X-ray Absorption Spectroscopy:
The extended region
XAS: X-ray Absorption
XAFS: X-ray Absorption Fine Structures
XANES: X-ray Absorption Near Edge Structures
NEXAFS: Near Edge X-ray Absorption Fine
Structure
EXAFS: Extended X-ray Absorption Fine
Structures
X-ray absorption spectroscopy
e
Photon
Eo element specific
ion
Io
I
hvf
hvop
t
Transmission: I  I o e  t ;
Yield spectroscopy:
t  ln( I o / I )
bonding local
Absorption symmetry structure
across an edge
t  Y (hv )
t  f (hv )  Y (hv )
What does  (hv) measure across an edge?
The modulation in  above the edge is the XAFS,
it contains information about the structure and
bonding of the absorbing atom in a chemical
environment
Edge jump: threshold of the core excitation
Near edge region: core to bound and quasi bound
state transition – multiple scattering effect dominates
Extended region above the threshold: core to
continuum transition, single scattering dominates,
interference of outgoing and backscattering waves
What is XAFS?
t
Near edge
EXAFS
Si(CH3)4
Origin of XAFS
hv (photon)
previously unoccupied
bound states
e- (photoelectron) wave
Free
core hole (K, L1, L2, L3 etc)
decay
atom
De-excitation/yield spectroscopy
hv  threshold energy of the core electron (binding energy)
Selection rule: electric dipole
Origin of EXAFS
e- with KE > 0 propagates as a wave with k ~ (KE)1/2. If it does not
encounter another atom in the vicinity, the (hv) varies
monotonically as a function of photon energy

Rydberg
edge
Diatomic molecule
hv (photon)
Transition to
LUMO, LUMO +
 XANES
(X-ray Absorption
Near Edge Structures)
hv > Eo (binding energy)
Outgoing wave (spherical wave)
Backscattering wave
(dominant for fast electrons)
If the outgoing and backscattered waves are in phase, the
interference is constructive, if they are out of phase, the
interference is destructive
Origin of EXAFS
The forward and backscattered waves will interfere constructively
and destructively at the absorbing atom, producing oscillations in
(hv)  EXAFS
constructive
EXAFS
destructive
LUMO
XANES
The interference modulates . Thus  depends on  of the outgoing
wave, the electronic structure (amplitude of the scattering atom and
the phase shift of both atoms) and the inter-atomic distance(R)
constructive interference
yields a maximum
absorbing atom
R
outgoing edestructive
wave
interference
yields a minimum
sample  atom
EXAFS:  ( k ) 
atom
Interference of outgoing
and backscattered wave
Kr
Br2
Brian M. Kincaid, "Synchrotron Radiation Studies of K-Edge X-ray Photo-absorption Spectra:
Theory and Experiment", Stanford University, 1975; Advisor: S. Doniach
EXAFS: the modern view
• EXAFS is the oscillations in  beyond the XANES region
• The oscillations arise from the interference of the outgoing
and the backscattered electron wave which modulates the
absorption coefficient (no neighboring atom, no EXAFS)
• Single scattering pathway dominates at high k (energy)
• The thermal motion modifies the amplitude of the EXAFS
• It is a short range phenomenon (~ 4 Å) (IMFP of electrons)
• It works for disorder systems (liquid, amorphous materials)
• EXAFS is additive - all absorber-scattering atom pairs
contribute to EXAFS additively
• The phase and amplitude are transferable for chemically similar
systems; they are also separable by Fourier Transform
• Theoretical phases and amplitudes are often used in analysis; they
can be generated in the FEFF code developed by J.J. Rehr of
the University of Washington
EXAFS of medium and high Z atoms
XANES
EXAFS
Conversion of kinetic energy E – Eo to wave vector k
0 1
k( A ) 
2

