Principles of EXAFS Spectroscopy

1. Principles of EXAFS Spectroscopy Sam Webb

2. Topics • Overview • Process and Experiment • Theory • Brief History • Derivation (simple) • Data Analysis • Data Reduction • Fourier Concepts • Data Modeling

3. What can I do with x-rays?

4. continuum unoccupied states EF filled 3d  1s core hole X-ray Absorption • X-rays are absorbed through the Photo-electric effect • Absorption occurs when incident X-rays are energetic enough to expel core-level electrons from atom • The atom is left in an excited state with an empty electronic level (core hole) • Any excess energy from the x-ray is given to the ejected photo-electron

5. X-ray Fluorescence • The excited core-hole will relax back to a “ground-state” by transition of a higher level electron in to the core hole. • This process emits a fluorescent x-ray. • The energy of the fluorescent x-ray is characteristic of the absorbing atom continuum unoccupied states EF 3d  1s core hole

6. X-ray absorption Coefficient • Intensity of an x-ray beam passing through a material of given thickness is determined by the absorption coefficient (m): I = I0e−μt • μ depends strongly on: • x-ray energy E • atomic number Z • density r • atomic mass A:

7. What is XAFS? • X-ray Absorption Fine Structure (XAFS) is the modulation of the x-ray absorption coefficient at energies near and above an x-ray absorption edge. • XANES: X-ray Absorption Near Edge Spectroscopy • EXAFS: Extended X-ray Absorption Fine Structure • Contain information about an element’s local coordination and chemical state. XAFS Characteristics: • local atomic coordination • chemical / oxidation state • applies to any element • works at low concentrations • minimal sample requirements Threshold Energy, E0

8. XAS is Element Specific

9. Absorption Edge Energies • As atomic number increases, threshold energies scale E0~Z2, absorption coefficient ~Z4. • All elements Z>18 have either a K- or L-edge between 3 and 35 keV, accessible at many synchrotrons

10. double-crystal monochromator ionization detectors beam-stop I2 I1 I0 “white” x-rays from synchrotron LHe cryostat reference sample sample collimating slits X-ray absorption spectroscopy (XAS) experimental setup • sample absorption is given by •  t = loge(I0/I1) • fluorescence is •  ~ If/I0 • reference absorption is • REF t = loge(I1/I2)

11. Theory Why do I see wiggles?

12. Discovery of X-Ray Absorption Fine Structure • First absorption edges noticed by Maurice de Broglie in 1913 • EXAFS in edge found ca. 1920 • Closest early theoretical explanation by Kronig • Utilized LRO in crystal structure to predict oscillatory “Kronig” structures. Not exact with experiments, but often “close” (1931) • Kronig structure in literature through the 1970s • In molecules, began to consider SRO and utilized backscattering photoelectrons in the phenomenon. • Started by Kronig (1932) • Advanced by Peterson (phase shifts-1936) • Kostarev (all condensed matter-1949) • Sawada (mean free path-1959) • Shmidt (disorder-1961) Coster and Veldkamp, Z. Phys. 70, 306 (1931).

13. Start of Modern Theory Sayers, Stern and Lytle, Phys. Rev. Lett. 71, 1204 (1971) • Utilized point scattering from neighboring atoms • Used Fourier analysis to solve for EXAFS • Transition from scientific curiosity, to quantitative tool • Aided by computers (256 kb)

14. EXAFS: Absorption by a Free Atom • Atom absorbs an x-ray of energy E, destroying core electron of energy E0 and creating a photo-electron of energy (E-E0). • Isolated atom has m(E) with a sharp step at the core binding energy and is smooth as a f(E) above the edge. • Once E is large enough to promote a core electron to the continuum, there is a sharp increase in absorption

15. EXAFS: Absorption & P-E Scattering • With another atom nearby, the ejected photo-electron can scatter from the neighboring atom. The back scattered P-E will interfere with itself. • EXAFS is an interference effect of the photoelectron with itself, due to the presence of neighboring atoms The amplitude of the back scattered P-E at the absorbing atom will vary with E, causing the oscillations in m(E) that are EXAFS

16. EXAFS

17. X-ray Absorption Fine Structure • Need to isolate the energy dependent oscillations in m(E): • Subtract the “atomic background” m0(E) • Divide by edge step Dm0(E0)

18.  EXAFS • Since EXAFS is an interference effect and depends on the wave-nature of the photo-electron, its convenient to think of the process in terms of photo-electron wavenumber (k) rather than energy (E). • EXAFS often weighted by k2 or k3 to amplify oscillations at high-k.

