An Introduction to Xray Absorption
Spectroscopy
Paul Fons
National Institute of Advanced Industrial Science & Technology
paul-fons@aist.go.jp

CopyrightThis document is copyright (c) 2015 by Paul Fons
These notes borrow heavily on the work of Matthew Newville and Bruce Ravel and in thesame spirit as they intended, these notes too are open to others for sharing provided thatproper attribution is given.
This work is licensed under the Creative Commons Attribution-ShareAlike License. To view a copy of this license, visit
http://creativecommons.org/licenses/by-sa/3.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford,
California 94305, USA.
to Share -- to copy, distribute, and transmit the work
to Remix -- to adapt the work
to make commercial use of the work
Attribution – You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of
the work).
Share Alike – If you alter, transform, or build upon this work, you may distribute the resulting work only under the same, similar or a compatible license.
Waiver – Any of the above conditions can be waived if you get permission from the copyright holder.
Public Domain – Where the work or any of its elements is in the public domain under applicable law, that status is in no way affected by the license.
Other Rights – In no way are any of the following rights affected by the license:
Your fair dealing or fair use rights, or other applicable copyright exceptions and limitations;•The author’s moral rights;•Rights other persons may have either in the work itself or in how the work is used, such as publicity or privacy rights.
Notice – For any reuse or distribution, you must make clear to others the license terms of this work.

2

What is XAFS

(X-ray Absorption Fine-Structure)• XAFS (also XAS) refers to the modulation of the x-ray
absorption coefficient near and above an absorptionedge.

• XAFS is elementally selective and can explore the local
atomic environment about the absorbing atom

3

XAFS Characteristics• elementally selective• local atomic coordination• chemical / oxidation state• applies to any element• works at low concentrations• small sample volumes (even monolayers)
4

GeTe “ideal” Structure• Rhombohedral symmetry but more easily
visualized as distorted rocksalt structure.

5

GeTe “ideal” Structure• Rhombohedral symmetry but more easily
visualized as distorted rocksalt structure.

5

GeTe Diffraction
long-bonds

short-bonds• Neutron powder diffraction shows long- andshort-bonds calculated from average structure
become equal at Tc

6

XAFS measurements• Analysis showed that rhombohedral distortion

existed up to and beyond Tc
XAFS is a local and Diffraction is a (longer) range probe
P. Fons, A. V. Kolobov, M. Krbal, J. Tominaga, K. Andrikopoulos, S. Yannopoulos, G. Voyiatzis, and T. Uruga. Phase transitions in
crystalline GeTe: Pitfalls of averaging effects. Phys. Rev. B, 82(15):155209, 2010.

7

XAFS Example
Complex Scattering Factor of Fe

15

10
f’’
5
Pre-Edge

0

0

2000

4000

Energy (eV)

6000

8000

Modulation of the x-ray absorption above the edge
constitutes the XAFS signal

8

XAFS Example
Complex Scattering Factor of Fe

15

10
f’’
5
Pre-Edge

0

0

2000

4000

Energy (eV)

6000

8000

Modulation of the x-ray absorption above the edge
constitutes the XAFS signal

8

XAFS Example

Pre-Edge

Modulation of the x-ray absorption above the edge
constitutes the XAFS signal

8

XANES vs. EXAFS• XANES X-ray Absorption Near-Edge Spectroscopy (0-40 eV)• EXAFS Extended X-ray absorption Spectroscopy (40-1000 eV)
XANES and EXAFS are labels for two parts of the same spectra.
They are distinguished due to the approximations used in theanalysis of data
XANES region: The energy of the photoelectron (PE) is small,the mean free path long (nm) and PE interactions strong
requiring more time consuming cluster calculations
EXAFS region: The PE energy is higher and scattering can betreated as a finite series of interactions that can be wellapproximated by the the EXAFS equations we will encounter.
9A visual interpretation of XAFS
Photon In
B

Photoelectron out
B

B

eA

A
B

B

For a s-state core level, an
outgoing spherical wave
moves out from theabsorbing atom.

The outgoing spherical wave is scattered by the (coloumb)potentials of neighbor atoms causing interference with itself.
The absorbing atom serves both as the source of thephotoelectron and the detector. In other words, the XAFS signal
reflects the local neighborhood about the absorbing atom.

