[Ifeffit] Fit in R and k space

[Matt Newville]

Hi Anatoly,

On 9/20/05, Frenkel, Anatoly <frenkel at bnl.gov> wrote:
> I personally like this argument in favor of r-space fitting over k-space fitting. I am pretty sure it 
> was made by Ed Stern:
> 
> R-space allows to think of EXAFS in terms of radial distribution of atomic positions relative to 
> the central atom. Peaks in r-space usually correspond to atomic positions and the regions 
> between the peaks - to the interstitial regions in the structure..

OK, so far I agree.  We use R-space because we can interpret the peaks
as being due to shells of atoms.....  Whether we do the _fit_ there is
another question.

> (of course, there are 1000 counter-examples, but we are talking about some 'average, well 
> behaving' system where a peak is NOT an atomic AXAFS, nor it is caused by a 
> Ramsauer-Townsend resonance, etc., and a valley is NOT a destructive interference between 
> neighboring shells' - AF).

I'd focus on the 1000 counter-examples as the 'average'.  The
electrons in the interstitial region should give hardly any EXAFS at
all (being relatively few electrons to begin with, and being well
distributed in  space, the scattering from them is very much washed
out).  As a result the intensity at the 'valleys' in |chi(R)| very
often are dominated by a) spectral bleeding from the scattering from
atomic cores (that is, 'normal EXAFS') and b) partial cancellations
due to destructive interference.  This can be seen easily  by plotting
the contributions from different shells, even for simple metal-oxides:
there is always bleeding of spectral components.

> We know, on another hand, that FEFF approximation electron charge density as spherical
> and the potential as muffin tin.  This approximation is not as accurate in the interstitial region
> between the atoms (where potential profile in real systems may be anything but flat) as it is 
> near the atoms. Thus, intuitively, FEFF should do better job fitting 'peaks' in r-space rather 
> than fitting the interstitial. 

This is essentially the same argument in the work of Zev Kvitky and
Bud Bridges  (PRB 64 214108) where they show systematic differences
between data and Feff (versions 6 and 7)  for Ag, Au, and a couple
other systems (most heavy enough to have a Ramsauer-Townsend resonance
in F(k), by the way).

I agree that the muffin-tin model could be more severely wrong where
the scattering is weakest.  I also believe that Kvitky et al argue, as
you do, for 'interstitial electrons'.  But I also believe they ignore
MS contributions (notably, the triangular paths), which would also
give small-but-non-zero scattering in the region between
"well-isolated shells".   I think the jury is still out for why "the
valley between the first and second shell is fitted poorly", but I
wouldn't rule out the notion that the approximations for the MS
contributions were not accurate enough.

>However, the r-factor and chi square do not really care. 

The R-factor does actually weight by the data so it is not
particularly sensitive to the parts of the spectra where |chi(R)| is
small.  So in a sense, it does care.
 
> Technically, if two models give similar r-factors and chi-square, the one that has better fit in 
> the peak region, the other - in the interstitial, one will examine the r-space fit and choose the 
> one that fits peaks better over the one that fits interstititial region better. 

Hmmm, maybe, but it seems fishy to me, as it's not clear what is
acceptable and what is not.

> However, this information, localized in r-space, is distributed in k-space. Thus, if you look at 
> the graph where data and fit are shown together, either in raw unfiltered k-space or in
> q-space, the misfit is difficult to interpret. It is, however, possible to interpret the misfit in 
> r-space, and that what often helps to refine the model or choose the better one.

Do you have an example of two models that give similar chi-square,
differ most strongly in the valleys in |chi(R)| and that you can
clearly distinguish from some other method?

Thanks, 

--Matt