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 x-rays?

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

unoccupiedstates

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

1s

3d

continuum

EF

core hole

unoccupiedstates

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:

3

4

AEZrm

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 L-edge between 3 and 35 keV, accessible at many synchrotrons

X-ray absorption spectroscopy (XAS) experimental setup

• sample absorption is given by m t = loge(I0/I1)

• fluorescence is m ~ If/I0

• reference absorption is mREF t = loge(I1/I2)

“white” x-rays from

synchrotron

double-crystal monochromator

collimating slits

ionization detectors

I0 I1 I2

beam-stop

LHe cryostatsample

reference sample

Theory

mk 2 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).

• 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

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

EXAFS

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)

)()()()(

00

0

EEEE

mmm

D

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.

20 )(2

EEmk e

0512.0 EEk

EXAFS: Simple Description• Simple model has ~ yscat• The photoelectron:

1. Leaves the absorbing atom2. Scatters from the neighbor3. 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

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– S0

2 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.

• 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!

3.06 Å

2.93 Å

1 2 3 4

-40

-20

0

20

40

16 U-Cu4 U-Pd12 U-Cu

r (Å)

FT o

f k3 (

k)U LIII edge

data fit

Amplitude envelope[Re2+Im2]1/2

Real part of the complex transform

12 U-Cu

4 U-Pd

16 U-Cu

Data Reduction: Fourier Transform

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:

– 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!

RkNind

DD

2

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…

XANES Region

Pre-edgeand Edge(XANES)

EXAFS (extended x-ray absorption fine structure)

XAS or XAFS

Energy

Abso

rptio

n Co

efficie

nt (m

u)

Electronic Information

Geometric Information

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,

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.