10

Photoelectric Effect• An x-ray is absorbed

via the photoelectric effect inwhich Ephoton induces a transition
from a core-level to a continuumstateContinuum
V

Energy

IV

M III
II
I

• The atom is left in an excited statewith an empty level (core hole)

III

L II
I

• Excess energy becomes kinetic

energy of ejected photoelectron
EKinetic = hν - EEdge

KI

A K-shell excitation
11

Recombination Processes
When a core-hole is created, the absorbing atom is in an
excited state and can return to the ground state via Auger
electron emission (AES) or X-ray fluorescence (XRF)
ContinuumContinuum

M
L

M
L

K

K

K

Continuum

M
L
K

K

XRF: photon out. Ephoton
is element specific andoffers a way to measure
XAFS signals selectively
and with great sensitivity

AES: Unlike XRF which has an (energy
dependent escape depth typically intens of µm in the hard x-ray region,
AES is a surface sensitive detection
technique12Linear Absorption Coefficient
The intensity of an x-ray beam passing through matter can be
described by

I = I0 eµtwhere I0 is the initial intensity, t is the sample thickness, and μ
is the x-ray absorption coefficient
μ depends on the x-ray
energy, the atomic number
Z of the absorber, thedensity ρ, and the atomic
mass A

µ

Z4
AE 3
13

Absorption in a Compoundz

The loss in a slice of thickness dz

dI =I(z)µdz

The number of absorption events W is
thus

W = I(z) atµ=dz

a dz = I(z)µ dz

a =

m NA

M

where
ρat = atomic number density
σa= atomic cross sectiona

μ for a compound canthus be calculated asj
at jµ(E) =
j14

EXAFS

Extended X-ray Absorption Fine Structure
We are interested in the interference function in the absorption
signal as it contains information about local structureµ(E) µ0 (E)(E) =µ(E0 )
By subtracting away the smooth atomic “bare atom” (µ0)
background and dividing by the edge step Δμ0 we can find theoscillations χ normalized to a single absorption event
3

5
2

0
1

-5
0

-10
11100

11200

11300

11400
E(eV)

11500

11600

11700

μ(E) and smooth μ0 for GeTe

100

200

300

400

500

600

μ(E) and smooth μ0 for GeTewith E0=11103 eV

15

EXAFS Equation
While the XANES region requires calculations, the EXAFS region
can be fit using the EXAFS Equation

Here Fi(k) and δ(k) encompass the photoelectric scattering propertiesof the ith neighboring atom. If we know F and δ, we can fit:
R the distance to the neighboring atom i
N coordination number of the neighboring atom
σ2 mean squared disorder
Both Fi(k) and δ(k) depend on the atomic number Z so the type ofatom surrounding the absorbing atom
16

XANES

increasing
valence

XANES region (absorption edge) can beused to determine chemical valence
Unlike EXAFS contains bond angle
information although not simply extracted
Spectra can be added to match
experimental spectra revealing the local
chemical environment

XANES edge for S compounds in
different oxidation states

For light elements XANES spectra canreveal the conduction band DOS

XANES spectra for Cr6+ and Cr3+

17

Measurement Techniques• We would like to measure a signal proportional to thenumber of x-rays absorbed• Transmission (directly measure the transmitted beam)• Fluorescence (measure the outgoing photon when thecore hole recombines: see slide 8)• Electron Yield/Conversion Electron Yield (measure the
Photoelectron/Auger electron signal)18

Transmission
I0

I1

Gas Ion Chambersµ(E)t =ln(I/I0 )• For concentrated samples, transmission is often best techniquefor XAFS, but proper sample preparation is required∋ 1< µ<3• Want• For Fe foil Δμ=1 requires 3.7 µm thick foil!

often solution is to make BN+sample powder pellet• samples must be uniform without pinholes• the grain size must be smaller than the absorption length19Why are Pinholes bad?
µ is defined by
I1 /I0 = e

I0

I1

where <> represents thespatial averageµ(E)tWhat is actually measured istµeff = ln I0 /I1 = ln eµ(E)t

If the thickness t is the same everywhere, the brackets
vanish and something proportional to µ(E) is measured
If t varies, the response is non-linear and notproportional to µ(E)20

Example
Transmission of each partA hypothetical sample consistsof 90% uniform material S with
10% pinholes.

The edge jump is 4 and µ
modulation is 0.1
The net transmission is 0.90

E < Eedge

E > Eedge

0

0

0.1

Tholes

1

1

1

TS

1

e-4

e-4.4

TS + 0.1

E < Eedge

Thole

E > Eedge

T
T

1
1

0.1 + 0.9e 40.116484

0.11105µ=log(T)02.15000

2.19778

0.1 + 0.9e 4.4

Instead of 𝜒~0.1, the calculated 𝜒 is ~0.048.
The spectra is “compressed”

21

Fluorescence Measurements
Fluorescence
Detector If

Ionization
Chamber I0• Concentrations down to the ppm levelcan be measured• Background from other signals candominate leading to dead time poor S/N• Sample non-uniformity big problem (Icorrection can fail)0• Self-absorption (concentrated samples)

X-ray Fluorescence Spectrum

can lead to effects similar to pinholes intransmission
22

Fluorescence Measurements• For a good fluorescence experiment

Energy Discrimination: Eliminate unwanted energy photons by physical orelectronic filter
Solid Angle: As emitted XRF photons are isotropic, maximize solid angle

An example of physical filtering is the use ofa so-called “Z-1” filter.