19. EXAFS: Simple Description • Simple model has c ~ yscat • The photoelectron: • Leaves the absorbing atom • Scatters from the neighbor • Returns to the absorbing atom With a spherical wave eikr/kr for the propagating photo-electron, and a scattering atom at distance r=R: Where the neighboring atom gives the amplitude f(k) and a phase shift d(k) to the scattered photo-electron

20. EXAFS: Developing the EXAFS Eqn • Combining terms… • For N atoms, and adding a thermal/static disorder of s2, giving a mean-square disorder in R: • A real system has neighbors at different distances and different types. Summing all these:

21. EXAFS: Additional Terms • Photo-Electron Mean-Free Path • P-E can scatter inelastically and may not return to absorbing atom • Core hole has finite lifetime, limiting how far the P-E can travel • Use damped wave-function: where l(k) is the mean-free path • Amplitude Reduction Term • Due to relaxation of all the other electrons in the absorbing atom to the core hole level • S02 is typically taken as a constant, 0.7< S02 <1.0 and multiplied into the XAFS c • Completely correlated with N! • Makes EXAFS amplitudes (and thus N) less precise than EXAFS phases (R)

22. EXAFS: The EXAFS Equation • The sum is over “shells” of atoms, or “scattering paths” for the P-E • If we know the scattering properties of the neighboring atoms: amplituitude, f(k) and phase-shift, d(k), as well as MFP l(k), we can solve for: • R: distance to atom • N: coordination number of atom • s2: mean-square disorder of atom distance • f(k) and d(k) depend on atomic number, so EXAFS sensitive to Z of neighboring atoms

23. EXAFS: Scattering and Phase Shift • f(k) peaks at different k and extends to higher-k for heavier elements. For very high Z, there is structure in f(k). • Heavy elements have sharp changes in d(k). • Both f(k) and d(k) can be accurately calculated by theory (FEFF). • f(k) and d(k) depend on Z. • In EXAFS, Z can be determined with ~±2. Fe and O can be distinguished, but Fe and Mn cannot.

24. EXAFS: Multiple Scattering • The sum over paths in the EXAFS equation includes shells of many atoms (1st shell, 2nd shell, 3rd shell, etc) but can also include multiple scattering paths. • MS paths are those in which the P-E scatters from more than one atom before returning to the central atom: • For MS paths, the total amplitude depends on the angles in the scattering path. The strong angular dependence of the scattering can be used to measure bond angles. • Triangle Paths with angles 45º < q < 135º are not strong, but can be a lot of them • Linear Paths with angles q≈ 180º are very strong, as the P-E can be focused through one atom to the next Multiple scattering is most important when the scattering angle is > 150º

25. Data Analysis What do I do with this data?

26. Data Reduction: Strategy • Take measured data to m(E) then to c(k): • Convert measured intensities to m(E). • Subtract a smooth pre-edge to get rid of background and absorption from other edges. • Normalize m(E) to unit step height to represent absorption of a single x-ray. • Remove a post-edge background function to approximate m0(E) to isolate the EXAFS c. • Identify the threshold energy, E0, and convert from E to k. • Weight the c(k) and Fourier transform from k to R space.

27. Data Reduction: Raw to m(E) I0 I

28. Data Reduction: Pre-Edge and Normalization • Pre-Edge • Subtract the background that fits the pre-edge region. Gets rid of absorption due to other edges in the sample. • Normalization • Estimate the edge step by extrapolating a simple fit above the edge to the edge

29. Data Reduction: XANES and E0 • XANES • The XANES portion has a rich structure. Can be used for fingerprinting and electronic structure. I like to normalize to a square, unit edge step • Derivative • Select E0 roughly as the energy with the maximum first derivative. Somewhat arbitrary, so will need to be refined. Needs to be fixed to a specific value if doing linear combination fitting of EXAFS.

30. Data Reduction: Post-Edge Background • Post-Edge Background • Don’t have a measured m0(E) or “atomic” EXAFS. • Approximate m0(E) with and adjustable smooth spline function • Can be dangerous! Too flexible spline will match the real m(E) and remove all oscillations! • Want a spline that matches the low frequency components of the EXAFS.

31. Data Reduction: c(k) and k-weighting • c(k) • Raw EXAFS usually decays rapidly with k and is difficult to assess by itself • Customary to weight the higher-k regions by multiplying by k2 or k3. • c(k) is composed of a series of sine waves, so take Fourier transform to convert from k to R-space. • To avoid FT “ringing” multiply by a windowing function

32. Data Reduction: Fourier Transform, c(R) • c(R) • The FT in FeO has 2 main peaks, one for Fe-O and one for Fe-Fe. • The Fe-O distance in FeO is 2.14 Å, but here appears to be 1.6 Å. This is due to the phase shift term: sin[2kR+d(k)]. • A shift of -0.5 Å is typical. • The FT makes c(R) complex. Usually only amplitude is shown, but there are really oscillations in c(R). • Both the real and imaginary parts are used in modeling and fitting.

33. Data Reduction: Fourier Transform 16 U-Cu 12 U-Cu 4 U-Pd 3.06 Å Amplitude envelope [Re2+Im2]1/2 Real part of the complex transform 2.93 Å • Peak width depends on back-scattering amplitude f(k), the Fourier transform (FT) range, and the distribution width of R, a.k.a. the Debye-Waller, s2. • Do NOT read this strictly as a radial-distribution function! Must do detailed FITS!

