a slope of DW and intercept of DG (3)1,1373 , thus providing a rapid visual estimate for the free energy of skiagite at this temperature (Fig. 1).The data at pressures greater than 45 kbar, disregarded in the analysis of Woodland & O'Neill (1993), are represented in Fig. 1 by values of p frw > 0Á1 and are clearly required in providing a sufficient range to determine a slope, and hence a value for DW.
Additionally, reaction (5) provides further endorsement for the skiagite free energy from the experiments of Woodland & O'Neill (1993).The equilibrium condition for reaction (5) is which allows simultaneous determination of both W frw,mt and the free energy of skiagite.Furthermore, reactions (3) and ( 5) are linearly related by the equilibrium (6) frw þ hc þ 2 coe ¼ alm, for which the data in Holland & Powell (2011) provide a constraining value of DG (6)1,1373 ¼ -32Á1 kJ.All three equilibria were therefore fitted to the experimental data simultaneously, using THERMOCALC (Powell & Holland, 1988) to generate an updated version of the Holland & Powell (2011) dataset.Thermal expansion and compressibility for all endmembers were incorporated (see Table 1).The best fit to all of the data yields Experimental data for reaction (3) fit the model remarkably well, yielding a value for the free energy at 1 bar and 1373 K of -69Á5 kJ, which is in near-perfect agreement with the earlier analysis of Woodland & O'Neill (1993).This is unsurprising, given that both studies use essentially the same value of W hc,mt ¼ 38 kJ, and at the pressures of the Woodland & O'Neill (1993) analysis (<45 kbar) the spinels are close to the binary hercynite-magnetite.However, the newly derived free energy for reaction (1) differs from that used by Gudmundsson & Wood (1995) by over 24 kJ, resulting in log K 1 values that are more positive by 1 log unit than those used in the oxygen barometer calibration by Gudmundsson & Wood (1995).Thus the apparent agreement between the derived barometer expression and the oxygen sensor experiments of Gudmundsson & Wood (1995) is fortuitous.Oxygen fugacities that are 1 log unit more oxidizing than those in the literature appear to be in very good agreement with the measured and calculated bulk Fe 2 O 3 contents of mantle peridotites (Jennings & Holland, 2015, fig. 14).The new free energies of reaction are found to be non-linear in pressure and temperature and are fitted with a simple polynomial whose coefficients are given in the Appendix.Because the magnitudes of the mixing energy terms in garnet, particularly the cross-site or reciprocal reaction terms, contribute significantly to the application of the barometer they require further discussion and assessment.It should be stressed that inclusion of reaction (5) makes no discernible difference to the derived skiagite free energy, but coupled with reaction (6) it does constitute a valuable endorsement of the internal consistency of the thermodynamic data and of the higher P experimental results.

Recalibration of the garnet mixing model
The magnitudes of the free energies of the following four reciprocal reactions strongly affect the garnet activities used in the barometer reaction (1): These four reactions represent the cross-site contributions to the garnet activities (see Appendix), such that 9) and W CaAlFeCrXY ¼ -DG (7) are the cross-site energies for garnet X and Y sites (e.g.Powell & Holland, 1993).Gudmundsson & Wood (1995) recognized that varying the garnet mixing parameters, other than for reactions ( 7) and ( 8), makes only small differences to results in calculated log f O2 for mantle peridotite garnets with low to moderate Cr contents.Reactions ( 9) and ( 10) become significant in Cr-rich garnets.Luth et al. (1990) also recognized that W Al,Fe3þ ¼ W gr,andr can, End-member names correspond to those of Holland & Powell (2011).H is the regressed enthalpy of formation from the elements at 1 bar and 298 K; sd(H) is one standard deviation on the enthalpy of formation; S is the entropy; V the volume (all properties at 1 bar and 298 K); a, b, c and d are the coefficients in the heat capacity polynomial dT À1/2 ; a 0 and K 0 are thermal expansion and bulk modulus at 298 K; K 0 ' is the first derivative of bulk modulus at 298 K. Units: H, kJ; S, J K -1 ; V, kJ kbar -1 ; C p , kJ K -1 ; a 0 , T -1 ; K 0 , kbar.It should be noted that C p (b) and a 0 need to be multiplied by 10 -5 .[See Holland & Powell (2011) for further information on the dataset including details of unchanged order-disorder parameters for sp, herc, mt, iron, q.] Data sources for end-members changed since Holland & Powell (2011) are as follows.ski: V, Woodland & O'Neill (1993); K 0 , K 0 ', Woodland et al. (1999); S, C p , a 0 , estimated from gr, andr and alm.alm, S, C p , Dachs et al. (2012).knor: S, reduced after Wijbrans et al. (2014).uv: S, V, Klemme et al. (2005); C p , estimated from gr, esk and cor; a 0 , as gr; K 0 , K 0 ', Leger et al. (1990).
Fig. 1.The data of Woodland & O'Neill (1993) used to calibrate skiagite properties, according to reaction (3).The plot is of where p frw is the proportion of ferrroringwoodite in spinel.The line has slope DW ¼ (W frw,mt -W hc,frw ) and intercept DG (3)1,1373 .Woodland & O'Neill (1993) used only the data for p frw < 0Á1 in their analysis (see text for further explanation).
if large in value, lead to significant changes.We find that uncertainties of 10 kJ in these three mixing parameters lead to changes of the order of 0Á5 log f O2 units, whereas similar uncertainties on all other mixing parameters produce barely perceptible differences.Although of minor impact, we prefer to include all the within-site mixing energies to avoid any small systematic bias to calculated log f O2 , but concentrate here on the important variables affecting calculated log f O2 .
From literature data and the dependent end-member relations (Powell & Holland, 1999) we may derive a complete set of internally consistent garnet mixing energies (see Appendix) involving skiagite and other garnet end-members as follows: and for a set involving andradite, the following additional energies may be calculated: A new measurement on the entropy (301 J K -1 ) of knorringite (knor, Mg 3 Cr 2 Si 3 O 12 ) has recently been determined by Wijbrans et al. (2014) that is slightly smaller than the value (317 J K -1 ) estimated by Holland & Powell (2011).We have elected to use these updated values in deriving the revised dataset and hence in reaction (9), even though the effects on the oxybarometer are negligibly small (less than 0Á01 in log f O2 ).It should be noted that, although garnets are asymmetric in their mixing properties (especially for Ca-Mg mixing), a simpler symmetric model has been fitted to garnets that are low in Ca relative to Mg, such as pyropes found in mantle peridotites.Use of the symmetric model has an imperceptible effect on calculated oxygen fugacity.
The equilibrium relation for reaction (1), rewritten as ÀDG ð1ÞP ;T RT À ln a 4 fa a fs a 2 ski ! may be used to determine oxygen fugacity using the expressions in the Appendix for the activities of fa, fs and ski and the free energy of reaction (1).

