EXAFS in theory: an experimentalist’s
guide to what it is and how it works
Corwin H. Booth
Chemical Sciences Division
Glenn T. Seaborg Center
Lawrence Berkeley National
Laboratory
Presented at the SSRL School on Synchrotron X-ray Spectroscopy Techniques in
Environmental and Materials Sciences: Theory and Application, June 2-5, 2009
Advanced EXAFS analysis and
considerations: What’s under the hood
Corwin H. Booth
Chemical Sciences Division
Glenn T. Seaborg Center
Lawrence Berkeley National
Laboratory
Presented at the Synchrotron X-Ray Absorption Spectroscopy Summer School ,
June 28-July 1, 2011
Topics
1. Overview
2. Theory
A. Simple “heuristic” derivation
B. polarization: oriented vs spherically averaged (won’t cover)
C. Thermal effects
3. Experiment: Corrections and Other Problems
A. Sample issues (size effect, thickness effect, glitches) (won’t cover)
B. Fluorescence mode: Dead time and Self-absorption (won’t cover)
C. Energy resolution (won’t cover)
4. Data Analysis
A.
B.
C.
D.
E.
F.
Correlations between parameters (e.g. mean-free path)
Fitting procedures (After this talk)
Fourier concepts
Systematic errors (won’t cover)
“Random” errors (won’t cover)
F-tests
X-ray absorption spectroscopy (XAS) experimental
setup
double-crystal
monochromator
ionization detectors
• sample absorption is given by
beam-stop
“white” x-rays
from
synchrotron
I0
collimating
slits
I1
I2
LHe cryostat
reference sample
sample
 t = loge(I0/I1)
• reference absorption is
REF t = loge(I1/I2)
• sample absorption in fluorescence
  IF/I0
•NOTE: because we are always
taking relative-change ratios,
detector gains don’t matter!
X-ray absorption spectroscopy
LIII, LII, LI
Xenon
filled 3d
1
-1
 (cm )
10
continuum
unoccupied
states
EF
0.1

M
K
0.01
1
10
100
E (keV)
From McMaster Tables
1s
core hole
• Main features are single-electron excitations.
• Away from edges, energy dependence fits a power law: AE-3+BE-4
(Victoreen).
• Threshold energies E0~Z2, absorption coefficient ~Z4.
X-ray absorption fine-structure (XAFS)
spectroscopy
e- : ℓ=±1
2.8
2.6
continuum
unoccupied
states
EF
UPdCu4
2.2
2.0
1.8
1.6
pre-edge
log(I1/I0)
2.4

~E0
filled 3d
U L3 edge
 2k 2
 E  E0
2me
1.4
1.2
16800 17000 17200 17400 17600 17800 18000 18200

occupied states
k  0.512 E  E0
E (eV)
E0: photoelectron
threshold energy
2p
EXAFS region: extended x-ray absorption
fine-structure
“edge region”: x-ray absorption near-edge structure (XANES)
near-edge x-ray absorption fine-structure (NEXAFS)
core hole
Data reduction: (E)(k,r)
2.8
“pre-edge”
subtraction
2.6
2.4
1.2
1.0
0.8
2.0
a
t+const
2.2
1.8
data
Victoreen-style background
1.6
1.4
 (k ) 
1.2
1.0
17000
17200
17400
17600
17800
E(eV)
1.4
18000
2
a  0
0
0.6
a=-pre
0.4
0
“post-edge”
subtraction
0.2
0.0
17000
2
 k
 E  E0
2me
17200
17400
17600
17800
18000
E(eV)
k  0.512 E  E0
data
10*derivative
0.7
20
10
3
k (k)
a
determine E0
0.0
U LIII edge
0
-10
-20
17100
17200
E(eV)
17300
0
2
4
6
8
-1
k (Å )
10
12
14
How to read an XAFS spectrum
12 U-Cu
4 U-Pd
16 U-Cu
3.06 Å
40
U LIII edge
3
FT of k (k)
20
2.93 Å
0
-20
data
fit
-40
1
2
Envelope = magnitude r (Å)
[Re2+Im2]1/2
•
•
3
4
Real part of
the complex
transform
 (k )   N i  g (r ) F (k , r ) sin( 2kr  ci )dr
i
g is a radial pair - distributi on function
Peak width depends on back-scattering amplitude F(k,r) , the Fourier transform
(FT) range, and the distribution width of g(r), a.k.a. the Debye-Waller s.
Do NOT read this strictly as a radial-distribution function! Must do detailed
FITS!
“Heuristic” derivation
• In quantum mechanics, absorption is given by “Fermi’s
Golden Rule”:
 ~ f ˆ  rˆ i
f  f0  f
 ~ f ˆ  rˆ i
2
2
 f 0  f ˆ  rˆ i
 f 0 ˆ  rˆ i
2
2
 i ˆ  rˆ f 0 f ˆ  rˆ i
 c.c.  h.o.t.

