Factors Affecting the Robustness of Data Inversion for Stable Isotope Measurement Using the Double Spike Method: Insights from Chromium Isotope Analysis

The double spike (DS) technique enables high-precision stable isotope ratio measurements by correcting mass-dependent fractionation occurring during instrumental analysis and potentially during sample preparation. It applies to isotope systems with four or more isotopes (and some with three). DS offers advantages over elemental doping and standard-sample bracketing (SSB), including internal mass bias correction, reduced requirements for matrix matching, and fewer repeated measurements. Despite its robustness, practical aspects of DS data inversion remain debated. Analysts may average within-cycle raw ratios before inversion (preinversion averaging) or invert each cycle then average the inverted results (postinversion averaging). Mass bias drift during an analysis can confound traditional outlier rejection on raw ratios, potentially biasing preinversion averaging. Prior work suggested both averaging schemes should agree if no outliers are rejected, but this had not been tested with extensive real data. This study uses a large chromium isotope dataset (N = 1116) to test impacts of pre- vs postinversion averaging on accuracy and precision, and uses synthetic data to evaluate effects of spike:analyte ratio, unresolved interferences (Fe–Ti–V), and deviations from assumed fractionation laws during instrument analysis and sample preparation.

DS methods for Cr isotopes are well established across ~20 laboratories, with numerous methodological reports and applications cited (e.g., Schoenberg et al., 2008; Bonnand et al., 2011; and others). DS is generally favored relative to SSB and elemental doping due to internal correction of mass bias and lower time costs. Klaver and Coath (2019) argued preinversion and postinversion averaging should yield identical results absent outlier rejection, a claim not previously tested with extensive empirical datasets. Natural variability in isotopic compositions of interfering elements (Fe, Ti, V) is documented in the literature and can impact interference corrections when natural ratios are assumed.

- Chromium isotope system and DS setup: Cr has four stable isotopes (50Cr, 52Cr, 53Cr, 54Cr). A double spike enriched in 50Cr and 54Cr is mixed with samples; the analyte composition is expressed using 53Cr/52Cr as δ53Cr relative to NIST SRM-979. The known 50Cr/54Cr of the spike allows determination of instrumental mass bias, which is then used to correct analyte ratios. - Interference handling: Unresolvable isobaric interferences include 54Fe on 54Cr and 50Ti and 50V on 50Cr. These are subtracted mathematically using measured 56Fe, 49Ti, and 51V signals, assumed natural isotope ratios (e.g., 54Fe/56Fe ≈ 0.06370279; 50Ti/49Ti ≈ 0.9585875; 50V/51V ≈ 0.00244702), and the instrument mass bias determined during analysis. Residual Fe–Ti–V after chromatography can require post-measurement corrections and introduce uncertainty due to natural isotopic variability of the interferents. - DS inversion framework: Instrumental mass bias is modeled with an exponential law, R_measured = R_true (m1/m2)^β. The mixture ratio R_true is expressed as a mixture of spike and analyte ratios with mixing proportion p. The analyte ratios are related to the standard via an exponential fractionation law characterized by γ. Combining these relations yields three nonlinear equations (for two non-DS ratios), with unknowns p, β, and γ. For four-isotope systems (like Cr), the system is solvable uniquely. - Iterative solution (nested iteration): Steps include (1) initial mass-biased Fe, Ti, V ratios using a trial β and assumed natural ratios; (2) subtract interferences on masses 50 and 54; (3) correct 50Cr/52Cr, 53Cr/52Cr, 54Cr/52Cr for mass bias with current β; (4a) compute spike 54Cr/50Cr from interference- and bias-corrected ratios (with analyte removed), (4b) update β by comparing computed spike ratio to the spike’s true ratio; (5) iterate steps 3–4 until β converges; (6) compute analyte 53Cr/52Cr; (7) compute analyte 50Cr/52Cr and 54Cr/52Cr from the chosen fractionation law; (8) iterate until analyte 53Cr/52Cr converges; (9) compute the spike–analyte mixing proportion p. Convergence thresholds for γ and β were set to 1E-7. - Calculation methods and datasets: Effects of pre- vs postinversion averaging were assessed using measured raw data from 1116 natural samples (raw data in Supporting Information). Other factors (spike:analyte ratio; interference levels and isotopic compositions; fractionation laws) were evaluated using synthetic data generated by (1) mixing spike and analyte of specified δ53Cr; (2) adding interferences; (3) applying mass bias; and (4) inverting to obtain δ53Cr. Errors are differences between input and inverted δ53Cr. A Monte Carlo approach (5000 runs) assessed combined uncertainties from Fe, Ti, and V isotopic variations across specified interference/analyte ratios and spike:analyte ratios.

