The double spike (DS) technique is a prevalent method in high-precision stable isotope ratio measurements, crucial across environmental, geological, medical, and forensic studies. This technique involves adding a tracer solution containing two enriched isotopes to samples. DS is valued for its ability to correct for mass-dependent isotope fractionation during both instrumental analysis (mass bias) and sample preparation, offering faster and more precise isotope ratio determinations compared to methods like elemental doping (ED) and standard-sample bracketing (SSB). Its advantage lies in tolerating analyte losses during purification if the DS is equilibrated beforehand, eliminating the need for strict matrix matching between standards and samples because mass bias correction is internal. However, the DS method's complex inversion calculations, requiring solving nonlinear equations to separate analyte and spike with high precision, lack consensus on optimal procedures. A key area of debate is data averaging: preinversion averaging (averaging raw data before inversion) versus postinversion averaging (averaging inverted data from individual cycles). While statistical tests are standard to remove outliers, mass bias drift can complicate this, potentially leading to biased results with preinversion averaging. Postinversion averaging is simpler with instruments having customizable data reduction software but presents challenges with others where it is more laborious and prone to errors. Previous claims that both averaging methods should yield the same results if outliers are not rejected lacked comprehensive testing. This study uses a large dataset (N = 1116) of chromium isotope data to compare preinversion and postinversion averaging, assessing the impact of outlier rejection. It also investigates the effects of other factors, such as variable spike:analyte ratios, interferences, and deviations from the exponential mass fractionation law, using synthetic data.

Numerous laboratories worldwide have established protocols for chromium isotope analysis using the double spike technique (refs 3, 12–26). Chromium possesses four stable isotopes: ⁵⁰Cr, ⁵²Cr, ⁵³Cr, and ⁵⁴Cr. In double spike analysis, ⁵⁰Cr and ⁵⁴Cr serve as spike isotopes, with ⁵³Cr/⁵²Cr expressing analyte isotope composition (δ⁵³Cr). Isobaric interferences (⁵⁴Fe on ⁵⁴Cr, ⁵⁰V and ⁵⁰Ti on ⁵⁰Cr) require mathematical subtraction from measured signals, introducing uncertainty due to potential deviations from assumed natural abundance ratios in real-world samples. Double spike data inversion involves solving nonlinear equations, often utilizing an exponential function to describe mass spectrometer mass bias. This necessitates iterative processes to refine mass bias and sample isotope ratios. Existing literature emphasizes the exponential law for mass bias, but alternative laws like equilibrium, power, and generalized power laws also exist (refs 8, 46, 47). Previous studies address aspects of double spike methodology; however, a comprehensive analysis encompassing the combined effects of pre/post-inversion averaging, spike ratios, interferences, and deviations from assumed fractionation laws remains lacking.

Chromium isotope analysis was performed using a double spike technique on multicollector mass spectrometers. The experimental procedures followed standard methods established in the literature, involving ion-exchange chromatography to separate Cr from interfering elements (Fe, Ti, V). The double spike inversion involved solving a system of nonlinear equations to determine mass bias and analyte isotopic composition. The study used a large dataset (N=1116) of chromium isotope data to evaluate the effects of preinversion versus postinversion averaging. Outlier rejection methods were applied to both raw and inverted data. The impact of varying spike:analyte ratios was explored using synthetic data, adjusting this ratio across a wide range (0.05–20). The effects of interference elements (Fe, Ti, V) were also investigated using synthetic data, considering the uncertainties in their isotopic compositions. The synthetic data calculations included: (1) calculation of the isotopic composition of the spike-analyte mixture, (2) addition of interferences, (3) application of mass bias, and (4) inversion using the iterative method. The deviation between input and inverted δ⁵³Cr values quantified the errors. Different mass fractionation laws (exponential, equilibrium, power, generalized power) were tested to assess their impact on the accuracy of the inverted δ⁵³Cr. A Monte Carlo simulation was employed to evaluate the combined effects of uncertainties in the isotopic compositions of interfering elements on the accuracy and precision of inverted δ⁵³Cr. Finally, the influence of sample preparation-induced isotope fractionation, under different Cr yields and deviating from the exponential law, was evaluated.