   0 


0
0
Note, this is the same as saying
this is the change in the
absorption per photoelectron
How is final state wave function modulated?
• Assume photoelectron reaches the
continuum within dipole approximation:
(ˆ  rˆ) 2
ˆ
How is final state wave function modulated?
• Assume photoelectron reaches the
continuum within dipole approximation:
 ikr
2 e
(ˆ  rˆ)
kr
How is final state wave function modulated?
• Assume photoelectron reaches the
continuum within dipole approximation:
 ikr
2 e
(ˆ  rˆ)
kr
e ikri c ( k )
(ˆ  rˆ)
kr
2
central atom phase shift c(k)
How is final state wave function modulated?
• Assume photoelectron reaches the
continuum within dipole approximation:
 ikr
2 e
(ˆ  rˆ)
kr
e ikri c ( k )
central atom phase shift c(k)
(ˆ  rˆ)
kr
 ikR i c ( k )
e
(ˆ  rˆ) 2
e  R /  ( k ) electronic mean-free path (k)
kR
2
How is final state wave function modulated?
• Assume photoelectron reaches the
continuum within dipole approximation:
 ikr
2 e
(ˆ  rˆ)
kr
e ikri c ( k )
central atom phase shift c(k)
(ˆ  rˆ)
kr
 ikR i c ( k )
e
(ˆ  rˆ) 2
e  R /  ( k ) electronic mean-free path (k)
kR
2
 ikR i c ( k )
e
(ˆ  rˆ) 2
kf ( , k )e  R /  ( k ) complex backscattering probability f(,k)
kR
How is final state wave function modulated?
• Assume photoelectron reaches the
continuum within dipole approximation:
 ikr
2 e
(ˆ  rˆ)
kr
e ikri c ( k )
central atom phase shift c(k)
(ˆ  rˆ)
kr
 ikR i c ( k )
e
(ˆ  rˆ) 2
e  R /  ( k ) electronic mean-free path (k)
kR
2
 ikR i c ( k )
e
(ˆ  rˆ) 2
kf ( , k )e  R /  ( k ) complex backscattering probability f(,k)
kR
 ikRi c ( k )
 ik ( R  r ) i c ( k ) i a ( k )
e
e
(ˆ  rˆ) 2
k f ( , k ) e  R /  ( k )
kR
kR
complex=magnitude and phase:
backscattering atom phase shift a(k)
How is final state wave function modulated?
ˆ
• Assume photoelectron reaches the
continuum within dipole approximation:
 ikr
2 e
(ˆ  rˆ)
kr
e ikri c ( k )
central atom phase shift c(k)
(ˆ  rˆ)
kr
 ikR i c ( k )
e
(ˆ  rˆ) 2
e  R /  ( k ) electronic mean-free path (k)
kR
2
 ikR i c ( k )
e
(ˆ  rˆ) 2
kf ( , k )e  R /  ( k ) complex backscattering probability kf(,k)
kR
 ikRi c ( k )
 ik ( R  r ) i c ( k ) i a ( k )
e
e
(ˆ  rˆ) 2
k f ( , k ) e  R /  ( k )
kR
kR
e i 2 kRi 2 c ( k )i a ( k )
2 R /  ( k )
Im(ˆ  rˆ)
f
(