Can be used with a detector with no energy
resolution or to reduce flux to reduce deadtime for energy discriminating fluorescence
detector4

Mn filter
Fe K α1
Fe K α23

Fe K edge

Filter and Lines

For example, the Fe Kα emission lines arelocated below the K-edge of the next “Z-1”element down, Mn.

Z-1 Filter Plot

2
1
0

6000

6500

7000

7500

Energy (eV)Fe Kα emission lines and a“Z-1” Mn filter
23

Experiment Design
Important considerations for designing an XAFS experiment include:
Monochromatic x-rays: Need x-rays with a small energy spread or bandwidth∆E ≈ 1 eV at 10 keV. Second harmonics of the monochromator need to beeliminatedLinear Detectors: The XAFS χ(k) ∼ 10−2 or smaller, so we need many photons anddetectors that are very linear in x-ray intensity (ion chambers). This usually means
using a synchrotron source.
Well-aligned Beam: The x-ray beam hitting the detectors has to be the same beamhitting the sample.
Homogeneous sample: For transmission measurements, we need a sample that is of
uniform and appropriate sample thickness of Δμ∼1 absorption lengths. It should befree from pinholes. Powders need to be very fine-grained.
Counting Statistics: μ(E) should have a noise level of about 10−3. That means weneed to collect at least ∼ 106 photons. For very low concentration samples, this may
require hours of counting time.
24

An EXAFS Experiment• To collect EXAFS data, we need to scan a monochromatic x-ray
beam through the edge. Typical ranges are below• In the EXAFS region, it is common to step in constant k• If errors are the result of Poisson statistics to have a 1%
statistical error will require √N/N=0.01

Typical Ranges for
XANES and EXAFS

RegionPre-edge
XANES
EXAFS

Start E (eV)
-200
-20
30

10,000 counts

End (eV)
-20
40
~800

Step (eV)50.25-0.50
0.05 Å-1

25

Free Atom Absorption

EnergyA transition can only occur if there isan empty state for the core electron
to move into.

EnergyUpon absorption of an x-ray with energy E0 by a free atom, a corehole is created with K.E. (E-E0). The core hole is filled by anotherelectron (fluorescence) or an Auger process occurs and another e- isejectedTransitions into the continuum lead tophotoelectron
large increases in absorption
V(x)μ increases sharply at E0 edge for a
free atom and monotonicallydecreases with increasing energy

x-ray
free atom

absorption
probability

26

Backscattering
The photoelectron propagates outward in the form of a sphericalwave (for k-shell holes) and parts of the wavefunction scatter off
surrounding atoms and return to the absorbing atom.
The backscattered photoelectron will then interfere with itself.

XAFS is thus caused by thephotoelectron interfering with itself
due to the presence of neighboring
atoms

EnergyEnergyThere must be empty states at (E-E0)
at the absorbing atom µ to be nonzero.
The amplitude of the backscatteredwavefunction at the absorbing atom photoelectron
and its variation with energy give rise
to oscillations in µ that are XAFS. V(x)x-rayabsorbingatom

scatteringatom

absorption
probability

27

EXAFS: Physics
Now let us find a simple form for χ. First we note that the initial state is acore state and we wish to find the effect of backscattering on the final state.
If we recall the definition for µ(E)µ(E) µ0 (E)(k) =µ(E0 )

H is the interaction. In the dipole
approximationis the final state which consists of aphotoelectron, a core hole, and no photon. The
photoelectron is scattered by the neighboring
atom.

EnergyNote that the initial state is a core state anda photon; this state is not changed by thepresence of a neighboring atom

Energyµ describes a transition between states and can be written in terms of
Fermi’s golden rule:

photoelectron

V(x)

x-ray
absorbingatom

scatteringatom

absorption
probability

28

EXAFS: Physics
EXAFS is a “final state effect”. As the initial state is deep and localized, itchanges little due to neighboring atoms. The final state does howeverchange due to backscattering from a neighboring atom. If is the final
state of an isolated atom, the final state isexpanding in µ to
second order gives:
as

where

implies,

If we take the initial state as a delta function δ(0), Χ becomes

Conclusion: is the part of the photoelectron wavefunction that isbackscattered to the origin by neighboring atoms
29

EXAFS: A simple picture
If, we can understandphotoelectron ψp’s pathby following theBB• ψp leaves the absorbing atom
• part of ψp scatters off a nearby atom
• returns to the absorbing atom
outgoing ψAB

BbackscatteringWith a spherical wave eikr/kr for the outgoing photoelectron and ascattering atom at distance r=R,

where scattering from the neighboring atom gives rise to the amplitude
f(k) and phase shift δ(k) to the photoelectron
30

