[Ifeffit] Lattice parameters: EXAFS vs. XRD
SCalvin at slc.edu
Fri Dec 25 12:04:20 CST 2009
Merry Christmas, everyone!
Yes, I'm pondering EXAFS on Christmas...
Here's an issue that I bet has been worked out, and I bet someone on
this list knows the result and where it's been published.
It's well known that the MSRD ("sigma squared") for EXAFS differs
substantially from the "Debye-Waller factor" in XRD, because the first
is the variance in the interatomic distance, and the second is the
variance in the atomic position relative to a lattice point.
But what about the lattice parameter implied by the nearest-neighbor
distance in EXAFS as compared to the lattice parameter found by XRD?
It is certainly true that in most materials, particularly highly
symmetric materials, the nearest-neighbor pair distribution function
is not Gaussian, and generally has a long tail on the high-r side.
(This is largely because the hard-core repulsion keeps the atoms from
getting much closer than their equilibrium positions.) So imagine a
set of atoms undergoing thermal vibrations around a set of lattice
points. For concreteness, let's consider an fcc material like copper
metal. The lattice points themselves are further apart than they would
be without vibration, sure, but that's not the question. The question
is whether the square root of two multiplied by the average nearest-
neighbor distance is still equal to the spacing between lattice points.
My hunch is that the answer is no, and that the EXAFS implied value
will be slightly larger. While the average structure is still closed-
packed, the local structure will not be. And in a local structure that
is not closed-packed, the atoms will occasionally find positions quite
far from each other, but will never be very close. In a limiting case
where melting is approached, it's possible to imagine an atom
migrating away from its lattice point altogether, leaving a distorted
region around the defect. While XRD would suppress the defect, EXAFS
would dutifully average in the slightly longer nearest-neighbor
distances associated with it.
Just to be clear, I am not talking about limitations in some
particular EXAFS model used in curve-fitting. For example,
constraining the third cumulant to be zero is known to yield fits with
nearest-neighbor parameters that are systematically reduced. In fact,
limitations like that mean the question can't be answered just by
looking at a set of experimental results: I can make my fitted lattice
parameter for copper metal go up or down a little bit by changing
details of a fitting model or tinkering with parameters that
themselves have some uncertainty associated with them, like the
photoelectron's mean free path. (Fortunately, this kind of tinkering
will affect standards and samples in similar ways, and thus don't
affect my confidence in EXAFS analysis as a tool for investigating
quantitatively differences between samples, or between samples and a
standard.) My question is about the ACTUAL pair distribution function
in a real fcc metal. To the degree it's a question about analysis,
it's about XRD:
"In an fcc metal should the expectation value of the nearest-neighbor
separation, multiplied by the square root of two, equal the lattice
spacing as determined by XRD?"
Sarah Lawrence College
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