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 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 + /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

Dear all,

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R is a scalar, not a vector: it has no direction, it is only length. There are no directional component to <R>, sigma^2, or other moments/cumulants of g(R).

Yes, I totally agree on that.

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There is no such thing as "parallel to R" or "perpendicular to R". I would point out that they use "u^2_perp" and "u^2_parallel", and that the diffraction values DO have directionality, while sigma^2 does not.

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This can be important if you want relate distances measured by XAFS to

Matt, on this point I feel there is a lot of confusion and misunderstanding, mainly due to different notation. The "parallel" and "perpendicular" things arise as *projections* on the distance between average atomic positions defined by crystallography and measured by diffraction. I attach a three-pages pdf which could make Fornasini's geometry and notation about local dynamics clearer. Just to be sure: Fornasini calls it C1*, John calls it sigma^1 and most of you call it simply R. They are the same thing: EXAFS distance! Furthermore, Fornasini's C2* is exactly your sigma^2. those

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measured by diffraction in systems with high thermal disorder (say, hot metals).

Actually most of Fornasini's work refers to systems at very low temperatures, down to 10K. The effect is particularly interesting in systems where XRD lattice parameter shows a negative expansion, while EXAFS distance increases...

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As a result, analysis of most real data suffers from various systematic sources of error such that the uncertainty on the fitted sigma^2 in the EXAFS equation is quite large.

Bruce, I agree that most of us are not concerned by all those subtle details. Most of experimental data are short, noisy, difficult to interpret, etc. However, Fornasini's work refers to very accurate temperature-dependent EXAFS measurements, at low temperatures, with excellent samples, in simple systems, etc... in order to detect very tiny variations.

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It is because XRD does not measure bond length but only the interplanar spacing

Just to say that this was already well known 50 years ago in crystallography. See for example W. R. Busing and H. A. Levy, Acta Crystallographica 17, 145 (1964) (ref 30 of the pdf attached). HTH Marco

Hi Marco, John, Thanks for the responses -- I think we're all in agreement, at least mostly. That is, I don't disagree with anything that Marco or John has said, and don't disagree with anything in the Fornasini paper or with Marco's attachment. My only complaint is there is the potential to cause confusion in the notation and terminology. For example, by setting <delta u_par> == 0 because any small change in average displacement along the line between lattice points would be taken up by L, one gets <R> ~= L + /2L. But then one then has to have the correct value for L at all measurements. This also might also lead one to conclude that only displacements perpendicular to L affect the XAFS. I suspect that some people are actually confused by this, which is why sigma_perp^2 and sigma_par^2 show up now and then in the literature. Of course, as Marco hinted at (by saying this is all in the literature and pre-dates EXAFS), <R> ~= L + /2L is really a way to convert diffraction results for L and into the interesting quantity <R>, to which diffraction is not very sensitive. On the other hand, the relation shows that, though EXAFS will give you <R>, it will not give you L. --Matt