Revised garnet oxybarometer reactions
The garnet mixing model above may be used directly in reactions such as (1) and (2) to determine oxygen fugacities of garnet-bearing peridotites.With the new thermodynamic data for skiagite it is possible to write 25 barometer reactions among the mineral end-members almandine (alm), skiagite (ski), andradite (andr), grossular (gr), pyrope (py), forsterite (fo), fayalite (fa), enstatite (en), ferrosilite (fs) and oxygen (O 2 ).However, with 10 end-members and six components (Ca-Mg-Fe-Al-Si-O), only 10 -6 ¼ 4 of these reactions are independent and provide all the information in the system.We choose the following four independent reactions, using data from the updated Holland & Powell (2011) dataset: where reactions (1) and ( 2) are the equilibria used in the barometers of Gudmundsson & Wood (1995) and Stagno et al. (2013), respectively.Both are now formulated on the same set of internally consistent thermodynamic data.It should be noted that reciprocal reaction (7) is one of the 25 reactions that may be written among these end-members and, because it is involved in the garnet mixing model, lends an added degree of internal consistency to the barometry.Each of the four reactions (1), ( 2), ( 11) and ( 12) furnishes a value of log f O2 that will be identical only in the case of perfect activity models, perfect thermodynamic data and perfect equilibrium in the mineral assemblage.However, an optimum log f O2 may be found from these reactions by least squares, analogous to the average pressure calculations of Powell & Holland (1988).In the least squares fit, each reaction carries different weight according to how well its thermodynamic data are known (via uncertainties and correlations among the free energies from the thermodynamic dataset, and in the uncertainties in the mixing parameters for garnet, orthopyroxene and olivine).
Additionally, the uncertainties in weight per cent of all oxides in garnet, olivine and orthopyroxene are propagated to contribute to the uncertainty of the activities in each reaction and hence its weight.In these calculations the errors are assigned as follows: in mineral analyses the uncertainty on each oxide weight per cent is taken as 1% relative with a minimum uncertainty of 0Á02 wt %.Uncertainty in Fe 2 O 3 for garnet is taken as 20% of the measured value [from scatter in experimental measurement and from corrections from 80 K to room temperature (Woodland & Ross, 1994)].
In olivine all Fe is taken as FeO, and Fe 2 O 3 in orthopyroxene is estimated as 0Á4 6 0Á3 wt %, based on the 14 tabulated measurements of Canil & O'Neill (1996).These assumptions are much more defensible than estimating Fe 2 O 3 from pyroxene stoichiometry, which produces large random variations that may far exceed the measured values of Canil & O'Neill (1996).The uncertainty on each interaction energy (including reciprocal terms) is assigned as 10% relative with a minimum uncertainty of 1Á0 kJ, based on typical calorimetric errors, and the uncertainties and correlations between the enthalpies of reaction are taken from the updated Holland & Powell (2011) dataset.Doubling or halving these error assumptions made little difference to calculated log f O2 values.Error propagation is used to determine the covariance matrix (V) for the log f O2 values from the four reactions (in column vector F) and the optimum f av is found by the least squares result where r 2 fit ¼ ð1 T V À1 1Þ À1 and 1 is a column vector of ones.
The appropriateness of averaging the four reactions in this way is provided by a v 2 test and the value of r fit , the mean weighted deviation ( ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi MSWD p ).If there is good agreement among the four equilibria within their mutual errors, the overall r fit is expected to be around 1Á0 or less [see the discussion by Powell & Holland (1994) for the analogous case of average pressure calculations].Values of r fit significantly greater than a cutoff value of 1Á61 [the maximum allowed by a v 2 test at the 95% confidence level for three degrees of freedom for the four independent reactions; see Powell & Holland (1994)] indicate that the barometers disagree sufficiently and averaging is inappropriate.This could occur either through disequilibrium, through analytical error, or simply by choosing an inappropriate temperature or pressure for the calculation.The correlations among the reactions frequently cause the calculated optimal log f O2 to differ considerably from a simple average of the four log f O2 values.Enlarging the assumed uncertainties on the input variables (e.g. chemical analyses, W values, etc.) will cause r fit to be smaller, so a failure of the v 2 test may be flagging that input uncertainties have been underestimated rather than pointing to disequilibrium in the sample.
Output is shown in Table 2 from program GtfO2 for a mantle garnet-harzburgite from the Finsch mine, Kaapvaal Craton (F5; Lazarov et al., 2009) that equilibrated at 54 kbar and 1150 C (Table 3).The output shows input mineral analyses for garnet, orthopyroxene and olivine, recalculated cations and their uncertainties, and calculated activities for end-members used in oxybarometry.The calculated Dlog f O2 (FMQ) values are -2Á69 6 0Á89, -2Á40 6 0Á90, -2Á45 6 0Á88 and -1Á95 6 0Á90 for reactions (1), ( 2), ( 11) and ( 12), respectively.The least squares optimum is -2Á63 6 1Á28, with a r fit value (labelled f in the output) of 1Á49.This is smaller than the cut-off of 1Á61 so combining the reactions is appropriate and the assemblage is reasonably well equilibrated.The fact that the v 2 test for internal consistency among the four independent reactions is passed    in the majority of natural samples investigated here confirms the internal consistency now attainable between the skiagite barometer (1) and the andradite barometer (2).Typical uncertainties for each barometer reaction lie in the range 0Á6-0Á9 log f O2 units, depending mainly on mineral compositional uncertainties, particularly in ferric iron content.If all reactions agree within mutual error then the overall uncertainty will be of this magnitude.However, if the barometers do not mutually agree then r fit will be greater than 1Á0 and the overall error will be enlarged through multiplication by r fit .Thus the minimum uncertainty on log f O2 will be of the order of 0Á6-0Á9 log units.
Figure 2a illustrates the experimental log f O2 data from Gudmundsson & Wood (1995) and Stagno et al.
(2013) compared with calculations from reactions (1) and ( 2).Calculated values tend to be somewhat scattered, but with uncertainties on calculated and experimental values being of the order of 2 and 1 log units, respectively, the agreement is deemed satisfactory.The calculated results tend to be higher at more reducing conditions but are in fairly good agreement at higher log f O2 , perhaps more so with the experiments of Stagno et al. (2013).It should be noted that only about 20% of the experimental samples of Gudmundsson & Wood (1995) and Stagno et al. (2013) pass the v 2 test, suggesting that the experimental runs may not have fully equilibrated.This should not be surprising, however, as experiments of short duration are likely to be less well equilibrated than natural samples.When the least squares results using all four equilibria (1), ( 2), (11)  (2015).Data sources for Slave Craton: Kopylova et al. (1999), Kopylova & Caro (2004), McCammon & Kopylova (2004) and Creighton et al. (2009).n.a., not applicable.
and ( 12) are compared with the experiments (Fig. 2b) the results are less scattered, illustrating the more robust estimation of log f O2 values in comparison with those from each of the equilibria.Figure 2c is a plot of the new calibration expressions for reactions (1) and ( 2) against the original barometer expressions of Gudmundsson & Wood (1995) and Stagno et al. (2013) and shows the fairly uniform relative offset of 1-1Á5 log units of the new calibrations.Given that Stagno et al.
(2013) used the Holland & Powell (2011) dataset, the offset relative to their expression must be caused by the different interaction energies in garnet, especially in the two reciprocal reactions ( 7) and ( 8) and the large value of W Al,Fe3þ used by them.The differences between the new calibrations and earlier studies for reactions (1) and ( 2) may be seen readily in Fig. 3.In Fig. 3a the separate effects of replacing the interaction energies (W CaAlFeFe3XY ¼ -DG (7) , W CaAlMgFe3XY ¼ -DG (8) and W Al,Fe3þ ) used by Gudmundsson & Wood (1995) by the new ones used in this study are illustrated clearly.Changing W Al,Fe3þ from zero to 2Á0 kJ has only a small impact whereas the other two have large and opposite effects on log f O2 .The opposing effects mean that changing W CaAlFeFe3XY and W CaAlMgFe3XY by the same amount (23 kJ) has only a negligible effect on log f O2 .The net effect of changing all three Ws together gives a result fortuitously identical to the original  (b) The same data but for the least squares calculation using reactions (1), ( 2), ( 11) and ( 12).The least squares results are less scattered than those for each of the barometers and in better agreement with the experiments.(c) Comparison of calculations using the new calibrations against the original result of Gudmundsson & Wood (1995) (GW) and that of Stagno et al. (2013) (Stag); this shows a consistent 1-1Á5 log unit offset.Gudmundsson & Wood (1995) expression, and thus demonstrates that the difference between our new calibration and the original barometer lies entirely with the revised Gibbs energy for reaction (1). Figure 3b shows a rather different situation for reaction (2) where the effects of W CaAlFeFe3XY and W Al,Fe3þ are small but that of W CaAlMgFe3XY is large.In contrast to reaction (1) changes of 23 kJ in both W CaAlFeFe3XY and W CaAlMgFe3XY move log f O2 in the same direction.Changing all three Ws simultaneously to the new values raises log f O2 by around 1 log unit and it is these W differences and not any change in Gibbs energy of reaction (2) that make our new calibration differ from that of Stagno et al. (2013).The internal inconsistency in the Stagno et al. (2013) barometer lies in their values for W CaAlFeFe3XY and W CaAlMgFe3XY , which do not agree with the free energies in the updated Holland & Powell (2011) dataset (Table 1) for reactions ( 7) and ( 8).Making both W CaAlFe- Fe3XY and W CaAlMgFe3XY larger barely affects reaction (1) because of the opposing senses of change, whereas reaction (2) is additively affected.Reaction (1) was well chosen by Gudmundsson & Wood (1995) in being more robust to uncertainties in mixing parameters.
The new barometer results are in fairly close agreement with the experimental data of Stagno et al. (2013) as shown in Fig. 2b.The lack of perfect agreement of the new calibrations with the high-pressure metal sensor techniques (Gudmundsson & Wood, 1995;Stagno et al., 2013) will be discussed after reassessment of the spinel oxybarometer.

