[Ifeffit] Lattice parameters: EXAFS vs. XRD

[Scott Calvin]

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