Revisiting the electron microprobe method of spinel-olivine-orthopyroxene oxybarometry applied to spinel peridotites

Samples analyzed

We analyzed 32 spinel samples, kindly provided by B. Wood, for major and minor elements by electron microprobe with the goal of determining Fe³⁺/∑Fe ratios. These spinels, which we refer to collectively as the “Wood spinels,” were previously examined by Wood and Virgo (1989), Bryndzia and Wood (1990), and Ionov and Wood (1992). Each has been previously analyzed by Mössbauer spectroscopy, which provides an independent estimate of the Fe³⁺/∑Fe ratio. The Wood spinels are separates from peridotites representing a diversity of major element compositions (Cr# = 0.04−0.57), Fe oxidation states (Mössbauer Fe³⁺/∑Fe = 0.058−0.32), and geological environments (continental and arc peridotite xenoliths, abyssal peridotites).

Four spinel peridotite samples from Hawaii were analyzed to test the individual contributions of each mineral phase (e.g., olivine and orthopyroxene in addition to spinel) to the total uncertainty in the f_O2 calculation. The Hawaiian samples are spinel lherzolite xenoliths from Salt Lake Crater, Oahu, originally collected by E. Dale Jackson and now part of the National Rock and Ore Collection at the Smithsonian Institution National Museum of Natural History. We also analyzed spinels from a harzburgite (BMRG08-98-2-2) dredged from the Tonga trench during the 1996 Boomerang cruise (Bloomer et al. 1996; Wright et al. 2000); the spinel in this sample has Cr# and MgO concentrations that depart significantly from the trend of the spinels used to correct Fe³⁺/∑Fe ratios.

Analytical methods

We analyzed spinel, olivine, and orthopyroxene at the Smithsonian Institution using a JEOL 8900 superprobe with five wavelength-dispersive spectrometers (WDS). Table 1 provides information on our primary standards, count times, and detector crystals, and additional information about the electron microprobe analyses is given in the supplementary Material^[1].

We analyzed spinels in two different types of analytical sessions. First, we analyzed the 32 Wood spinels without correcting Fe³⁺/∑Fe ratios (sessions S1-S3, Table^[1]. 2 and supplementary^[1]. Table S1) to determine the range of compositions present, to reveal compositional systematics, and to check for intra- and intergranular heterogeneity of the samples. Out of the 32 Wood spinels, we chose 7 as correction standards (hereafter, the correction set; selection criteria given in Results). The correction set spinels are the standards that we use to determine the Fe³⁺/∑Fe correction to be applied to a given analytical session. In this second type of analytical session the correction set was analyzed along with another subset of the Wood spinels treated as unknowns (hereafter, the validation set) and the Tongan and Hawaiian spinels (sessions A1–A4 and B1–B4, supplementary^[1]. Table S2).

In sessions S1–S3, we analyzed one to six individual grains of each of the Wood spinels. From each grain, we analyzed 3–10 points, depending upon the total number of grains analyzed (supplementary^[1]. Table S1), except for the samples made from crushed Mössbauer powders. We collected only one analytical point on each grain of these powdered samples. We analyzed secondary standards of chromite, Cr-augite, and hypersthene (Jarosewich et al. 1980, 1987) every 2–4 h during the session to monitor instrumental drift.

In sessions A1–A4 and B1–B4, we collected three analytical points on each sample in the correction set both before and after analyzing unknowns, corresponding to re-analysis of the correction standards after 12–14 h. We analyzed secondary standards at regular intervals as described above. In these sessions, individual analyses were discarded when totals fell outside the range 97–101%; this range is asymmetrical around 100% because we expect samples with high Fe³⁺ to give relatively low totals when total Fe is calculated as FeO. We also excluded individual analyses that contained SiO₂ > 0.3 wt% to avoid analyses that may have sampled surrounding silicate material. To ensure that each session could be considered separately in terms of intersession reproducibility, the filament was turned down and allowed to cool for at least 12 h and then saturated again at the start of each new session, even when these sessions occurred on consecutive days.

We analyzed the major-element compositions of olivine and orthopyroxene from the four Hawaiian xenoliths in a single analytical session by analyzing the core compositions of 10 different grains of each mineral in each sample (Table 3).

The spinel Fe³⁺/ΣFe ratio correction method

We applied the W&V89 correction to spinels analyzed in sessions A1–A4 and B1–B4. We calculated Fe³⁺/∑Fe ratio and Cr# of each of the correction set spinels measured both at the beginning and end of the analytical session. We calculated the Fe³⁺/∑Fe ratios of the spinels by normalizing the spinel cation proportions to 3 total cations, treating all Fe as Fe²⁺, and then adjusting the Fe³⁺/Fe²⁺ ratio to balance the charge deficiency or excess (Stormer 1983). Occasionally, spinels with low Fe³⁺/∑Fe ratios gave a small positive charge excess, which was balanced by allowing negative contributions to Fe³⁺.

We calculated ΔFe³⁺/∑Fe^Möss-EPMA for each measurement and determined the best fit line through all measurements of the correction set to determine slope and intercept (Wood and Virgo 1989):

We used the resulting slope and intercept to correct the calculated Fe³⁺/∑Fe ratios of all other spinels measured during that session. To maintain consistency in data processing, we applied this correction irrespective of whether AFe³⁺/∑Fe^Moss-EPMA and Cr# were strongly correlated. We explain the rationale for this procedure in the discussion.