EXAFS: A simple picture
Combining all terms, we get the expression for a single scattering atom

If N atoms are present with a thermal and static disorder of σ2 (whichgive the mean square displacement in R, we find

In a real structure, there are neighbors at different distances and a sum
must be taken

31

EXAFS: A simple picture
To get the expression,
We used a spherical wave (eikr/kr) for the photoelectron, but we musttake into account inelastic scattering processes. To do this, we include a
damping function representing that limits how far a photoelectron can
travel and still return to interfere coherently
Here λ is the mean free path (andincludes core-hole lifetime effects)
This leads to our first version of the EXAFS equation

32

Mean Free Path

The mean free path λ depends on k, but λ < 25 Å for the
EXAFS range. The λ term and R-2 lead to EXAFS being ashort range probe.
Note that λ and R-2 get large for low k (e.g. the XANES
region)33

Amplitude Reduction Term
The one remaining correction term in the EXAFS equation is the
Amplitude Reduction Factor or S02 . This term is due to therelaxation of the other electrons in the atom due to the presence of acore hole.

Here theterm accounts for the relaxation of the N-1 electrons
left behind with the core hole relative to the same electrons in theunexcited atom
. In fitting the EXAFS equation, S02 is often taken
as a constant 0 < S02 < 1.
While great strides have been taken into calculations of λ and S02, typically
S02 is still taken as a constant for the absorbing atom in fitting data.
This and other issues make the accuracy in determining
R (phases) better than N (amplitudes)

34

The EXAFS Equation
The EXAFS equation thus takes the form

The sum is taken over all “shells” of atoms or “scattering paths” forthe photoelectron.
If we know the scattering properties of: fi(k) and δ(k) as well as λ(k),we can determine:
N the coordination number about the absorbing atom
R the distance to the nearest atom
σ2 the mean-squared disorder of the corresponding neighbor distanceAs the amplitude f(k) and phase-shift δ(k) depend on atomic
number, XAFS is also sensitive to the Z of the neighboring atom.
35

Scattering amplitude f(k) and phaseshift δ(k) examples
The scattering amplitude f(k) and phase-shift(k) depend on
atomic number and can be calculated.

The scattering amplitude f(k) peaks at
different k-values and goes to larger k-values
as Z increases. Note the structure in f(k) for
heavy elements like Pb.
The phase-shift δ(k) changes abruptly forlarge Z. The Z of a neighboring atom cantypically be determined ±3 using phase-shifts.
Accurate values for f(k) and δ(k) can be calculatedusing first-principles code such as FEFF9.

36

Using feff to calculate f(k) and δ(k)• Feff is a real-space Greens function code that solves thescattering problem and computers the effective scattering
amplitude fi(k) and phase-shift δi(k) a given path i.

• Input to Feff (version 9 at the moment) consists ofspecifying the location of atoms in space, their (Z) types,
and the absorbing atom. The resulting f(k),δ(k), and λ foreach path are saved in files feff0001.dat, feff0002.dat, etc.

• Recent fitting programs use feff directly to model EXAFS
data.

• A structure close to unknown is sufficient for feff input
and fitting allows the refinement of distances andcoordination numbers.

37

Multiple Scattering
The path sum in the EXAFS equation can include manyshells (1st, 2nd, 3rd nearest-neighbors). For paths beyond
the nearest neighbor path, multiple-scattering paths canbecome important
singlescattering

triangular focused multiple
paths
scattering paths

For multiple scattering paths, theamplitude depends on scattering angles.
For triangular paths with 45 <θ< 135°
not strong, but may be many in number.

For nearest-neighbor analysis, multiple
scattering is rarely needed.

For linear-paths, with angles θ~180°, arevery strong: the photoelectron can befocused from one atom to the next

Multiple scattering strongest for θ>150°
38

Fitting Strategy• Convert measured intensities to µ(E)• Subtract a smooth pre-edge function to eliminate backgroundfrom instruments or other edges• Normalize µ(E) to go from 0 to 1 so that it represents theabsorption of one x-ray photon.

