Ratios determined from counting a subset of atoms in a sample are positively biased relative to the true ratio in the sample (Ogliore et al. 2011). The relative magnitude of the bias is approximately equal to the inverse of the counts in the denominator of the ratio. SIMS studies of short-lived radionuclides are particularly subject to the problem of ratio bias because the abundance of the daughter element is low, resulting in low count rates. In this paper, we discuss how ratio bias propagates through mass-fractionation corrections into an isochron diagram, thereby affecting the inferred initial ratio of short-lived radionuclides. The slope of the biased isochron can be either too high or too low, depending on how it is calculated. We then reanalyze a variety of previously published data sets and discuss the extent to which they were affected by ratio bias. New, more accurate, results are presented for each study. In some cases, such as for 53Mn-53Cr in pallasite olivines and 60Fe-60Ni in chondrite sulfides, the apparent excesses of radiogenic isotopes originally reported disappear completely. Many of the reported initial 60Fe/56Fe ratios for chondrules from ordinary chondrites are no longer resolved from zero, though not all of them. Data for 10Be-10B in CAIs were only slightly affected by bias because of how they were reduced. Most of the data sets were recalculated using the ratio of the total counts, which increases the number of counts in the denominator isotope and reduces the bias. However, if the sum of counts is too low, the ratio may still be biased and a less-biased estimator, such as Beale’s estimator, must be used. Ratio bias must be considered in designing the measurement protocol and reducing the data. One can still collect data in cycles to permit editing of the data and to monitor and correct for changes in ion-beam intensity, even if total counts are used to calculate the final ratio. The cycle data also provide a more robust estimate of the uncertainties from temporal variations in the secondary ion signal.
Secondary ion mass spectrometry (SIMS) is an important tool for understanding short-lived isotopic systems and for constraining early solar system chronology. The SIMS technique, like many other analytical techniques, is fundamentally a sampling experiment: a subsample of a parent population is measured to estimate certain parameters of the source population (e.g., an isotope ratio). It is typically assumed that these sampled ratios are unbiased estimates of the true isotope ratios in the object. When the number of counts of the denominator isotope is large, this assumption is generally safe, but when the number of counts is low, the expectation value of the ratio calculated from the measurement can be significantly higher than the true ratio in the object (see e.g., Pearson 1910). Count rates of the denominator isotope during SIMS measurements of short-lived radionuclide systems are often low, particularly when parent/daughter elemental ratios are high, making the isotope ratios susceptible to ratio bias. In this paper, we refer to the bias as the expectation value of the measured isotope ratio minus the true ratio in the object. Ogliore et al. (2011) discuss the issue of ratio bias as it applies to SIMS measurements. They show that positive bias in isotope ratios inferred from counting data can be significant and can result in incorrect inferences about the objects. The relative bias (the bias divided by the true ratio) in ratio estimation is approximately equal to the inverse of the number of counts in the denominator (assumed to be Poisson distributed). For example, if the number of total counts of the denominator isotope is 200, the bias of the estimated ratio will be 5‰. The more counts of the denominator isotope, the smaller the bias and the closer the estimated ratio is to the true ratio that the investigator seeks to measure.
Ratio bias is particularly insidious in SIMS measurements of short-lived radionuclide systems where statistical bias from low counts in the denominator isotope can produce a correlation similar to an isochron with a positive slope (Fig. 1a). The bias increases as the number of denominator counts decreases. In the example in Fig. 1a, the counts of 55Mn have been held constant for each set of data, so the x-axis of the plot is effectively 1/52Cr, producing a perfect correlation. In a real system, the 55Mn counts will also vary, which weakens the correlation. But, for many natural systems, the parent/daughter element ratio is controlled primarily by variations in the daughter element abundance and the system approaches the modeled case. As we will show below, unrecognized ratio bias can easily be interpreted as evidence for the presence of a short-lived nuclide when the sample formed.
Averaging the ratios from many cycles of a measurement is especially prone to ratio bias, because the counts obtained over the measurement are divided up amongst the individual ratios, resulting in a lower number of counts for each ratio. For instance, if a measurement is partitioned into 100 cycles, the ratio calculated by the mean of these 100 ratios will have a relative positive bias about 100 times larger than the ratio determined from dividing the total counts of the numerator isotope by the total counts of the denominator isotope. Increasing the number of ratios (while maintaining the same number of counts per ratio) improves the statistical uncertainty, but it does not decrease the bias (Fig. 1b). If one averages 300 ratios with a mean of 200 counts in the denominator of each ratio, the estimated ratio will have an expectation value 5‰ greater than the true value and a statistical uncertainty of ±6.6‰ (the highest gray point in Fig. 1a). However, if one averages 900 ratios, the estimated ratio will still be 5‰ larger than the true ratio even though the uncertainty has decreased to ±3.8‰ (Fig. 1b). Totaling the counts before calculating the ratios will lower the bias. However, either method can be significantly biased if the counts in the denominator are low. The data for many of the published SIMS studies of short-lived radionuclide systems have been calculated using the mean of the ratios, implying that the published ratios may be significantly affected by statistical bias.