Compositions of the wood spinels and selection of correction and validation standards

We present the uncorrected EPMA analyses of the 32 Wood spinels in Table 2^[1]. We used these measurements to look for compositional systematics in the entire set of Wood spinels and to select the samples for the correction and validation sets. Figure 1 shows uncorrected Fe³⁺/∑Fe ratios determined by EPMA compared to Fe³⁺/∑Fe ratios determined by Mössbauer (Wood and Virgo 1989; Bryndzia and Wood 1990; Ionov and Wood 1992). The uncorrected Fe³⁺/∑Fe ratios determined by EPMA analysis correlate with the Fe³⁺/∑Fe ratios determined by Mössbauer spectroscopy (r² = 0.86). The whole data set fits the 1:1 line well; however, closer inspection reveals that agreement between the two methods varies from session to session. For example, analyses from session S1 plot consistently above the 1:1 line, and analyses from session S3 plot below. This underscores the assessment of Wood and Virgo (1989) that EPMA is sufficiently precise, but insufficiently accurate to be used without correction. Among the Wood spinels, we find no systematic relationship in the sample average compositions between the Cr# of the spinels and their Fe³⁺/∑Fe ratio, but Cr# and MgO (r² = 0.93) are negatively correlated (Fig. 2).

Figure 1

Uncorrected electron microprobe analyses of Wood spinels from sessions S1–S3. Sample-average uncorrected Fe³⁺/∑Fe ratios determined by EPMA in sessions S1–S3 (Table 2) plotted against Fe³⁺/∑Fe ratios determined by Mössbauer spectroscopy. Vertical bars show the range of compositions for a given sample across all grains measured (supplementary^[1]. Table S1), indicating the degree of intergranular heterogeneity exhibited by a sample. Circles represent samples chosen for the correction set, and triangles represent samples chosen for the validation set. All other Wood spinels are represented with diamonds.

Figure 2

Compositional range of spinels included in the correction and validation sets. Fe³⁺/∑Fe ratios by Mössbauer are from Wood and Virgo (1989), Bryndzia and Wood (1990), and Ionov and Wood (1992). MgO and Cr# of the correction set and validation set are EPMA measurements from this study (Table 2), while values for the correction standards used by Wood and Virgo (1989) are as reported in that study. The correction set used in this study spans a similar range of Cr# and MgO (b) as the correction standards used by Wood and Virgo (1989), and Fe³⁺/∑Fe ratios span a larger range (a). Taking all of these data together, Fe³⁺/∑Fe ratio is not correlated with Cr# (r² = 0.003) or MgO (not shown, r² < 0.001), and MgO and Cr# are highly correlated (r² = 0.93).

From the Wood spinels, we selected seven samples for the cor-rection set and six samples for the validation set. These samples are indicated in Table 2^[1]. The selection criteria for the correction and validation sets are given in the supplementary^[1] Material.

Uncorrected and corrected Fe³⁺/∑Fe ratios of the validation set spinels

Compositions of the validation set spinels were determined by averaging the replicate analyses from analytical sessions A1–A4 and B1–B4 (Table 4^[1]). Full results of all analytical sessions are presented in the supplementary^[1] Material (supplementary^[1] Table S2). Figure 3 shows several illustrative examples of uncorrected and corrected Fe³⁺/∑Fe ratios of the correction and validation set spinels compared with Fe³⁺/∑Fe ratios determined by Mössbauer, and Figure 4 shows the relationship between ΔFe³⁺/∑Fe^Möss-EPMA and Cr# from these sessions. Below, we discuss the implications of these results for the accuracy and precision of spinel Fe³⁺/∑Fe ratios determined by EPMA.

Figure 3

Examples of the W&V89 correction applied to spinels from several independent analytical sessions. The first column (a, c, and e) shows uncorrected Fe³⁺/∑Fe ratios by EPMA of the correction set spinels measured at the start of each session and at the end of each session, along with validation set spinels analyzed in between. These are plotted against their published Fe³⁺/∑Fe ratios measured by Mössbauer. The second column (b, d, and f) shows the same measurements after correction of the EPMA data using the Wood and Virgo (1989) method. Uncorrected EPMA analyses from session A1 plot around the 1:1 line (a), and the W&V89 correction causes only imperceptible changes to the corrected Fe³⁺/∑Fe ratios (b). Uncorrected EPMA analyses from session B3 (c and d) are offset from the 1:1 line and display a relatively high degree of scatter around the trend with Mössbauer data (c). The W&V89 correction decreases scatter in the data and shifts it upward so that the corrected data lie on the 1:1 line (d). Uncorrected EPMA analyses from session B4 (e and f) are offset from the 1:1 line but are relatively tightly clustered along a linear trend with the Mössbauer data (e). The W&V89 correction shifts Fe³⁺/∑Fe ratios onto the 1:1 line (f).

Accuracy and precision of the correction method

Since the Wood group presented their correction method and measurements of spinel peridotite f_O2 (Wood and Virgo 1989; Bryndzia and Wood 1990; Ionov and Wood 1992), the application of this correction method to spinel peridotite fO₂ studies has been sporadic. Some groups adopted the Wood and Virgo (1989) approach (e.g., Ionov and Wood 1992; Woodland et al. 1992; Luhr and Aranda-Gómez 1997; Parkinson and Pearce 1998; Parkinson and Arculus 1999; Bryant et al. 2007; Wang et al. 2007, 2008; Dare et al. 2009), while others presented data with no correction (e.g., Ballhaus 1993; Qi et al. 1995; Fedortchouk et al. 2005; Canil et al. 2006; Foley et al. 2006; Nasir et al. 2010; Wang et al. 2012).