,
k
)
e
kR2
2
complex=magnitude and phase:
backscattering atom phase shift a(k)
final interference modulation per point atom!
Other factors
• Allow for multiple atoms Ni in a shell i and a distribution function
function of bondlengths within the shell g(r)
 (k )  S02  N i  (ˆ  rˆ) 2 f ( , k ) e 2 r /  ( k ) g (r )
i
1
g (r ) 
e
2 
where

sin 2kr  2 c (k )   a (k )
dr
2
kr
( r  Ri ) 2
2 2
and S02 is an inelastic loss factor
 (k )  S  N i (ˆ  rˆ) 2 f ( , k ) e 2 r /  ( k ) e 2 k 
2
2
0
i
Requires curved wave scattering, has rdependence, use full curved wave theory:
FEFF
2
sin 2kr  2 c (k )   a (k )
kr 2
Assumed both harmonic potential AND
k<<1: problem at high k and/or  (good to
k of about 1)
Some words about Debye-Waller factors
 (k )  S02  N i  (ˆ  rˆi ) 2 f i ( , k ) e 2 r /  ( k ) gi (r )
i
sin 2kr  2 c (k )   i (k )
dr
2
kr
• Harmonic approximation: Gaussian
g i (r ) 
Pi
Pj

1
e
2 
σ  (Pi  Pj )  Pi  Pj  2Pi Pj
2
ij
2
2
 Pi
2
 Pi
2
 2 Pi Pj
σ  U  U  2U iU j
2
ij
•
2
i
2
i
2
•
( r  ri ) 2
2 2
(non-Gaussian is
advanced topic:
“cumulant
expansion”)
Ui2 are the position mean-squared
displacements (MSDs) from diffraction
Important: EXAFS measures MSD differences in position
(in contrast to diffraction!!)
Lattice vibrations and Debye-Waller factors
• isolated atom pair, spring constant 
mr=(1/mi+1/mj)-1
Classically…
1
1 2
2
E  mr v   x
2
2
1
1
 E  mr v 2   x 2 
2
2
  x 2    x 2   2
with   mr 2
Quantumly…
1
 E  n   
2
 E  k BT at high T
 
2
E

k BT



2 
at low T

Some words about Debye-Waller factors
• The general formula for the
variance of a lattice vibration is:
 2j 

2mr
d

  