REVISED SPINEL OXYBAROMETER EQUILIBRIA
The advantage of solving several reactions simultaneously by the least squares method makes it desirable to extend this approach to spinel oxybarometry.The following three independent equilibria are used: The first of these three equilibria was calibrated by Bryndzia & Wood (1990) and has been used widely since.We use the activity model for spinel from Bryndzia & Wood (1990) coupled with thermodynamic data from Holland & Powell (2011).The free energies for (13) are virtually indistinguishable from those given by Bryndzia & Wood (1990).It is important to note that (13) is rather sensitive to the activity of fs.This is significant because the mol fraction of fs in mantle orthopyroxenes is very small and poorly determined (especially with uncertain Fe 2 O 3 content) and because the earlier calibrations assumed ideal mixing.Reaction ( 14) is more resistant to uncertainty as (1) it does not depend on the fs end-member, (2) it involves fo and en endmembers, which have large and better determined activities in olivine and opx, and (3) the Mg-poor silicate end-member is fa, the activity of which is more reliably estimated than that of fs.When averaging log f O2 from ( 13), ( 14) and ( 15) it is reaction (14) that has smallest error and dominates the calculation.Non-ideality in olivine is taken directly from Gudmundsson & Wood (1995), who used a slightly smaller value of W fo,fa than Bryndzia & Wood (1990).Non-ideality in orthopyroxene (see Appendix) makes a small but significant difference to the results, raising calculated log f O2 for reaction (13) by around 0Á2 log units.Wood (1990, fig. 2c) showed that an offset of 0Á2 log units would produce an almost perfect fit of the barometer equation ( 13) with his experiments.Thus the new calibration here is in excellent agreement with the 1 bar oxygen sensor measurements.The equations for free energies of reactions ( 13), ( 14) and ( 15) are presented in the Appendix, and computer programs will be made available (from links at http://www.esc.cam.ac.uk/directory/tim-holland) to perform the error propagation of the uncertainties in chemical analyses, the thermodynamic data and the mixing properties of the phases in a least squares optimization of log f O2 .
An important finding of our new calibrations is that, in comparison with earlier parameterizations, the multireaction spinel barometry yields log f O2 values that are more reducing by around 0Á5 log units.This results from the dominance of the new barometer reaction ( 14) with its smaller uncertainty and the addition of non-ideal mixing in orthopyroxene.The implications of this and application of both garnet and spinel barometry to mantle peridotites will be explored in the next section.

CAUSES OF DIFFERENCES IN OXYBAROMETERS
There may be several factors that cause differences between the new calibrations of garnet oxybarometers and earlier work.One is the error in the earlier calculation of skiagite free energy as discussed above.Although it is possible that the new skiagite free energy may be uncertain, the high level of agreement between calculations on all three reactions (3), ( 5) and ( 6) with the phase equilibrium experiments of Woodland & O'Neill (1993) at all pressures from 27 to 90 kbar at 1100 C suggests that the new data are reliable.The fact that the combined results of four independent reactions, via least squares, yield consistent results within error for well-equilibrated natural samples and most of the experimental samples of Stagno et al. (2013) also suggests that the thermodynamic data and phase equilibrium studies on the end-members used are in good agreement.In this context an error of 12 kJ on skiagite free energy, as noted above, would correspond to an error of 23 kbar on the breakdown pressure of skiagite at 1100 C as measured experimentally by Woodland & O'Neill (1993).Their experiments are most unlikely to be in error by that amount.
One possible explanation for the discrepancy may be in the use of oxygen sensor techniques at elevated pressures.The spinel barometer reaction ( 13) is in excellent agreement with the 1 atm oxygen sensor measurements of Wood (1990), as shown above, but the garnet equilibria are in less good agreement with the higher pressure oxygen sensor measurements of Gudmundsson & Wood (1995) and Stagno et al. (2013).It would appear also that the new calibrations are in much better agreement with the measurements at higher than lower oxygen fugacity, and this might reflect the difficulty of measuring accurately the ferric iron content in garnets at very low concentrations.