Ballhaus et al. (1991) questioned whether Fe³⁺/∑Fe ratios determined from EPMA require correction, and additionally suggested that such a correction may introduce additional error. Their argument was based in part on a compilation of logfO₂ calculated from analyses of spinels in peridotites and basalts, including spinel Fe³⁺/∑Fe ratios that had been measured both by Mössbauer spectroscopy and EPMA. They found that differences in calculated f_O2 seldom varied by greater than 0.4 log units and concluded that correcting EPMA data was unnecessary; however, plotting calculated logf_O2 rather than spinel Fe³⁺/∑Fe ratios disguises the true effects of the uncertainty in the EPMA measurements. Uncertainty in spinel Fe³⁺/∑Fe ratio has a decreasing influence on calculated f_O2 as Fe³⁺/∑Fe ratio increases (Ballhaus et al. 1991; Parkinson and Arculus 1999). Plotting Fe³⁺/∑Fe ratios determined by EPMA and Mössbauer across several studies of natural spinels (Fig. 5) shows that disagreement between the two methods can be substantial. In aggregate, the uncorrected Fe³⁺/∑Fe ratios from previous studies (Fig. 5a) are offset to low Fe³⁺/∑Fe ratios compared to Mössbauer (ΔFe³⁺/∑Fe^Möss-EPMA = 0.022 ± 0.049, 1σ). This may indicate a common analytical bias between laboratories. Bias to low values of uncorrected Fe³⁺/∑Fe ratios could be caused by an overestimate of cations with valence ≥3 or an underestimate of cations with valence ≤2. Possible sources of this bias include treating all Cr as trivalent when a significant fraction may be divalent (Lucas et al. 1988) and omission of divalent minor cations, such as Zn, from the analysis. Corrected Fe³⁺/∑Fe ratios from these same studies are in closer agreement with Mössbauer Fe³⁺/∑Fe ratios and are more evenly distributed around the linear trend between the two measurements (ΔFe³⁺/∑Fe^Möss-EPMA = −0.007 ± 0.021, 1σ; Fig. 5b).

Figure 4

Relationship between Cr# and ΔFe³⁺/∑Fe^Möss-EPMA in uncorrected analyses of the correction set spinels. Cr# and ΔFe³⁺/∑Fe^Möss-EPMA are the measured parameters that contribute directly to the W&V89 correction and the same analytical sessions are shown as in Figure 3. In session A1 (a), Cr# and ΔFe³⁺/∑Fe^Möss-EPMA are uncorrelated (r² = 0.01) and ΔFe³⁺/∑Fe^Möss-EPMA is near zero, so the W&V89 correction makes negligible adjustments to the Fe³⁺/∑Fe ratio, as expected given that the uncorrected data already overlapped the Mössbauer values. In session B3 (b), Cr# and ΔFe³⁺/∑Fe^Möss-EPMA are correlated (r² = 0.81), with slope and intercept both significantly different from zero. The W&V89 correction shifts Fe³⁺/∑Fe ratios of all samples upward and Cr-poor spinels are adjusted more than Cnrich spinels. In session B4 (c), Cr# and ΔFe³⁺/∑Fe^Möss-EPMA are poorly correlated (r² = 0.10) with slope near zero but an intercept significantly different from zero; consequently, the W&V89 correction shifts all Fe³⁺/∑Fe ratios upward by a nearly constant correction factor.

Figure 5

Literature compilation of spinel Fe³⁺/∑Fe ratios measured by Mössbauer spectroscopy and calculated from EPMA. Uncorrected spinel Fe³⁺/∑Fe ratios calculated from EPMA analyses of natural peridotite- and basalt-hosted spinels plotted against Fe³⁺/∑Fe ratios of the same spinels analyzed by Mössbauer spectroscopy (a). Uncorrected spinel Fe³⁺/∑Fe ratios by EPMA are biased to low Fe³⁺/∑Fe, with a mean ΔFe³⁺/∑Fe^Möss-EPMA of 0.022 ± 0.049 (1σ). After correction by the W&V89 method, Fe³⁺/LFe ratios deviate less from the 1:1 line and are more evenly distributed around it (b), with a mean ΔFe³⁺/∑Fe^Möss-EPMAof −0.007 ± 0.021 (1σ).

Ballhaus et al. (1991) also argued that EPMA analyses of spinel should be corrected only if non-stoichiometry is suspected. But the Wood and Virgo (1989) correction does not imply nonstoichiometry of the spinel sample; it corrects for error in the determination of Fe³⁺/∑Fe ratios that results from forcing an imperfectly analyzed composition into a perfect stoichiometric calculation. The corrected compositions in this study are still stoichiometric (supplementary^[1] Table S2) with total cations between 2.99 and 3.01 when calculated on a four oxygen basis.

We tested the accuracy and precision of the W&V89 correction using our replicate analyses of spinels from the validation set, BMRG08-98-2-2, and Hawaiian xenoliths analyzed in sessions A1–A4 and B1–B4 (supplementary^[1]. Table S2, Figs. 3 and 4). If we assume that the Mössbauer analyses of the Wood spinels accurately represent their Fe³⁺/∑Fe ratios, then we can use the validation set to demonstrate that the W&V89 correction improves the accuracy of Fe³⁺/∑Fe ratios determined by EPMA. Figure 6 shows the averages and 1σ ranges of both corrected and uncorrected Fe³⁺/∑Fe ratios of the validation set spinels measured across all analytical sessions (Table 4). Corrected Fe³⁺/∑Fe ratios are distributed closely around the 1:1 line with an average offset of +0.004 from the Mössbauer Fe³⁺/∑Fe ratios. The uncorrected Fe³⁺/∑Fe ratios of these spinels are offset below the 1:1 line by an average of −0.031 from the Mössbauer Fe³⁺/∑Fe ratios. Similar to the offset in the literature data described above (Fig. 5a).

Figure 6

Mean Fe³⁺/ΣFe ratios of the validation set spinels measured by Mössbauer spectroscopy and calculated from EPMA. Mean uncorrected (a) and corrected (b) Fe³⁺/ΣFe ratios by EPMA (Table 4) were calculated by taking the unweighted average of the mean Fe³⁺/ΣFe ratios of all analytical sessions (A1–A4 and B1–B4; Supplementary1 Table S2). Error bars are 2 st.dev.