The most effective way to avoid ratio bias is to ensure that there are enough counts of the denominator isotope in each cycle of the measurement. In cases where this cannot be achieved, calculating ratios from the total counts will usually suffice to eliminate the effect of ratio bias. When the counts per cycle are very low, Beale’s estimator, a method that takes into account the total counts of the isotopes and the correlation between the numerator and denominator isotopes, will provide more accurate estimated ratios, as it is less biased (Ogliore et al. 2011). Coath and Steele (2013) proposed another ratio estimator with low bias. Determining the number of counts necessary to avoid biased ratios depends on the accuracy required to clearly observe the desired effects (e.g., excesses in radiogenic isotopes).
In this paper, we discuss the effect of ratio bias on isochron slopes, and we report recalculated results for the data published in Hsu et al. (1997) and Hsu (2005) on 53Mn-53Cr systematics of pallasites, in Tachibana and Huss (2003) and Guan et al. (2004) on 60Fe-60Ni systematics of sulfides from ordinary and enstatite chondrites, and in Tachibana et al. (2006) and subsequent abstracts from the University of Hawai’i on 60Fe-60Ni systematics of silicates from ordinary chondrites. We also recalculated some of the data reported in Mishra et al. (2010) on 60Fe-60Ni systematics of silicates from ordinary chondrites, and data for 10Be-10B systematics of CAIs from CV chondrites published by MacPherson et al. (2003). We corrected for possible biases in the published isotopic ratios by calculating the ratios using the total counts instead of averaging the ratios from each cycle. For the 10Be-10B data, we also used Beale’s estimator to calculate the ratios. The data reported here should be used in place of those reported in the original publications. We hope that re-evaluating these datasets will encourage other researchers to revisit their data. The new data will provide much needed clarification on the abundances of short-lived radionuclides in the solar system.
Bias in Isochrons
The radiogenic excess of the daughter isotope is determined by calculating the ratio of the daughter isotope to a normalizing isotope of the same element and then correcting for instrumental mass fractionation and intrinsic mass fractionation in the sample. To make an isochron, the mass-fractionation-corrected isotope ratios are plotted as a function of the parent/daughter elemental ratio and the error-weighted regression, the isochron, is computed. The slope of the isochron gives the initial abundance of the radioactive parent (e.g., Faure and Messing 2005). To predict the effect of ratio bias in the slope of the isochron, it is necessary to understand how the ratio bias propagates through the entire analysis. Here, we describe the correction for instrumental and natural mass fractionation and determine how the isochron slope is affected by ratio bias.
Propagation of Ratio Bias through the Mass Fractionation Correction
In a system with only one isotope ratio available (e.g., 10Be-10B and 53Mn-53Cr), the correction for mass fractionation is done “externally.” (The 53Mn-53Cr system is typically treated as a one-ratio system because only 52Cr and 53Cr are free from interferences, leaving one ratio for the mass fractionation correction.) The correction is done by comparing the measured ratio in the sample to the measured ratio in a standard of similar mineralogy and known isotopic composition. The difference between the ratio measured in the standard and the true composition of the standard corresponds to the instrumental mass fractionation, and this difference is applied to the ratio for the unknown, using an appropriate mass-fractionation law. If the ratios for the standard and unknown are unbiased, any resulting difference between the true ratio for the standard and the corrected ratio for the unknown is considered an isotope anomaly. If the anomaly is an excess of the daughter isotope of a short-lived radionuclide, and if the excesses among several measurements correlate with the parent/daughter elemental ratio, there is evidence for the former presence of the short-lived radionuclide in the sample, and its abundance at the time the sample formed is given by the slope of the isochron. If the isotope ratio for either the standard or the unknown is biased, the calculation of excess radiogenic daughter and the slope of the isochron are compromised. Typically, the standard has a higher abundance of the daughter element than does the unknown. Therefore, if the data are gathered in the same way for both, ratio bias will affect the measured ratio for the unknown more than for the standard and the difference between the biases would appear as an excess of the radiogenic daughter isotope.
When two isotope ratios are available (e.g., 26Al-26Mg, 60Fe-60Ni systems), the correction for mass fractionation can be done “internally.” One ratio, the one that does not include the radiogenic daughter isotope, is used to estimate the mass fractionation, and the fractionation inferred from that ratio is applied to the other ratio using an appropriate mass-fractionation law. The difference between the mass-fractionation-corrected ratio and the normal isotope ratio is the isotope anomaly. As in the case described above, if this anomaly is an excess of the daughter isotope and is correlated with the parent/daughter elemental ratio, there is evidence for the former presence of the short-lived radionuclide, and its abundance at the time the sample formed is given by the slope of the isochron.
Figure 2 describes schematically the effect of ratio bias on the inferred radiogenic excess for the 26Al-26Mg and 60Fe-60Ni systems. Figure 2a shows magnesium isotopes plotted as delta values with 24Mg as the normalizing isotope. The normal isotopic ratios, without mass fractionation or isotopic anomalies, plot on the horizontal flat line. Two cases of mass-dependent fractionation are illustrated by the gray symbols and solid lines. Here, we arbitrarily define excesses of the heavy isotopes relative to normal magnesium as a positive mass fractionation and excesses of light isotopes as negative mass fractionation. The biases introduced into δ25Mg and δ26Mg from low counts in 24Mg will be the same. The open symbols in Fig. 2 represent the additional effect of a positive statistical bias on the mass-fractionated ratios. If an internal mass-fractionation correction is done using the biased 25Mg/24Mg ratio, represented by the dashed lines, the inferred value for δ26Mg will be negative. This will decrease any real excess of radiogenic 26Mg inferred for the measurement. However, if one does an external mass-fractionation correction, the system will behave like the 10Be-10B and 53Mn-53Cr systems and any bias will be positive, increasing the inferred excess of radiogenic 26Mg. In general, because the 24Mg count rate is approximately 10 times higher than the count rates of 25Mg and 26Mg, ratio bias is not a significant issue in the 26Al-26Mg system, but this should be verified for each new measurement.
