[Ifeffit] Cumulant expansion fittings

[Matt Newville]

Sorry this is so long, and I certainly believe that everyone in this
conversation understands these issues well.  But I do think that there is
confusion in the literature (not Fornasini's work) and so some
potential for confusion for novices too.

Ignoring 3 body correlations probed by multiple scattering, XAFS is
sensitive only to R, the distance between the absorbing atom and scattering
atom.  R is a scalar, not a vector: it has no direction, it is only length.
There is no such thing as "parallel to R" or "perpendicular to R".  Bond
direction only becomes a useful idea when you include a third atom (or
more, say a whole crystal lattice).  Single Scattering XAFS is not
sensitive to this third (or more) atom, and is not sensitive to any changes
in the relative orientation of the two atoms to the rest of the system.

XAFS samples a distribution of distances, g(R).  sigma^2 is second moment
of g(R).  There are no directional component to <R>, sigma^2, or other
moments/cumulants of g(R).

If you imagine a central atom fixed in space and a neighboring atom that
has a fixed distance from it but can rotate freely about that atom on a
spherical shell, XAFS will see no disorder: sigma^2 will be zero.  This
closely describes an organic ligand for a molecule in solution.  The
molecule can tumble and rotate and take any orientation relative to the lab
frame (or x-ray polarization vector) but the interatomic distance is well
defined, and there will be a strong XAFS signal with a relatively small
sigma^2.  XAFS is sensitive only to displacements in R.

I generally like Fornasini's work, and agree with the basic physics (well,
maybe it's geometry) of the result John cites, but I might use different
notation and terms. I would point out that they use "u^2_perp" and
"u^2_parallel", and that the diffraction <u^2> values DO have
directionality, while sigma^2 does not.

The general topic here is "how can one relate XAFS values for R and sigma2
to diffraction results for spacing between lattice planes and points"?,
part of a recurring theme in the literature to relate temperature dependent
changes in XAFS R to thermal expansion coefficients measured as bulk
properties of materials or as the temperature dependence of the distance
between lattice planes as measured by diffraction.

Imagine two lattice points separated by a distance L, and two atoms
vibrating isotropically around each lattice point, such that the average
position is exactly at the lattice point, but that it samples off-lattice
points with some Gaussian distribution.  Assume that the displacements of
the two atoms are completely uncorrelated.  That is, you have two fuzzy
little distributions for atoms around fixed lattice points:

         O -- O
	 1    2

Consider a snapshot with atom 1 displaced "up" (+y direction) from it's
lattice point by distance d, and atom 2 displaced by the same distance:

 Atom2      sign of (R-L)   R^2
 center      +             L^2 + d^2
 up    (+y)  0             L^2
 down  (-y)  +             L^2  + (2d)^2
 right (+x)  +             (L+d)^2 + d^2
 left  (-x)  -             (L-d)^2 + d^2
 in    (+z)  +             L^2 + 2 * d^2
 out   (-z)  +             L^2 + 2 * d^2

If these are sampled uniformly, the average R is greater than L.
Considering all the displacements for atom 1, you'll have something like
      <R> ~= L + <d^2>/2R

This is just due to the triangle inequality, and is basically the result
Fornasini et al give.  It means that the <R> from XAFS will be longer than
the distance between lattice points measured by diffraction.  It also means
the temperature dependence of <R> may be different from that of L.  This
can be important if you want relate distances measured by XAFS to those
measured by diffraction in systems with high thermal disorder (say, hot
metals).   And it is precisely because XAFS is sensitive to R alone.

--Matt