We can assess the improvement in precision of Fe³⁺/∑Fe ratios achieved by using the W&V89 correction by quantifying the intersession variability of Fe³⁺∑Fe ratios that we calculated for the validation set spinels. Values of 1σ around mean uncorrected Fe³⁺∑Fe ratios of the validation set spinels vary from ±0.029 to ±0.065 (Table 4) with a mean of ±0.049. Values of 1σ around mean corrected Fe³⁺∑Fe ratios of the validation set spinels vary from ±0.012 to ±0.032 (Table 4) with a mean of ±0.023, which suggests greater than a factor of two increase in precision when using the W&V89 correction. The variations in 1σ around these averages are not random. Intersession variability in corrected and uncorrected Fe³⁺/∑Fe ratios is greater in samples with lower total Fe. The stoichiometric calculation produces a concentration of Fe³⁺ with uncertainty resulting from the accumulated analytical errors from each element propagated through the stoichiometric calculation. When this error is propagated through the calculation of the Fe³⁺/∑Fe ratio, it scales with the total concentration of Fe. We approximate errors on corrected Fe³⁺/∑Fe ratios by dividing the error on Fe³⁺ by the amount of total Fe. We demonstrate this relationship by plotting the magnitude of 1σ variations in corrected Fe³⁺/∑Fe ratios from validation set and Hawaiian spinels as a function of the inverse of total Fe per 3 formula cations (Fig. 7a). A line fit through the origin gives the following relationship:

where 1σ₃ describes the magnitude of 1 st.dev. around a corrected spinel Fe³⁺/∑Fe ratio, 0.006 is the magnitude of 1σ variation around the mean molar Fe³⁺ concentration per 3 formula cations, and X_∑Fe is the molar concentration of total Fe per 3 formula cations. This relationship describes an error envelope for corrected Fe³⁺/∑Fe ratios that depends on the total concentration of Fe in the spinel. Figure 7b shows the error envelope with measurements of Fe³⁺/∑Fe ratios from the validation set, Hawaiian, and Tongan spinels presented as deviations from the mean. Equation 3 describes the precision of the EPMA method of determining Fe³⁺/∑Fe ratios when using the W&V89 correction.

Figure 7

Relationship between analytical precision of spinel Fe³⁺/ΣFe ratios and total concentration of Fe for spinels from the validation set, Hawaiian xenoliths and Tonga. Magnitude of 1 st.dev. in corrected Fe³⁺/ΣFe ratios measured across all sessions (A1–A4 and B1–B4, Supplementary1 Table S2) for validation set and Hawaiian spinels as a function of the inverse of the multi-session average total Fe concentration on a 3 cation basis (a). In black is the best fit line through the origin (r² = 0.57). Tonga sample BMRG08-98-2-2 was not included in this fit because it does not plot near the global trend in Cr#-MgO (Fig. 8). The equation for this line is given in the text (Eq. 3), and we use it to calculate precision in our corrected measurements of Fe³⁺/ΣFe ratios of unknown spinel samples. Deviations of each session average (Supplementary1 Table S2) from their respective multisession means (Table 4) plotted as a function of total Fe concentration on a 3 cation basis (b). The 1σ error envelope is calculated using Equation 3.

It is likely that the accuracy of uncorrected Fe³⁺/∑Fe ratios and the precision of Fe³⁺/∑Fe ratios corrected using the W&V89 method can be improved further by analyzing additional minor elements. As we described above, our uncorrected spinel Fe³⁺/∑Fe ratios measured by EPMA and those in previous studies skew low compared to Mössbauer Fe³⁺/∑Fe ratios, consistent with under-sampling of divalent cations. We examined the spinel data set hosted in the GEOROC database (http://georoc.mpch-mainz.gwdg.de/georoc/, accessed 30 June 2016). In 139 peridotite-hosted spinels (those with host rock listed as peridotite, lherzolite, harzburgite, or dunite) with both major and trace firstrow transition elements given, only three transition elements have average concentrations in excess of 100 ppm: V (620 ± 390), Co (270 ± 220), and Zn (960 ± 690). Forty-two of these samples have measured V, Co, and Zn, and in that subset V is positively correlated with both Co (r² = 0.54) and Zn (r² = 0.33; supplementary^[1]. Fig. S1). In all but 4 of these spinel samples (Co+Zn) is greater than V; therefore, it is likely that by excluding these elements from the analysis, we have introduced a bias that could lead to uncorrected Fe³⁺/∑Fe ratios that are underestimated by 0.003 to 0.013 (see supplementary^[1]. Materials for calculations). The W&V89 correction accounts for the systematic underestimation of Fe³⁺/∑Fe, but variations in V, Co, and Zn in standard and unknown spinels may lead to diminished precision of the corrected Fe³⁺/∑Fe ratios on the order of 0.01 if these elements are not analyzed.

Underlying mechanics of the wood and virgo correction

Although the W&V89 correction demonstrably improves agreement between measurements of spinel Fe³⁺/∑Fe ratios by Mössbauer and EPMA, no studies by the Wood group nor any subsequent studies have explained how the method works in detail. Wood and Virgo (1989) report that ΔFe³⁺/∑Fe^Möss-EPMA is generally linearly related to Cr#; they suggest a simple linear relationship can be determined in each EPMA analytical session by comparing EPMA analyses of a set of spinel standards that have been characterized by Mössbauer. We explore two complications in this section: (1) How should correction proceed when this linear relationship is not observed, even when agreement between EPMA and Mössbauer measurements of the standards is poor? (2) How does this correction, based only on two measured elements, work to correct a multi-element analysis? The first question was raised by Luhr and Aranda-Gómez (1997), who reported that their EPMA measurments of spinel ΔFe³⁺/∑Fe^Möss-EPMA and Cr# were not correlated. We also do not observe a correlation between ΔFe³⁺/∑Fe^Möss-EPMA and Cr# in some of our analytical sessions (e.g., session B4, Fig. 4). The second question was addressed briefly by Wood and Virgo (1989), who suggested that Al was subject to greater systematic errors than other elements, at least in their own data set; therefore a correction relying on Cr# addressed the greatest source of uncertainty. As we demonstrate below, systematic error of any element in the EPMA analysis can lead to increases in the magnitude of AFe³⁺/∑Fe^Möss-EPMA, but the W&V89 correction can still correct for these biases. That is, the correction works, even when bias is introduced by elements other than Cr or Al, largely because of a correlation between Cr# and MgO in the global spinel peridotite data set (Fig. 8); however, this means that the correction may be less effective for samples that fall off the MgO-Cr# trend. This aspect of the W&V89 correction needs to be accounted for by workers analyzing off-trend spinels, either by adjusting estimates of uncertainty or by choosing correction standards that are compositionally relevant to their particular sample set.

