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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
Date created: 7/20/2001 |
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
,
for a test with size 0.10,
the detection rate (of inhomogeneity)
exceeds 99 percent and
the estimated standard deviation
closely tracks