Figure 2a can also be used to illustrate the situation where the radiogenic daughter isotope is between the two isotopes used for the fractionation correction (e.g., if 25Mg were radiogenic). If the 26Mg/24Mg ratio is used for the mass fractionation correction, the inferred degree of mass fractionation will be lower (still higher than the true value), and the fractionation-corrected 25Mg/24Mg would have a positive bias.
Figure 2b illustrates the nickel isotopes plotted as delta values normalized to 61Ni. Again, mass fractionation is shown by the solid lines and gray symbols. The bias introduced by low counts of 61Ni is the same for δ60Ni and δ62Ni. In this case, if the mass fractionation correction is done using the measured δ62Ni, δ60Ni will be overcorrected (dashed lines). The inferred excess in δ60Ni will be the sum of the bias in the 60Ni/61Ni ratio and that in the 62Ni/61Ni ratio. Because 61Ni is the least abundant isotope being measured, the ratio bias will be large in an absolute sense as well. For the 61Ni normalization, an external fractionation correction will be less biased (all other things being equal) because the bias only contributes once to the final ratio.
Figure 2c illustrates the nickel isotopes normalized to 62Ni. This case is analogous to that for the magnesium isotopes (Fig. 2a). A bias in the 61Ni/62Ni ratio will result in an over correction for mass fractionation in the 60Ni/62Ni ratio, which in turn will produce a negative isotope anomaly in δ60Ni. However, once again, if an external mass fractionation correction is used, any bias in the 60Ni/62Ni ratio will be positive. For a given measurement, the total bias in δ60Ni after an internal mass-fractionation correction introduced by normalizing to 61Ni will be larger in magnitude than that introduced by normalizing the 62Ni for two reasons: 1) the abundance of 61Ni is approximately three times lower than the abundance of 62Ni, and 2) the effect of bias on the mass fractionation correction is twice as great in the 61Ni normalization.
Ratio Bias in Isochron Fitting
Once the excess of the radiogenic isotope is determined, the abundance of the radiogenic isotope relative to that of the normalizing isotope of that element is plotted as a function of the parent/daughter elemental ratio (see Figs. 3–9 below). As both ratios have analytical uncertainties, the equation of the fitted line is determined by solving the weighted least-squares problem. Generally, this is solved by the York method (York 1966) or an equivalent formulation (e.g., Ludwig 2003). The effects of biased isotope ratios on an isochron can differ significantly, depending on the isotope system. We consider several cases explicitly below, but this treatment is not intended to be comprehensive.
Consider the simplest case where the isotope ratios are biased due to low denominator counts and there are no complications from an internal mass-fractionation correction. Consider further that the isochron is being constructed from several measurements of a single mineral with different parent/daughter elemental ratios. In this case, the differences in parent/daughter ratio will be due almost exclusively to the variable abundance of the daughter element. An isochron plot is then effectively a plot of the isotope ratio versus the inverse of the daughter-element abundance. In this case, the bias will correlate precisely with the elemental ratio and an array with a positive slope will be generated without any contribution from an excess of the radiogenic isotope. This is observed in some of the 53Mn-53Cr data discussed below.
If the isochron is produced from several minerals with different compositions, the effect of bias on the regression will be similar. Except, in this case, the correlation will be less precise than in the situation with a single mineral discussed above. There will be a general correlation because the abundance of major elements typically varies by a factor of two to three while the abundance of a trace element can vary by orders of magnitude.
We now turn to the case where ratios are corrected internally for mass fractionation. For the 26Al-26Mg system, isochrons are typically generated from several minerals with different Al/Mg ratios. A slightly negative bias in the 26Mg/24Mg ratio will correlate with the inverse of the 24Mg counts during the measurement. The resulting effect on the isochron will depend on the mineral composition and the measurement conditions, but will typically lower the inferred 26Al/27Al initial ratio. Fortunately, for most 26Al-26Mg measurements, the count rates are sufficiently high that the slight negative bias on the ratios is not significant.
For the 60Fe-60Ni system where the normalizing isotope is 61Ni, bias in the isochron can be a severe problem, both because of the low number of counts and because the internal mass-fractionation correction amplifies the effect, as described above. In many samples measured to date, the spread in Fe/Ni ratios that permit calculation of an isochron comes from abundance variations in the trace-element nickel from spot to spot in the same mineral. In this case, the isochron plot will be similar to a plot of the nickel isotope ratio versus the inverse of the nickel abundance, and the array will have a well-correlated positive slope due to significant ratio bias. However, if the data are reduced using 62Ni as the normalizing isotope, the bias will be slightly negative and the isochron slope will be lower than the true value or negative if there is no radiogenic nickel. We observe this behavior in most of the 60Fe-60Ni data we review below.
