Hi Michel, On Tue, 20 Sep 2005, Michel Schlegel wrote:
Pardon my poor knowledge of EXAFS fitting and reliability of the results, but more generally, I wondered if EXAFS fitting in raw unfiltered k-space is always relevant.
I would not call your knowledge of EXAFS fitting "poor". I think you've pretty much got every point exactly right. For the original question (why prefer R-space v. k-space or even Q-space for fitting), the difference is only important when there are spectral content (k- and R-range) that you want to ignore. The content you'd want to ignore is most commonly the high-R shells that you don't (yet?) have a model for. That is, if your model could account for all spectral features, fitting in k- and R-space would be equivalent. For the more common case in which the model is more limited than the data, using R-space makes it very easy to specify which components to ignore. Fitting in Q-space is nearly identical to R-space, expect for an issue you point out later.... Being able to limit the spectral content in this way is entirely to get a good measure of the fit statistics. When fitting in original k-space, you cannot say "fit the first shell and ignore the fact that I'm not modelling the second shell". As Michel points out, the limited k-range and the physics of EXAFS does mean there is 'spectral bleeding', so that the frequencies (R-values) for a single 'shell are not perfectly sharp. Looking at any plot of |chi(R)| it is pretty obvious that the peaks are not delta functions, or even particularly sharp. As Michel put it "the contribution from non-modelled distant shells would affect the structural parameters from the modelled shells". This is unavoidable, but is also generally a small effect. The best things to do are to be aware of this possibility and try to model any significant further shells than those you're really willing to say you've got right. This is why we prefer R-space to Q-space and the older approach of 'Fourier Filtering'. It is often difficult and sometimes impossible to really isolate the 'First Shell' from the 'Second Shell', which can make it dangerous to compare isolated 'First Shells' from systems with different 'bleeding' of higher shells. Michel also wrote:
Finally, another issue - for which I would not lay out my neck, though - is the noise. In EXAFS, the signal to noise increases with k, and of course fitting in the raw k space is another way of dealing conveniently with the noise -or relative uncertainty, as we may name it. However, the FT is a way of filtering out some of this noise, and so maximise the signal from a shell. Likewise, fitting in filtered q-space may yield more accurate results, because some of the noise is filtered out. However, I concur that somehow the fact that the uncertainty on the high-q part of the filtered contribution os greater than in the low q-part should be implemented somehow.
The FT does not actually filter out noise. It can be used to filter out the highest frequencies, and so the sharp spikey spectra that oftens shows up in k-weighted chi(k) at high k (where the noise is larger than the signal). Sadly, this is the part of the noise we care the least about: there is also noise with the same frequency (hopefully lower amplitude!) as the signals we're analyziing. That's not removed by ignoring the high frequency components. In general, I would say that fitting in R-space is slightly preferred over fitting in Q-space and much preferred over fitting in k-space. The only real reason it is much preferred to k-space fitting is that you can systematically ignore shells that you are not modelling. Cheers, --Matt