Matt is right - and the triangular inequality is a good way to think about it. Fornasini and Dalba, as well as Ed Stern in J. Phys. paper, estimated this term more accurately: 2*sigma2/R should be added to the Delta R to account for the transverse vibration and obtain the difference between the bond distance in the unknown sample and the standard (experimental or theoretical). However, if sigma2 is of the order of 0.01 A2, and 1NN distance is of the order of 2.8 A as in gold, platinum or palladium, this correction is only 0.007 A, which is smaller than thermal expansion in many materials (that can be on the order of 0.02-0.06 A, depending on the temperature range, bond strength (or Debye/Einstein temperature). Of course, since sigma2 varies with temperature linearly in classical limit (at temperatures great than Einstein temperature), this correction is temperature dependent. That means, at 300 K it may be 0.007 A2 (and the correction is 0.005 A), and at 400 K it may be 0.01 A2 (and the correction is 0.007 A). However, if the thermal expansion (R2-R1) is measured with and without this correction, the difference is only 0.002 A, which is for sure comparable or most likely less than the error bar in R. Thus, transverse vibrations, ideally, should be included for more accurate bond length determination. If they are small, the one-dimensional model of Frenkel-Rehr is a reasonable approximation. Anatoly Anatoly Frenkel Yeshiva University -----Original Message----- From: ifeffit-bounces@millenia.cars.aps.anl.gov [mailto:ifeffit-bounces@millenia.cars.aps.anl.gov]On Behalf Of Matt Newville Sent: Wednesday, October 05, 2005 5:23 PM To: XAFS Analysis using Ifeffit Subject: Re: [Ifeffit] delR versus thermal expansion A and first cumulant C1* Leandro, This is a topic that seems to be slightly controversial. I don't want mis-represent the varying views/opinions, but today is a 'at the beamline' kind of day. So this reply is less well planned (coherent??) and a little more 'off the cuff' than I'd like it to be. Perhaps we start a wiki page about this discussion?? Also, everything below is 'in my opinion': The first thing is that EXAFS is sensitive to R. Period. It seems obvious, but it's sometimes easy to forget. As a result, I think notions like "perpendicular to the lattice planes" doesn't get you very far. Also, don't forget that the crystallography is an amazingly powerful technique for studying solids, and that the macroscopic thermal expansion parameter (dL/L)/dT is often determined from the crystallographically (atomic-scale) measurement (dA/A)/dT. (A= lattice parameter, L= bulk distance, T= temperature). To first approximation, dR/R from EXAFS is similar to dA/A from diffraction. Many people have equated these. In detail, and especially at high temperature, they are not the same. OK, so on to your questions:

Whenever we fit an EXAFS spectrum with Artemis/IFEFFIT, we get a /delR/ value. As far as my (very) limited comprehension goes, /delR/ is the difference between the theoretical (FEFF8) bond length, /R0/, and the result of the best fit to my data, /R/, which gives the experimental bond length. So, this should be the /first cumulant/ of my distance distribution, right?

Yes. Cumulants are simply one way to describe a distribution function of a variable (for us, R). They're especially convenient for functions that are exponential in that variable, and that's why they're often used in EXAFS. They're most useful when the distribution is near normal (or Gaussian). For complicated distributions, they don't work so well. delR is the first cumulant, C1, and is equal to the first moment of the distribution. It is the displacement from the starting center value, R0, Reff, etc. There is actually a subtlety in getting delR=C1 from EXAFS, as the EXAFS is not simply exponential in R, but also has a 1/R^2 term and R dependence in the mean-free-path term. These can be dealt with (and are dealt with in Ifeffit/Feffit), so that the delR, sigma2, third, and fourth *are* the cumulants of the atomic pair distribution. But that's not your question (yet??).

... when I came across some papers (P. Fornasini and G. Dalba in [PRL82, 4240, 1999; PRB70, 174301, 2004] and E. A. Stern in [J.Phys.IV 7, C2-137,1997]) where an issue is raised; it is said that the thermal evolution of the first cumulant /C1/* of the real distance distribution in a crystal (which was said to be equivalent to /delR/) is NOT equal to /A/, because in a crystal there are vibrations perpendicular to the bond direction which are not considered in the one-dimensional model. It is argued that the temperature dependence of these vibrations perpendicular to the bond direction give raise to a positive shift of the minimum of the effective pair potential, while the net thermal expansion /A/ accounts only for changes due to the asymmetry of the potential.

