Optimal double and triple spiking for high precision lead isotopic measurement

Conventional lead isotope ratio measurements by thermal ionization mass spectrometry (TIMS) are currently limited to external precisions of ∼0.1% (2σ) due to variations in instrumental mass discrimination, the magnitude of which can only be assessed using external standard or duplicate measurements. The goal of the double-spike technique, by contrast, is to circumvent this fractionation problem entirely by determining the mass discrimination factor ϵ directly for each sample. The unknown factor can be obtained after performing a second measurement on a sample aliquot to which a double spike of known, calibrated composition has been added. Practical applications of the double-spike technique up until now have used a mixed 207Pb–204Pb tracer with considerable success, but there appears to have been little or no attempt to optimize the composition of the tracer used. An optimal tracer composition must fulfil two criteria: (1) the ratio of the internal, measured errors to a posteriori errors—the error magnification factors—should be as small as possible; (2) the propagated errors should not be strongly influenced by the spike/sample ratio of the second, mixture run—the latter is important since sample size cannot, in general, be rigorously controlled in advance. Of equal importance to the tracer composition itself is the choice of the three mass-balance equation set used to solve for ϵ; geometrically, this is equivalent to defining a mixing line in a particular 3D isotope space. Potential spike compositions were examined in four such 3D isotope spaces, corresponding to cases where 208Pb,207Pb,206Pb and 204Pb in turn are common, denominator isotopes of the ratios on the three orthogonal axes. Within each isotope space there exist three possible double-spike families. Tracers composed of three stable isotopes of lead—so-called `triple spikesʹ—are also considered, of which there are three families per isotope space. These spike compositions were evaluated for a typical common lead composition, at first using geometric methods, and then refined by determining error magnification factors using an ion statistical error model. It is shown that the 207Pb–204Pb double spike used in previous studies, where ϵ is determined in 208,207,206Pb/204Pb isotope space, is one of the least suitable to use. Optimal tracer compositions for determining ϵ appear to be restricted to two families: The first is a family of 207Pb–204Pb–206Pb triple spikes with 207Pb/204Pb of around unity and the isotope space has 206Pb as the denominator isotope; the second family consists of 204Pb–207Pb double spikes, solved in 208,206,204Pb/207Pb isotope space. Both of these families yield error magnification factors on a posteriori, corrected 206Pb/204Pb ratios as low as ∼1.2, and are less than 2 for all but highly underspiked and overspiked mixture compositions.