APPLICATIONS
A reliable method of calculating f O2 values for mantle peridotites is important for assessing melting, metasomatism and fluid speciation.The use of the oxybarometers presented here requires the measurement of major element oxides for olivine, orthopyroxene, garnet and/ or spinel, as well as a PT estimate for each xenolith.Because some literature data provide only the Mgnumber for olivine and orthopyroxene, the application programs developed here allow for these to be entered in place of a full analysis, and approximate fo, fa, en and fs activities are assigned.It is very important to have an accurate Fe 2 O 3 analysis for garnet as this greatly affects the activity of skiagite, and is the last remaining hurdle in accurately calculating f O2 for garnet peridotites.

Variability in the oxidation state of the lithospheric mantle
We have applied the new garnet and spinel oxybarometer calibrations presented above to published data from four mantle xenolith suites.The samples come from a range of tectonic settings-the Baikal Rift (Vitim Volcanic Field; Goncharov & Ionov, 2012) and three major global cratons: Siberian (Udachnaya; Goncharov et al., 2012;Yaxley et al., 2012), Kaapvaal (Woodland & Koch, 2003;Lazarov et al., 2009;Hanger et al., 2015) and Slave (Diavik;McCammon & Kopylova, 2004;Creighton et al., 2009).All these samples have Fe 3þ /Fe total ratios for pyrope garnets that were measured by Mo ¨ssbauer spectroscopy, Fe K-edge XANES or the flank method, which currently provide the most accurate values.For internal consistency, we have taken the published analyses of mineral chemistry and recalculated pressures and temperatures using the PTmantle program (Nimis & Gru ¨tter, 2010).The following combinations of thermobarometers (with associated errors) were used for the xenolith suites: for garnet lherzolites we used the opx-gt barometer of Nickel & Green (1985; P NG85 ; 63 kbar) and the cpx-opx solvus thermometer of Taylor (1998; T Ta98 ; 631 C); for garnet harzburgites P NG85 was used with the opx-gt thermometer of Nimis & Gru ¨tter (2010; T NG10 ; 650 C); for spinel lherzolites we have followed Goncharov et al. (2012) and extrapolated temperatures, calculated using T Ta98 , to the conductive geotherm that was estimated from garnet-bearing samples.We have used our new oxybarometer programs to give revised values of log f O2 relative to the fayalitemagnetite-quartz (FMQ) buffer.For the FMQ buffer we use the expression of O'Neill (1987) with a volume correction from Holland & Powell (2011), as given in the Appendix.Carbon phase stability has then been assessed using log f O2 buffers, calculated using an updated version of the Holland & Powell (2011) dataset, and the relevant conductive geotherms for the various tectonic settings calculated using the GeoTherm program (Mather et al., 2011) for each xenolith suite.A summary of the results is given in Table 3.

Baikal Rift (Vitim Volcanic Field)
Spinel-and garnet-bearing mantle peridotites occur in the Vitim Volcanic Field, which is situated to the SE of the Siberian Craton.Our recalculated PT estimates for 36 xenoliths analysed by Goncharov & Ionov (2012) show that spinel-only-bearing samples occupy almost the whole depth range (45-65 km) sampled by Vitim magmas (Table 3 and Fig. 4).Because Vitim peridotites contain both garnet and spinel they offer a rare insight into the accuracy of f O2 estimates provided by our new independent oxybarometers.For these samples, our spinel oxybarometer gives Dlog f O2 (FMQ) from -0Á43 to þ0Á20 (av.-0Á17), whereas our garnet oxybarometer gives Dlog f O2 (FMQ) in the range -1Á79 to -0Á14 (av.-1Á22).For spinel-only-bearing Vitim peridotites Dlog f O2 (FMQ) estimates range from -1Á29 to þ0Á52 (av.-0Á42) and in the spinel-absent garnet-bearing peridotites Dlog f O2 (FMQ) ranges from -1Á82 to -0Á41 (av.-1Á22).A Fig. 4. Plot of pressure vs temperature estimates for off-craton Vitim peridotites from the Baikal Rift zone, using mineral chemistry from Ionov & Wood (1992) and Goncharov & Ionov (2012), together with the thermobarometers of Nickel & Green (1985), Taylor (1998) and Nimis & Taylor (2000).A model conductive geotherm of 61Á4 mW m -2 has been calculated to best fit the PT estimates for garnet-bearing samples in the xenolith suite, using the depth to the Moho of Suvorov et al. (2002) and the program described by Mather et al. (2011).The diamondgraphite transition (Kennedy & Kennedy, 1976)  comparison of these new values with the published estimates of Goncharov & Ionov (2012) shows that there is a change in spinel-based estimates by -0Á2 Dlog f O2 (FMQ) units to more reducing conditions, whereas the garnet-based oxybarometry has increased f O2 estimates by þ1Á3 Dlog f O2 units.As a consequence the D log f O2 (FMQ) values calculated using our new oxybarometers for all samples are in remarkable agreement (Figs 5 and 6), which contrasts with the distinct oxidation states for different assemblages proposed by Goncharov & Ionov (2012).Furthermore, there is no clear variation in Dlog f O2 with depth for the Vitim garnet and spinel peridotites.The wide range in Dlog f O2 (FMQ) (-1Á82 to 0Á52) over a small pressure range (20-25 kbar) may relate to variable extents of metasomatism by ascending carbonatitic melts at the carbonated peridotite solidus 'ledge' (e.g.Eggler, 1974;Wyllie & Huang, 1976).