 2k BT ) 
 j ( ) coth
where j() is the projected density of
modes with vibrational frequency 
• Einstein model: single frequency
• correlated-Debye model:
quadratic and linear dispersion
cD = c kcD
Poiarkova and Rehr, Phys. Rev. B 59, 948 (1999).
A “zero-disorder” example: YbCu4X
0.014
60
Ag K edge YbCu4Ag
Pd K edge UCu4Pd
0.012
3
0.008
2
2
20
FT of k (k)
X-Cu
0.010
 (Å )
40
data
0.006
fit
0.004
0
0.002
-20
0.000
0.010
-40
X-Yb
0.008
1
2
3
4
2
0
 (Å )
-60
X
Tl
In
Cd
Ag
cD(K)
Cu
Yb
2
r (Å)
static2(Å)
Cu
Yb
X/Cu
interchange
X
S02
Tl
0.89(5)
230(5)
230(5)
0.0004(4)
0.0005(5)
4(1)%
In
1.04(5)
252(5)
280(5)
0.0009(4)
0.0011(5)
2(3)%
Cd
0.98(5)
240(5)
255(5)
0.0007(4)
0.0010(5)
5(5)%
Ag
0.91(5)
250(5)
235(5)
0.0008(4)
0.0006(5)
2(2)%
0.006
0.004
0.002
0.000
0
50
100
150
200
250
300
T (K)
AB2(T)=static2+F(AB,cD)
J. L. Lawrence et al., PRB
63, 054427 (2000).
XAFS of uranium-bacterial samples
15
U LIII edge
Bacteria
10
3
FT of k (k)
5
0
-5
data at 298 K
fit
-10
-15
0
1
2
3
4
3
FT of k (k)
r (Å)
14
12
10
8
6
4
2
0
-2
-4
-6
-8
-10
-12
-14
•
Salleite
U LIII edge
•
•
•
30 K
300 K
0
1
2
r (Å)
3
4
XAFS can tell us whether uranium and
phosphate form a complex
U-Oax are very stiff cD~1000 K
U-Oax-Oax also stiff! Won’t change with
temperature
U-P much looser cD~300 K: Enhance at
low T w.r.t. U-Oax-Oax!
Fitting the data to extract structural information
•
Fit is to the standard EXAFS equation using either a theoretical calculation or an
experimental measurement of Feff
 (k )  S02  N i  (ˆ  rˆi ) 2 f i ( , k ) e 2 r /  ( k ) gi (r )
i
•
•
•
sin 2kr  2 c (k )   i (k )
dr
2
kr
Typically, polarization is spherically averaged, doesn’t have to be
Typical fit parameters include: Ri, Ni, i, E0
Many codes are available for performing these fits:
— EXAFSPAK
— IFEFFIT
• SIXPACK
• ATHENA
— GNXAS
— RSXAP
FEFF: a curved-wave, multiple scattering EXAFS
and XANES calculator
•
•
The FEFF Project is lead by John Rehr and is very widely used and trusted
Calculates the complex scattering function Feff(k) and the mean-free path 
TITLE
CaMnO3 from Poeppelmeier 1982
HOLE 1
1.0
POTENTIALS
*
ipot
z
0
22
1
8
2
20
3
22
ATOMS
0.00000
0.00000
0.00000
-1.31250
1.31250
1.31250
-1.31250
0.00000
-2.62500
-2.62500
0.00000
Mn K edge
(
6.540 keV), s0^2=1.0
label
Mn
O
Ca
Mn
0.00000
-1.85615
1.85615
0.00000
0.00000
0.00000
0.00000
1.85615
1.85615
-1.85615
1.85615
0.00000
0.00000
0.00000
1.31250
-1.31250
1.31250
-1.31250
-2.62500
0.00000
0.00000
2.62500
0
1
1
1
1
1
1
2
2
2
2
Mn
O(1)
O(1)
O(2)
O(2)
O(2)
O(2)
Ca
Ca
Ca
Ca
0.00000
1.85615
1.85615
1.85616
1.85616
1.85616
1.85616
3.21495
3.21495
3.21495
3.21495
Phase shifts: functions of k
10
• sin(2kr+tot(k)): linear part of (k)
will look like a shift in r slope is
about -2x0.35 rad Å, so peak in r
will be shifted by about 0.35 Å
Co-O, Co K edge
2c