• Remove a smooth post-edge background function toapproximate µ0(E) to determine χ(k)• Convert E to k-space• Assign a k-weighting function to χ(k) and the Fourier Transformto real space• Isolate the χ(k) for a particular “shell” by Fourier filtering
Next: Model f(k) and δ(k) to fit χ(k)39

Data Reduction:Transmission
Converting raw measured intensities (in transmission) to µ(E)

I0

I

50x10

3

2200

I1 (counts)

I0 (counts)

40
2100

30

20

2000

10

1900
11.0

11.5

12.0

3

11.0

12.5x10

11.5

Energy (eV)

3

12.0

12.5x1012.0

12.5x10

Energy (eV)-ln(I/I0)
For XAFS in transmission-0.6

ln(I0/I1)

-0.8

-1.0

-1.2

-1.4
11.0

11.5

3

Energy (eV)40

Software• Some free data processing programs
A comprehensive system for processing and analyzing X• Demeter
ray Absorption Spectroscopy data. (Linux, Windows, Mac-version:

available)• http://bruceravel.github.io/demeter/• Athena (data processing) Artemis (fitting)• Macintosh version iXAFS• installable using the demeter package frommacports http://www.macports.org (installdemeter)• Ifeffit mailing list (artemis, athena, demeter)
41

Data Reduction: Athena
GeTe at 300 K @bl01b1
3µpre-edgepost-edge

xμ(E)

2

1

Δμ0
0

Pre-Edge Subtraction
Pre-Edge subtraction removes
the effect of other edges/
background effects

-1
11000

11200

11400

11600

Energy (eV)Normalization

normalized xµ(E)2Post-edge subtraction
determines Δμ0 so the EXAFS
event is for a single x-ray
photon

1

0
11000

11200

11400

11600

Energy (eV)42

Data Reduction: Athena3µ0 spline
I1

Post-Edge Backgroundµ0(E) is approximated by asmooth spline with no Fouriercomponents in the EXAFS
region (k-space)

xμ(E)

2

1

0

-1
11000

11200

11400

11600

Energy (eV)k2 Weighted χ Function
After subtraction of spline
approximation to µ0(E), convert
to momentum units

0.5

0

-0.5

0

5

10

15

43

Data Reduction: Athenak-weighted χ(k): k2χ
χ(k) is composed of sine waves
from individual paths so we willtransform to R-space. Use aapodization window to avoid“ringing” effects

0.5

0

-0.5

0

5

10

15

0.6

0.3

0.2

0

0

1

2

3

4

5

6

χ(R)
The Fourier Transform of k2χ
has two main peaks. The spectra
are shifted with respect to thetrue bond lengths due to thephase-shift δ(k). A shift of 0.5Åis typical44

Data Reduction: Athena
0.6

0.3

0.2

0

0

1

2

3

4

5

6

χ(R) is complex. Often only the magnitude is shown.
χ(R) is not a radial distribution function, but an interference function
In data fitting, both the real and imaginary parts are used
45

The Information Content of EXAFSThe number of parameters obtainable from a single set of data is limited:

2 k R
N⇠
⇡
Here Δk and ΔR are the k- and R-ranges of the data used. For typicalranges of k=[3,12]Å-1 and R=1,3] Å, there are about 11.5 statisticaldegrees of freedom or parameters that can be fit from EXAFS.
The “Goodness of Fit” statistics and corresponding confidence in themeasured parameters need to reflect this limitation.
It is often important to constrain parameters R, N, and σ2 for different
paths or even different data sets (different T, element edges etc.)
We can use chemical plausibility to weed out bad results orincorporate additional known data (coordination, Bond Valence Modeletc. ) to improve the quality of the fit.

46

Polarization Dependence• The (ȇ・r ) of the dipole approximation
leads to an orientation dependence of χthat depends on the type of edge

• K,L Edges
1

•

2
3
cos
✓
L2 L3 Edges (assuming d-like final state)
7p
2p

1
2
1 + 3 cos ✓2

For systems with symmetry lower thancubic, polarization can be used as a tool
to check for the anisotropy of astructure

3p

12

p

5p

2
5.

3

12

3

5p

p

K,L1s-state
L2,L3 p-state

4.

4

p

4

p

3.

6

6
11 p

2.

p

12

1.

12

p

0.

0

13 p

23 p

!
E

12
7p

12
11 p

6

6
5p
4

7p
4p
3

5p
17 p
12

3p

19 p
12

3

4

47

A Fitting Example

48

A Fitting Example

49

A Fitting Example

50

A Fitting Example

51

Experimental Considerations• In what follows a brief introduction to some of thecomponents of a beamline will be shown

52

BL01B1• Si double monochromator (111), (311), or (733)• Focusing harmonic rejecting mirrors• automatic adjustment of beam location with energy• energy range 4.5-110 keV
53

Bl01B1 Energy Dependence• The energy dependence of x-rays from bl01b1 as
predicted by the program SPECTRA (http://
radiant.harima.riken.go.jp/)54Absorption Edges
K and L3 edges•• Elements with Z>18 have either a105

K Edge Energies ∝ Z2

104
Log Edge Energy HeVLK or an L-edge accessible between
3 and 35 keV

L3-edges

1000

100

10

• Soft x-ray XAFS possible as well
• Edges with E<3 keV require special

Elements up to Z=93

1
0

20

40

60

80

Atomic Number

K and L3 edges
30 000Elements up to Z=53

25 000Edge Energy HeVL

preparation (vacuum) while edges
over E>35 keV can suffer fromreduced cross section, Compton
scattering, and multielectron effects.
(3 keV<E<35 keV Hard X-ray
region easy to measure in)K-edges

K-edges

20 000
15 000
10 000L3-edges

5000
0
0

10

20

30

Atomic Number

40

50

55

Harmonic Rejection• As the monochromator lets through x-rays that satisfy

the Bragg condition (λ=2d sinθ) not only λ, but λ/2, λ/3(higher harmonics) are transmitted as well.