 (2m /  2 )( E  E o )  0.263E (eV )
k  0.513 E
absorption
1.0
t
0.5
0.5
0.0
0.4
d(t)/dE
0.3
derivative
Where Eo is the threshold energy
(usually the point of inflection of
the edge jump), E is the photon
energy.
Thus 50 eV above the threshold
corresponds to a k value of
3.63 Å-1
k = 2 Å-1 corresponds to
15.2 eV above threshold
Ge K-edge
1.5
0.2
0.1
0.0
-0.1
11080
Eo
11100
11120
11140
Photon Energy (eV)
11160
Extracting EXAFS data from absorption spectrum
334.6 cm2/g  = 1 (334.6-44.2 )8.9
= 2584.56cm-1 density
44.2 cm2/g
1.4
Ni K-edge
1.2
of Ni
1.0
1.0
0.8
0.8
Ni foil
0.6
Normalized
edge jump to 1
0.4
Absorption
absorption = ln (Io/I)
1.2
converted to k
space k ~0.513
(E)0.5
0.6
0.4
0.2
0.2
0.0
8200
8400
8600
8800
9000
9200
9400
8200
8400
8600
Photon Energy (eV)
  o
(k ) 
o
X(k)
0.04
0.02
0.00
-0.02
-0.04
-0.06
-0.08
2
4
6
8
10
-1
k(A )
12
14
16
18
Absorption
0.10
0.06
9000
9200
9400
Photon Energy (eV)
0.12
0.08
8800
1.14
1.12
1.10
1.08
1.06
1.04
1.02
1.00
0.98
0.96
0.94
0.92
0.90
spline fit
to atomic
Background
o
2
4
6
8
10
-1
k (A )
12
14
16
18
The EXAFS formula
0.12
0.10
A(k)
0.08
0.06
(k)
0.04
0.02
0.00
-0.02
-0.04
-0.06
-0.08
2
4
6
8
10
12
14
16
18
-1
k(A )
 N S 2 f ( k , )e 2 R j / e ( k ) e 2 k  j
j 0
 (k )   
kR 2j
j 

2
 A( k )  ( k )
2

 sin[2 kR j   ( k )]


EXAFS formula and relevant parameters
The EXAFS equation
 N S f ( k , )e
j

 (k )  
2
kR

j
j

2
0
2 R j / e ( k )
e
 2 k 2 2j

 sin[2 kR j   ( k )]


amplitude
 A( k )  ( k )
phase
Nj: coordination number of atom type j, e.g. TiCl4, NCl = 4
So2: amplitude reduction term (shake-up, shake-off loss)
f(, k): atomic scattering amplitude, e.g. f(, k) for single scattering
2 R j /  e ( k )
2 k 2 2j
: amplitude damping terms due to electron
e
, e
escape depth e and mean displacement due to thermal motion j,
Rj : interatomic distance, (k) = 2a(k)+ b(k) : phase function with
contributions from the absorber a(k) and the backscatterer b(k)
Relationship between (E) and (k):
A simplified EXAFS equation
2
'
We recall ( E )   i H  f
i  i
'

H
f  f
Initial state wavefunction,
describing the core electron
Interaction, H’ = ekr ~ 1 (dipole)
Final state wavefunction
describing the photoelectron,
sensitive to chemical environment
where all the actions are
Probing matter with SR, versatile light
Footnote : interaction of light with a bound electron in an atom
What is light ? (wave and particle dual behavior)
E = hv = hc/

energy
frequency(s-1)
E vector (polarization)
propagation
c: speed of light
c = 3 x108 ms-1
h: Planck’s constant
h = 6.626 x10-34 Js
5
E field interacts with electron charge,
B field interacts with magnetic dipole moment
For a plane wave B and E have equal strength in free space
Which interaction is more important?
Footnote, cont’
E field interacts with electron charge
The interaction energy U  e ·V = e E dr
potential
Interaction energy ~
eEa o
distance
electric field
If we take the radius of the
atom as ao = Bohr’s radius
B field interacts with magnetic dipole moment
The interaction energy U ~ (eћ/mc) ·B
magnetic field
gyromagnetic ratio:
magnetic dipole moment /angular momentum
Footnote, cont’
Relative strength of magnetic/electric interaction
U mag
U elect
e
B
B
 mc 
eEa o mcEa o
2 2
e
1
ao 
 2
mc 
Since
e2
1
And fine structure constant  

