[Ifeffit] Asymmetric error bars in IFeffit

[Scott Calvin]

Hi all,

I'm puzzling over an issue with my latest analysis, and it seemed like  
the sort of thing where this mailing list might have some good ideas.

First, a little background on the analysis. It is a simultaneous fit  
to four samples, made of various combinations of three phases.  
Mossbauer has established which samples include which phases. One of  
the phases itself has two crystallographically inequivalent  absorbing  
sites. The result is that the fit includes 12 Feff calculations, four  
data sets, and 1000 paths. Remarkably, everything works quite well,  
yielding a satisfying and informative fit. Depending on the details,  
the fit takes about 90 minutes to run. Kudos to Ifeffit and Horae for  
making such a thing possible!

Several of the parameters that the fit finds are "characteristic  
crystallite radii" for the individual phases. In my published fits, I  
often include a factor that accounts for the fact that a phase is  
nanoscale in a crude way: it assumes the phase is present as spheres  
of uniform radius and applies a suppression factor to the coordination  
numbers of the paths as a function of that radius and of the absorber- 
scatterer distance. Even though this model is rarely strictly correct  
in terms of morphology and size dispersion, it gives a first-order  
approximation to the effect of the reduced coordination numbers found  
in nanoscale materials. Some people, notably Anatoly Frenkel, have  
published models which deal with this effect much more realistically.  
But those techniques also require more fitted variables and work best  
with fairly well-behaved samples. I tend to work with "messy" chemical  
samples of free nanoparticles where the assumption of sphericity isn't  
terrible, and the size dispersion is difficult to model accurately.

At any rate, the project I'm currently working on includes a fitted  
characteristic radius of the type I've described for each of the  
phases in each of the samples. And again, it seems to work pretty  
well, yielding values that are plausible and largely stable.

That's the background information. Now for my question:

The effect of the characteristic radius on the spectrum is a strongly  
nonlinear function of that radius. For example, the difference between  
the EXAFS spectra of 100 nm and 1000 nm single crystals due to the  
coordination number effect is completely negligible. The difference  
between 1 nm and 10 nm crystals, however, is huge.

So for very small crystallites, IFeffit reports perfectly reasonable  
error bars: the radius is 0.7 +/- 0.3 nm, for instance. For somewhat  
larger crystallites, however, it tends to report values like 10 +/-  
500 nm. I understand why it does that: it's evaluating how much the  
parameter would have to change by to have a given impact on the chi  
square of the fit. And it turns out that once you get to about 10 nm,  
the size could go arbitrarily higher than that and not change the  
spectrum much at all. But it couldn't go that much lower without  
affecting the spectrum. So what IFeffit means is something like "the  
best fit value is 10 nm, and it is probable that the value is at least  
4 nm." But it's operating under the assumption that the dependence of  
chi-square on the parameter is parabolic, so it comes up with a  
compromise between a 6 nm error bar on the low side and an infinitely  
large error bar on the high side. Compromising with infinity, however,  
rarely yields sensible results.

Thus my question is if anyone can think of a way to extract some sense  
of these asymmetric error bars from IFeffit. Here are possibilities  
I've considered:

--Fit something like the log of the characteristic radius, rather than  
the radius itself. That creates an asymmetric error bar for the  
radius, but the asymmetry the new error bar possesses has no  
relationship to the uncertainty it "should" possess. This seems to me  
like it's just a way of sweeping the problem under the rug and is  
potentially misleading.

--Rerun the fits setting the variable in question to different values  
to probe how far up or down it can go and have the same effect on the  
fit. But since I've got nine of these factors, and each fit takes more  
than an hour, the computer time required seems prohibitive!

--Somehow parameterize the guessed variable so that it does tend to  
have symmetric error bars, and then calculate the characteristic  
radius and its error bars from that. But it's not at all clear what  
that parameterization would be.

--Ask the IFeffit mailing list for ideas!

Thanks!

--Scott Calvin
Sarah Lawrence College
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