Phase shift (rad)
5
a
0
• Both central atom and
backscattering atom phase shifts
are important
-5
0
5
10
15
20
-1
k(Å )
"CoCaO3"
R=1.85 Å
R=3.71 Å
50
3
Magnitude of FT of k (k)
60
40
30
• Luckily, different species have
reasonably unique phase and
scattering functions (next slide)
20
10
0
• Can cause CONFUSION:
sometimes possible to fit the wrong
atomic species at the wrong
distance!
0
1
2
r (Å)
3
4
Species identification: phase and magnitude
signatures
-3
60
-4
50
-5
40
-6
30
-7
FT of k (k)
20
-8
a
10
3
-9
-10
-11
-12
0
-10
-20
-30
-13
-14
-40
Co-O
Co-C
-15
-50
-16
-60
-17
0
5
10
15
0
20
1
2
k (Å )
• Magnitude signatures then take over
• Rule of thumb is you can tell difference in species
within Z~2, but maintain constant vigilance!
Magnitude of FT of k (k)
• Note Ca (peak at 2.8 Å) and C have nearly the
same profile
4
1.2
Co-Mn
Co-Co
Co-Ba
1.0
3
• First example: same structure, first neighbor
different, distance between Re and Ampmax shifts
3
r (Å)
-1
0.8
0.6
0.4
0.2
0.0
0
5
10
r (Å)
15
20
More phase stuff: r and E0 are correlated
• When fitting, E0 generally is allowed to float (vary)
• In theory, a single E0 is needed for a monovalent absorbing
species
• Errors in E0 act like a phase shift and correlate to errors in
R!
consider error  in E0: ktrue=0.512[E-(E0+)]1/2
for small , k=k0-[(0.512)2/(2k0)]
eg. at k=10Å-1 and =1 eV, r~0.013 Å
• This correlation is not a problem if kmax is reasonably large
• Correlation between N, S02 and  is a much bigger problem!
Information content in EXAFS
• k-space vs. r-space fitting are equivalent if each is done
12 U-Cucorrectly!
4 U-Pd
16 U-Cu
20
10
U LIII edge
20
FT of k (k)
0
3
3
k (k)
40
U LIII edge
-10
0
-20
data
fit
-40
-20
0
2
4
6
8
-1
k (Å )
10
12
14
1
2
3
4
r (Å)
• r-range in k-space fits is determined by scattering shell with highest R
• k-space direct comparisons with raw data (i.e. residual calculations) are
typically incorrect: must Fourier filter data over r-range
• All knowledge from spectral theory applies! Especially, discrete sampling
Fourier theory…
Fourier concepts
• highest “frequency” rmax=(2k)-1 (Nyquist frequency)
eg. for sampling interval k=0.05 Å-1, rmax=31 Å
• for Ndata, discrete Fourier transform has Ndata, too! Therefore…
FT resolution is R=rmax/Ndata=/(2kmax), eg. kmax=15 Å-1, R=0.1 Å
• This is the ultimate limit, corresponds to when a beat is observed in two
sine wave R apart. IF YOU DON’T SEE A BEAT, DON’T RELY ON
THIS EQUATION!!
0.2
1.0
2 2
0.0
exp(-2k  )sin(2kr)
0.1
0.5
sin(2kr)
r=2.0, =0.1
r=2.1, =0.1
average
-0.5
sin(2k2)
sin(2k2.1)
average
-1.0
0
5
10
k
15
20
0.0
-0.1
-0.2
0
5
10
k
15
20
More Fourier concepts: Independent data points
•
Spectral theory indicates that each point in k-space affects every point in r-space.
Therefore, assuming a fit range over k (and r!):
 r  r2  r1

