Discussion of the FeS2 EXAFS Analysis Example

This talk provides the post-mortem of my favorite teaching example for EXAFS data analysis using Artemis. At the end of the talk, there typically are a number of questions and details that did not get covered during the presentation. This talk attempts to tie up those loose ends.

Transcript

1. FeS2 EXAFS The post-mortem on an Artemis demonstration Bruce Ravel Synchrotron Methods Group, Ceramics Division Materials Measurement Laboratory National Institute of Standards and Technology & Local Contact, Beamline X23A2 National Synchrotron Light Source July 3, 2012 FeS2 EXAFS 1 / 12
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3. The amplitude parameter The amplitude parameter evaluates to something around 0.7 in the FeS2 fit. This is at the low end of what is expected1 for an S2 0 parameter. Lots of things are correlated with amplitude: 1 Coordination number, although this is a pure standard, so it is unlikely that coordination numbers are different from what we expect 2 Sample preparation: I do not know the provenance of these data. (They were taken from an on-line XAS data library.2 ) If the sample was not homogeneous, that would attenuate the amplitude3 by the “pinhole efffect”. 3 Again, without knowing the provenance, I cannot comment on the linearity of the detectors or any other aspect of the measurement. Conclusion A result of ∼ 0.7 for amplitude seems acceptable. FeS2 EXAFS 3 / 12 1. G.G. Li, F. Bridges, & C.H. Booth X-ray-absorption fine-structure standards: A comparison of experiment and theory, Phys. Rev. B 52:9 (1995) pp 6332-6348. DOI:10.1103/PhysRevB.52.6332 2. http://cars9.uchicago.edu/ newville/ModelLib/search.html 3. K.-Q. Lu & E.A. Stern, Size effect of powdered sample on EXAFS amplitude, Nuclear Instruments and Methods 212:1-3 (1983) pp 475-478, DOI:10.1016/0167-5087(83)90730-5
4. The σ2 constraint on the 2nd and 3rd shell S Here we see the contribution in k of the scattering from the 6 S atoms in the 2nd shell and the 2 S atoms in the 3rd shell. These shells are separated in distance by 0.15 ˚ A, which is just enough to have them contribute almost completely out of phase. This is the reason that the σ2 parameter for the 3rd shell is so unreliable (indeed, negative when floated independently). The fit was relatively insensitive to that parameter because it could reduce the 2nd shell σ2 to compensate for the unphysically small σ2 from the 3rd shell. Conclusion While it is certainly unphysical to constrain these two σ2 parameters, the fit is more defensible with this constraint. FeS2 EXAFS 4 / 12
5. That σ2 constraint examined in detail Plot the data along with a VPath (i.e. the sum of two or more regular paths) constructed from the 2nd and 3rd shell S atoms. def ss3 = ss2 Number of variables : 6 Chi-square : 6104.705744295 Reduced chi-square : 493.543240341 R-factor : 0.009268899 ss2 = 0.00332806 # +/- 0.00130826 ss3 := 0.00332806 # [ss2] guess both ss2 and ss3 Number of variables : 7 Chi-square : 5756.383603039 Reduced chi-square : 506.316510008 R-factor : 0.009218088 ss2 = 0.00270523 # +/- 0.00164548 ss3 = 0.00014725 # +/- 0.00367061 correlation: ss3 & ss2 --> 0.8050 FeS2 EXAFS 5 / 12
6. The σ2 constraints on the MS paths The σ2 parameters for the three paths involving collinear MS among the absorber and the 1st shell S atoms are all correct.1 The σ2 parameters for the non-collinear MS paths are rather hokey approximations. The problem is that we don’t have a good model to account for the effects on σ2 of all the legs of the path nor of the disorder in scattering angle. I worry about introducing a new fitting parameter to account for a rather small effect in the data. We need to approximate. Assertion The σ2 constraints for the triangle MS paths are non-physical ap- proximations, but are a better solution than floating one or more new parameters in the fit. FeS2 EXAFS 6 / 12 1. E.A. Hudson et al., Polarized x-ray-absorption spectroscopy of the uranyl ion: Comparison of experiment and theory, Phys. Rev. B 54 (1996) pp. 156-165 DOI:10.1103/PhysRevB.54.156