c 137
U mag
U elect
B mc 2 2
1

  
2
mcE e
137
Electric interaction is more than two orders of magnitude
stronger than the magnetic interaction
Footnote, cont’
Dipole approximation
The photon field in the electromagnetic wave is a plane wave
A  { Ao e
i ( k r   t )
}real
If the wavelength is larger than the diameter of the atom, the eikr
term can be replaced with 1 in the expansion (small atom approx.)
e
ikr
1
1
2
 1  i ( k  r )  (ik  r )  (ik  r ) 3     
2!
3!
The term (k·r)n is called electric 2(n+1) pole and magnetic
2n -pole transition. E.g.: n = 0, electric dipole and magnetic
monopole, n = 1, electric quadrupole and magnetic dipole etc.
Footnote, cont’
Dipole approximation
The interaction Hamiltonian
e  
H 
A P
mc
vector potential · momentum
Same as ( ·r) discussed earlier
can be replaced with
e
H 
Ao P cost
mc
Time dependent
perturbation
 f  f  f 0  f scatt
Final state in
EXAFS
free atom neighboring atom
( E )   i H '  f
2
 i H ' f 0  f scatt
2
Expanding  we get
( E )  i H ' f 0
2
 (1 
i H ' f scatt
f 0 H ' i
i H ' f 0
2
*
 complex conjugate)
This is the only term that is sensitive to the environment
( E )  0 ( E )[1   ( E )]
'

 ( E )  i H f scatt  i f scatt
 ( E )  i H ' f scatt  i f scatt
EXAFS involves the change in the photoelectron wavefunction
due to back scattering.
Since the initial state is localized spatially ( function), We have
initial
final at r~ 0
 ( E )    (r ) scatt (r )dr ~  scatt (0)
scattering
Vanishes if there is
no neighboring atom
Thus EXAFS is the modulation of the photoelectron
wavefunction at the absorbing atom resulting from its
backscattering from neighboring atoms
EXAFS of a diatomic molecule
Scattering angle
De Broglie wave
e-
 ikx
eikr 
  A e  f (  ) 
r 

plane wave
hv
Out going photoelectron
spherical wave
eikr
fo 
r
(r  )
spherical wave
Backscattering photoelectron
spherical wave carrying info.
of amplitude and phase of
the scattering atom
ikr
e
f ( , k )(ei ( k ) )
r
Since
 ( E ) ~  scatt (0)
We can build a simple model to account for EXAFS
by considering the following events
[ref. Newville, XAFS workshop
http://cars9.uchicago.edu/xafs_school/]
• outgoing photoelectron (spherical wave)
• scattering of the electron wave by the
neighboring atom
• returning of the backscattered wave to the absorbing
atom
For the spherical photoelectron wave, eikr/kr,
with a neighboring atom at a distance R,
we get
ikR
ikR
e
e
i ( k )
(k ) 
 2k  f ( , k )  (e ) 
 c. c.
kR
kR
amplitude
phase-shift
The EXAFS of an atom in a diatomic molecule is
 f ( , k )  ei ( 2 kR  ( k )) 
 (k )   Im

2
kR


f ( , k )
 (k ) 
sin 2kR   (k )
2
kR
amplitude for 180o
backscattering
phase shift has both
absorbing and backscattering atom
contributions
backscattering
atom
absorbing
atom
R
the Debye-Waller (DW) factor
Since the molecule vibrates, leading to an instantaneous
variation of the bond length, the effect is represented
by the mean square displacement, 2 , an amplitude
damping term, the Debye-Waller (DW) factor,
e
2 k 2 2
DW term