Siberian Craton (Udachnaya)
We have used the published data of Yaxley et al. (2012) for 18 samples of garnet peridotite and those of Goncharov et al. (2012) for 37 samples of spinel-and garnet-bearing peridotites entrained by the Udachnaya kimberlite from the Siberian Craton with our new oxybarometer calibrations.These xenoliths were entrained from a large depth interval (40-210 km; Table 3 and Fig. 7).Spinel-and garnet-bearing samples last equilibrated at depths of 40-100 km and 90-210 km, respectively.Only a few samples (Table 3) contain both aluminous phases.For the Udachnaya spinel peridotites estimates of Dlog f O2 (FMQ) range from -1Á71 to -0Á09 (av.-0Á70).For the deeper garnet peridotites Dlog f O2 (FMQ) ranges from -4Á93 to -1Á02 (av.-2Á01).A comparison of our new values with the published estimates of Goncharov  Goncharov & Ionov (2012), based on the oxybarometer of Gudmundsson & Wood (1995) (GW95) and the oxybarometer of Bryndzia & Wood (1990) (BW90).Buffer reactions relative to FMQ (fayalite-magnetite-quartz) were calculated from the Holland & Powell (2011) dataset and are as follows: WM, wu ¨stite-magnetite; IW, iron-wu ¨stite; D/GCO, diamond/graphite-CO; EMOD/G, enstatite-magnesite-olivine-diamond/graphite.MBL as in Fig. 4. The published Dlog f O2 (FMQ) values, together with our recalculated values (Fig. 8), indicate that a number of xenoliths that previously plotted in the diamond stability field now plot at more oxidized conditions on the carbonate stability side of the enstatite-magnesite-olivinediamond/graphite (EMOD/G) buffer.These more oxidized values agree with the findings of experimental work by Stagno et al. (2013), which also suggest that the Gudmundsson & Wood (1995) oxybarometer provides f O2 estimates that are too low.The Udachnaya mantle xenoliths show a clear Dlog f O2 (FMQ) versus depth relationship (Fig. 8).Over a depth range of 165 km, the Dlog f O2 (FMQ) values change from 0Á0 at the top of the lithospheric mantle to -3Á0 near the base.This gives a lithospheric mantle Dlog f O2 (FMQ) gradient of c. 0Á18 log units per 10 km, which is lower by 0Á07 log units per 10 km than the gradient estimated by Goncharov et al. (2012).
A comparison (Fig. 10) of the f O2 values calculated by Lazarov et al. (2009), who used the calibration of Gudmundsson & Wood (1995) as corrected by Woodland & Peltonen (1999), with recalculated values for the same xenoliths using our new oxybarometers indicates a shift in f O2 estimates of þ1Á0 Dlog f O2 (FMQ) units.The same shift in f O2 estimates is also seen for the other Kaapvaal xenoliths.Moreover, garnet lherzolites from across the entire craton give a lithospheric mantle gradient of 0Á24 Dlog f O2 (FMQ) per 10 km, whereas garnet harzburgites indicate a lithospheric gradient of 0Á29 Dlog f O2 (FMQ) per 10 km, which is similar to the estimate of Lazarov et al. (2009).Reassuringly, both diamond-bearing samples plot in the diamond stability field.The relatively large spread of f O2 that is observed at depth in the Kaapvaal  lithosphere may reflect variable metasomatic enrichment over short length scales.

Slave Craton (Diavik Mine)
Analyses of mineral phases present in 69 garnetbearing mantle peridotites from Diavik Mine, central Slave Craton have been used to calculate final equilibration pressures and temperatures with the P NG85 barometer and the T Ta98 thermometer (Table 3 and Fig. 11).Creighton et al. (2009) presented two sets of PT estimates for these same xenoliths: one using the Brey & Ko ¨hler (1990) barometer (P BKN ) and thermometer (T BKN ); the other using the thermometer of O'Neill & Wood (1979; T OW ) in combination with P BKN .The PT estimates using the P BKN and T OW combination were lower than the P BKN and T BKN combination.Our recalculations agree more closely with the P BKN and T BKN combination.Based on these calculations, most of the xenoliths were entrained from 150 to 190 km with a few additional samples from 125 to 135 km.
Using our new oxybarometers, estimates of Dlog f O2 (FMQ) for Diavik garnet peridotites range from -3Á73 to þ0Á03 (av.-1Á68), which is an increase of þ1Á4 Dlog f O2 (FMQ) units compared with the previous Gudmundsson & Wood (1995) calibration.This range in f O2 occurs in the lower half of the cratonic lithosphere, and has had the effect of moving half of the xenolith samples out of the diamond stability field, crossing the wu ¨stitemagnetite (WM) and EMOD/G buffers to more oxidizing conditions (Fig. 12).The large range in f O2 for the lithospheric mantle beneath the central Slave Craton is similar to that for the Siberian Craton (Udachnaya; Fig. 8).

Small-scale variability of f O 2 : an example from the Kaapvaal Craton
The variations in f O2 that we have described above represent changes in oxidation state of the lithospheric mantle over large depth intervals.Localized interactions between percolating metasomatizing melts and existing mineral phases, immediately prior to or during entrainment, may also cause micro-scale changes in f O2 .Although high-quality garnet Fe 2 O 3 data are limited for such samples, Berry et al. (2013) made observations and calculations on a single zoned garnet from a Kaapvaal mantle peridotite, using the compositions of the core and the rim to deduce their separate PT and f O2 conditions.They assumed that the core of the garnet was in equilibrium with the surrounding orthopyroxene, and calculated a pressure of 47 kbar (P NG85 ) and temperature of 1060 C with the Canil (1994) thermometer.Our PT recalculation using P NG85 and T Ta98 for both Fig. 9. Pressure vs temperature estimates for peridotites from the Kaapvaal Craton.Circles, spinel-bearing samples; diamonds, garnet-bearing samples.Mineral analyses are from Woodland & Koch (2003) and Lazarov et al. (2009).Pressures and temperatures were estimated using the thermobarometers of Nickel & Green (1985), Taylor (1998) and Nimis & Gru ¨tter (2010).A model conductive geotherm of 45Á6 mW m -2 has been calculated to best fit the PT estimates from the Finsch xenolith suite, along with the depth to the Moho (Nair et al., 2006) using the program described by Mather et al. (2011).The estimated thickness of the mechanical boundary layer (MBL) is 204 km.This is in reasonable agreement with published findings for Finsch (Gibson et al., 2008;Lazarov et al., 2009).The diamond-graphite transition (Kennedy & Kennedy, 1976) is shown, along with a greyscale indication of the Dlog f O2 (FMQ) value for each xenolith.Fig. 10.Variation of Dlog f O2 (FMQ) vs pressure for Finsch and other Kaapvaal garnet-peridotites: (a) using the mineral analyses of Woodland & Koch (2003) and Lazarov et al. (2009), and the oxybarometer calibration from this study; (b) published P and f O2 values from Woodland & Koch (2003) and Lazarov et al. (2009).Buffer reactions are relative to FMQ as in Fig. 5.It should be noted that Lazarov et al. (2009) did not publish f O2 values for all 28 xenoliths.MBL as in Fig. 9.
garnet core and rim oxide data from Berry et al. (2013) gave revised estimates of 50Á6 kbar and 1061 C for the core, and 50Á0 kbar and 1059 C for the rim.Although identical to the results of Berry et al. (2013), it should be noted that the PTmantle program (Nimis & Gru ¨tter, 2010) suggests orthopyroxene-clinopyroxene disequilibrium errors for both estimates, and a garnet-pyroxene disequilibrium error for the core calculation.
Our new oxybarometer calibration gives estimates of Dlog f O2 (FMQ) ¼ -1Á59 for the garnet core and -0Á57 for the rim compositions.This increase in Dlog f O2 (FMQ) of þ0Á8 for the core and þ0Á5 for the rim places the rim firmly in the carbonate stability field whereas the core moves to the EMOD/G buffer, making diamond stability questionable.Nevertheless, the oxybarometer presented here, and in earlier versions (Gudmundsson & Wood, 1995;Woodland & Peltonen, 1999), relies on the assumption that the phases used in the calculation were in equilibrium.It seems likely that either the core or the rim of the garnet, being compositionally different, was not in equilibrium with the xenolith assemblage.This is most probably due to cryptic metasomatism of the garnet rim, either immediately prior to or during entrainment.

