Absolute isotope ratios - Analytical solution for the determination of calibration factors for any number of isotopes and isotopologues

Absolute isotope ratios cannot be determined directly by mass spectrometry. The measured ion intensities are biased by many effects like mass fractionation or discrimination and amplifier gain. To correct the ion intensities, calibration factors are needed. These calibration factors can be derived by comparing the known isotope ratios of a reference material and its measured intensity ratios, provided that these materials have been characterized traceable to the International System of Units (SI). As mentioned above, absolute isotope ratios are not directly available but are necessary for calibration. Thus, if there is no well-characterized reference material or it is out of stock, another solution for this problem must be found. Nier [Nier, 1950] already presented a solution for this problem through gravimetrically prepared mixtures back in 1950. Mana and Rienitz [Mana and Rienitz, 2010] derived an analytical solution for the needed calibration factors for a system of three isotopes from this idea. Until then, calibration factors could only be determined iteratively. This paper shows how calibration factors can be derived from gravimetrically prepared mixtures for a system of any number of isotopes. In contrast to iterative methods, the method presented here is not affected by possible poor convergence or local minima. The subsequent uncertainty propagation is much more straightforward being another big advantage. Two examples are given to demonstrate the overall validity of this approach. In the first example, this approach is applied to the seven-isotope system of mercury. In the second, using carbon dioxide, it is shown that this method can also be applied to a system of twelve isotopologues. Additionally, we present an EXCEL (R) application GIMiCK, which allows every user to calculate calibration factors for any number of isotopes and isotopologues.