Evaluation strategies and uncertainty calculation of isotope amount ratios measured by MC ICP-MS on the example of Sr

Abstract

This paper critically reviews the state-of-the-art of isotope amount ratio measurements by solution-based multi-collector inductively coupled plasma mass spectrometry (MC ICP-MS) and presents guidelines for corresponding data reduction strategies and uncertainty assessments based on the example of n((87)Sr)/n((86)Sr) isotope ratios. This ratio shows variation attributable to natural radiogenic processes and mass-dependent fractionation. The applied calibration strategies can display these differences. In addition, a proper statement of uncertainty of measurement, including all relevant influence quantities, is a metrological prerequisite. A detailed instructive procedure for the calculation of combined uncertainties is presented for Sr isotope amount ratios using three different strategies of correction for instrumental isotopic fractionation (IIF): traditional internal correction, standard-sample bracketing, and a combination of both, using Zr as internal standard. Uncertainties are quantified by means of a Kragten spreadsheet approach, including the consideration of correlations between individual input parameters to the model equation. The resulting uncertainties are compared with uncertainties obtained from the partial derivatives approach and Monte Carlo propagation of distributions. We obtain relative expanded uncertainties (U rel; k = 2) of n((87)Sr)/n((86)Sr) of < 0.03 %, when normalization values are not propagated. A comprehensive propagation, including certified values and the internal normalization ratio in nature, increases relative expanded uncertainties by about factor two and the correction for IIF becomes the major contributor.

Fig. 1

Flowchart specifying the individual data evaluation steps and their associated uncertainty contributors

Fig. 2

Uncertainty contributor of precision to U (k = 2) of n(⁸⁷Sr)/n(⁸⁶Sr) for different IIF correction approaches calculated for a series of standards, related to the total measured voltage for Sr with polynomial trend line

Fig. 3

Uncertainty contribution (k = 2) of blank correction to the uncertainty of n(⁸⁷Sr)/n(⁸⁶Sr), determined using standards with variable Sr concentration and one instrumental blank for the different IIF correction approaches

Fig. 4

n(⁸⁷Sr)/n(⁸⁶Sr), corrected for IIF by approach 1, for Sr-Rb standard mixtures with increasing Rb/Sr. Uncertainty bars reflect U(k = 2). The red solid line corresponds to the certified value of n(⁸⁷Sr)/n(⁸⁶Sr) in NIST SRM 987, the red dotted lines to the limits of its certified range (95 % CI)

Fig. 5

Expanded uncertainty contributor (k = 2) of the correction for residual Rb to the uncertainty of n(⁸⁷Sr)/n(⁸⁶Sr) versus the Rb/Sr voltage ratio measured in standards. There is no significant difference depending on the IIF correction approach

Fig. 6

n(⁸⁷Sr)/n(⁸⁶Sr) corrected by IIF correction approach 3 (SSB-Zr) for a series of standard with varying Zr/Sr ratio given as voltage ratio of both most abundant isotopes. Error bars correspond to expanded uncertainties (k = 2) without propagation of the uncertainty of the certified value. Measured voltages for both isotopes range from 2 to 8 V. The red solid line corresponds to the certified value of n(⁸⁷Sr)/n(⁸⁶Sr) in NIST SRM 987, the red dotted lines to the limits of its certified range (95 % CI)

Fig. 7

Absolute contributions of corrections to the expanded variance of n(⁸⁷Sr)/n(⁸⁶Sr) [= U ² (k(U) = 2)] for one wood digest sample using different IIF correction approaches (1, 2, 3); (A) without, (B) with propagation of uncertainty of estimated natural range (1B) or certified values (2B, 3B). Numbers on top of bars show corresponding relative expanded uncertainties U _rel (k = 2)

Fig. 8

n(⁸⁷Sr)/n(⁸⁶Sr) of two different wood digest samples evaluated by three different approaches with error bars showing U (k = 2) without propagation of normalization values and certified values. Parts a and b show different wood samples, which differ in their observed δ(⁸⁸Sr/⁸⁶Sr): +0.39 ‰ and –0.60 ‰ for samples a and b, respectively

Fig. 9

Hypothetical calculation of the effect of MDF in nature [expressed as δ(⁸⁸Sr/⁸⁶Sr)] on n(⁸⁷Sr)/n(⁸⁶Sr) depending on the IIF correction approach; assuming MDF follows Russell’s model. The straight line depicts a hypothetical original radiogenic Sr isotope ratio (prior MDF, e.g., the geogenic source of Sr) of 0.7090. After MDF has occurred, the isotopic composition of the sample can be found on the dashed line depending on its new δ(⁸⁸Sr/⁸⁶Sr). The black bullets represent the hypothetical result for [n(⁸⁷Sr)/n(⁸⁶Sr)]_internal and the open diamonds show the hypothetical results for [n(⁸⁷Sr)/n(⁸⁶Sr)]_SSB for the same hypothetical samples. Error bars are estimated expanded uncertainties: U _rel = 0.02 % (k = 2)