rmax 
2 k
N ind 
2

 k  k 2  k1
Nk 
kmax
k
 rmax 
 1  2
 1
 r

 r  k  2  h.o.t. "Stern's rule" EXAFSresult
Stern, Phys. Rev. B 48, 9825 (1993).
Booth and Hu, J. Phys.: Conf. Ser. 190, 012028(2009).
•
Fit degrees of freedom =Nind-Nfit
•
Generally should never have Nfit>=Nind (<1)
•
But what does this mean? It means that:
For Nfit exceeding Nind, there are other linear combinations of Nfit that produce
EXACTLY the same fit function
not so Advanced Topic: F-test
200
5000 simulations
Stern=14.0
2
Frequency ( )
150
2
 -distribution
with =14.1±0.1
100
50
0
0
10
20
30
40
2

• F-test, commonly used in crystallography to test one fitting model versus
another
F=(12/1)/(02/0)0/1R12/R02
(if errors approximately cancel)
alternatively: F=[(R12-R02)/(1-0)]/(R02/0)
• Like 2 , F-function is tabulated, is given by incomplete beta function
• Advantages over a 2-type test:
— don’t need to know the errors!
Statistical tests between models: the F-test
12 / 1
F 2
 0 / 0
R
(y
i
 yi )
f
i
2
( 12   02 ) /(1  0 )
F
 02 / 0
 R  2  (n  m)
For tests between data with
F   1   1
~constant error models that
 R0 
 b
share degrees of freedom,
nm b
  P( F  Fb,n m, )  1  I 2 [
, ] Hamilton test with b=1-0
2 2
where is theconfidenceleveland I x [ y,z] is an incompletebeta function
• F-tests have big advantages for data sets with poorly defined
noise levels
— with systematic error, R1 and R0 increase by ~constant,
reducing the ratio (right direction!)
F-test example testing resolution in EXAFS:
knowledge of line shape huge advantage
30
Sn=4.0
20
0
3
k (k)
10
-10
-20
-30
0
5
10
15
20
•
•
•
•
•
•
•
6 Cu-Cu at 2.55 Å, 6 at 2.65 Å
kmax=/(2R)=15.7 Å-1
1 peak (+cumulants) vs 2 peak fits
Fits to 18 Å-1 all pass the F-test
Sn=50 does not pass at 15 Å-1
Sn=20 does not pass at 13 Å-1
Sn=4 does not pass at ~11 Å-1
-1
k (Å )
40
50
30
40
Sn=20
Sn=50
30
20
20
k (k)
10
0
3
3
k (k)
10
-10
0
-10
-20
-20
-30
-30
-40
-40
0
5
10
15
-1
k (Å )
20
-50
0
5
10
15
-1
k (Å )
20
Unfortunately, systematic errors dominate
60
Example without systematic error:
kmax=13 Å-1,0=6.19, 1=7.19
R0=0.31
R1=0.79
=0.999  passes F-test
-1
40
3
FT of k (k)
20
k=2.5-12 Å
r=1.5-3 Å
Sn=20
=5.2,7.2
=96.2%
0
-20
Re
Mag
1p
2p
-40
-60
1.5
2.0
2.5
r (Å)
3.0
Example with systematic error at
R=2%:
kmax=13 Å-1,0=6.19, 1=7.19
R0=0.31+2.0=2.31
R1=0.79+2.0=2.79
=0.86  does not pass
Finishing up
• Never report two bond lengths that break the resolution rule
• Break Stern’s rule only with extreme caution
• Pay attention to the statistics
Further reading
•
•
•
•
•
Overviews:
— B. K. Teo, “EXAFS: Basic Principles and Data Analysis” (Springer, New York,
1986).
— Hayes and Boyce, Solid State Physics 37, 173 (1982).
— “X-Ray Absorption: Principles, Applications, Techniques of EXAFS, SEXAFS
and XANES”, ed. by Koningsberger and Prins (Wiley, New York, 1988).
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)
Further reading
• Thickness effect: Stern and Kim, Phys. Rev. B 23, 3781 (1981).
• Particle size effect: Lu and Stern, Nucl. Inst. Meth. 212, 475 (1983).
• Glitches:
— Bridges, Wang, Boyce, Nucl. Instr. Meth. A 307, 316 (1991); Bridges,
Li, Wang, Nucl. Instr. Meth. A 320, 548 (1992);Li, Bridges, Wang, Nucl.
Instr. Meth. A 340, 420 (1994).
• Number of independent data points: Stern, Phys. Rev. B 48, 9825 (1993);
Booth and Hu, J. Phys.: Conf. Ser. 190, 012028(2009).
• Theory vs. experiment:
— Li, Bridges and Booth, Phys. Rev. B 52, 6332 (1995).
— Kvitky, Bridges, van Dorssen, Phys. Rev. B 64, 214108 (2001).
• Polarized EXAFS:
— Heald and Stern, Phys. Rev. B 16, 5549 (1977).
— Booth and Bridges, Physica Scripta T115, 202 (2005). (Self-absorption)
• Hamilton (F-)test:
— Hamilton, Acta Cryst. 18, 502 (1965).
— Downward, Booth, Lukens and Bridges, AIP Conf. Proc. 882, 129
(2007). http://lise.lbl.gov/chbooth/papers/Hamilton_XAFS13.pdf
Further reading
• Correlated-Debye model:
— Good overview: Poiarkova and Rehr, Phys. Rev. B 59, 948 (1999).
— Beni and Platzman, Phys. Rev. B 14, 1514 (1976).
— Sevillano, Meuth, and Rehr, Phys. Rev. B 20, 4908 (1979).
• Correlated Einstein model
— Van Hung and Rehr, Phys. Rev. B 56, 43 (1997).
Acknowledgements
• Matt Newville (Argonne National Laboratory)
• Yung-Jin Hu (UC Berkeley, LBNL)
• Frank Bridges (UC Santa Cruz)