Relative effects of P and ferric ratio of garnet on calculated log f O 2
The relative effects of pressure and ferric ratio (f ¼ Fe 3þ / P Fe) of garnet on log f O2 are investigated for a particular peridotite sample, along an imposed mantle geotherm.For this example, sample Y17 from Udachnaya (Siberian Craton, Table 3) was selected.The lithospheric geotherm of 50Á4 mW m -2 from Fig. 7 was used and these conditions are represented as T ( C) ¼ 252 þ 17Á39P (kbar).Total Fe in sample Y17 garnet was maintained, but the ferric ratio f was varied from 0Á03 to 0Á12, a range typical of mantle xenolith samples (e.g.Woodland & Koch, 2003, fig. 3).Compositions of olivine and orthopyroxene were kept constant, a procedure that leads to imperceptible error.Because Fe and Mg in garnet depend on the garnet-olivine exchange equilibrium, the Fe/Mg ratio was adjusted at each temperature and pressure along the geotherm using the values of K d from O'Neill & Wood (1979).Concentrations of other elements in garnet were held constant.Figure 13 shows the results of applying the multi-reaction oxybarometry to Y17 in the range 30-60 kbar along the geotherm, for four chosen values of f.As expected, calculated values of log f O2 decrease with increasing pressure, with the smallest f yielding the most reduced  Kopylova (2004) and Creighton et al. (2009) were used with the thermobarometers of Nickel & Green (1985), Taylor (1998) and Nimis & Gru ¨tter (2010).A model conductive geotherm of 45Á2 mW m -2 has been calculated to best fit the PT estimates from the xenolith suite, along with the depth to the Moho (Bank et al., 2000) Kopylova (2004) and Creighton et al. (2009), based on the oxybarometers of Bryndzia & Wood (1990) (BW90) and Gudmundsson & Wood (1995) (GW95).Buffer reactions as in Fig. 5. values.It is well known (e.g.Woodland & Koch, 2003) that f increases with temperature (and hence pressure via the imposed geotherm) such that the deepest samples will be less reduced than shown by extrapolating along lines of constant f.The shaded region in Fig. 13 corresponds to the range in f from fig. 3 of Woodland & Koch (2003) and has a shallower slope than the constant f isopleths.In these calculations the garnet composition was not varied, except for allowing changes to the ferric iron via the parameter f and adjustment of Fe/ Mg ratios at constant Ca to satisfy garnet-olivine equilibrium.Garnet in sample Y17 contains 4Á8 wt % Cr 2 O 3 , a value slightly smaller than the median of the observed range in this study (maximum around 12 wt % Cr 2 O 3 ).Exchanging Cr 2 O 3 for Al 2 O 3 in garnet to cover the range 0-12 wt % Cr 2 O 3 was found to make less than 0Á1 log unit log f O2 difference to the results shown.
Also shown in Fig. 13 are calculated curves for three assigned values of Fe 2 O 3 for the Y17 bulk composition, using the thermodynamic model of Jennings & Holland (2015), to indicate the effects of changing the bulk ferric iron content.The constant bulk composition slopes are significantly shallower than the f isopleths and flatten off to almost constant log f O2 at high pressures.They also match the slope of the shaded region reasonably well.It is important to keep in mind the very different nature of the information in Fig. 13: the isopleths of f are based on oxybarometry on a natural sample, whereas the constant ferric bulk composition curves are predictions from a thermodynamic model using the same sample bulk composition.The measured log f O2 values for Udachnaya, as seen in Fig. 8, suggest a change from -1 to -3 log units over this pressure range, in good agreement with the shaded region in Fig. 13.A comparison with fig.14 of Jennings & Holland (2015) suggests that a mantle with a composition similar to fertile peridotite KLB-1 (Takahashi, 1986), with around 0Á2 wt % Fe 2 O 3 , can satisfactorily explain the bulk of mantle xenolith data, whereas depleted mantle peridotites may be characterized by slightly lower bulk Fe 2 O 3 contents.Values of log f O2 in natural samples appear to be closely controlled by relatively constant composition.The variation in f from 0Á03 to 0Á12 at any pressure along the geotherm in Fig. 13 leads to changes of around 2Á5 log f O2 units.

CONCLUSIONS
Assuming that all phases are in equilibrium within a spinel-and garnet-bearing peridotite, it is expected that f O2 estimates from the two independent spinel-based and garnet-based oxybarometers will give the same value.Our revision of the spinel-and garnet-based oxybarometers has changed estimates of f O2 for both oxybarometers.As detailed above, spinel-based estimates are now more reduced, whereas garnet-based estimates are more oxidized, leading to a reduced discrepancy between the two methods by around 2Á0 Dlog f O2 units.Figures 5, 6 and 8 show the effect that recalibration has had on f O2 estimates in resolving the discrepancy between spinel-and garnet-based oxybarometers.In Fig. 6 all the spinel-and garnet-bearing peridotites now cluster around the wu ¨stite-magnetite (WM) and enstatite-magnesite-olivine-diamond/graphite (EMOD/ Fig. 13.Variation of Dlog f O2 (FMQ) vs pressure along the mantle geotherm from Fig. 7 for peridotite sample Y17 from Udachnaya (Siberian Craton, Table 3) using four values of Fe 3þ / P Fe in garnet.Details are discussed in the text.The shaded field corresponds to the range in measured mantle Fe 3þ / P Fe in garnet from Woodland & Koch (2003) and the dashed lines are calculated for the Y17 bulk composition, with three Fe 2 O 3 contents using the thermodynamic model from Jennings & Holland (2015).
Fig. 14.A summary plot comparing recalibrated oxygen fugacities with earlier studies for all peridotites in this study.For spinel-bearing peridotites (circles) the oxygen fugacities are 0-1 log units more reduced than those estimated by the Bryndzia & Wood (1990, BW90) barometer; for garnet-bearing peridotites the largest change is relative to the Gudmundsson & Wood (1995, GW95) barometer (black diamonds), which shows more reduced oxygen fugacities by 0-2Á5 log units, whereas the barometer of Stagno et al. (2013, Stagno13) lies closer to the present results (grey diamonds) but is displaced to more reducing conditions by 0Á2-1Á5 log units, values that are within the combined uncertainty of measurements and barometers.
G) buffers.Further conclusions from this work are as follows.
1. Recalibration of the Gudmundsson & Wood (1995) skiagite oxygen barometer using (a) the recalculation of skiagite free energy derived from the experimental data of Woodland & O'Neill (1993) and (b) the recalibration of the garnet mixing model has shifted garnet-based oxybarometer f O2 estimates to more oxidized values by c. 0Á70-1Á5 Dlog f O2 (FMQ) units (Fig. 14).This shift to more oxidized conditions has moved f O2 estimates away from the iron-wu ¨stite (IW) buffer, and hence away from the highly reducing conditions necessary for metal saturation at the base of the lithosphere.2. Several xenoliths previously thought to have originated from the diamond stability field may have experienced more oxidizing conditions, placing them above the WM and EMOD/G buffers, where carbonate is the stable carbon phase.Known diamondiferous xenoliths from Finsch Mine remain within the diamond stability field (within error) (Fig. 10).

