[Ifeffit] Cumulant expansion fittings

[Bruce Ravel]

On Thursday 22 January 2009 09:50:26 pm Patrick Kluth wrote:
> Hi Matt,
>
> >Of course, XAFS *is* a one-dimensional probe, not a three-dimensional one.
> >At least ignoring for the moment the angular dependence of multiple
> >scattering, XAFS is sensitive to g(r) only.  Sadly, this is sometimes
> >forgotten in the literature, and one sees attempts to distinguish
> >"sigma^2_perpendicular" and "sigma^2_parallel", which is a good sign of a
> >paper that is complete nonsense.
>
> I don`t agree with you in this point. To my understanding, because
> EXAFS is a one-dimensional probe and it measures the average over
> instantaneous inter-atomic distances, it is only sensitive to the
> motion relative to each other along the bond-direction - and thus
> measures sigma^2_parallel. Combined with XRD which measures the the
> atomic motion averaged over all directions one can then extract a
> "sigma^2_perpendicular". If we are talking about the same literature
> this point is well elaborated in there and I don`t think it is
> nonsense. Please correct me if I am wrong.

I think I would have used somewhat less strong language than Matt, but
I do agree with his essential point.  In the other response to
Patrick's posting, John gives an overview (following Fornasini's
excellent work) of the math behind the whole "sigma_parallel" and
"sigma_perpendicular" business.  As John says, the cumulant is

  sigma^1 = <u_par + u_perp^2/2R>

and when you square that there are additional terms beyond (u_par)^2.
Were EXAFS strictly 1D, that would be the only part of sigma^2.  In a
3D world, the other terms exist.

However, most of us are working in the EXAFS trenches:
  1. Our samples are never homogeneous on all length scales.
  2. Most materials of (catalytic or environmental science or
     materials physics or whatever) interest have significant inherent
     structural disorder.
  3. The theory that we use to analyze our data makes assumptions and
     approximations.  (Although rumor has it that Feff is scheduled to
     be perfect around Feff 13 or so ... I can hardly wait! :-P )

As a result, analysis of most real data suffers from various
systematic sources of error such that the uncertainty on the fitted
sigma^2 in the EXAFS equation is quite large.  In special cases, one
might be able to control those sources of error adequately such that
the measurement uncertaintly is not large compared to the additional
terms in sigma^2.  However, few of us are working on problems like
that.  In most cases, a shortcoming of the physical model or an
inhomogeneity of the sample is the likely cause of an oddly large
sigma^2 term.  An attempt to interpret a measured sigma^2 in terms of
u_perp probably won't be statistically supportable.

B




-- 

 Bruce Ravel  ------------------------------------ bravel at bnl.gov

 National Institute of Standards and Technology
 Synchrotron Methods Group at NSLS --- Beamlines U7A, X24A, X23A2
 Building 535A
 Upton NY, 11973

 My homepage:    http://xafs.org/BruceRavel
 EXAFS software: http://cars9.uchicago.edu/~ravel/software/exafs/