1. Principles of EXAFS Spectroscopy
Sam Webb

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

3. What can I do with x-rays?
SMW
Wilhelm Röntgen's first "medical" X-ray, of his wife's hand, taken on 22 December 1895

4. 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 1s
filled 3d
continuum EF 
core hole
unoccupied
states

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. X-ray absorption spectroscopy (XAS) experimental setup
“white” x-rays from synchrotron
double-crystal monochromator
collimating slits
ionization detectors I0 I1 I2
beam-stop
LHe cryostat
sample
reference sample
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
SMW

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) and d(k) depend on Z.
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).
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. What do I do with this data?
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)

28. Data Reduction: Pre-Edge and Normalization
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
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
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
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)
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. Real part of the complex transform
Data Reduction: Fourier Transform
Amplitude envelope
[Re2+Im2]1/2
Real part of the complex transform
12 U-Cu
4 U-Pd
16 U-Cu
3.06 Å
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. What do these wiggles mean?
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= Å-1 and R= 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=blue fit=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=blue fit=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=blue fit=red

41. That odd part in the front of my EXAFS…
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. No simple equation for XANES
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 octahedra distort, 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: Historically important: History
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). (
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)
(International XAS society)
(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.