34. Data Modeling What do these wiggles mean?

35. Data Modeling: FEFF • Can calculate f(k), d(k), and l(k) easily using FEFF • Take input of x,y,z coordinates of a physical structure and the central absorbing atom • Outputs files containing the calculation for each scattering path – can be a LOT of output files • These files can be utilized by many analysis programs • FEFFIT • ATHENA & ARTEMIS • SIXPACK • WINXAS • EXAFSPAK • Others • A structure that is close to the expected one can be used to generate a FEFF model, then used in the analysis program to refine distances, coordination numbers, etc.

36. Data Modeling: Information Content • Number of parameters that can be reliably determined from data is limited: • For typical data range k=3.0-12.0 Å-1 and R=1.0-3.0 Å, there are ~11.5 fit parameters that can be determined • Fit degrees of freedom =Nind-Nfit • Generally should never have Nfit>=Nind (<1) • This means that for every fit parameter exceeding Nind, there is another linear combination of the same Nfit parameters that produces EXACTLY the same fit function. • Important to constrain parameters or use chemical knowledge to help model Use as much information about the system as possible!

37. Data Modeling: 1st Shell of FeO • FeO has rock-salt structure • Calculate f(k) and d(k) using FEFF based on a a guess of structure, with Fe-O distance R=2.14 Å in a regular octahedral coordination • Use the calculated functions to refine values of R, N, s2, and E0 to match experiment

38. Data Modeling: 1st shell of FeO • k-space • Clearly shows there is another component! • R-space • Fit to the magnitude of c(R) does not look great, but definitely have the right phases as seen in Re[c(R)] data=bluefit=red

39. Data Modeling: 2nd Shell of FeO • Results are consistent with the know values for crystalline FeO: • 6 O at 2.13 A • 12 Fe at 3.02 A data=bluefit=red

40. Data Modeling: 2nd Shell of FeO • Fe-Fe EXAFS extends to higher-K than Fe-O • Even in simple system, some overlap of shells in R-space • Helps the fit significantly • Better agreement in both magnitude c(R) and Re[c(R)] data=bluefit=red

41. XANES That odd part in the front of my EXAFS…

42. XANES Region EXAFS (extended x-ray absorption fine structure) Pre-edge and Edge (XANES) Geometric Information Absorption Coefficient (mu) Electronic Information Energy XAS or XAFS

43. XANES Interpretation • EXAFS equation breaks down at low-k and mean free path goes up. No simple equation for XANES • XANES can be described qualitatively in terms of what electronic states the P-E can fill: • Coordination chemistry Octahedral, tetrahedral, distorted • Molecular Orbitals p-d hybridization, crystal field • Band-structure density of states • Multiple Scattering multiple P-E bounces • XANES calculations becoming more accurate and easier. Can explain what orbitals and/or stuctural characteristics give rise to certain features. Use of DFT also getting better. • Quantitative XANES using 1st principle calculations are rare, but becoming very possible,

44. XANES: Oxidation State and Coordination Chemistry • XANES of Cr(III) and Cr(VI) show a dramatic dependence on oxidation state and coordination chemistry • For ions with partially filled d shells, the p-d hybridization change dramatically as octahedradistort, and is very large for tetrahedral coordination • This gives a dramatic pre-edge peak – caused by absorption to a localized electronic state (1s to 3d)

45. XANES Fingerprinting • Since theory is not “easy”, often use XANES in fingerprinting analysis • Use series of model compounds and perform linear combination fits • Can use for phase and oxidation state measurements

46. XANES Summary • XANES is a larger signal than EXAFS • XANES can be done at lower concentrations and with samples that are less than ideal. • XANES interpretation is easy for crude analysis • Linear combination to previously measured model compounds is often sufficient • Information on electronic structure and coordination • Full theoretical analysis of XANES is more difficult than EXAFS • But the situation and theory is progressing… • And will be discussed in the next talk!

47. Further reading • Overviews: • B. K. Teo, “EXAFS: Basic Principles and Data Analysis” (Springer, New York, 1986). • Hayes and Boyce, Solid State Physics 37, 173 (192). • Historically important: • Sayers, Stern, Lytle, Phys. Rev. Lett. 71, 1204 (1971). • History • Lytle, J. Synch. Rad. 6, 123 (1999). (http://www.exafsco.com/techpapers/index.html) • Stumm von Bordwehr, Ann. Phys. Fr. 14, 377 (1989). • Theory papers of note: • Lee, Phys. Rev. B 13, 5261 (1976). • Rehr and Albers, Rev. Mod. Phys. 72, 621 (2000). • Useful links • xafs.org (especially see Tutorials section) • http://www.i-x-s.org/ (International XAS society) • http://www.csrri.iit.edu/periodic-table.html (absorption calculator)

48. Acknowledgements • Matthew Newville (APS) • Serena DeBeer George (SSRL) • Corwin Booth (LBNL) • SSRL • DOE, Office of Basic Energy Sciences • DOE, Office of Biological and Environmental Research, ERSD • SMB program supported by the NIH, NCRR, Biomedical Technology Program, and the DOE, OBER.