Exponential isotope fractionation in TIMS can be linearized by binomial expansion. The linearization can be used to model a particular calculation of the isotopic composition of an element with at least three isotopes by ID-MS technique (here applied to strontium single-spike ID). In the equivalence of the spiking ratio for all the isotopic ratios in a sample-spike mixture, the true values of the mixture can be expressed in terms of the measured values and of the instantaneous relative deviations ?1, ?2, ?3 of the measured values of 87Sr/86Sr, 84Sr/86Sr and 88Sr/86Sr ratios from the respective true values. The instantaneous relative deviations of the 84Sr/86Sr and 88Sr/86Sr ratios can be expressed in terms of the relative deviation of the 87Sr/86Sr ratio at the same instant, and of increments which are functions of the instantaneous fractionation factor f and of the natural logarithm of the mass ratio 84Sr/ 87Sr and 88Sr/ 87Sr, respectively. A system of two linear equations can therefore be written in the two unknowns ?1 and f , which can be solved to calculate the other two relative deviations ?2, ?3 . Since the instantaneous relative deviations can also be expressed as the products between the instantaneous fractionation factor and the natural logarithm of the mass ratio between the isotope at the numerator and the isotope at the denominator, respectively, the pairs ?i, ln(mi/m86) for the three ratios fall on a straight line which passes through the origin, the angular coefficient giving the instantaneous fractionation factor f. In this calculation, we have used the true values of the isotopic ratios in the sample. Actually, these values are unknowns. Nevertheless, we can scan values for these ratios, generate systems of equations, calculate the relative deviations ?1, ?2, ?3 , plot the pairs ?i, ln(mi/m86) to define straight lines, and calculate how far each straight line passes from the origin. We can therefore identify the straight line which passes closest to the origin, and the values of 84Sr/86Sr, 87Sr/86Sr and 88Sr/86Sr we have used in that calculation should be the values closest to the respective values in the natural sample. The isotopic composition of a spike enriched in 84Sr has been determined using the method proposed by Cavazzini (2005). An algorithm has been designed and simulated calculations have been performed, which have shown that the 87Sr/86Sr of the sample is determined within 0.004 - 0.006% from the true value. The algorithm has been used to determine the isotopic composition of a modern coral (Cladocora Cespitosa) and of the NIST standard reference material 611.
Linearization of the exponential law of isotope fractionation: a minimization technique to determine the isotopic composition of elements by isotope dilution mass spectrometry
Cavazzini Giancarlo
2015
Abstract
Exponential isotope fractionation in TIMS can be linearized by binomial expansion. The linearization can be used to model a particular calculation of the isotopic composition of an element with at least three isotopes by ID-MS technique (here applied to strontium single-spike ID). In the equivalence of the spiking ratio for all the isotopic ratios in a sample-spike mixture, the true values of the mixture can be expressed in terms of the measured values and of the instantaneous relative deviations ?1, ?2, ?3 of the measured values of 87Sr/86Sr, 84Sr/86Sr and 88Sr/86Sr ratios from the respective true values. The instantaneous relative deviations of the 84Sr/86Sr and 88Sr/86Sr ratios can be expressed in terms of the relative deviation of the 87Sr/86Sr ratio at the same instant, and of increments which are functions of the instantaneous fractionation factor f and of the natural logarithm of the mass ratio 84Sr/ 87Sr and 88Sr/ 87Sr, respectively. A system of two linear equations can therefore be written in the two unknowns ?1 and f , which can be solved to calculate the other two relative deviations ?2, ?3 . Since the instantaneous relative deviations can also be expressed as the products between the instantaneous fractionation factor and the natural logarithm of the mass ratio between the isotope at the numerator and the isotope at the denominator, respectively, the pairs ?i, ln(mi/m86) for the three ratios fall on a straight line which passes through the origin, the angular coefficient giving the instantaneous fractionation factor f. In this calculation, we have used the true values of the isotopic ratios in the sample. Actually, these values are unknowns. Nevertheless, we can scan values for these ratios, generate systems of equations, calculate the relative deviations ?1, ?2, ?3 , plot the pairs ?i, ln(mi/m86) to define straight lines, and calculate how far each straight line passes from the origin. We can therefore identify the straight line which passes closest to the origin, and the values of 84Sr/86Sr, 87Sr/86Sr and 88Sr/86Sr we have used in that calculation should be the values closest to the respective values in the natural sample. The isotopic composition of a spike enriched in 84Sr has been determined using the method proposed by Cavazzini (2005). An algorithm has been designed and simulated calculations have been performed, which have shown that the 87Sr/86Sr of the sample is determined within 0.004 - 0.006% from the true value. The algorithm has been used to determine the isotopic composition of a modern coral (Cladocora Cespitosa) and of the NIST standard reference material 611.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