Revision of spinel oxybarometry in combination with
our new garnet oxybarometer calibration reduces the difference between spinel-and garnet-based f O2 estimates for mantle peridotites.The revised spinel oxybarometer now gives slightly more reduced f O2 values, by c. 0Á7 Dlog f O2 units (Fig. 14).4. Peridotites with coexisting spinel and garnet from the Vitim Volcanic Field (Baikal Rift, Russia) now show similar f O2 values for the two independent methods, but whether they should be identical remains questionable.5. Introduction of multi-reaction oxybarometry for spinel and garnet peridotites increases the robustness of the estimation process and allows assessment of the possible disequilibrium in mantle samples.6. Accurate and reliable measurement of Fe 2 O 3 , from Fe 3þ / P Fe ratios, remains the last hurdle in reliable oxygen fugacity calculation, as the amount of Fe 2 O 3 greatly affects the activity of skiagite, and hence the f O2 value calculated.

APPENDIX Calculation of reaction (4)
At 1373 K the free energy of reaction ( 4 for which DG 3 ¼ -69Á3 kJ, as explained in the text. First the Gibbs energy of skiagite is derived from reaction (3) and data from Holland & Powell (2011) tabulated below; The Gibbs energies for alm, mt, hc, q, O 2 at 1373 K are listed below, where HP11 is Holland & Powell (2011), HP90 is Holland & Powell (1990), RH is Robie & Hemingway (1995), HONA is Holmes et al. (1986).andgive (using HP11) Using the data in the older HP90 dataset yields an almost identical result (DG 4 ¼ 970Á4 kJ).This is significantly different from the value for DG 4 (981Á0 kJ) given by Woodland & O'Neill (1993) from the same starting point of DG 3 ¼ -69Á3 kJ.Repeating the calculation of Woodland & O'Neill (1993) using the data tabulated above (they used HONA for Fe & O 2 , HP90 for all other phases) yields DG 4 ¼ 967Á2 kJ, very similar to the calculation using the new Holland & Powell (2011) dataset.
As can be seen in the table above, the dataset values of Holland & Powell (1990, 2011) are in very good agreement with those of Robie & Hemingway (1995) and Holmes et al. (1986) for phases with available calorimetric data.Gudmundsson & Wood (1995) calculated the free energy of reaction (1) using the following equilibria: ).However, using the correct value for DG 4 of 969Á0 kJ in place of 981Á0 kJ changes DG 1 by 24 kJ and hence log K 1 by 0Á91 log units.

Garnet barometer equilibria
The following four reactions: have been calibrated on the basis of this work and the updated dataset of Holland & Powell (2011; see Table 1) and expressed as equations of the form reproduce the full calculations to within 100 J [or 300 J for ( 12)] over the range 0-100 kbar and 1300-1800 K.

Spinel barometer equilibria
The following three reactions: have been calibrated from the updated dataset of Holland & Powell (2011) (see Table 1) and free energies (in Joules) expressed as equations of the same form as above, and are valid for the range 0-40 kbar and 1300-1800 K.
Log f O2 values are given relative to the FMQ buffer, which is taken from O'Neill (1987) and pressurecorrected using the volumes from Holland & Powell (2011); the expression used is (with P in kbar and T in K)

Garnet mixing model
Garnets with formula (Ca,Mg,Fe,Mn) 3 (Al,Fe 3þ ,Cr) 2 Si 3 O 12 are here described with the following six independent end-members: gr (Ca 3 Al 2 Si 3 O 12 ), alm (Fe 3 Al 2 Si 3 O 12 ), py ).Five compositional variables are required to describe the variations in composition, and are taken as the site fractions on X and Y sites: x ¼ X X Fe , c ¼ X X Ca , m ¼ X X Mn , f ¼ X Y Fe 3þ , z ¼ X Y Cr .This leads to the remaining two dependent site fractions as X X Mg ¼ 1 -x -c -m, X Y Al ¼ 1 -z -f.The end-member proportions p i , are given by p py ¼ 1-x -c-m The ideal activities may be written out using mixingon-sites as and the non-ideal activity coefficients may be found from the macroscopic symmetric formalism as RT ln c a ¼ À X i X j>i ðp 0 i À p i Þðp 0 j À p j ÞW ij in which p k is the proportion of end-member, k, in the phase, p 0 k is the value of p k in pure a, and W ij is the macroscopic interaction parameter for the ij binary.The summations are over an independent set of endmembers chosen to represent the composition of the phase.So, for example, the ski activity coefficient would be RT ln c ski ¼ Àp py p alm W py;alm À p py p gr W py;gr Àp py p sps W py;sps À p py p uv W py;uv þp py ð1 À p ski ÞW py;ski À p alm p gr W alm;gr Àp alm p sps W alm;sps À p alm p uv W alm;uv þp alm ð1 À p ski ÞW alm;ski À p gr p sps W gr;sps Àp gr p uv W gr;uv þ p gr ð1 À p ski ÞW gr;ski Àp sps p uv W sps;uv þ p sps ð1 À p ski ÞW sps;ski þp uv ð1 À p ski ÞW uv;ski and, for the dependent end-member andr ( gr þ skialm), the activity may be simply determined from RT ln a andr ¼ RT ln a gr þ RT ln a ski À RT ln a alm À DG ð7Þ where DG (7) ¼53Á8 þ 0Á0017T -0Á068P kJ (see text).

Garnet mixing energies
For garnet mixing, the within-site and reciprocal energy terms used are given below (units kJ, K, kbar) including references (lower-case roman numerals) to their derivation.
The cross-site terms are determined here as follows.W CaAlFeFe3XY comes from the discussion in the text and uses our free energy of skiagite in conjunction with the data of Holland & Powell (2011).It should be noted that the cross-site Ws here have opposite sign to the free energies of the reactions in the text.W CaAlMgFe3XY is made identical to W CaAlFeFe3XY as done by Gudmundsson & Wood (1995) and is equivalent to assuming a zero energy for FeFe þ MgAl ¼ MgFe þ FeAl.This latter was estimated by Ottonello et al. (1996) as a small value, within error of zero on inspection of their other estimates.A value for W CaAlMgCrXY was determined from the updated Holland & Powell (2011, see Table 1) dataset, and that for W CaAlFeCrXY was taken to be identical.The two cross-site terms involving Mn are given much smaller values based on the fact that calderite Mn 3 Fe 3þ 2 Si 3 O 12 is stable to much lower pressures (it is found in natural blueschists) than knorringite or skiagite.Oxybarometry is not sensitive to W CaAlMgCrXY , W CaAlFeCrXY , or the Mn cross-site terms.
This set allows us to determine the complete set of macroscopic Ws presented in the text, using the relations given below.
When considering andradite, the additional W terms used in the text are defined as

