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3.2.3 Detection and Quantification of Isotopic Ratio Inhomogeneity

Kevin J. Coakley

Charles Hagwood

Hung-kung Liu

Statistical Engineering Division, ITL

David S. Simons

Surface and Microanalysis Science Division, CSTL

Most chemical elements in nature are multi-isotopic; i.e. they exist in several atomic forms with the same number of protons but different number of neutrons in their nuclei. Geologic and biological processes can alter the isotopic ratio of particular isotopes in a sample. Also, isotopic ratios can be intentionally altered by enrichment schemes. Materials with constant isotopic ratios are said to be isotopically homogeneous. In an inhomogeneous material, the isotopic ratio varies from location to location.

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. The difference between the measured and predicted value is divided by an estimate of its standard deviation. The approximate standard deviation of the prediction error is computed by the propagation of the errors method. To estimate the spatial standard deviation of the isotopic ratio, we equate the sum of squared standardized 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 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. We find that 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 estimated standard deviation closely tracks the actual standard deviation.


Figure 13: Sample histograms corresponding to simulated data where the isotopic ratio has a binary distribution. The standard deviation of the mixture distribution ${\sigma}_r$ varies from 0 to 0.0006. The solid lines correspond to the values of the two isotopic ratios in the mixture. Mixing fractions are 0.95 and 0.05. For $\sigma_r > 0.0002$, for a test with size 0.10, the detection rate (of inhomogeneity) exceeds 99 percent and the estimated standard deviation ${\hat{\sigma}}_{r}$ closely tracks ${\sigma}_r$.

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Date created: 7/20/2001
Last updated: 7/20/2001
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