Conventional EXAFS Analysis Steps

1. Introduction to X-ray Absorption Spectroscopy:Data AnalysisK. Klementiev, Alba synchrotron - CELLS K. Klementiev - Introduction to data analysis

2. K. Klementiev - Introduction to data analysis Conventional EXAFS Analysis Steps • Instrumental corrections (energy calibration, deglitching etc.) • Subtraction of pre-edge background • Conversion from energy dependence to wave number dependence (E → k) • Construction of the post-edge background 0 • Normalization data to edge step • kw-weighting • Fourier transform, with optional back Fourier transform • Determination of experimental errors in EXAFS signal • Calculation of theoretical scattering amplitudes and phases (ATOMS, FEFF) or extraction from reference experimental spectra • Fitting data in k-space or r-space using theoretical or experimental scattering amplitudes and phases • Determination of fitting errors The next slides show the above steps as screenshots of programs ATOMS, FEFF, VIPER and XANES dactyloscope. See the corresponding manuals.

3. K. Klementiev - Introduction to data analysis Conventional EXAFS Analysis Steps • Instrumental corrections: energy calibration derivative of absorptionspectrum of a foil measuredsimultaneously (15 spectra are aligned in energy) • Instrumental corrections: deglitching

4. K. Klementiev - Introduction to data analysis Conventional EXAFS Analysis Steps • Subtraction of pre-edge background pre-edge background • Conversion from energy dependence to wave number dependence: Positioning of E0 at the 1st derivative maximum of 

5. K. Klementiev - Introduction to data analysis Conventional EXAFS Analysis Steps • Construction of the post-edge background 0 1) through varied knots These knots are varied in Yto achieve no FT signal here 2) as smoothing spline

6. K. Klementiev - Introduction to data analysis Conventional EXAFS Analysis Steps • Fourier transform, with optional back Fourier transform

7. K. Klementiev - Introduction to data analysis Conventional EXAFS Analysis Steps • Determination of experimental errors in EXAFS signal by FT noise at high r: by uncertainty in 0 knots: here the error bars (noise) and  envelope (signal)refer to the right Y-axis here the experimental errors are determinedthrough high-r FT... … but be cautious with end jumps in  function! They contribute to FT alsoin the high-r portion.

8. K. Klementiev - Introduction to data analysis Conventional EXAFS Analysis Steps • Calculation of theoretical scattering amplitudes and phases (ATOMS, FEFF) Atoms by Bruce Ravel. I use WebATOMS (Google it). There is a searchable database of input data for ATOMS. For example, this input was generated from the database: Visit ATOMS and FEFF web-pages for more information! It generates input file for FEFF6 or FEFF8(save it as feff.inp):

9. K. Klementiev - Introduction to data analysis Conventional EXAFS Analysis Steps • Calculation of theoretical scattering amplitudes and phases (ATOMS, FEFF) I normally use the following way: 1) create a new folder for the new compound 2) put feff.inp and feff.exe (temporarily) there 3) run FEFF 4) move feff.exe to its usual location Remember to edit feff.inp beforehand! change the last '0' to '3' to get feffNNNN.dat files Remove or comment out SCF, XANES, FMS, LDOS: you do not need them for EXAFS. Put the 2nd parameter of HOLE as less than 0.1; then S02 will be calculated. Ready in a second…

10. K. Klementiev - Introduction to data analysis Conventional EXAFS Analysis Steps • Fitting data An example of the simplest fitting in VIPER (see VIPER manual and examples for more)

11. K. Klementiev - Introduction to data analysis Conventional EXAFS Analysis Steps • Fitting data. Continued • What if the coordination number is too low (for a reference spectrum)? • Return to FEFF calculations. Edit the feff.inp file: • * ixc [ Vr Vi ] • EXCHANGE 0 0 “some value here” • Repeat fitting with the new feff files. Typically, 1-2 iterations are enough to get the right coordination number within the confidence limits. • What to do with S02? • See VIPER manual for a discussion.

12. K. Klementiev - Introduction to data analysis Conventional EXAFS Analysis Steps • Determination of fitting errors The number of independent points in the data: Degree of freedom for P varied parameters: The figure of merit: Postulated: the variate 2 obeys the 2-distribution law with  degrees of freedom. This can be used to determine i. In the simplest case (without a priori knowledge) the posterior probability P(p|d) exp(2/2). The 2nd central moments give the fitting errors. See the VIPER manual/tutorial for more explanations. On one hand, this topic is quite broad. The fitting errors depend on how you treat the experimental uncertainties. They may also include a priori knowledge. Its relative weight in comparison to the data can be considered differently. On the other hand, many EXAFS works neglect the fitting errors completely. They report some 'standard' errors, like dr = 0.02 Å and dN = 10%, independently of data quality, signal strength or correlations. How deeply you want to go into the error analysis depends on you and on the referee of your publication.

