Kevin J. Coakley
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
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 root mean square value.
To estimate the spatial standard deviation of
isotopic ratio, we
the sum of
to its approximate expected value.
The approximate expected value is
obtained by a bootstrap resampling method.
null distribution of the
estimated spatial standard deviation of the isotopic ratios,
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
Results are consistent with the hypothesis
We simulate data corresponding to
a sample where the isotopic ratio has
a binary distribution.
that when the standard
deviation of the
exceeds twice the 86th percentile of the null distribution,
detection of inhomogeneity is almost certain.
the estimated standard deviation closely tracks
the actual standard deviation.
To clarify results,
is expressed as a function of a scaled spatial standard deviation.
The scaling factor is the sampling
error associated with the esimated spatial mean value of the
We submitted a paper
Chemometrics and Intelligent Laboratory Systems.
Figure 16: Sample histograms corresponding to simulated data where the isotopic ratio has a binary distribution. The standard deviation of the mixture distribution 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 .
Date created: 7/20/2001