Biases in EPMA can lead to two different effects on calculated Fe³⁺/∑Fe ratios of a group of spinels for which independent Mössbauer Fe³⁺/∑Fe ratios are available. The whole set of EPMA-derived Fe³⁺/∑Fe ratios may be offset from the EPMA-Mössbauer 1:1 line. Alternatively, scatter around the linear trend between Mössbauer and EPMA Fe³⁺/∑Fe ratios may increase as spinels with different compositions are variably affected by measurement bias. These two effects are not mutually exclusive.

When the correction set yields a constant offset (e.g., sessions B1 and B4, Fig. 3e), uncorrected Fe³⁺/ΣFe ratios of the correction and validation set spinels plot mostly below the 1:1 line and ΔFe³⁺/ΣFe^Möss-EPMA and Cr# are also uncorrelated (r² ≤ 0.10; Fig. 4c). In such instances, EPMA-derived Fe³⁺/ΣFe ratios display systematic bias, but it is not clear that a correction scheme that relies on a correlation between ΔFe³⁺/ΣFe^Möss-EPMA and Cr# should be applied when those parameters are uncorrelated. Luhr and Arranda-G ómez (1997) observed just such a scenario when they attempted to apply the W&V89 correction to their own spinel peridotite xenolith suite. They chose not to apply the W&V89 correction and instead determined the average value of ΔFe³⁺/ΣFe^Möss-EPMA for all their correction standards, which did not vary with Cr#. They then made a single, constant-value adjustment to the calculated Fe³⁺/ΣFe ratio of all samples run during that session. This method of correction can improve Fe³⁺/ΣFe ratio accuracy in the case where the offset between EPMA and Mössbauer is roughly constant for all samples; however, devising a separate correction for such a case is unnecessary. The W&V89 correction contains this same functionality: when ΔFe³⁺/ΣFe_Möss-EPMA and Cr# are uncorrelated, the slope of the best-fit line will be approximately zero, and the W&V89 correction functions as a constant Fe³⁺/ΣFe ratio offset correction. The two correction methods are, in effect, equivalent.

The W&V89 correction is valuable because it addresses the differential effects of measurement bias on spinels of variable composition. Hence, the correction can also decrease scatter in EPMA measurements of Fe³⁺/ΣFe ratios when these data show large deviations from the linear trend with Fe³⁺/ΣFe ratios by Mössbauer. Sessions A3 (Figs. 3a and 3b) and B3 are examples of this effect. The initial analyzed. Wood and Virgo (1989) explain their choice of Cr# as the compositional parameter for correcting Fe³⁺ΣFe ratios because, among the major elements, Al₂O₃ concentrations determined by EPMA were the most affected by the choice of matrix correction schemes. This explanation does not account for systematic biases of elements other than Al and Cr that may result from imperfect analysis of primary standards.

MgO is a major element in peridotitic spinels and is just as likely as Al or Cr to be the root of systematic offsets of calculated Fe³⁺ΣFe ratios from ideality. Interestingly, the W&V89 correction is able to correct for bias in the MgO analysis because MgO and Cr# are correlated in the spinels studied by the Wood group (Wood and Virgo 1989; Bryndzia and Wood 1990; Ionov and Wood 1992) and in peridotitic spinels globally (Figs. 2 and 8). The correlation between MgO and Cr# in peridotitic spinels is an expected consequence of the dependence of the Fe²⁺-Mg exchange coefficient between olivine and spinel on the Cr concentration in the spinel (Irvine 1965; Wood and Nicholls 1978). This relationship in the spinel data increases the effectiveness of the W&V89 correction when unknown spinel samples overlap with the compositional range of the correction standards, but the correction may not be as effective for samples that fall significantly off that compositional trend. In the Supplementary ^[1] Materials, we present calculations that demonstrate how the W&V89 correction improves the accuracy of spinel Fe³⁺/ΣFe ratios when elements other than Cr and Al are biased during the analysis. These calculations also demonstrate that corrected Fe³⁺/ΣFe ratios from spinels that depart from the MgO-Cr# trend of the correction set, such as Tonga sample BMRG08-98-2-2, are subject to degraded precision.

In summary, the W&V89 correction can effectively correct for systematic biases in EPMA derived Fe³⁺/ΣFe ratios, even when the correlation between ΔFe³⁺/ΣFe_Möss-EPMA and Cr# is weak or absent. The W&V89 method also corrects for bias in measured elements other than Al₂O₃ and Cr₂O₃, due predominantly to the correlation of spinel Cr# with MgO. Precision of corrected Fe³⁺/ΣFe ratios decreases for samples that fall outside the compositional range of the correction standards used, but accuracy is no worse than if the correction had not been applied. This effect could be mitigated by choosing a different set of correction standards that are compositionally similar to the unknowns being analyzed.

