[Ifeffit] Lattice parameters: EXAFS vs. XRD

Frenkel, Anatoly frenkel at bnl.gov
Fri Dec 25 14:37:01 CST 2009

Big deal...
We have final exams on Christmas...
Anatoly Frenkel
Yeshiva University


From: ifeffit-bounces at millenia.cars.aps.anl.gov on behalf of Scott Calvin
Sent: Fri 12/25/2009 1:04 PM
To: XAFS Analysis using Ifeffit
Subject: [Ifeffit] Lattice parameters: EXAFS vs. XRD

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?"

--Scott Calvin
Sarah Lawrence College
Ifeffit mailing list
Ifeffit at millenia.cars.aps.anl.gov

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