The effect of ratio bias on an isochron slope can be approximated mathematically using certain simplifying approximations and data from any given measurement. The online supplement derives the relevant equations and provides an example for determining the bias in the isochron of an actual dataset. We also provide an example of the complete calculation of ratios and isochron for the 60Fe-60Ni systematics of an E-chondrite sulfide.
Recalculating Previously Published Sims Data
We have access to the original data from each of the studies discussed in this paper; these studies were carried out by a number of coworkers in collaboration with Gary Huss. To the extent possible, we corrected the measured data and edited the data in the same way as they were edited originally. In most cases, we were able to reproduce the original published numbers. We recalculated the final ratios using the following prescription:
1) Data were corrected for deadtime and detector background and were edited, as much as possible, as they were in the original paper.
2) If data were collected in monocollection mode by peak jumping (most of the data), time interpolation was applied in the same way as in the original data analysis.
3) The same mass-fractionation corrections were applied.
4) The means of the individual ratios and the standard error of the means were calculated.
5) The counts from cycles for each isotope were summed and the results were used to calculate the isotope ratios.
6) The final reported ratios are those calculated from the total counts, and the uncertainties are the standard errors of the mean of the individual ratios (we discuss the uncertainties further below).
A robust estimate of our measurement uncertainties is required to understand the cosmochemical implications of our work. A given measurement has different types of uncertainty: 1) statistical uncertainty due to counting statistics of the secondary ions, also called Poisson noise; 2) variations in the secondary ion signal not attributable to counting statistics that occur on timescales significantly shorter than the measurement (e.g., varying parent/daughter elemental ratio, noise bursts); and 3) systematic uncertainty that decreases the accuracy of the measurement due to phenomena like nonlinearity in the detector, inaccurate deadtime correction, or imperfect background measurements. We cannot evaluate the systematic uncertainties from the data, so we will not consider them further here. The statistical uncertainty of a ratio (1) is well known and is easily derived from the standard error propagation equation. If the numbers of counts for isotopes in the ratio are reasonably large and the numerator and denominator counts are uncorrelated and normally distributed, the statistical standard deviation of the ratio r = y/x is approximately:
When calculating ratios by summing the counts of one isotope and dividing by the summed counts of another, the above expression represents the minimum total uncertainty of the measurement. An additional contribution from sources other than Poisson noise (such as those discussed in #2, above) must be accounted for. In this work, we estimate the contribution from other sources of uncertainty by looking at the data collected in the individual cycles during the measurement.
Isotope data are typically collected in a number of short cycles to monitor the effects of drift in signal strength, changes in mass fractionation, etc. If drifts in signal strength can be adequately sampled, they can be corrected for relatively easily with time interpolation. After editing the data to remove highly anomalous cycles and correcting for drift, the variability among individual ratios is a good estimate of the total uncertainty from sources 1 and 2 described above. The standard error of the measured ratios is a reasonable measure of how well the mean of the ratios is known when the data are sampled from a single parent distribution with a well-defined mean. Even though the individual cycle ratios can be significantly biased when the denominator counts are low, in most cases, the variability of the data is independent of whether or not the data are biased (Ogliore et al. 2011). It is therefore appropriate to continue to use the standard error of the measured ratios as an estimate of the total uncertainty (from 1 and 2) in the final ratio.
After a study of phosphates in the Springwater pallasite that appeared to show evidence of live 53Mn when the pallasites formed was published (Hutcheon and Olsen 1991), the Caltech ion probe group carried out a study of the 53Mn-53Cr system in pallasite olivines. The olivine measurements appeared to show excesses of 53Cr that correlated with the 55Mn/52Cr ratio, and the inferred (53Mn/55Mn)0 ratios for the olivines when they formed were (0.6–2.0) × 10−5 (Table 1; Hsu et al. 1997; Hsu 2005). These results were consistent with those found for phosphates in Springwater (Hutcheon and Olsen 1991) and implied that pallasites must have cooled much more quickly than generally believed, calling into question the origin of pallasites at the core-mantle boundary of the parent asteroids (e.g., Hsu et al. 1997; Hsu 2005). However, subsequent measurements of olivine in pallasites by other techniques and by higher transmission ion probes were unable to confirm high (53Mn/55Mn)0 ratios (Lugmair and Shukolyukov 1998; Tomiyama and Huss 2005; Tomiyama et al. 2007).
Table 1. (53Mn/55Mn)0 ratios for pallasite olivines.
For this paper, we have re-reduced the data gathered by Hsu et al. (1997), which was published in Hsu (2005). We recalculated the isotope ratios using total counts, instead of the mean of the ratios. The data were reduced and edited in exactly the same way as in the original work; the only change was the way the final ratios were calculated. The results are shown in Fig. 3 and Table 1. In all measurements, the 53Cr/52Cr was lower in the recalculated results (Fig. 3). When the newly calculated data were regressed on a 53Mn-53Cr isochron diagram, all evidence for extinct 53Mn disappeared (Fig. 3, Table 1). The olivine data reported by Hsu et al. (1997) and Hsu (2005) are all biased and the inferred initial ratios are artifacts of that bias. The complete recalculated data set is available in the supporting information. Data reported by Tomiyama and Huss (2005) and Tomiyama et al. (2007) were not significantly affected by ratio bias because, in those studies, the count rates for the chromium isotopes were much higher than in the Caltech study (i.e., 52Cr counts per cycle were between 2,200 and 22,000 for Tomiyama et al.  and between 100 and 800 for the Caltech study). With the re-reduced data, none of the ion probe studies show evidence of in situ decay of 53Mn in pallasites.