Effect of the matrix correction scheme

Wood and Virgo (1989) investigated the effect of the choice of matrix correction schemes on EPMA measurements of spinels. They found that different matrix correction schemes led to systematic differences in the calculated magnetite activity in spinel compositions before applying the W&V89 Fe³⁺/ΣFe correction. We have reprocessed the analyses from analytical sessions A1-A4 using the PAP matrix correction (Pouchou and Pichoir 1986; Supplementary^[1] Table S3), also considered by Wood and Virgo (1989). We find that, compared to the ZAF correction we have used throughout our study, the PAP correction results in systematically lower Al₂O₃ (averaging 1.4 rel%), FeO (0.4%), and MgO (0.6%) and systematically higher Cr₂O₃ (0.7%). Uncorrected Fe³⁺/ΣFe ratios determined using the PAP procedure are 0.001 to 0.015 lower than those ratios determined using the ZAF procedure, and the difference in Fe³⁺/ΣFe ratio by these two matrix corrections is correlated with Cr# (r₂ = 0.975). Despite this systematic offset, the magnitude of the difference in uncorrected Fe³⁺/ΣFe ratios is small, even in Al-rich samples, compared to the variation caused by session-to-session differences in the primary standardization. After correcting the Fe³⁺/ΣFe ratios using the W&V89 method, the PAP and ZAF procedures yield differences in Fe³⁺/ΣFe (ZAF-PAP) between -0.002 and 0.001. When the W&V89 correction is used, effects of the matrix correction on the Fe³⁺/ΣFe ratio are negligible.

Calculation of f_O2 from the analyses of spinel, olivine, and orthopyroxene

Ultimately, the goal of determining the Fe³⁺/ΣFe ratios of spinels from peridotites is to estimate the f_O2 of equilibration. We calculate f_O2 following Mattioli and Wood (1988) and Wood and Virgo (1989) using the following equation for logf_O2:

where P is pressure in bars, T is temperature in K, Mg#=XMg0l/(XMg0l,XFe01);andXMg0l,XFe01 are the mole fractions of Mg and Fe in olivine; XFeM1,XFeM2X_M1, X_Me2 are the mole fractions of Fe in the two orthopyroxene octahedral sites calculated following Wood and Banno (1973); and aFe3O4Sp1 is the activity of the magnetite component in spinel. We discuss the calculation of aFe3O4Sp1 in the following section. Temperature and pressure are required to calculate f_O2. We calculate temperature using the spinel-olivine Fe-Mg exchange thermometer of Li et al. (1995) and, unless otherwise specified, we follow Bryndzia and Wood (1990) and Wood et al. (1990) in assuming a pressure of 1.5 GPa.

Equation 4 does not appear in the above form in any of the Wood group papers, and there is some confusion in the literature about how different versions of the formula arose. Commonly, the version given in Wood et al. (1990) and Wood (1991) is cited, which includes a term for thef_O2 of the quartz-fayalite-magnetite (QFM) buffer:

As discussed by Herd (2008), this version of the f_O2 equation requires the QFM formulation of Myers and Eugster (1983):

(Subtracting Eq. 6 from the first two terms in Eq. 4 yields the second two terms in Eq. 5.) Herd (2008) also suggests that the proper way to calculate absolute f_O2 from Equation 5 is to use Equation 6 as written, without a pressure term, to calculate log f_O2(QFM)P,T. This is incorrect because the pressure term in Equation 5 is derived from the difference between the pressure dependences of the spinel-olivine-orthopyroxene buffer and QFM. Mattioli and Wood (1988) describe a method by which the pressure dependence of the spinel-olivine-orthopyroxene buffer can be approximated from standard state molar volumes of the phases by assuming Mg# = 0.90 in each of the silicates and a magnetite proportion of 0.02 in spinel. The resulting coefficient [ΔV/(2.303 ·R) = 0.0567] appears in both Wood and Virgo (1989) and Bryndzia and Wood (1990). From this, we can determine the P/T coefficient in Equation 5 by subtracting the pressure dependence of the QFM buffer [ΔV/(2.303 ·R) = 0.0936, using the standard state molar volumes given in Robie et al. (1995)] from Equation 4.

Using Equation 6 to calculate logfO₂(QFM)_P,T for use in Equation 5 with no pressure term leads to a 0.6 log unit underestimation of absolute f_O2 at 1150 °C and 1 GPa. Substituting a parameterization of QFM into Equation 5 other than that of Myers and Eugster (1983) leads to systematic errors in f_O2. For example, replacement with O’Neill (1987) results in a 0.15 log unit underestimation of absolute f_O2 at 1150 °C and 1 GPa. We avoid this confusion by using Equation 4 to calculate log(f_O2)_P,T before calculating f_O2 relative to the QFM reference buffer. Except where otherwise indicated, we report oxygen fugacity relative to QFM using the parameterization of Frost (1991):

We calculate magnetite activity in spinel, aFe3O4Sp1 using the MELTS Supplemental Calculator (Sack and Ghiorso 1991a, 1991b; http://melts.ofm-research.org/CalcForms/index.html) with a slight modification to the calculation of spinel components. We describe this modification and justify our choice of the MELTS Supplemental Calculator for calculating aFe3O4Sp1 in the Supplementary Materials^[1].

Uncertainty in the f_O2 calculation contributed by the EPMA analysis

Uncertainty in f_O2 calculated from spinel peridotite oxybarometry depends on the accuracy and precision of the three compositional variables in Equation 4, Mg#^O1, (XFeM1⋅XFeM2)Opx,andaFe3O4Sp1, resulting from analysis by EPMA as well as on the uncertainty in the temperature and pressure of equilibration. We estimated uncertainty contributed by the compositional terms from repeated analysis of secondary standards. The uncertainty in f_O2 from the olivine analysis increases with Mg#^O1 from ±0.04 log units at Mg#^O1 = 0.85 to ±0.14 log units at Mg#^O1 = 0.95, and the orthopyroxene analysis contributes an additional ±0.04 log units. We provide a complete description of how these uncertainties were calculated in the Supplementary Materialy^[1].

