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., ^{10}Be-^{10}B and ^{53}Mn-^{53}Cr), the correction for mass fractionation is done “externally.” (The ^{53}Mn-^{53}Cr system is typically treated as a one-ratio system because only ^{52}Cr and ^{53}Cr 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., ^{26}Al-^{26}Mg, ^{60}Fe-^{60}Ni 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 ^{26}Al-^{26}Mg and ^{60}Fe-^{60}Ni systems. Figure 2a shows magnesium isotopes plotted as delta values with ^{24}Mg 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 δ^{25}Mg and δ^{26}Mg from low counts in ^{24}Mg 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 ^{25}Mg/^{24}Mg ratio, represented by the dashed lines, the inferred value for δ^{26}Mg will be negative. This will decrease any real excess of radiogenic ^{26}Mg inferred for the measurement. However, if one does an external mass-fractionation correction, the system will behave like the ^{10}Be-^{10}B and ^{53}Mn-^{53}Cr systems and any bias will be positive, increasing the inferred excess of radiogenic ^{26}Mg. In general, because the ^{24}Mg count rate is approximately 10 times higher than the count rates of ^{25}Mg and ^{26}Mg, ratio bias is not a significant issue in the ^{26}Al-^{26}Mg 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 ^{25}Mg were radiogenic). If the ^{26}Mg/^{24}Mg 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 ^{25}Mg/^{24}Mg would have a positive bias.

Figure 2b illustrates the nickel isotopes plotted as delta values normalized to ^{61}Ni. Again, mass fractionation is shown by the solid lines and gray symbols. The bias introduced by low counts of ^{61}Ni is the same for δ^{60}Ni and δ^{62}Ni. In this case, if the mass fractionation correction is done using the measured δ^{62}Ni, δ^{60}Ni will be overcorrected (dashed lines). The inferred excess in δ^{60}Ni will be the sum of the bias in the ^{60}Ni/^{61}Ni ratio and that in the ^{62}Ni/^{61}Ni ratio. Because ^{61}Ni is the least abundant isotope being measured, the ratio bias will be large in an absolute sense as well. For the ^{61}Ni 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 ^{62}Ni. This case is analogous to that for the magnesium isotopes (Fig. 2a). A bias in the ^{61}Ni/^{62}Ni ratio will result in an over correction for mass fractionation in the ^{60}Ni/^{62}Ni ratio, which in turn will produce a negative isotope anomaly in δ^{60}Ni. However, once again, if an external mass fractionation correction is used, any bias in the ^{60}Ni/^{62}Ni ratio will be positive. For a given measurement, the total bias in δ^{60}Ni after an internal mass-fractionation correction introduced by normalizing to ^{61}Ni will be larger in magnitude than that introduced by normalizing the ^{62}Ni for two reasons: 1) the abundance of ^{61}Ni is approximately three times lower than the abundance of ^{62}Ni, and 2) the effect of bias on the mass fractionation correction is twice as great in the ^{61}Ni 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 ^{53}Mn-^{53}Cr 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 ^{26}Al-^{26}Mg system, isochrons are typically generated from several minerals with different Al/Mg ratios. A slightly negative bias in the ^{26}Mg/^{24}Mg ratio will correlate with the inverse of the ^{24}Mg 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 ^{26}Al/^{27}Al initial ratio. Fortunately, for most ^{26}Al-^{26}Mg measurements, the count rates are sufficiently high that the slight negative bias on the ratios is not significant.

For the ^{60}Fe-^{60}Ni system where the normalizing isotope is ^{61}Ni, 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 ^{62}Ni 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 ^{60}Fe-^{60}Ni 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 ^{60}Fe-^{60}Ni systematics of an E-chondrite sulfide.