The first paper to claim clear evidence for the presence of 60Fe in chondritic materials was Tachibana and Huss (2003b). In this paper, (60Fe/56Fe)0 ratios of (1–2) × 10−7 were reported for troilite from unequilibrated chondrites Bishunpur (LL3.15) and Krymka (LL3.2) (Table 2). The initial 60Fe/56Fe ratio for the solar system estimated from the troilite data range from (2.8–4.0) × 10−7. The results were apparently confirmed by Mostefaoui et al. (2003, 2004, 2005), who found even higher initial ratios for sulfides from Semarkona (LL3.0) using NanoSIMS. Tachibana and Huss (2003b) calculated the nickel-isotope ratios using the mean of the ratios from the individual measurement cycles. The published ratios were normalized to 61Ni because the uncertainty for ratios normalized to 62Ni is systematically larger than for ratios normalized to 61Ni due to the mass fractionation correction.
Table 2. (60Fe/56Fe)0 ratios for troilites from unequilibrated ordinary chondrites.
bThese values differ slightly from those in Tachibana and Huss (2003b) because the analysis done for the current paper did not take into account the correlated component of the errors. The data in this table differ only in the method of ratio calculation and are directly comparable.
1.10 ± 0.32b
0.16 ± 0.37
0.12 ± 0.40
0.21 ± 0.63
1.06 ± 0.66b
−0.04 ± 0.79
−0.28 ± 1.17
−0.04 ± 1.36
1.28 ± 0.67b
0.48 ± 0.77
0.36 ± 1.16
0.37 ± 1.34
1.82 ± 0.78b
−0.23 ± 0.96
−0.61 ± 1.34
−0.12 ± 1.67
1.64 ± 0.93b
−0.12 ± 0.96
−0.71 ± 1.57
−0.12 ± 1.65
We re-reduced the Tachibana and Huss (2003b) data using the total counts. The data were reduced and edited exactly as they were for the original publication; only the method of calculating the ratios changed. The initial (60Fe/56Fe)0 ratios reported in Table 2 are slightly different than those in the original publication because we did not take into account the correlated component of the uncertainties in the current work. This makes for a fair comparison between the two data-reduction methods in Table 2. We found that the 60Ni/61Ni ratios are distinctly lower and the 60Ni/62Ni ratios are marginally higher than the original values calculated from the means of the measured ratios (Fig. 4), as expected (Fig. 2). For the recalculated data, the inferred (60Fe/56Fe)0 ratios are unresolved from zero, independent of the normalizing isotope (Table 2; Fig. 4). We conclude that the initial ratios reported by Tachibana and Huss (2003b) were strongly affected by ratio bias and that there is no longer evidence for the presence of 60Fe when these samples formed. The complete recalculated data set is available in the online material.
Guan et al. (2007) measured the 60Fe-60Ni and 53Mn-53Cr systems in sulfides from enstatite chondrites to further constrain the initial abundance of 60Fe in the solar system and to investigate possible correlations between 60Fe-60Ni and 53Mn-53Cr systems. Some of these results were also reported in various abstracts (Guan et al. 2003, 2004). Sulfides from ALHA77295, MAC 88136, and Qingzhen (all EH3) were measured using a Cameca ims-6f at ASU. Large variations were found in the (60Fe/56Fe)0 and (53Mn/55Mn)0 ratios of (2–20) × 10−7 and (2–7) × 10−7, respectively, but there was no clear correlation between 60Fe-60Ni, and 53Mn-53Cr systems. These authors also found that the 60Ni/61Ni ratios did not correlate with the 56Fe/61Ni ratios. Although the 60Fe-60Ni system may be disturbed, they concluded that there was clear evidence for the presence of 60Fe in these sulfides.
Isotope ratios for these measurements were originally calculated using the mean of the ratios. We recalculated some of the data using total counts (Tables 3 and 4). Although we were not able to precisely match the published data in our recalculations, our values using the means of the ratios are similar to the published ratios. The counts per cycle for 52Cr in sphalerite for these measurements range from 20 to 80, so the ratios calculated from the mean of individual ratios are too high. Using total counts, 53Cr/52Cr ratios were lower (Fig. 5) and the inferred (53Mn/55Mn)0 ratios dropped, although there is still clear evidence for the former presence of 53Mn in one of the sulfides (Table 3).
Table 3. (53Mn/55Mn)0 ratios for sulfides from enstatite chondrites.
bThese initial ratios do not match the published values because we were unable to fully duplicate the original data reduction. The data in this table differ only in the method of ratio calculation and are directly comparable.
4.56 ± 0.75b
3.42 ± 0.83
1.75 ± 0.50b
0.41 ± 0.58
Table 4. (60Fe/56Fe)0 ratios for sulfides from enstatite chondrites.
bThese initial ratios do not match the published values because we were unable to fully duplicate the original data reduction. The data in this table differ only in the method of ratio calculation and are directly comparable.