• At bl01b1, higher harmonics are rejected using a mirror,
for a fixed wavenumber k, x-rays will not reflect forangles greater than the critical angle αc.
c =fundamental λ is reflected but higher
harmonic λ/2 and above are not

2 =

4

r0
k

ρ number density of coating
r0 electron scattering length=2.82 x 105 Å
k = incident wavevector
56

Measurement Techniques
x-raysI0

sample

I1• The intensity of the synchrotron varies with time. It

must be normalized. Typically this is done by using an ion
chamber which measures the current of ionized gas
between two capacitors.

• To optimized statistics without too much signal

attenuation, the I0 chamber should operate at about
15% attenuation. I1 should be about 85%.
Hephaestus can help.
At 12 keV, a 140 mm Ion Chamber
will absorb 15% of the incoming
signal if a N2/Ar mixture in a ratioof 85:15 is used.
57

Absorption Length• Absorption Length µ is
• distance over which the incident intensity decreases
by a factor of 1/e (37%)• an important factor in designing EXAFS experiments
and should be calculated

sample calculation for GeTe
ρ=6.212 gm/cm3, fi(Ge)= fi(Te) =1/2
above edge @11.4 keV σGe=22,340 Barns, σTe=22,380 Barns
In[342]:=

s@GeD = 185.359 cm 2 ë gm; s@TeD = 105.629 cm 2 ë gm;

In[344]:=m = I6.212 gm ë cm 3 M Hf@GeD s@GeD + f@TeD s@TeDL
835.838

Out[344]=cm

In[345]:=absorptionlength = 104

Out[345]=

11.964 mm

mm 1
cm m

where a Barn is 10-24 cm2
The 1/µ length is about 10 µm
58Absorption Length• One can also calculate the absorption length using Hephaestus(part of the Horae suite).
http://cars9.uchicago.edu/ifeffit/Downloads• or http://macports.org “port install demeter”

59

Absorption Lengths andParticle Size
• It is important to keep particle size
below one absorption length• This leads to sample areas that areopaque and gaps that distort the“white line” and affects σ2 as well

• Samples should be uniform on thelength scale of the absorption length(small particles must be used)

MnO spectra as a function ofparticle size (10% leakage)
60

X-ray Fluorescence• X-ray fluorescence is highly sensitive• A good sample is:• dilute (to avoid self absorption)• homogeneous (I will not divide0

out if sample is non-uniform)• If there is only one strong

Lytle-Heald Detector(not energy sensitive)

EnergyDispersive
Analyzer(Ge SSD)

fluorescence, a Lytle-Heald detector
can be used (ion-chamber)• If several elements are present, anenergy dispersive (Ge SSD) can beused, but dead time can be a
problem.
61

X-ray Fluorescence (cont.)• Self-absorption can significantly distort

Lytle-Heald Detector(not energy sensitive)

signal.

• Correction possible

Athena/Demeter offer correction
Booth and Bridges, Physica Scripta T115, 202 (2005)• Dead time of detector can distortfeatures (e.g. white line)

EnergyDispersive
Analyzer(Ge SSD)• Correction possible• Diffraction (e.g. from single crystal
substrates) can be lead to strong
glitches• can be reduced by averaging signal
while rotating sample slightly

62

Disorder in EXAFS and diffraction
In diffraction, we define single center
displacement parameters u
u21 = (R1

R1 )2 , u22 = (R2

u2

u1

R2 )2

In EXAFS, we define correlated displacement parameters also
known as Mean Squared Relative Displacement (MRSD)
2
12 =

(R2

= u21 + u22

If |u1 | = |u2 |

R1 )

(R2

R1 )

2

=

2 (u1 · u2 ) = u21 + u22

(R2

(R1

R1 )

2u1 u2 uˆ1 · uˆ22= 2u2 , if uˆ1 · uˆ2 = 0

2= 0, if uˆ1 · uˆ2 = 1

2

R2 )

= 4u2 , if uˆ1 · uˆ2 =163

Spatial Resolution (Beats)

Spatial resolution

The ability to resolve two distances “close” to each other.
R1

R1

OR
R2

R2

R = (R1 + R2 ) /2

 = R /2

R

Intuition: To resolve, ΔR needs to be a significant fraction of the smallest
Intuition: To
resolve needs R to be a significant fraction of smallest p.e.
photoelectron
wavelength
wavelength:

2
min

R 
=
=
4
4k max 2k max

kmax 
2R

Pioneering
Science andTechnology

De Broglie 2k=Office of Science
U.S. Departmentof Energy

64Spatial Resolution (Beats)

65

Features of XANES• Pre-edge region• Features caused by dipole transitions to (empty)
bound states.