13. K. Klementiev - Introduction to data analysis Conventional XANES Analysis Steps • Instrumental corrections (energy calibration, deglitching etc.) • Subtraction of pre-edge background • Normalization data to edge step • Principal component analysis (PCA), target transformation (TT) • Fitting by a linear combination of reference spectra (if TT was successful) XANES dactyloscope:

14. K. Klementiev - Introduction to data analysis Principal Component Analysis PCA answers the question “How many independent spectra?”The given here explanation of the PCA/TT methods is not standard but I believe is more transparent. For standard derivation based on SVD see [1a] S. R. Wasserman, J. Phys. IV France7 (1997) C2-203, which followed [1b] [1b] E. R. Malinowski. Anal. Chem. 49 (1977) 606 [2] T. Ressler, et al., Env. Sci. & Technol.34 (2000) 950 1) N spectra  data matrix 2) make DTD (NN matrix)and find its eigen-pairs i and ei. Always: 1iN|eiei| 3) for M linearly independent data vectors, this sum can be truncated at i=M and still 1iM|eiei|. 4) compare D with DiM|eiei|. If they coincide within noise then truncate further. Finally, M gives the number of independent vectors. 5) alternatively, compare the cut-off part Di=M+1N|eiei| with noise. How strongly it may deviate from noise? Different spectra have different noise; which noise then to take? For the answers, see the manual of XANES dactyloscope. 6) Finally, either you know the noise and infer the number of independent spectra or, inversely, you assume the number of independent spectra and estimate the noise.

15. K. Klementiev - Introduction to data analysis PCA: test example 1 This example has two independent spectra (thick red and blue) and a spectrum constructed as the average of the two plus normal noise with nominal σ = 0.005 (thick magenta). The thin cyan curves are the PCA-transformed curves. How many independent spectra we have within this set? One needs 2 PCs to reproduce all 3 spectra If we had all 3 spectra independent, the noise level had to be within these bounds with 95% probability.

16. K. Klementiev - Introduction to data analysis PCA: test example 2 This example has 10 spectra which are all artificially constructed from a single spectrum with added 10 various realizations of normal noise of nominal σ = 0.005. One needs 1 PC to reproduce all 10 spectra We have our noise within these bounds and thus we can say that with 95% probability there is 1 PC. If we had our noise within these bounds (~1!) then we could tell that there is 0 PCs and all the data are just noise.

17. K. Klementiev - Introduction to data analysis Target Transformation With TT one can answer the question whether a particular spectrum (material) represents a mixture of the basis spectra (materials). When answered positively, one can try fitting by a linear combination of the basis spectra. 1) Construct basis matrix B from N basis spectra. Make BTB (NN matrix)and find its eigenvalues i and eigenvectors ei 2) If the basis spectra are independent then (BTB)-1 exists. 3) The matrix is an orthogonal projector to the basis space (it is equal to its square, check this). Hence, if a spectrum S is a linear combination of the basis spectra then B(BTB)-1BTS = S and inversely. 4) In practice one checks if B(BTB)-1BTS coincides with S within noise. How strongly the difference is allowed to deviate from noise? For the answer, see the manual of XANES dactyloscope.

18. K. Klementiev - Introduction to data analysis Linear combination fitting: real example 1. Basis spectra: precursor (Pd I) and metallic Pd particles. 2. The spectrum “catalyst 2” shows coincidence with its target transformation. 3. Linear combination fitting of XANES of “catalyst 2” by the two basis spectra. 4. The found linear combination (the same coefficients) found for the XANES spectra is then successfully applied to EXAFS (see the curves below in k- and r-space). Palladium Nanoparticles immobilized on Mesoporous Silica Support – New Efficient Catalysts for Aerobic Alcohol Oxidation in Supercritical Carbon Dioxide Z. Hou et al., J. of Catalysis258 (2008) 315

19. K. Klementiev - Introduction to data analysis Conclusion • Both XANES and EXAFS require careful attention to the data quality. If necessary, one has to perform instrumental correction: energy calibration and/or deglitching. • EXAFS requires preliminary calculations of the scattering amplitudes and phases with checking them on reference spectra. • EXAFS fitting typically requires careful attention to the error analysis. • There are two main classes of applications of XANES: • Fingerprint analysis: presence/absence of pre-edge peaks, edge shift, white line height (see the introductory lecture). This analysis doesn’t require sophisticated methods. • Factor analysis (PCA, TT) and linear combination fitting. This analysis requires some mathematics and statistics and careful attention to the estimation of noise. This part has been considered in the present lecture. The examples can be found among the example projects of XANES dactyloscope program and in its manual. In the practical session we will follow all the above analysis steps.