We are able to relate uncertainty in the calculation of spinel Fe³⁺/ΣFe ratios to uncertainty in log⁡(aFe3O4Sp1) calculated from the MELTS Supplemental Calculator through a logarithmic relationship described in the Supplementary Material (Supplementary^[1]Fig. S4). Following Parkinson and Arculus (1999) and Ballhaus et al. (1991), Figure 9a shows the relationship between log⁡aFe3O4Sp1 and relative f_O2, and Figure 9b shows how relative f_O2 varies with spinel Fe³⁺/ΣFe ratio. The uncertainty in relative f_O2 contributed by the spinel analysis is asymmetrical and increases with decreasing Fe³⁺/ΣFe ratio. At Fe³⁺/ΣFe = 0.10 the error in f_O2 is (+0.3/–0.4) log units (1a), while at Fe³⁺/ΣFe = 0.35 the error is ±0.1 log units (Fig. 9b). Figure 9 also shows an error envelope for the precision of the Fe³⁺/ΣFe ratio measurement with no correction (dotted lines). Uncertainty in log⁡(aFe3O4Sp1) approximately doubles for spinel analyses that have not been corrected using the W&V89 method. This is particularly important for spinels with Fe³⁺/ΣFe ratios <0.10, which may have uncertainties in f_O2 in excess of a log unit when uncorrected.

Figure 8

Relationship between MgO concentration and Cr# of natural peridotite spinels. Samples are separated by tectonic setting: abyssal peridotites (n = 743) from the compilation of Warren (2016): (Prinz et al. 1976; Hamlyn and Bonatti 1980; Dick and Bullen 1984; Michael and Bonatti 1985; Shibata and Thompson 1986; Dick 1989; Bryndzia and Wood 1990; Johnson et al. 1990; Juteau et al. 1990; Komor et al. 1990; Bonatti et al. 1992, 1993; Cannat et al. 1992; Johnson and Dick 1992; Snow 1993; Constantin et al. 1995; Arai and Matsukage 1996; Dick and Natland 1996; Ghose et al. 1996; Jaroslow et al. 1996; Niida 1997; Ross and Elthon 1997; Stephens 1997; Hellebrand et al. 2002a, 2002b; Brunelli et al. 2003; Hellebrand and Snow 2003; Seyler et al. 2003, 2007; Coogan et al. 2004; Workman and Hart 2005; Morishita et al. 2007; Cipriani et al. 2009; Warren et al. 2009; Brunelli and Seyler 2010; Dick et al. 2010; Warren and Shimizu 2010; Zhou and Dick 2013; Lassiter et al. 2014; Mallick et al. 2014; D’Errico et al. 2016), continental xenoliths not associated with subduction (n = 154): (Wood and Virgo 1989; Ionov and Wood 1992; Woodland et al. 1992), and supra-subduction zone for xenoliths and seafloor drilled samples from subduction-related settings (n = 85): (Wood and Virgo 1989; Canil et al. 1990; Luhr and Aranda-Gomez 1997; Parkinson and Pearce 1998). MgO and Cr# are correlated in the global data set (solid line, r² = 0.82, slope = .15.0 ± 0.2, intercept = 21.46 ± 0.07, 1σ). The slope defined by the correction set used in this study (dashed line, r² = 0.94, n = 7, slope = .11.6 ± 1.4, intercept = 21.5 ± 0.4, 1σ) is shallower, as is the line defined by the Wood spinels (not shown, r² = 0.92, n = 32, slope = .12.7 ± 0.7, intercept = 21.80 ± 0.17, 1σ). Also shown are Hawaiian xenoliths and Tonga peridotite BMRG08-98-2-2 from this study.

Figure 9

Effect of activity of magnetite in spinel on the calculation of relative f_O2. Calculated logf_O2 relative to the quartz-fayalite-magnetite buffer (AQFM, Frost 1991 calibration) using all input parameters from sample 114923-57 at 1038 °C and 1.5 GPa and varying the value of log⁡aFe3O4Sp1 while holding Mg#^O1 and (XFeM1⋅XFeM2)Opx constant (a). The dashed lines show ±1α error on the corrected EPMA measurement of spinel Fe³⁺/ΣFe ratio calculated using Equation 3. Dotted lines show ±1α error on the uncorrected EMP measurement of spinel Fe³⁺/ΣFe ratio assuming a twofold increase in uncertainty for uncorrected measurements (see text). The increased uncertainty in f_O2 at low activities of magnetite has been demonstrated previously by Ballhaus et al. (1991) and Parkinson and Arculus (1999). The dependence of logf_O2 (ΔQFM) on Fe³⁺/ΔFe ratio rather than activity of magnetite (b).

Temperature and pressure both enter into the calculation of f_O2 from spinel-olivine-orthopyroxene equilibria and must be determined through thermobarometry, or some suitable temperature and pressure must be assumed. We have calculated equilibration temperatures of the Hawaiian xenoliths using the spinel-olivine Fe-Mg exchange thermometer of Li et al. (1995), and we assume a pressure of 1.5 GPa. Li et al. (1995) did not evaluate the standard error of their thermometer using an independent validation data set, so we estimated uncertainty from the standard deviation in our own measurements of the Hawaiian xenoliths. For each Hawaiian xenolith, we calculated temperatures from the average spinel composition from each analytical session (Supplementary^[1] Table S2) and the sample average olivine compositions in Table 3. The standard deviation in calculated temperature for these samples varies from 25 to 82 °C. To be conservative, we used ±80 °C as our temperature uncertainty to explore the effects of temperature error on the f_O2 calculation.

The effect of temperature on calculated f_O2 is compositionally dependent as temperature enters into the f_O2 calculation both explicitly in Equation 4 and in the calculation of aFe3O4Sp1. Increasing temperature leads to a decrease in calculated f_O2 relative to QFM, but this effect is greater when spinel Cr# is lower (Supplementary^[1] Fig. S5). The magnitude of uncertainty due to temperature is also a function of temperature, such that samples with colder equilibration temperatures have greater uncertainty in f_O2. The temperature uncertainty of ±80 °C contributes about ±0.1 log units of uncertainty to the f_O2 calculated for low-Cr# sample 114923-41 at its calculated equilibration temperature (1118 °C). Temperature uncertainty for a similar peridotite with a colder equilibration temperature of 700 °C (e.g., as appropriate for supra-subduction zone peridotites; Parkinson and Pearce 1998), would contribute >0.2 log units of uncertainty to the f_O2 calculation.