The study revealed several crucial factors influencing the accuracy and precision of double spike isotope analysis. Preinversion averaging of raw isotope ratios, even with outlier rejection, introduced errors ranging from -0.05% to +0.05%, a mean of 0.00%, and a standard deviation of 0.020%. These errors are significant considering modern mass spectrometers' precision. The maximum errors were correlated with both the standard error of the mean (SEM) of δ⁵³Cr and signal intensities, highlighting the greater impact of outlier removal when measurement scatter is larger. Preinversion averaging also overestimated the 2SEM of inverted δ⁵³Cr by as much as 0.1%. Postinversion averaging was consistently superior. Variation in spike:analyte ratios between 0.05 and 20 had insignificant effects (<0.001%) on inverted δ⁵³Cr accuracy, but lower ratios required more iterations for convergence. Uncertainties in the isotopic compositions of interfering elements (Fe, Ti, V) significantly affected inverted δ⁵³Cr accuracy. The errors caused by Fe and Ti were substantial, especially at lower spike:analyte ratios. The effect of V was limited due to its low natural abundance. The combined effect of Fe, Ti, and V uncertainties was evaluated using Monte Carlo simulations. Results indicated that with a spike:analyte ratio above 0.5 and low interference:⁵²Cr ratios (⁵⁶Fe/⁵²Cr < 0.2, ⁴⁹Ti/⁵²Cr < 0.04, ⁵¹V/⁵²Cr < 1), uncertainties in inverted δ⁵³Cr due to these interferences were below 0.02%. Deviations of the instrumental mass bias from the assumed exponential law introduced large errors but these were considered unlikely in most situations, as they were constant over time and would cancel out between samples and standards. Sample preparation-induced fractionation could also cause significant errors, especially when Cr recovery was very low (<70%). Achieving >70% Cr yield effectively minimized these errors.

The findings directly address the research question regarding optimal data processing strategies in double spike analysis. The superiority of postinversion averaging over preinversion averaging is clearly demonstrated, providing practical guidelines for improving the accuracy and precision of isotope measurements. The impact of spike:analyte ratio, interference elements, and deviations from assumed fractionation laws on inverted δ⁵³Cr values highlights the need for careful experimental design and rigorous data processing. The observed correlations between errors and factors like SEM and signal intensities reveal the subtle interplay of various factors influencing data quality. The use of synthetic data allowed for systematic investigation of individual parameters, providing clear recommendations for minimizing errors in the double spike technique. The Monte Carlo simulation effectively quantified the combined effects of multiple uncertainty sources, offering a valuable approach for risk assessment in similar studies. This work improves understanding of the double spike method, contributing to higher confidence in isotope analysis results across diverse applications.

This study emphasizes the importance of postinversion averaging for accurate and precise double spike isotope analysis. Maintaining an optimal spike:analyte ratio and minimizing interference from Fe, Ti, and V through careful sample preparation are crucial for reducing errors. Monitoring convergence during iterative calculations and ensuring high Cr yields are also critical. The results are generalizable to other double-spike-compatible isotope systems. Further research could focus on developing more sophisticated models for mass bias correction or investigating the impact of different sample preparation techniques on isotope fractionation.

The study primarily focused on chromium isotope analysis. Although the findings are expected to generalize to other elements, validation with different isotope systems is needed. The synthetic data analyses relied on assumptions about the distribution of isotopic compositions of interfering elements. The real-world distribution might differ slightly, which could affect the conclusions. The Monte Carlo simulation considered only mass-dependent fractionation during sample preparation, and other fractionation processes may influence the results. The study focused on specific interferences; others may need to be considered for different sample types.