Hi Ian, I think that Scott, John, and Shelly answered some of your questions about XAFS Debye-Waller factors, and gave many of the standard answers about how to deal with these in the analysis of real XAFS data. Here are some more thoughts on this:
1. Are phenomenological models like the correlated Debye model and Einstein model appropriate? It seems that Debye temperatures are hard to find for oxides.
The Einstein model asserts that there is a single, dominant (or effective) vibrational mode between two atoms, with a vibrational amplitude related to the Einstein temperature. For first shell EXAFS, this is almost always an appropriate model. But, if you don't know what that Einstein temperature is, you have to treat it as an unknown. Einstein / Debye temperatures are hard to find for metal oxides because most bulk measurements of thermal properties don't fit with these simple models. That doesn't mean that the metal-near-neighbor-oxygen bond doesn't have a single, dominant (or effective) vibration, just that the other vibrational modes of the solid are important in the total thermal behavior. But a result of the Einstein model that may be important for your application is that sigma2 is linear in T over most of the temperature range.
2. Is there an equivalent value for ss2 in x-ray crystallography data that I can refer to for comparison in these materials? What should I look for in reviewing published x-ray crystallographic data?
Crystallographic Debye-Waller Factors are generally unrelated to XAFS Debye-Waller Factors. Many people have tried to make this relation. In my maybe-not-so-humble-as-it-should-be opinion, they're all doomed to fail. They're very different views of thermal and static disorder. One is with respect to the near-neighbors, the other with respect to "the fixed stars" of the crystal. On the other hand, molecular vibrational information from IR, Raman, NMR, Mossbauer, etc, also estimate bond strengths and disorder in bond length, and are more like the XAFS Debye-Waller Factors in their 'localness'. As others have said, XAFS cares about the vibrational mode between the two atoms, and so is most strongly related to the optical phonon modes. In many of these other vibrational spectra, these modes can be selected and their amplitudes extracted. I don't think much (or enough) effort has been put into relating vibrational measurements to XAFS DWFs. It seems like an interesting and potentially useful approach. It's been more common for people to try to fit the experiment into the Einstein model by taking temperature dependent data well below room temperature or by coming up with models to relate crystallographic and XAFS DWFs. I think these are not appropriate for most systems, including yours.
3. Is the temperature dependence reported for the ss2 for some metals (like Cu and Al) similar to their respective oxides?
No, not at all. Metallic bonds are generally weak, and so tend to have larger sigma2 than metal-oxygen bonds.
For example, in Debye-Waller factor calculations reported for Al metal by R.C.G Killean (in J.Phys.F: Metal Phys., v. 4, pg. 1908, 1974), the ss2 changes by a factor of three between 25C and 300C. Would I expect a factor of three increase in crystalline zeolites (AlO4 structural units).
Nope. The Al-O bond strength is much higher than Al-Al (Al melts at 660C, Al2O3 at 2050C).
4. Specific to my work: I have studied Al containing oxides at temperatures between 25 and 300 C. I would like to quantify coordination changes by doing EXAFS fitting of the Al K-edge data. The EXAFS was taken at temperature and I observe no change in the broadening of the data. Likewise, the fitting shows no sensitivity in the Debye-Waller factor. It is nearly constant at 0.001 A over the temperatures of interest. Since ss2 and the coordination number (CN) are correlated, I would like a way to bind the error on the fitted CN by modeling real physical changes in the Debye-Waller factors. Do you have any suggestions?
As others mentioned, you could model the temperature dependence. You can probably model sigma2 as simply as: sigma2(T) = sigma2_off + T * sigma2_slope With T either in C or K. That is, since you're probably well below the Einstein temperature and bond-breaking temperature, sigma2 is probably linear in T. The offset term incorporates the static disorder as well as the thermal disorder at your lowest temperature. Then the sigma2 values for all your temperature dependent data get mapped to two variables (sigma2_off and sigma2_slope). If coordination number is assumed to be independent of temperature, you'll have three parameters to fit the amplitudes of all your temperature-dependent data. It sounds like you might find an appreciable static disorder and a very small slope. That could be right, but would indicate a very strong bond. Hope that helps, --Matt