Pressure is not well constrained for spinel peridotites due to the absence of a strongly pressure dependent reaction (MacGregor 2015). It is common for spinel peridotite oxybarometry studies to assume a single pressure for the f_O2 calculation (e.g., Bryndzia and Wood 1990; Wood et al. 1990; Ballhaus 1993). We follow Wood et al. (1990) in choosing 1.5 GPa, which is roughly the center of the pressure range of spinel stability. Logf_O2 decreases linearly with increasing pressure (Eq. 4), and each 0.25 GPa of pressure uncertainty leads to about ±0.1 log units uncertainty in f_O2.

Hawaiian xenolith f_O2

The replicate analyses of the Hawaiian xenoliths allow for an additional check on our estimated uncertainty in the f_O2 calculation. Figure 10 shows Fe³⁺/ΣFe ratios, aFe3O4Sp1, and f_O2 relative to QFM calculated for each individual analysis of each of the Hawaiian spinels. We calculated aFe3O4Sp1 and f_O2 using the spinel-olivine exchange temperature particular to each spinel analysis. For each sample, measurements of relative f_O2 from each analytical session fall well within the estimated error from all other measurements of that sample (Fig. 10a). This broad overlap is partly due to the use of only a single measurement of olivine and orthopyroxene from each sample, eliminating two sources of potential variation. Figure 10b, which shows variation in the calculated aFe3O4Sp1, shows that error bars in log⁡aFe3O4Sp1 for all measurements of a given sample are also overlapping. This suggests that we have suitably propagated uncertainty in spinel Fe³⁺/ΣFe ratios to uncertainty in aFe3O4Sp1. Although samples with greater spinel Fe³⁺/ΣFe ratios also record greater f_O2 relative to QFM, estimates of aFe3O4Sp1 based on independent measurements of any given sample can increase, decrease, or remain constant with increasing spinel Fe³⁺/ΣFe ratio (Fig. 10b). This highlights that the temperature estimation contributes significantly to the calculation of f_O2.

Figure 10

Relative f_O2 and activity of magnetite in each of the four Hawaiian xenoliths and comparison between Wood (1991) and Ballhaus et al. (1991) formulations of the spinel-olivine-orthopyroxene oxybarometer. Logf_O2 (ΔQFM; a) and log⁡aFe3O4Sp1 (b), calculated from corrected Fe³⁺/ΣFe ratios, for each session in which a given spinel was analyzed (Supplementary^[1] Table S4 and S5). The olivine and orthopyroxene compositions in Table 3 were used for all f_O2 calculations. Uncertainty in logf_O2 (ΔQFM) includes contributions from analytical uncertainty on each phase (a). Uncertainty in log⁡aFe3O4Sp1 was determined as described in the Supplementary^[1] material. Uncertainty in corrected Fe³⁺/ΣFe ratios was calculated using Equation 3. f_O2 relative to the QFM buffer (Frost 1991), calculated for each measurement of the four Hawaiian spinel lherzolite xenoliths (c). Relative f_O2 on the x-axis was calculated using the Equation 4 and the MELTS Supplemental Calculator (Sack and Ghiorso 1991a, 1991b) to calculate aFe3O4Sp1. Relative f_O2 on the y-axis was calculated following the methodology of Ballhaus et al. (1991). Error using the Ballhaus et al. (1991) method was estimated by propagating through the Ballhaus et al. (1991) oxybarometer our estimates of uncertainty in spinel Fe³⁺/ΣFe ratio and olivine Mg#.

The four Hawaiian xenoliths analyzed in this study record relative oxygen fugacities between QFM+0.15 (+0.34/–0.37) and QFM+0.98(+0.24/–0.25) (Table 5, Fig. 10) at their equilibration temperatures and 1.5 GPa, which is slightly more oxidized than the mean f_O2 recorded by abyssal peridotites from spreading centers (Bryndzia and Wood 1990; Wood et al. 1990). These relative oxygen fugacities also depend on temperature and pressure, and the quoted uncertainties do not reflect the additional uncertainties associated with our choice of pressure and temperature. Further contextualization of these results is beyond the scope of this communication, and will be discussed along with a larger data set in a future publication.

Comparison of the Wood (1991) and Ballhaus et al. (1991) parameterizations of the spinel peridotite oxybaromater

Although we have chosen to use the Wood (1991) version of the spinel-olivine-orthopyroxene oxybarometer to calculate f_O2(Eq. 4), numerous other studies use the Ballhaus et al. (1991) version of the oxybarometer. These two parameterizations are commonly considered to be interchangeable (e.g., Woodland et al. 1992; Ballhaus 1993; Canil et al. 2006). Herd (2008) demonstrated that relative f_O2 calculated using the Ballhaus et al. (1991) method is systematically lower than relative f_O2 calculated with the Wood (1991) method. Luhr and Arranda-Gómez (1997) similarly found that f_O2 calculated using Ballhaus et al. (1991) was on average 0.8 log units below those calculated using Wood (1991). We compare the two parameterizations in Figure 10c using the Hawaiian xenolith data. Results from the two methods are correlated (r₂ = 0.82). Consistent with Herd (1008), relative f_O2 calculated using the Ballhaus et al. (1991) method is 0.7 to 1.3 log units lower than results from the Wood (1991) method. The two methods cannot be considered directly comparable, and comparisons of peridotite f_O2 data between studies using different f_O2 parameterizations requires that sufficient analytical data be provided to allow for recalculation using either method.