Detection and quantification of isotopic ratio inhomogeneity

We quantify the spatial variation of the ratio of two isotopes within a material based on Secondary Ion Mass Spectrometry (SIMS) data. At many spatial locations, a detector counts each of two isotopes of a chemical element. At each location, we predict the less abundant isotope count in terms of the measured value of the more abundant isotope count and the estimated mean isotopic ratio. At each location, the expected value of the prediction error depends on the difference between the actual isotopic ratio at that location, and the estimated spatial mean value of the isotopic ratio. To get weighted residuals, the difference between the measured and the predicted value is divided by an estimate of its root mean square value. To estimate the spatial standard deviation of the isotopic ratio, we equate the sum of the squared weighted residuals to its approximate expected value. The approximate expected value is obtained by a bootstrap resampling method. Based on the estimated null distribution of the estimated spatial standard deviation, we test the hypothesis that the isotopic ratio is constant throughout the sample. To check the validity of our methods, we analyze SIMS data collected from a homogeneous chromium sample. Results are consistent with the hypothesis of homogeneity. We simulate data corresponding to a sample where the isotopic ratio has a binary distribution; when the standard deviation of the binary distribution exceeds twice the 86th percentile of the null distribution, detection of inhomogeneity is almost certain. Further, the expected value of the estimated standard deviation closely tracks the actual standard deviation.