R
f ( , k )e
 (k ) 
kR2
2 k 2 2
sin 2kR   (k )
DW = 1 when  = 0, when  > 0, DW < 1, thus
vibration contributes to the damping of the amplitude
For a multi-atom system with coordination number Nj
 (k )  
N j f j ( , k )e
j
2 k 2 2j

sin 2kRj   j (k )
2
j
kR

We next consider the inelastic
scattering of the photoelectron,
the escape depth,  (k)
contributing to a amplitude loss, e
e
2 R j /  ( k )
 (k )  
j
N j f j ( , k )e
2 k
2
j
kR
 2j
2
e
2 R j /  ( k )
k = 0.513E

sin 2kRj   j (k )
Free mean path term

The last term to consider is the manybody effect term associated
with shake-up and shake-off, contributing to amplitude loss. This is
to be distinguished from the inelastic loss, which takes place after the
electron has left the atom. This term is often denoted as “So2”
S  
2
o
N 1
f
fully relaxed

N 1
o
2
un-relaxed
So2 = 1 in the absence of manybody effect (single particle
approximation); in reality, it has a constant in k for modest k
values, ranging from 0.7 ~ 0.9 (depending on the chemical
environment)
For more detailed information visit the following site and
down load course materials from the 2009 XAFS workshop
http://cars9.uchicago.edu/xafs_school/
Summary: The EXAFS formula
 N S f ( k , )e
j
(k )   
2
kR

j
j

2
0
Coordination
no., Nj
Manybody,
So2
Free mean
path e –2Rj/(k)
DW
exp[-2k22]
2 R j / e ( k )
e
 2 k 2 2j
Amplitude, A(k)
Atomic scattering
Factor, f (k,)
Interatomic
distance, R

 sin[2 kR j   ( k )]