8.5 ± 6.1b
1.8 ± 9.0
0.4 ± 10.0
1.8 ± 14
6.1 ± 2.1b
−4.2 ± 2.9
−6.7 ± 3.6
−4.0 ± 4.7
For the Fe-Ni system, counts per cycle for 61Ni ranged from 4 to 100 (approximately three times higher for 62Ni). The 60Ni/61Ni ratios calculated from total counts are lower, while the 60Ni/62Ni ratios were similar or slightly higher than those determined from means of the ratios (Fig. 6; Table 4), as expected (Fig. 2). The (60Fe/56Fe)0 ratios for these sulfides are no longer resolved from zero; there is no longer clear evidence for the presence of 60Fe in these sulfides. Data for the individual measurements are available in the online material.
Measurements of sulfides in unequilibrated chondrites apparently showed that the 60Fe-60Ni systematics is easily disturbed by secondary processing (Guan et al. 2004b, 2007). Because of this, Huss and Tachibana (2004) decided to look at Fe-rich silicates, which should be less susceptible to metamorphic disturbance. They found that Fe-rich pyroxene can have very high Fe/Ni ratios, particularly the fine-grained radiating pyroxene chondrules. Huss and Tachibana (2004) reported the first data from a chondrule to show evidence of 60Fe. They measured a radiating pyroxene chondrule using the ASU Cameca ims 6f. To confirm this measurement, they re-measured the chondrule using the Cameca ims 1270 at the Geological Survey of Japan. The two data sets gave very similar results, giving them confidence that the data were robust and correct (Table 5). Tachibana et al. (2005, 2006) measured some additional pyroxene-rich chondrules from Semarkona (LL3.00) and Bishunpur (LL3.15) and found evidence of 60Fe in all of them. They reported (60Fe/56Fe)0 ratios of (2–4) × 10−7 for the chondrules (Fig. 7; Table 5) and inferred that the initial ratio for the solar system was in the range of (0.5–1.0) × 10−6. The 61Ni counts per cycle for BIS21, which gave the highest (60Fe/56Fe)0 ranges from 10 to 100 for an average of approximately 200 cycles.
Table 5. (60Fe/56Fe)0 ratios for chondrules from unequilibrated ordinary chondrites.
bSome of these values differ slightly from those in Tachibana et al. (2006) because the analysis performed for the current paper did not take into account the correlated component of the errors. The data in this table differ only in the method of ratio calculation and are directly comparable.
SMK 1-4 (ASU)
4.2 ± 1.7
1.0 ± 2.0
1.3 ± 3.1
0.8 ± 2.3
2.6 ± 1.0
0.6 ± 1.1
0.2 ± 1.7
0.6 ± 1.9
2.7 ± 1.0b
0.7 ± 1.1
0.2 ± 1.6
0.7 ± 1.8
2.0 ± 1.1a,b
−1.3 ± 1.5
−4.7 ± 2.0
−1.2 ± 2.6
2.8 ± 2.3a,b
−0.2 ± 2.8
−1.4 ± 3.8
−0.2 ± 4.7
4.4 ± 2.5a,b
−1.7 ± 2.7
−3.5 ± 4.0
−1.6 ± 4.6
We recalculated the data reported by Huss and Tachibana (2004) and Tachibana et al. (2005, 2006) using total counts. As expected, the 60Ni/61Ni ratios are lower than the previously published values, while the 60Ni/62Ni ratios are higher than those determined using mean of ratios (Fig. 7). The inferred (60Fe/56Fe)0 ratios for the chondrules are also significantly lower. Once again, the choice of normalizing isotope for the nickel isotope ratios (61Ni or 62Ni) does not change the result, although it does change the uncertainty for the reasons described above.
In this section, we review 60Fe-60Ni data gathered at the University of Hawai’i and reported prior to our learning about ratio bias. Data for seven chondrules from Semarkona and Bishunpur were reported by Tachibana et al. (2007). They reported (60Fe/56Fe)0 ranging from (1–2) × 10−7. Six of these chondrules were also measured for 26Al-26Mg systematics (Huss et al. 2007). A subset of these data along with new data obtained using the Cameca ims 4f at the Physical Research Laboratory (PRL) in Ahmedabad, India were subsequently published by Mishra et al. (2009, 2010). Mishra et al. (2010) inferred an initial 60Fe/56Fe ratio for the early solar system of approximately 4 × 10−7 (using the revised half-life of 2.6 Myr for 60Fe; Rugel et al. 2009).
We re-reduced the data gathered at the University of Hawai’i using total counts (Table 6; Fig. 8). Counts per cycle for 61Ni among the low-Ni points ranged from 10 to 250 (approximately three times higher for 62Ni) for typically 200 cycles. We found that the 60Ni/61Ni ratios are lower and the 60Ni/62Ni ratios are similar to or slightly higher than the original values (Fig. 8). The inferred (60Fe/56Fe)0 ratios are also lower. None of the seven chondrules from Tachibana et al. (2007) show evidence for the former presence of 60Fe (Table 6). The Al-Mg data gathered at UH were also recalculated using total counts, and these data were not found to be significantly biased. The data for the other chondrules measured at PRL were not available for this paper. Although the Fe-Ni data were reduced using total counts (Mishra, personal communication), we suspect that they may also be affected by bias because they were collected using the ims 4f, which has significantly lower transmission than the ims 1280.
Table 6. (60Fe/56Fe)0 ratios for chondrules from unequilibrated ordinary chondrites.
eThis value differs slightly from the one published in Tachibana et al. (2009) because we were not able to completely reproduce the original data reduction.