Orthopyroxene mixing
Orthopyroxene non-ideality has only minimal impact on garnet oxybarometry, but affects the spinel barometers significantly, raising the typical calculated log f O2 for reaction (13) by around 0Á2 log units.Here we modify the mixing model of Green et al. (2012) and Jennings & Holland (2015), simplifying it by making it symmetric, by ignoring Fe-Mg ordering between M2 and M1 sites and by taking all non-ideal interactions as contributed only by the end-members en, fs, di, mgts.The di endmember refers to orthorhombic diopside (CaMgSi 2 O 6 ) and mgts to Mg-Tschermak pyroxene (MgAlAlSiO 6 ).
For the en and fs end-members the ideal activities are given by The powers of 1 2 (rather than two) come about because the entropy of mixing on the tetrahedral sites is taken as a quarter that of full disorder, to help mimic shortrange order between M and T sites (Green et al., 2012).Non-ideality is expressed as a regular solution, as discussed above for garnet, with the following parameters (in kJ): W mgts;di ¼ 75Á 0 -0Á 94P :

Olivine mixing
Olivine mixing is represented as in Gudmundsson & Wood (1995), with ideal activities for fo and fa given as Mg X M1 Mg and a ideal fa ¼ X M2 Fe X M1 Fe and non-ideality expressed as a regular solution, taking W fo,fa ¼ 7Á4 kJ (Gudmundsson & Wood, 1995).

V
C The Author 2016.Published by Oxford University Press.1199 et al.

Fig. 3 .
Fig.3.The effect on calculated log f O2 of varying three significant garnet W mixing parameters.(a) The calculations on reaction (1) using the calibration ofGudmundsson & Wood (1995) on the x-axis and similar calculations varying W parameters on the y-axis.Circles, W AlFe3þ changed to 2Á0 kJ; diamonds, W CaAlMgFe3XY changed by þ23 kJ; squares, W CaAlFeFe3XY changed by þ23 kJ; crosses, all three changes.(b) The equivalent calculations on reaction (2) using the calibration ofStagno et al. (2013).Symbols as for (a).[Note the cumulative effects of W CaAlMgFe3XY and W CaAlFeFe3XY for reaction (2) as opposed to the compensating effects in reaction (1).]

Fig. 2 .
Fig. 2. Calculated log f O2 compared with experiments.(a) The experiments of Gudmundsson & Wood (1995) (GW) and of Stagno et al. (2013) (Stag).Upward-facing triangles are for reaction (1) and downward-facing triangles are for reaction (2).(b)The same data but for the least squares calculation using reactions (1), (2), (11) and (12).The least squares results are less scattered than those for each of the barometers and in better agreement with the experiments.(c) Comparison of calculations using the new calibrations against the original result ofGudmundsson & Wood (1995) (GW) and that ofStagno et al.  (2013) (Stag); this shows a consistent 1-1Á5 log unit offset.
Fig.4.Plot of pressure vs temperature estimates for off-craton Vitim peridotites from the Baikal Rift zone, using mineral chemistry fromIonov & Wood (1992) andGoncharov & Ionov  (2012), together with the thermobarometers ofNickel & Green (1985),Taylor (1998) andNimis & Taylor (2000).A model conductive geotherm of 61Á4 mW m -2 has been calculated to best fit the PT estimates for garnet-bearing samples in the xenolith suite, using the depth to the Moho ofSuvorov et al. (2002) and the program described byMather et al. (2011).The diamondgraphite transition(Kennedy & Kennedy, 1976) is indicated, along with a greyscale indication of the Dlog f O2 (FMQ) value for each xenolith.The data indicate that the base of the mechanical boundary layer (MBL) under the Vitim Volcanic Field is relatively shallow at 82 km (26Á5 kbar), which is consistent with a previous estimate based on the composition of the host lavas (85 km; Johnson et al. 2005).

Fig. 7 .
Fig. 7. Pressure vs temperature estimates for Udachnaya peridotites (Siberian Craton).Mineral analyses used for PT estimation are from Goncharov et al. (2012) using the thermobarometers of Nickel & Green (1985), Taylor (1998), Nimis & Taylor (2000) and Nimis & Gru ¨tter (2010).A model conductive geotherm of 50Á4 mW m -2 was calculated to best fit the PT estimates from the xenolith suite, along with the depth to the Moho (Suvorov et al., 2006) using the program described by Mather et al. (2011).The estimated mechanical boundary layer (MBL) thickness is 202 km.The diamond-graphite transition pressure (Kennedy & Kennedy, 1976) is indicated, along with a greyscale indication of the Dlog f O2 (FMQ) value for each xenolith.

Fig. 11 .
Fig. 11.Pressure vs temperature estimates for Diavik peridotites (Slave Craton).Mineral analyses from McCammon & Kopylova (2004) and Creighton et al. (2009) were used with the thermobarometers of Nickel & Green (1985), Taylor (1998) and Nimis & Gru ¨tter (2010).A model conductive geotherm of 45Á2 mW m -2 has been calculated to best fit the PT estimates from the xenolith suite, along with the depth to the Moho (Bank et al., 2000) using the program described by Mather et al. (2011).The estimated thickness of the mechanical boundary layer (MBL) beneath the central Slave Craton is 211 km.The diamond-graphite transition pressure (Kennedy & Kennedy, 1976) is indicated, along with a greyscale indication of the Dlog f O2 (FMQ) value for each xenolith.It should be noted that McCammon & Kopylova (2004) assumed a pressure of 30 kbar for the spinel-only peridotites.
Fig. 11.Pressure vs temperature estimates for Diavik peridotites (Slave Craton).Mineral analyses from McCammon & Kopylova (2004) and Creighton et al. (2009) were used with the thermobarometers of Nickel & Green (1985), Taylor (1998) and Nimis & Gru ¨tter (2010).A model conductive geotherm of 45Á2 mW m -2 has been calculated to best fit the PT estimates from the xenolith suite, along with the depth to the Moho (Bank et al., 2000) using the program described by Mather et al. (2011).The estimated thickness of the mechanical boundary layer (MBL) beneath the central Slave Craton is 211 km.The diamond-graphite transition pressure (Kennedy & Kennedy, 1976) is indicated, along with a greyscale indication of the Dlog f O2 (FMQ) value for each xenolith.It should be noted that McCammon & Kopylova (2004) assumed a pressure of 30 kbar for the spinel-only peridotites.

Table 1 .
Thermodynamic data used in this study

Table 2 .
Example output from program GtfO2; sample F5, for P ¼ 54Á4 kbar and T ¼ 1150 C

Table 3 .
Pressure, temperature and oxygen fugacity estimates for mantle xenoliths from the Baikal Rift, Siberian Craton, Kaapvaal Craton, and Slave Craton