7. The fourth shell S Because the σ2 for the 4th shell S atom is so large, we see no improvement to the fit by introducing this scatterer. Why is its σ2 so large? That’s hard to say without help from theory, but clearly the relative positions of the absorber and this rather distant atom have a large thermal disorder. Conclusion It is safe to exclude this scatterer from the fit. Indeed, the fit is improved by not having its frail σ2 parameter in the fit. It would be interesting to measure this material at 10 K to see if the signal from this distant atom could be observed. FeS2 EXAFS 7 / 12
8. The remaining MS paths Nine of the first 15 paths from the calculation were included in the fit. The remaining 6 paths are MS paths with small amplitudes. We got a sensible fit with a model which excluded these paths. It would be a good exercise to figure out a sensible parameterization of their σ2 s, include them in the fit, and determine if the fit is improved by having them. Conclusion It was safe to exclude these paths, but this should be verified by examining the fits with and without those paths. FeS2 EXAFS 8 / 12
9. The parameterization of ∆R FeS2 is a cubic crystal. In this case, there are only two parameters that determine the locations of all the atoms in the cluster – the lattice constant a and the position of the S atom in the unit cell. For now, we neglect the effect of the position of the S atom. Why is the parameterization that sets ∆R = α · Reff acceptible for all paths? The distance between any two atoms in a cubic crystal is some geometrical factor multiplied by the lattice constant. That factor depends on the positions of the atoms in the unit cell, but is a pure number. Thus, from the calculation, deff (i, j) = Cij · a0 for any two atoms i and j We consider an isotropic expansion (or contraction) of the unit, which is reasonable for a cubic lattice that does not undergo a phase transition. So a = (1 + α) ∗ a0. dij =deff (i, j) + ∆d(i, j) =Cij · a =Cij · (1 + α) · a0 =Cij · a0 + Cij · α · a0 ∴ ∆d(i, j) =Cij · α · a0 =α · deff (i, j) Conclusion α · deff works for all legs of any SS or MS path in a cubic crystal (if there are no internal degrees of freedom). The R of a path is the sum of d for each leg, thus ∆R for a path is the sum of ∆d for each leg. This trick is only valid for a cubic crystal. FeS2 EXAFS 9 / 12
10. Improving on the parameterization of ∆R In the crystal data for FeS2, the S atom is at position (0.384, 0.384, 0.384), or (3 8 + δ, 3 8 + δ, 3 8 + δ), where δ = 0.009. The effect of changing δ can be incorporated into the math expressions for ∆R for any path that includes a S atom. Doing so is beyond the scope of this document. Exercise for the reader Examine the ‘feff.inp’ file for FeS2 . Think about how to incorpo- rate the effect of δ into a fit. FeS2 EXAFS 10 / 12
11. Correlations We have a pretty robust set of parameters in our fit. Only two of the correlations are above 60%. ∆E0 and α This correlation is about 86%. That is reasonable. Those are the only two parameters effecting the phase of the fit. This is a common level of correlation for such parameters. 1st shell σ2 and amplitude This correlation is about 81%. Again, this is pretty common for two things that have such an effect on overall amplitude of the fit. Conclusion The correlations we see are within acceptable limits. FeS2 EXAFS 11 / 12
12. The happiness “parameter” Always remember Happiness is a semantic parameter and should NEVER be reported in a publication – NEVER! We have decades of knowledge of how the parameters of an EXAFS fit should behave. “Happiness” attempts to encode that general knowledge into a single, non-statistical, entirely semantic parameter. The R-factor should be small. An R-factor below 0.02 gives no penalty. Above that, the penalty scales linearly to some maximum. A penalty is assessed if more than 2/3 of the number of independent points are used. A penalty for each Path with a negative S2 0 or σ2 value. A penalty for each E0, ∆R, or σ2 path parameter that is “too big”. A penalty is assessed for each correlation above 0.95. A penalty is assessed for each non-zero restraint. The evaluation of the happiness is tunable via configuration parameters. FeS2 EXAFS 12 / 12