- Preinversion averaging errors: Using 1116 Cr isotope analyses, preinversion averaging introduced errors ranging from approximately −0.05 to +0.05 (units as reported), with mean 0.00 and SD ≈ 0.020 when raw outliers were rejected; when outliers were not rejected, SD ≈ 0.016 with the same mean. Maximum errors scale with the true analytical precision (2SEM) by roughly ±1.5× and correlate with signal intensity. - Uncertainty inflation under preinversion: Preinversion propagation overestimates 2SEM of inverted δ53Cr when mass bias drifts during analysis, on average by ~0.02 and up to ~0.1, with positive correlation to the 2SEM of raw ratios. - Spike:analyte ratio: Across ratios from 0.05 to 20, accuracy effects on inverted δ53Cr are negligible (<0.001). Lower spike:analyte ratios require more iterations to converge, emphasizing the need to monitor convergence for nonoptimal spiking. - Interference isotopic composition uncertainties: • Fe: Deviations in δ56Fe/54Fe from assumed values can significantly bias δ53Cr, with larger errors at higher 56Fe/52Cr and lower spike:analyte ratios (e.g., at 56Fe/52Cr = 1 and δ56Fe/54Fe = −3.5‰, error ≈ −0.12). • Ti: Variations in δ49/50Ti can cause notable errors (e.g., with 49Ti/52Cr = 0.2 and δ49/50Ti = −1.2‰, error ≈ 0.08), amplified at lower spike:analyte ratios. • V: Given low natural 50V abundance, δ51/50V variability has limited effect when spike:analyte > 0.5. - Monte Carlo synthesis: If 54Cr_spike/52Cr_sample ratio > 0.5 and interferences are kept low (56Fe/52Cr < 0.2; 49Ti/52Cr < 0.04; 51V/52Cr < 1), the uncertainty in inverted δ53Cr due to unknown Fe, Ti, V isotopic compositions is generally <0.02. - Fractionation law assumptions: • Instrumental mass bias: Deviations from the exponential law can cause large errors in inverted δ53Cr, though these are likely mitigated by periodic standard bracketing if instrument behavior is stable over time. • Sample preparation: If chromatography-induced fractionation deviates from the assumed law, errors increase at low Cr yields. Modeling an extreme scenario with highly fractionated lost Cr (7.6%) indicates achieving >70% Cr yield limits errors in inverted δ53Cr to <0.02.

The study demonstrates that the choice of averaging scheme materially affects DS inversion outcomes in multi-cycle measurements. Preinversion averaging biases results and inflates uncertainties due to mass bias drift in raw ratios, whereas postinversion averaging leverages DS correction cycle-by-cycle, yielding more accurate δ53Cr and realistic precision estimates. Synthetic tests clarify operational thresholds to minimize interference-driven errors: ensure sufficient spiking (54Cr_spike/52Cr_sample > 0.5) and keep Fe–Ti–V interference-to-52Cr ratios low (56Fe/52Cr < 0.2; 49Ti/52Cr < 0.04; 51V/52Cr < 1). Monitoring convergence becomes critical at low spike:analyte ratios. While deviations from the exponential mass bias law can be problematic, routine analysis of standards can track and correct such effects if they are stable over time. For sample preparation, high Cr recovery reduces risks from deviations in fractionation laws, safeguarding the fidelity of inverted δ53Cr. These insights enhance protocol robustness for Cr and other DS-compatible systems.

Postinversion averaging of multi-cycle DS Cr measurements is superior to preinversion averaging because (1) preinversion can introduce errors up to ~1.5× the internal analytical precision and (2) it overestimates uncertainties by ~0.02–0.1. Accurate results are supported by adequate spiking (54Cr_spike/52Cr_sample > 0.5), low Fe–Ti–V interference levels (56Fe/52Cr < 0.2; 49Ti/52Cr < 0.04; 51V/52Cr < 1), and careful convergence monitoring at low spike:analyte ratios. Potential errors from deviations in instrumental mass bias law can be mitigated by periodic standard analyses, while preparation-related deviations can be minimized by achieving >70% Cr recovery. Although focused on Cr, these conclusions generalize to other DS-enabled isotope systems, enabling high accuracy and precision through postinversion averaging and control of spike and interference ratios and preparation yields.

- Many factor assessments (spike:analyte ratio effects; interference impacts; fractionation law deviations) relied on synthetic data; real-world complexities could introduce additional variance. - The large errors predicted from deviations in instrumental mass bias law may be less applicable when frequent bracketing standards effectively remove constant biases; the study notes this mitigation. - Sample preparation error estimates depend on modeled scenarios (e.g., Rayleigh-type fractionation and an extreme 7.6% fractionation of lost Cr); actual fractionation behavior may vary among protocols and matrices. - Recommended thresholds for interference and spiking are empirically motivated from simulations and may require adjustment for different instruments or detector configurations.