[Ifeffit] Cumulant expansion fittings

Scott Calvin SCalvin at slc.edu
Wed Jan 21 13:42:17 CST 2009

Thanks, Matt--you said that much more clearly than I did.

I'd add that, personally, I avoid using cumulants to account for  
unresolved features when possible.

For example, suppose that a central metal atom is coordinated to six  
oxygen atoms in an octahedral arrangement. But suppose also that I  
have a "hunch" that the octahedron may be stretched along one axis, so  
that two of the oxygen atoms are a bit further away than the other  
four. This hunch may come from other reports in the literature, other  
structural probes, chemical modeling, etc. Depending on how much  
usable data I have in k-space and in how small the splitting is, the  
metal-oxygen scattering may appear as only one peak in the magnitude  
of the Fourier transform.

One approach would be to use cumulants to try to handle the  
"disorder"--the radial distribution function is not symmetric, after  
all, and while I might not have the resolution to show splitting, the  
shape and position of the peak and its sidebands will be affected.

A different approach would be to use two different paths, one for the  
equatorial atoms and one for the axial atoms, with appropriate  
coordination numbers. Then create a guessed parameter for the  
separation in distance between the two paths, and constrain the paths'  
MSRD's ("sigma2's") to be the same. Mathematically, this is only  
introducing one new free parameter, just as using a third cumulant  
introduces one new free parameter.

The cumulant method has the advantage that it makes no judgements  
about what is causing the anharmonicity.

The multiple-path method has the advantage that if the model is  
correct, it may provide a somewhat better fit, and it also provides  
information that may be of use in understanding the system (such as a  
value for the difference in average distance to atoms at the two kinds  
of sites). The disadvantage is that it can be misleading: the fit is  
"working" because it has a new variable to play with that models the  
anharmonicity, and a statistically good fit is not evidence that the  
particular model is correct. For example, just as anharmonicity of a  
single path can be used to approximate multiple unresolved paths,  
multiple unresolved paths can also be used to approximate the  
anharmonicity of a single path!

So my personal approach is this. If I have a hunch as to what kind of  
splitting might be going on, I model that splitting, keeping in mind  
that a good fit in such a circumstance is not enough, on its own, to  
say that my model is physically correct. If I don't have a good hunch,  
I use cumulants. And, of course, if I suspect there is no splitting,  
but rather a path that is intrinsically anharmonic (e.g. all  
coordinated atoms at the same average distance, but the thermal  
variation about that average distance is not symmetric), then I also  
use cumulants.

I hope that was helpful.

--Scott Calvin
Sarah Lawrence College

On Jan 21, 2009, at 1:27 PM, Matt Newville wrote:

> Umesh, Scott, Anatoly,
> The real question was "can one use the fourth cumulant without the  
> third
> cumulant in fitting (XAFS)"?  The answer is: Yes.  As Scott and  
> Anatoly
> suggested, doing that may not make a great physical model for g(r),  
> but
> perhaps Umesh has a good reason to try this.
> My advice is to try it and see if you get a fourth cumulant that is  
> clearly
> non-zero.
> My experience (and all the experiences I've heard of) is that the  
> fourth
> cumulant rarely matters.  This is probably related to the idea that  
> the
> cumulant expansion diverges for very disordered systems and you would
> either need many higher order cumulants to describe such a  
> distribution or
> are much better off using a finite set of atomic distances with some  
> model
> for the weighting of the different distances.  The first option  
> (using many
> cumulants) is impractical, and probably computationally dangerous  
> (as you'd
> quickly explore issues with computer precision of floating point  
> numbers).
> The second option (often called a "histogram" approach in the Ifeffit
> world, and modeled somewhat after The GNXAS Approach) has been used
> successfully a number of times.
> --Matt

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