[Ifeffit] Meaning of sigma^2 and real space resolution in XAFS

[Scott Calvin]

Hi all,

This got marked spam and rejected when I sent it from home this weekend... :)

At 12:48 AM 6/30/2007, you wrote:
>Hello,
>
>Thanks Bruce and Matt for the previous comments©
>I have the following questions and I would appreciate comments about them:
>
>My Questions are:
>1- Can we consider sigma^2 of a single 
>scattering path to be a measure of Debey Waller factor?

If you mean the Debye-Waller factor from x-ray 
diffraction, the answer is no. The EXAFS sigma^2 
is the variance in the absorber scatterer 
distance, the XRD sigma^2 is the variance in the 
position of an atom relative to its lattice point 
in the crystal. Thus, IF the motion of the 
absorber and scatterer were uncorrelated, it 
would be reasonable to think that the EXAFS 
sigma^2 would be twice the XRD Debye-Waller 
factor. For nearest-neighbors, though, this is a 
terrible assumption, as the atoms will tend to 
move in a correlated fashion (put another way, 
acoustic phonons dominate over optical ones). 
Thus the EXAFS sigma^2 for nearest-neighbors will 
usually be much less than the XRD Debye-Waller factor.

Further confusing matters is terminology. In 
EXAFS, sigma^2 is sometimes called the 
Debye-Waller factor...but this Debye-Waller 
factor is not the same as that from XRD! I've 
also seen "Debye-Waller factor" used to include 
all information about the shape of the 
distribution of the absorber-scatterer distance, 
so that sigma^2 is "the harmonic part of the 
Debye-Waller factor," but that seems to be an older usage.

>2- Is sigma^2 a measure of ¥or directly proportional to¤ strain in a bond?

Hmm...I'd say no, although I think I know what 
you're getting at. One major contributor to 
sigma^2 is thermal vibrations, which of course 
increase with temperature. I don't think that's 
generally categorized as "strain" (although I'm 
wary of the differing definitions that terms such 
as this can have in different specialties!). 
Another contributor to sigma^2 is variations in 
absorber-scatterer distance due to local defects, 
surface effects, etc., generally lumped together 
under the term "static disorder." This term 
itself can be a little misleading, as, e.g., a 
nanoparticle can have a sigma^2 that varies in a 
perfectly regular way with distance from the 
surface--we call that static disorder, even 
though it is not particularly disorderly. :) At 
any rate, what material scientists call "strain" 
can show up as static disorder. But even in that 
case, note that if all the bonds in a sample were 
strained, sigma^2 would not reflect it--it's only 
when some are strained and some are not that it shows up.

So yes, I've seen (knowledgeable) people point at 
a sigma^2 and say "I think that's high because of 
strain." But that's a kind of short-hand for what's going on.

>3- When doing first shell fitting in Artemis, 
>The actual bond length is: r_eff + dr +- delta ¥dr¤©
>Are the error bars associated with dr values 
>¥delta¥dr¤ of about 0©001 A¤ realistic? ¥i©e© 
>0©001 A is too small to detect by XAFS¤©

Your formatting didn't come through very well, 
but I'll give it a shot. Ifeffit (and thus 
Artemis) do a pretty good job of putting a lower 
bound on delta dr, in my opinion. Why a lower 
bound? Because there are all sorts of systematic 
errors that could be coming in, e.g. an iffy 
model--such as a first-shell only fit for a 
crystalline material, since there's often a bit 
of spectral leakage from more distant shells.

I'd be very surprised if Artemis ever gave a 
0.001 A delta dr, though, and absolutely shocked 
if it did so on a first-shell only model. Are you 
saying that it did do that for you?

>4- What determines the real space resolution in 
>XAFS measurements other than K_max?

k_min, of course.

To answer the gist of your question more fully, 
though, I have to get on my soapbox. :)

"Resolution" is often taken to mean something 
other than it does. Resolution criteria literally 
mean the point at which you could be expected to 
distinguish between two peaks. It is NOT the 
limit to the precision of information that can be 
retrieved by a clever person.

Huh?

Let's take an example of, oh, a proposal that 
there are two slightly different (but consistent) 
nearest-neighbor distances in some material. If 
those distances differ by less than the 
resolution implied by the k-space range, then 
they will not show up as two peaks in the Fourier 
transform. Instead, they'll show up as one broad 
peak; i.e. a peak with an anomalously large 
sigma^2. If a multiple-shell fit was performed 
and splitting was known to be a possibility, this 
anomalously large sigma^2 could be taken as 
indirect evidence for the splitting. In fact, if 
there were some way to estimate sigma2's a priori 
(I'll grant you that's a big if!), then the broad 
peak could be modeled as two scattering paths, 
with the splitting as a free parameter. In that 
case, EXAFS could give information about 
splitting to a greater precision than the resolution of the measurement.

A lot of the lore about the capabilities of EXAFS 
suffers from a similar lack of subtlety. Joe 
Woicik recently gave a very interesting talk at 
the NSLS Users' Meeting. It is generally bandied 
about that EXAFS cannot distinguish scatterers 
that differ in atomic number by less than 5. This 
is kind of true--in a mindless first-shell 
fit--because in those cases del_r and sigma^2 
will usually just slop up the differences in 
phase and amplitude due to the different atomic 
number of the scatterer. But if you really really 
knew del_r and sigma^2 a priori, even a 
difference of one atomic number stands out like a 
sore thumb! The upshot is that if you model, say, 
zinc scatterers as if they were iron, the fit 
will probably "work," but you'll be introducing 
substantial systematic errors into your fitted 
variables. A corollary is that in the case of a 
highly-constrained fit, you may be able to do 
considerably better than the usual +/- 5 rule.

The bottom line of this little speech is that the 
oft-quoted limits of EXAFS analysis are usually 
referring to a fit in which no a priori 
information is used. If you know (or suspect) 
anything about the system and incorporate that 
into your model, greater precision is sometimes 
possible. Of course, the results are then only as 
good as the a priori information; the whole thing 
could be a house of cards ready to collapse. But 
in favorable cases, multiple lines of evidence 
can be used to construct a convincing scenario...

Hope that helped!

--Scott Calvin
Sarah Lawrence College