Revised exponential model for mass bias correction using an internal standard for isotope abundance ratio measurements by multi-collector inductively coupled plasma mass spectrometryw Douglas C. Baxter, a Ilia Rodushkin, a Emma Engstro¨m b and Dmitry Malinovsky b Received 8th December 2005, Accepted 22nd February 2006 First published as an Advance Article on the web 10th March 2006 DOI: 10.1039/b517457k An internal standard (IS) can be used to account for moderate, matrix-related shifts in mass bias using multi-collector inductively coupled plasma mass spectrometry through the empirical, linear relationship between measured isotope abundance ratios for different elements in ln-ln space. Unfortunately, erroneous mass bias corrected isotope abundance ratios may be returned by the model, requiring artificial adjustment of the true isotope abundance ratio of the IS. Although inadequate correction for peak tailing has been convincingly used to explain this problem, our analysis of the literature describing the development of the mass bias correction model using an IS reveals the presence of a source of systematic error. The origin of this error is purely mathematical and is eliminated in the revised model presented, in which mass bias corrected isotope abundance ratios are independent of the isotopic composition of the IS. An expression for computing the total combined uncertainty in the corrected ratio, incorporating contributions from the linear model, the isotopic reference material, and measurements of analyte element and IS in the sample, is also derived. Introduction Although application of multi-collector inductively coupled plasma mass spectrometry (MC-ICP-MS) is growing ra- pidly, 1,2 there is no current consensus on how best to deal with the problem of mass bias. While affecting all forms of MS, the need to correct accurately for mass bias in MC-ICP- MS is particularly acute due to: (i) the magnitude of the effect, often several % per atomic mass unit; and (ii) the frequent need to quantify minor variations, at the sub-permil (%) level, in isotopic compositions. One empirical strategy is the use of an internal standard element to facilitate mass bias correction by external normal- ization. Originally proposed by Longerich et al., 3 this strategy has evolved as a result of important contributions by Mare´- chal et al. 4 and Woodhead. 5 Specifically, measures are taken to ensure variable mass bias over the day, and each isotope ratio of interest is calibrated individually. 4,5 Consequently, accuracy and precision comparable to those furnished by the double spike technique and thermal ionization mass spectrometry have been claimed. 5,6 Such claims, on the other hand, have been seriously challenged by the results of other investiga- tions. 6,7 In this work, a revised empirical model for mass bias correction in MC-ICP-MS using the internal standard method is presented. Statistical analysis is utilized to determine an optimal, yet straightforward, data treatment protocol for minimizing uncertainty magnification during mass bias cor- rection. Theory Mass bias correction using an internal standard To correct for instrumental mass bias, the measured isotope abundance ratio of analyte (subscript X) in an isotopic reference material (RM), r X,RM , is related to the true value, R X,RM , via the so-called K-factor r X;RM ¼ R X;RM K ; K ¼ R X;RM r X;RM ð1Þ The determined K-factor can then be applied to correct subsequent isotope abundance ratio measurements on samples for mass bias R X,corr = K  r X,sample (2) At present, the most popular functional form for K is arguably the exponential model of Russell et al. 8 K ¼ m 2 m 1   f ð3Þ where m 1 and m 2 are the masses of the lighter and heavier isotopes, respectively, and f is a parameter that must be determined to describe the effect of mass bias on the measure- ment. This can be achieved by analyzing an isotopic reference material for the element of interest, mixed with an internal standard, at regular intervals during the measurement session; the same internal standard is added to all samples as well. a Analytica AB, Aurorum 10, SE-977 75 Lulea ˚, Sweden. E-mail: douglas.baxter@analytica.se b Division of Applied Geology, Lulea ˚ University of Technology, SE-971 87 Lulea ˚, Sweden w Electronic supplementary information (ESI) available: data bias table, equations, data, comments on eqn. 17 and delta values. See DOI: 10.1039/b517457k This journal is  c The Royal Society of Chemistry 2006 J. Anal. At. Spectrom., 2006, 21, 427–430 | 427 TECHNICAL NOTE www.rsc.org/jaas | Journal of Analytical Atomic Spectrometry