• Local geometry about absorbing atom

• Absorption Edge
• ionization threshold to continuum states• Larger oxidation state (chemical shift) moves edge up
in energy• XANES region is characterized by multiple scattering.

Includes information about bond angles and distances.
Need to use simulation (Feff9) to understand structure66

Selection Rules
XANES probes unoccupied electron states• Dipole selection ruleΔl (orbital angular momentum) = ±1, Δj (total angular momentum) = ±1

• Edges• K-edge and L Edges: s (l=0) ➞ p (l=1)• L , L -Edges: p(l=1) ➞ d(l=2) or s(l=0)• Note that states can mix hence by using the12

3

selection rules XANES can be used as a probe of
hybridizationnote that quadrupole transitions are possible but typically have about 1/100 of the strength of
dipole transitions
67

K-edge Pre-Edge Peaks• In a metal, a K-edge transition implies a s ➞ p liketransition• For Octahedral coordination site iscentrosymmetric implying no p-d mixing (onlyweak quadrupole transitions)• p-d mixing becomes possible with distortion
from octahedral point symmetry• tetrahedral site symmetry, largest pre-edge
peaks

68

Using XANES with mixtures• The coherence length (the distance over which themultiple scattering occurs) in XANES is on the order of
1 nm• If the sample is homogeneous over distances of 1 nm,
then different regions of the sample will add
incoherently
XANES(mixture) = Σfi XANESi

Example of the decomposition of PtO1.6
Example: The decomposition of PtO1.6
All XANES spectra collected repeatedly during the thermal
decomposition process could be fit by a weighted sum of Pt and thestarting compound PtOx demonstrating that no intermediate phases
were involved. The figure on the left shows the metal fraction allowing
the kinetics of the reaction to be explored

Kolobov et al., Appl. Phys. Lett. 86, 12 (2005), 121909.

69

XANES Summary• The XANES signal is significantly larger than EXAFS andexhibits little temperature dependence.

• XANES spectra can be used as a fingerprint and linear

combination type analyses are easily carried out. Simple
selection rule considerations can give considerable
insight into the bonding in a sample.

• The multiple scattering present in XANES spectra makeit more difficult to treat theoretically, but the real-space
multiple scattering code Feff (now at version 9) allows
calculation of theoretical spectra using only thecoordinates and types of atoms in a cluster.
http://leonardo.phys.washington.edu/feff/

70

XANES SoftwareThis list is inevitably incomplete• Multiple Scattering Code
• FEFF9 from http://leonardo.uwash.edu• EXCURVE http://ccpforge.cse.rl.ac.uk/gf/project/ Free

ES
N
XA FS
EXA

excurv/• Density-Functional Theory• Wien2K all electron code http://wien2k.at• CASTEP plane wave code
e
e
FMNES
http://neel.cnrs.fr/spip.php?rubrique1007
r
F
•

ES
N
XA FS
EXA
ES
N
XA FS
EXA
ES
N
XA FS
EXA

71

Quick Fit• If time allows, we shall review a quick fit of GeTe at the

Ge edge. (an more complete tutorial would take several
hours)• goal is to give a bird’s eye overview of the fitting
process• Software: artemis and athena (horae suite)• Steps• Athena: Background processing: using a spline to
subtract background and obtain 𝛘 function• Artemis: Calculate theoretical paths for a model
structure and fit to experimental 𝛘 data72

Athena• Artemis is the Greek goddess of wisdom• Read in 200K GeTe data taken at Ge K-edge• with plugin can read SPring-8 or KEK data
directly• Fit linear pre-edge line to pre-edge region andpolynomial spline to Normalization range• Subtract to get 𝛘Athena
Edge E0

pre-edge
markers

Normalization Range

Edge Step Normalization
Raw µ• Use a spline function with no real
space contributions above 1.4 Å• The E value will be used to0determine to zero of k

• Subtract off spline to generate 𝛘• Save 𝛘 in a file for processing withathenaEdge step
normalized µAthenaRaw µ(E)

Fourier Transform𝛘(R)𝛘 Output

Reverse Fourier Transfor
𝛘(q)Athena• Many bells and whistles to Athena• deglitching• linear component analysis (XANES)• principle component analysis (XANES)• self absorption corrections (fluorescence)• multi-electron effects removal• calculate differences between spectra• show many spectra and work in groups• and many more features …
77

Multiple Data Sets

• Analyze multiple datasets to see• experimental trends with external variable, e.g. T• compare and adjust Athena processing parameters