1.9 ± 1.1a,b
0.0 ± 1.3
−0.1 ± 1.9
0.1 ± 2.1
3.2 ± 1.6a,b
0.1 ± 1.9
0.0 ± 2.8
0.1 ± 3.1
0.5 ± 4.2a,b
0.6 ± 3.2
−3.9 ± 7.1
0.6 ± 5.4
1.7 ± 1.1a,b
0.6 ± 1.2
0.5 ± 1.9
0.6 ± 2.0
2.0 ± 1.9a
0.2 ± 2.0
0.1 ± 3.0
0.3 ± 3.4
0.5 ± 1.0a
−0.5 ± 1.1
−0.4 ± 1.6
−0.5 ± 1.8
1.2 ± 0.9a
0.4 ± 1.1
−0.2 ± 1.7
0.4 ± 1.8
1.42 ± 0.21
1.14 ± 0.22
0.88 ± 0.32c
0.94 ± 0.33
0.95 ± 0.42
0.63 ± 0.44
0.54 ± 0.64c
0.62 ± 0.68
2.76 ± 0.44
2.35 ± 0.44
2.36 ± 0.68c,e
2.36 ± 0.67
4.5 ± 1.3
2.1 ± 1.2d
1.1 ± 1.9
2.1 ± 1.9
Multicollection measurements of 60Fe-60Ni systematics of chondrules from Krymka (LL3.2) were reported by Tachibana et al. (2009) (Table 6). The inferred initial (60Fe/56Fe)0 ratio for these shows the least amount of change in nickel-isotope ratios because the count rates for 61Ni were relatively high. For chondrule KRM 3-1, which has the highest 56Fe/61Ni ratios, the average counts per cycle range from 200 to 400 for 200 cycles. The data were originally reduced using the mean of ratios, but normalized to 62Ni instead of 61Ni. As 62Ni has more counts, using it as the normalizing isotope generally gives results that are less biased. When reduced using total counts, the three Kyrmka chondrules give resolved initial ratios ranging from (0.6 ± 0.4) × 10−7 to (2.4 ± 0.4) × 10−7 (Table 6; Fig. 9). As Tachibana et al. (2009) mention, the correlation between the nickel-isotope ratios and Fe/Ni ratio for chondrules KRM 3-11 and KRM 3-9 is weak (both have χν2 > 3) and suggests the Fe-Ni system in some of these chondrules is disturbed. There is evidence for excess 60Ni at the 2σ level in these chondrules, but one cannot extract a robust estimate of the initial (60Fe/56Fe)0 ratio for when these chondrules formed.
Semarkona chondrule, DAP-1, has been analyzed several times (Huss et al. 2010; Telus et al. 2011a). The most recent data set, measured in multicollection mode at UH, is shown in Table 6 and Fig. 9. The counts per cycle for 61Ni range from 40 to 1000 for 200 cycles. When the data are calculated using total counts, the inferred initial (60Fe/56Fe)0 ratio for DAP-1 is (2.1 ± 1.2) × 10−7 (χν2 = 2.5) (Table 6; Fig. 9). Again, the relatively weak correlation suggests that the Fe-Ni system for this chondrule is disturbed and cannot give an accurate estimate of the true (60Fe/56Fe)0 for this chondrule.
The 60Fe-60Ni data for all of these chondrules are available in the online material.
Another isotope system that gives low count rates in SIMS measurements is the 10Be-10B system (t1/2 = approximately 1.4 × 106 yr; Chmeleff et al. 2009). MacPherson et al. (2003) published 10Be-10B data for seven type A calcium-aluminum inclusions (CAIs) from CV chondrites. All showed clear evidence for 10Be, but the initial abundances of 10Be did not correlate tightly with the initial abundances of 26Al in the inclusions.
The data in this study were collected over as many as 300 cycles, with 10B and 11B counted for 10 seconds and 3 seconds, respectively. The count rates were low enough that many cycles had zero counts. This meant that the data could not be reduced using the mean of the individual cycle ratios. So the counts of each isotope were totaled and the 9Be/11B and 10B/11B ratios were calculated from the total counts. The total number of counts of 11B used to calculate the 10B/11B ratios ranged from 3000 to 2.4 × 105 counts in pyroxene and 35 to approximately 8,500 counts in melilite. According to Fig. 1 of Ogliore et al. (2011), the positive bias in the published (10B/11B)0 ratios should be insignificant for pyroxene, but up to approximately 30‰ for melilite. To check the bias, we recalculated all of the ratios from MacPherson et al. (2003) using the Beale’s estimator (Beale 1962; Ogliore et al. 2011), which should reduce the bias to significantly less than a part per million.
The recalculated 10B/11B ratios were from approximately 0.3‰ to approximately 42‰ lower than the published ratios (excesses relative to normal boron can be as much as 1200‰). New regressions for each inclusion sometimes resulted in very slight revisions in the inferred (10Be/9Be)0 ratios for these CAIs (Table 7). The new results do not change the conclusions of the MacPherson et al. (2003) paper in any significant way. The recalculated data are given in the online material (Table 7).
Table 7. (10Be/9Be)0 ratios for Type A CAIs from CV chondrites.
bA mistake in the original data reduction resulted in an inferred initial ratio of (0.75 ± 0.19) × 10−3, rather than the value listed in the table.
