Double-spike inversion for three-isotope systems

Double spiking is conventionally used to make accurate determinations of natural mass-dependent isotopic fractionations for elements with four or more stable isotopes. Here we document a methodology which extends the effective application of double spiking to three isotope systems. This approach requires making a mixture with isotope ratios that lie on a 'critical curve' where the sample - double-spike mixing line and the tangent to the instrumental mass-bias curve are coincident. Inversion of the mixing equations for such a mixture leads to a solution for the sample fractionation which is independent (to first order) of the uncertainty in the instrumental mass-bias and, hence, independent of any mass-dependent artefacts in the measurement such as those produced by residual matrix not completely removed by prior chemical purification. In practice, mixtures can be made which yield an accuracy conservatively estimated to be ~ 0.005?/amu. The precision of the method is explored as a function of double-spike composition for Mg, Si and K isotope systems. We show that for Mg and Si measurement precision is not compromised by the compositions of viable critical mixtures nor by uncertainty magnification during inversion of the equations. Thus, double spiking provides a valuable means to obtain robust, high precision isotopic measurements of Mg and Si. For K, however, the low abundance of K in the optimal critical mixture places a significant practical limitation on the application of double spiking to analyses of this element.