Error-systematics of determining elemental isotopic abundance ratios by the molecular ion beam method: a case study for the simultaneous isotopic analysis of lithium and boron as Li2BO2+

Abstract

The simultaneous isotopic analysis of lithium and boron by the Li2BO2+ ion beam method involves measurements of two different molecular abundance ratios (say, Rj+/-delta(j) and Rk+/-delta(k)), and subsequently extensive calculations to arrive at the analyte isotopic ratios (say, L and Y). It is not presently known how the measurement errors (delta(j) and delta(k)) are transformed into the errors of analysis (deltaL and deltaY). This work addresses this question from fundamental considerations. In the literature, the calculations are sometimes simplified using Ri formulae based on Li2B16O2+ ions and then applying correction factors for the actual Li2BO2+ ions, but this procedure is not generally applicable. We show how equations based on true Li2BO2+ ions (with full isotopic variations of all the constituent elements) can be solved, and illustrate the procedure with several examples. These studies show that accuracy of analysis depends not only on the accuracies of measurements, delta(j) and delta(k), but also on the particular isotopic Li2BO2+ ion-pairs (j and k) used as the monitor pairs. Moreover, this dependence is shown to be different for the different isotopic ratios (L and Y) to be determined simultaneously. Therefore, proper selection of monitor molecular pairs is a requirement for avoiding larger (propagated) errors in the analysis. Similar arguments would, in fact, apply to any arbitrarily chosen case of determining two or an even greater number of isotopic abundance ratios (Ei's) by the molecular ion beam method, irrespective of whether the different analyte ratios, Ei's, relate to a single multi-isotopic element, or different elements.

Error magnification factors in the isotopic analysis of oxygen as O: possibility of determining true isotopic abundance ratios

“…either or both of δ Y and δ Z will be non‐zero). Moreover, as already demonstrated in the cases of some other examples of MIBM analysis,34,, 37 the errors of isotopic analysis (δ Y and δ Z ) are never expected to be the same as the experimental errors (δ j and δ k ). That is to say that, although it is always the experimental errors (either or both of δ j and δ k as non‐zero) which and only which can cause the results of analysis to be imperfect (either or both of δ Y and δ Z to have non‐zero values), the magnitudes of δ Y and δ Z will be controlled also by the data processing involved in analysis 34,.…”

“…Then, it can be shown that, for any given experimental errors (δ j and δ k ), the errors of analysis (errors of solutions δ Y and δ Z ) take the following forms:34 …”

“…The atomic analysis of oxygen is most conveniently carried out using one of its molecular ions (e.g. $\hbox{O}_{2}^{+}$ or CO + or $\hbox{CO}_{2}^{+}$ ) rather than its, as expected, atomic ions as the mass spectrometric monitor species,1–33 and the technique of analysis is referred to generally as the molecular ion beam method (MIBM) 34. That is, in practice, the actual abundance measurements are carried out on the chosen isotopic molecular ions instead of the required isotopic O + or O − ions of the sample (oxygen) to be analyzed.…”

“…That is to say that, although it is always the experimental errors (either or both of δ j and δ k as non‐zero) which and only which can cause the results of analysis to be imperfect (either or both of δ Y and δ Z to have non‐zero values), the magnitudes of δ Y and δ Z will be controlled also by the data processing involved in analysis 34,. 37 In other words, solutions of Eqns 1 and 2 not only give the desired results (Y and Z values), but also transform the experimental errors (δ j and δ k ) into the corresponding errors of analysis, δ E 's (here E includes both Y and Z), through multiplication factors:34,, 37 where the parameters $[\hbox{MF}]_{\rm j}^{\rm E}$ and $[\hbox{MF}]_{\rm k}^{\rm E}$ , which are referred to as the multiplication factors for the individual errors δ j and δ k , respectively, can be described as characteristic properties of Eqns 1 and 2, i.e. they are governed by variables and constants which define Eqns 1 and 2 34…”

“…Also, we can simplify Eqn. 4 as:34,, 37 where the parameter $[\hbox{REMF}]_{\rm jk}^{\rm E}$ is referred to as the error magnification factor of analysis (or the combined error multiplication factor), and is defined34 to be (numerically) equal to the ratio of actual error in analysis to the experimental errors as: …”