(0.76 ± 0.16) × 10−3
(0.76 ± 0.16) × 10−3
(0.58 ± 0.19) × 10−3
(0.57 ± 0.19) × 10−3
(0.53 ± 0.17) × 10−3
(0.53 ± 0.17) × 10−3
(0.73 ± 0.19) × 10−3b
(0.72 ± 0.19) × 10−3
(0.67 ± 0.24) × 10−3
(0.67 ± 0.28) × 10−3
(0.48 ± 0.17) × 10−3
(0.48 ± 0.17) × 10−3
(0.30 ± 0.12) × 10−3
(0.29 ± 0.13) × 10−3
Discussion and Conclusions
We have reviewed and re-reduced many of the data sets collected by Gary Huss and his collaborators over the years that could have been subject to bias. As we have shown, many SIMS studies of short-lived radionuclide systems (e.g., 26Al-26Mg, 10Be-10B, 53Mn-53Cr, and 60Fe-60Ni) have unintentionally generated biased data due to the low number of counts in the denominator of the isotope ratios. The re-reduced data presented in this paper should be used instead of the originally published data in all future work. We hope that this paper will prompt others to review their previously reported data and to republish corrected data in cases of serious bias.
A key question is how to evaluate whether or not the ratio bias is significant for a particular study. Cosmochemists are used to evaluating whether or not two measurements are different by looking to see if their uncertainties overlap. If two independent measurements have overlapping uncertainties, they are considered indistinguishable at a certain confidence level. It is not valid to use the same test to compare ratios calculated by different methods from the same data to evaluate whether or not the bias is significant. Figure 10 shows a ratio calculated from the mean of 100 ratios compared with a ratio calculated from the same data using total counts. The mean of 100 ratios shows a 20‰ positive bias compared with the mean of total counts and the statistical error on both ratios is 23‰ (2σ). The comparison to be made is not between the bias and the error, but between the bias and the size of the effect that is significant in the study. For example, if the solid line represents the isotopic composition of a mantle reservoir and our hypothesis is that our sample was derived from that reservoir, we would reject our hypothesis at the 2σ level if we calculated the data from the mean of the ratios, but we would not reject the hypothesis if we calculated the ratio from total counts. If a bias of 20‰ does not change the conclusions of the work, then it can be considered insignificant. But if the bias dominates the isotope effects seen in the work, as in many examples shown in this paper, then it must be dealt with even if it is smaller than the 2σ measurement uncertainties. The bias is a systematic shift, not a random fluctuation.
For most of the data sets in this paper, we have used total counts to calculate less-biased estimators of the true isotope ratios. But even total counts can lead to biased isotope ratios. Analysts must monitor the number of counts acquired during each measurement and decide if the bias in the calculated ratios is small enough to be insignificant (cf. Ogliore et al. 2011). If there are not enough counts in the denominator to assure a sufficiently unbiased result, there are other less-biased estimators available. Ogliore et al. (2011) discuss the Beale’s estimator (Beale 1962), which we applied to the 10Be-10B data reported by MacPherson et al. (2003). Beale’s estimator noticeably reduced the bias in some of the data, which was initially reduced using total counts. Coath and Steele (2013) discuss another way to handle the bias introduced by a small number of counts.
For future studies, experimental design must take into account the issue of ratio bias. The goal must be to have enough counts of the denominator isotope so that ratio bias is insignificant, i.e., much smaller than the total uncertainty of the measured ratio. Data can still be collected using a large number of measurement cycles to evaluate the performance of the mass spectrometer, heterogeneities in the sample, and electronic noise. The data can be edited, and time interpolation can be applied as before. But if the number of counts per cycle is too low (see Ogliore et al. 2011), the data for all cycles should be totaled before calculating the ratios to minimize the bias. If the number of total counts is still too low, a more sophisticated analysis is required to estimate accurate ratios (e.g., Beale’s estimator). Uncertainties calculated from variations among individual ratios are still appropriate to estimate the total uncertainty in the measurement, as discussed above.
As we discussed in the Bias in Isochrons section, ratio bias can propagate into the slope of the isochron in different ways. For example, when an external mass fractionation correction is applied to biased ratios (e.g., 10Be-10B and 53Mn-53Cr systems), the resulting isochron is typically positively biased. On the other hand, if an internal mass fractionation correction is applied to biased ratios (e.g., 26Al-26Mg and 60Fe-60Ni systems), the isochron can be either positively or negatively biased, depending on the ratio used for normalization. For instance, in the 60Fe-60Ni system, biased ratios can produce a large positive bias in the isochron when the normalizing isotope is 61Ni or a small negative bias in the isochron when the normalizing isotope is 62Ni (e.g., Fig. 2). The tables presented above show that when the data are reduced properly, there is no systematic difference in the results as a function of normalizing isotope. This emphasizes the importance of internal consistency checks during data reduction.
Acknowledgments— We thank Trevor Ireland, associate editor Ian Lyon, and an anonymous reviewer for helpful reviews. This work was supported by NASA grant NNX11AG78G to G. R. Huss and by NESSF Fellowship NNX11AN62H to M. Telus. This is Hawai’i Institute of Geophysics and Planetology publication No. 1986 and School of Ocean and Earth Science and Technology publication No. 8737.