Principles of EXAFS
Spectroscopy
Sam Webb
Topics
• Overview
– Process and Experiment
• Theory
– Brief History
– Derivation (simple)
• Data Analysis
– Data Reduction
– Fourier Concepts
– Data Modeling
What can I do with xrays?
X-ray Absorption
• X-rays are absorbed
through the Photoelectric effect
• Absorption occurs when
incident X-rays are
continuum
energetic enough to
expel core-level
unoccupied
electrons from atom
states
EF
• The atom is left in an
filled 3d
excited state with an
empty electronic level
(core hole)

• Any excess energy from
the x-ray is given to the
ejected photo-electron
1s
core hole
X-ray Fluorescence
• The excited corehole will relax back
to a “ground-state”
by transition of a
higher level electron continuum
unoccupied
in to the core hole.
states
EF
• This process emits a
3d
fluorescent x-ray.
• The energy of the
fluorescent x-ray is

characteristic of the
absorbing atom
1s
core hole
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:
m
rZ 4
AE
3
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
XAS is
Element Specific
Absorption Edge Energies
•As atomic number
increases, threshold
energies scale E0~Z2,
absorption coefficient
m~Z4.
•All elements Z>18
have either a K- or Ledge between 3 and
35 keV, accessible at
many synchrotrons
X-ray absorption spectroscopy
(XAS) experimental setup
double-crystal
monochromator
ionization detectors
beam-stop
“white” x-rays
from
synchrotron
• sample absorption is given by
m t = loge(I0/I1)
• fluorescence is
m ~ If/I0
• reference absorption is
mREF t = loge(I1/I2)
I0
collimating
slits
I1
I2
LHe cryostat
reference sample
sample
Theory
Why do I see wiggles?
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).
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)
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).
•
Once E is large
enough to
promote a core
electron to the
continuum, there
is a sharp
increase in
absorption
• Isolated atom has m(E) with a sharp step at the core binding
energy and is smooth as a f(E) above the edge.
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.
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
• EXAFS is an interference effect of the photoelectron
with itself, due to the presence of neighboring atoms
EXAFS
X-ray Absorption Fine
Structure
• Need to isolate the energy dependent
oscillations in m(E):
 (E) 
m ( E )  m0 ( E )
Dm 0 ( E0 )
– Subtract the “atomic background” m0(E)
– Divide by edge step Dm0(E0)
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).

k
2me ( E  E0 )
2
k  0.512 E  E0
• EXAFS often weighted by k2 or k3 to amplify oscillations
at high-k.
EXAFS: Simple Description
• Simple model has  ~ yscat
• The photoelectron:
1. Leaves the absorbing atom
2. Scatters from the neighbor
3. Returns to the absorbing
atom
With a spherical wave eikr/kr for the propagating photoelectron, 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
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:
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 
– Completely correlated with N!
– Makes EXAFS amplitudes (and thus N) less
precise than EXAFS phases (R)
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
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.
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
Data Analysis
What do I do with this data?
Data Reduction: Strategy
•
Take measured data to m(E) then to (k):
1. Convert measured intensities to m(E).
2. Subtract a smooth pre-edge to get rid of
background and absorption from other
edges.
3. Normalize m(E) to unit step height to
represent absorption of a single x-ray.
4. Remove a post-edge background function to
approximate m0(E) to isolate the EXAFS .
5. Identify the threshold energy, E0, and
convert from E to k.
6. Weight the (k) and Fourier transform from k
to R space.
Data Reduction: Raw to m(E)
I0
I
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
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.
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.
Data Reduction:
(k) and k-weighting
•
(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.
– (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
Data Reduction:
Fourier Transform, (R)
•
(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 (R)
complex. Usually only
amplitude is shown, but
there are really
oscillations in (R).
– Both the real and
imaginary parts are used
in modeling and fitting.
Data Reduction:
Fourier Transform
12 U-Cu
4 U-Pd
16 U-Cu
12 U-Cu
40
U LIII edge
3
FT of k (k)
20
0
2.93 Å
-20
data
fit
1
2
Amplitude envelope
[Re2+Im2]1/2
•
3.06 Å
4 U-Pd
-40
•
16 U-Cu
3
r (Å)
4
Real part of the
complex transform
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!
Data Modeling
What do these wiggles mean?
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.
Data Modeling:
Information Content
• Number of parameters that can be reliably determined
from data is limited:
N ind 
2DkDR

– 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!
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
Data Modeling: 1st shell of FeO
• k-space
– Clearly shows there is
another component!
• R-space
– Fit to the magnitude of
(R) does not look
great, but definitely
have the right phases
as seen in Re[(R)]
data=blue
fit=red
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
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 (R) and Re[(R)]
data=blue
fit=red
XANES
That odd part in the front of my EXAFS…
Absorption Coefficient (mu)
XANES Region
Pre-edge
and Edge
(XANES)
EXAFS (extended x-ray
absorption fine structure)
Geometric
Information
Electronic
Information
Energy
XAS or XAFS
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
Molecular Orbitals
Band-structure
Multiple Scattering
Octahedral, tetrahedral, distorted
p-d hybridization, crystal field
density of states
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,
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)
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
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!
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)
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.