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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