A basic tenant of EPMA is that the sample material must be homogeneous for the matrix correction to work in a meaningful way. But to be strictly correct here, the sample needs to only be locally homogeneous, on the scale of the interaction volume. With that caveat aside, my current research project features some challenging materials I wish to analyse for bulk composition, and I was hoping someone could point me in a helpful direction. It's more like a sampling problem than an EPMA, but we'll see...
The material of interest is mostly iron, with trace metals on the order of 100-5000 ppm (w/w). All of the trace metals behave differently depending on their siderophile nature, and some disperse very nicely throughout the iron, while others congregate in small blebs 1-10um across, while others still show a tendency to do both. The most interesting (for my work) trace elements present are heavy volatile metals, such as Ag, In, Cd, Bi, Pb, and Tl. Running some monte carlo simulations in Casino show small (<1um deep) interaction volumes at 20, and even 25 KV, so while there will be some mixing for some analysis points, on the whole, I think moving forwards we can assume that the probe is sampling the two populations separately and relatively well.
However, my question is whether anyone has any thoughts on the best way to recover the bulk composition of this material, especially when working close to the detection limit.
For clarity, let's just consider an arbitrary trace element of interest:
Essentially, the material contains some kind of bi-modal distribution, with relative proportions in the two phases and some kind of distribution between them.
If the abundance in both phases is above detection then the whole distribution is above detection, and taking an average of all analytical points matches the bulk average of the material. This works as expected, provided the material is sampled enough times. There may be a tail extending below the detection limit, but it's probably insignificant.
If the abundances in both phases are all below detection, then everything can be trivially discarded.
When the abundance in the bulk matrix is below detection, but in the higher abundance blebs it is above detection, then we have truncated the distribution at an arbitrary point. As a consequence averaging the remaining distribution over estimates (by about 2x, but 1.5x is more typical) the bulk composition for this trace element if the below-detection results are thrown away, which most people would do without thinking about it.
By this point most people would tell me to increase the counting times, crank the beam current, triple-check everything, lower the detection limits and do it all again!
However...
Is anyone aware of any strategy that would enable me to salvage the analyses done so far and model the below-detection numbers in some kind of half-meaningful way.
I made a simple stats sampling model today (just stats, no PENEPMA) that shows in general that the following strategy kind of works:
- Considering one trace element in a set of analyses (say 100 points) on the same material.
- Replace all below-detection values with half the detection limit for that element.
- Take the average.
- If the average is below detection, discard the dataset for this element, it's probably no good.
- If a certain proportion of the measurements are below detection, then discard the dataset, it's probably no good. I don't have a strategy for determining this percentage: it feels like 1-10% would be OK, and anything over 50% would definitely not be OK. My simple stats model for two Gaussian distributions shows that as expected it mostly depends on the relative proportions of the concentrations in the two phases, but also importantly on the width of each distribution: broad distributions are truncated more by the detection limit and the over estimation in the average value is the worst.
Is this just a complete waste of time? It feels like it, and seems much too arbitrary for my liking.
Any thoughts, recommended reading, or advice will be very much appreciated, thank you,
Ash
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