GeTe Structure
The rhombohedral
structure can be
more easily bevisualized as a nonprimitive facecentered cubic cellblue = 3.158 Å
green = 2.843 ÅArtemis• Artemis is the Greek goddess of the hunt
• Use for fitting against theoretical (feff) standards• We will read in the 200K background subtracted GeTedata from Athena• Fit using two feff models• GeTe polycrystalline matrix• Ge (small amount of second phase)80Artemis Layout
datasets

feffcalculations

Plotting Tools• Artemis can fit one or more datasets
simultaneously (multiple edges,
multiple temperatures, multiple
pressures, etc.)• older version of feff, feff6 (for EXAFS
paths) built in - good for EXAFS but
not for XANES

paths to plot

• fit EXAFS equation81Artemis• Clicking on the Add buttonin the dataset box allows
importing of dataprocessed by Athena• the data is displayed

selected
path

Fitting
Paths• It also cause a new fitting
dialog box to open inwhich paths to be fitted
can be added

selected
path parameters• The fitting paths have tobe generated by a feff
calculation (next)

Each dataset has a corresponding Fit window82Artemis• Clicking on the Add tab in
the feff calculations brings
up a dialog box for
specifying the structure• Running atoms generates afeff input file• make changes ifnecessary83Artemis• Clicking on the Add tab in
the feff calculations brings
up a dialog box for
specifying the structure• Running atoms generates afeff input file• make changes ifnecessary83Artemis• running the feff
calculation• feff calculates

important paths

path plotting
options• Refine structureusing these paths• Next we will

specify parameters
for each path fromthe EXAFS
equationscattering
paths

84

Artemis• running the feff
calculation• feff calculates

important paths• Refine structureusing these paths• Next we will

specify parameters
for each path fromthe EXAFS
equation

84Artemis: Fitting paths
feff result
window

datasetwindow
Fitting
Paths

Drag and Drop

Artemis: FEFF calculations• In this case, we need a second model for Ge
precipitates

86

GeTe
model

Artemis: FEFF calculations

Ge
model

• In total we have three paths in our model:• 2 from GeTe
• 1 from Ge

86Artemis Fitting Paths• In our example we have• carried out two feff calculations• GeTe• use the two NN paths from this• Ge
• use the NN path from this• Next step is to set up the fitting expressions
87Artemis: path visualization• We can for example plot the

GeTefeff calc

magnitude of first three paths
for GeTe in real space using theplotting controls• Note the phase shift results inthe path lengths being smaller
than in the actual structure• We can do the same thing for
the Ge feff modelGe
feff calc

88Artemis: Fitting Parameters
Selected Path
Fitting Domain
Fitting Weights

Fitting Parametersof selected path

Fitting parameters
Label
N
S02
ΔE
Δr
σ2

ΔE represents diff between energy zero of feff and expt90Define expressions for paths• For each path
• we define math expressions
for each EXAFS equation
variable• These variables can be thenbe varied either individual orby mathematical expressions
containing multiple variables

Label
N
S02
ΔE
Δr
σ2

coordination
error in energy zero
distance
disorder91

GDS panel• The GDS (guess, define, set)
panel allows• defining a guess to bevaried• a bounds restraint• a penalty function• all entries can be

Options

potentially complicatedalgebraic expressions
92

Artemis: Fit result• Fit in real space from 1.8 to 3.2 Å: r-factor = 0.00373• Display the Real part of the complex function χ as wellas the magnitude93Artemis: Summary• We have fit a 200K GeTe EXAFS spectra out to 1st NN• There are many different options we have not touched
upon in the software suite due to time limitations• Some additional things we could do• Temperature dependence (multiple data sets)• Fitting simultaneously at more than one edge
• Including multiple scattering paths• many more
94

Resources• Books••

XAFS for Everyone by Scott Calvin, CRC Press 2013.

•

X-Ray Absorption: Principles, Applications, Techniques of EXAFS, SEXAFS and
XANES (Chemical Analysis: A Series of Monographs on Analytical Chemistry and Its
Applications) by D. C. Koningsberger and R. Prins, John Wiley and Sons 1988.

•

Elements of Modern X-ray Physics by Jens Als-Nielsen and Des McMorrow, 2nd Ed.
Wiley 2011.

Introduction to XAFS: A Practical Guide to X-ray Absorption Fine Structure
Spectroscopy by Grant Bunker, Cambridge 2010.

• Review Articles•

Theoretical approaches to x-ray absorption fine structure by J. J. Rehr and R. C. Albers,
Rev. Mod. Phys. 72 (3) 621-654 (2000)• Internet•

http://www.xafs.org International XAFS Organization
This site has links to tutorials by synchrotron organizations and other useful
information including an online magazine. Membership is free.
95