I haven't looked at their paper in a long time, but I think that Fornasini and Dalba have the math and basic explanation right. I don't so much like the 'perpendicular' v. 'parallel' distinction, but others do.

So, the main questions hammering my head are: - does /delR/ include any contribution from vibrations perpendicular to the bond direction? Can really /delR/ and /C1/* be the same quantity?

I think the answer to the first is 'No'. delR is the change in average bond length, and that's it. For the second, delR and C1 really are the same quantitiy.

- is the thermal evolution of /delR/ equal to the net thermal expansion /A/? Or should some correction for perpendicular vibrations be added to relate both quantities?

No. If you imagine two atoms vibrating independently about lattice points separated by distance A, you will be able to convince yourself that <R> >= A, just from the triangle inequality. In a simple approximation, the differnce is proportional to sigma2/R. The important result is that delR/R is not equal to delA/A. Hope that helps, or at least keeps the conversation going.... --Matt _______________________________________________ Ifeffit mailing list Ifeffit@millenia.cars.aps.anl.gov http://millenia.cars.aps.anl.gov/mailman/listinfo/ifeffit

Matt, first of all, thank you very much for your reply. It was pretty coherent and did help to settle some conceptual mismatchs that were bugging me. As for starting a wiki on this, looks like a good idea to me. Specially if there are more people interested on the topic.

Hope that helps, or at least keeps the conversation going....

Following your suggestion above, I'd like to make some more questions, if you don't mind.... ;)

To first approximation, dR/R from EXAFS is similar to dA/A from diffraction. Many people have equated these. In detail, and especially at high temperature, they are not the same.

Ok, that's understood. I guess I didn't express myself propelry in the previous message, sorry. The thing is that Anatoly and John define their thermal net expansion as a(T)=<r-r0>, r0 being the minimum of the pair-potential and r the bond length at a given temperature. This made me think that their a(T) could be directly related to dR/R from EXAFS instead of dA/A from diffraction. Does this sound correct to you?

delR is the first cumulant, C1, and is equal to the first moment of the distribution. It is the displacement from the starting center value, R0, Reff, etc.

There is actually a subtlety in getting delR=C1 from EXAFS, as the EXAFS is not simply exponential in R, but also has a 1/R^2 term and R dependence in the mean-free-path term. These can be dealt with (and are dealt with in Ifeffit/Feffit), so that the delR, sigma2, third, and fourth *are* the cumulants of the atomic pair distribution. But that's not your question (yet??).

Well, I think you already saw where I am trying to go... I was pretty sure that the values of sigma2 and C3 I was getting from the fits were the "real" cumulants of the atomic pair distribution, but I wasn't so sure about delR, because of the 1/R^2 and lambda corrections. I believed they would be there, and now you reassured me. The thing is, I have these temperature dependent EXAFS spectra from 10 to 300K to analyze. If I try to analyze each at a time, I get reasonably values with small errors for sigma2, which are insensitive to small variations on the fitting conditions. But the values obtained for delR and C3 are pretty wonky, have big errors and vary a lot with small variations on the fitting conditions (like k-weights, windows, E0 values,...). So, I was trying to make a multiple dataset fit, with all of them together. And I was trying to do it the same way you (Matt) and some other people did, which involves writing the cumulants as functions of the pair potential constants and the temperature. This way, the number of free parameters is drastically reduced, as well as the uncertainties on the determination of delR and C3. But, as my data is not on the "high T" limit (unlike most people), I cannot use the same equations that you have used. I was using Anatoly's equations instead [PRB48, 585, 1993], where there is no direct mention to a first cumulant, only to the linear thermal expansion given by a(T)=<r-r0>. Looking at some other papers (like [Troger et al., PRB49, 888 (1994)], [Yokoyama, JSR6, 323 (1999)], [Van Hung and Rehr, PRB56, 43, 1997]), I saw people treating the thermal evolution of the first cumulant as being given by a(T). So, my question now is: - How reasonable it is to represent the thermal evolution of delR by Anatoly and John's a(T)? Should any corrections be added when relating this two quantities? Sorry for another over-sized messaged and thank you very much for your attention. Cheers, Leandro