Phase, (k)
Interatomic
distance, Rj
absorbing atom
phase shift a (l)
backsacttering
atom phase
shift b
K-edge l =1, p wave
The behavior of EXAFS parameters in k space
The phase ( k )  sin[2kR j   ( k )]
For K, L1 edge
ab ( k )  a ( k )  b ( k )   ,
  1, p wave (   1, dipole)
For L 3,2 edge
ab ( k )  a ( k )  b ( k )
  2,0 (   1, dipole)
ab : Phase function of the absorber-backscatterer
al ; Phase shifty of the outgoing e- from the absorbing atom
b : Phase shift of the neighboring atom
Note: the phase shift difference between K and L3-edge is 
Eo
Hexamethyl disilane, Si K & L-edge
K-edge
ab (k )  al 1 (k )  b (k )  
L1-edge
flip
a b (k )  al 1 (k )  b (k )  
L2,3-edge
ab (k )  al 2,0 (k )  b (k )
Max Min,
phase shifted by 
Theoretical phase functions
Absorbing atom
phase-shifts al
for the K-edge,
l = 1, p - wave
For low and medium z
elements, ab =  + k
is a good approximation
Backscattering
Atom phase
shifts b
Si-C =Si(a)+C(b)
Pd-Ti =Pd(a)+Ti(b)
0
( k )  sin[2kR j   ( k )]
Sin(2kR)
2kR
R1
R2
The large the R, the shorter the period in k space
( k )  sin[2kR j   ( k )]
When (k) ~  + k is included, the separation between
oscillation maximum increases in k
Solid curve: (k) = 0
Dashed curve  (k) =  + k
The behavior of EXAFS parameters in k space
Amplitude, A(k) = Atomic scattering
Factor, f (k,)
modified by
Coordination
no., Nj
Manybody,
So2
Free mean
path e –2r/(k)
DW
exp[-2k22]
constant
~ constant, 0.7 – 0.9
Damps A(k)
as k increase
Element
specific
Atomic scattering factor, f (k,)
 ( k )  f ( k ,  ) sin[2kR j  ( k )]
Backscattering
( = ) amplitude
With max in the
low k region
f(k, )•
sin[2kR +(k)]
f
f’
Backscattering
( = ) amplitude
With max in the
medium k region
f ’(k, )•
sin[2kR +(k)]
Theoretical backscattering amplitude, f(k, )
B.K. Teo and P.A. Lee J. Amer. Chem. Soc. 101 2815(1979)
Teo and Lee (old
data) produce
qualitatively the
same shape of
A(k) but the
max/min occurs at
higher k
FEFF
developed by
J.J. Rehr,
(best calculation)
f ( k , )e
 2 R j / e ( k )
sin[2kR j   ( k )]
f ( k , )e 2 k  sin[2 kR j   ( k )]
2
2
larger 
NaCl
Inelastic Mean Path (A)
250
Lab XPS
200
SiO2
150
Si
100
GaAs
Au
50
0
0
2000
4000
6000
8000
Electron Kinetic Energy (eV)
10000
Damps the A(k) more significantly in low k Damps the A(k) more significantly in high k
XAFS measurements
• Transmission
• Total electron yield (vacuum)
• Partial electron yield (vacuum)
• Ion yield (vacuum)
• Fluorescence yield
• Photoluminescence yield
• Photoconductivity (dielectric liquid)
Yield
(soft x-ray)
Assumption: Yield  t (not always the case)
 t
Yield  f (hv) I o (1  e )
For a thin sample:
Yield  f (hv ) I o [1  (1  t  21! ( t ) 2   ]  t
Transmission
t
It
Io
Incident flux
Transmitted flux
Beer-Lambert law
Transmission
Absorption
I t  I o e  t
I a  I o  I t  I o (1  e  t )
 Io 
t  ln  To be measured
 It 
X-ray detectors
Ionization chamber
300 to 500 V
typical
I
Iinc
t
voltage to
frequency converter
An Ion chamber is filled with inert gas ( He, Ar, N2 etc. ). The
absorption of x-rays produces ion pairs, typically, it requires ~30 eV
(W value) to produce 1 ion pair at 1 atm pressure (e.g. 27 eV for Ar
and 33 eV for N2). Thus for Ar, a 2700eV photon produces 100 ion
pairs. The flux I emerging from an Ar ion chamber is
current amplifier
e   ( Ar ) t
I ( photons / s)  iw( Ar )
eE (hv)(1  e   ( Ar ) t )
current
photon energy
Electron yield detectors
bias V (50 V)
V to F
e
ADC
Analog The specimen current (total electron yield)
(current)
(Auger)
to digital
(counts)
hemispherical
e
analyzer, energy
converter
selected
e
Channeltron (TEY)
X-ray Fluorescence yield
Nondispersive
• Scintillation counter
• Ion chamber (Lytle detector) (with filters)
Moderate energy resolution
Solid state detectors (Ge, Si)
High energy resolution
WDX detector (crystal monochromator)
hv
S.Helad, APS workshop 2005
~10 eV
0.012
0.006
Ce L3-edge
0.008
FLY
Intensity
Ce L 
0.004
0.010
Ce L
Fluorescence
X-rays
0.006
Green Titanite
0.004
0.002
0.002
Brown Titanite
0.000
0.000
2.1
Ce in Titanite
2.2
2.3
2.4
Wavelebgth
2.5
2.6
5710 5720 5730 5740 5750 5760 5770
Photon Energy (eV)
EXAFS Analysis
I. Data reduction: extract EXAFS  from the absorption 
a) Pre-edge background removal
1.2
0.6
1.0
0.8
0.4
EJ
pre-edge points are fitted
to a victoreen C3+D4
Absorption
absorption = ln (Io/I)
Ni metal
0.6
0.4
0.2
0.2
0.0
8200
8400
8600
8800
9000
Photon Energy (eV)
9200
9400 8200
8400
8600
8800
9000
Photon Energy (eV)
Subtract pre-edge background, divided by EJ
9200
9400
b) Determine Eo and convert data to k space
1.2
1.2
1.0
1.0
0.8
1.2
0.6
Eo = 8333 eV
Absorption
Absorption
0.8
1.0
0.8
0.6
0.4
0.4
0.6
0.4
0.2
0.2
0.2
0.0
-0.2
8300
0.0
8320
8340
8360
8380
8400
0.0
Photon Energy (eV)
0
8400 8600 8800 9000 9200
200
0.08
0.06
0 1
0.00
  o
(k ) 
o
-0.04
-0.06
-0.08
2
4
6
8
10
-1
k(A )
12
14
16
18
Spline fit, 3rd order
polynomial, 3 sections
1.05
Absorption
0.02
k ( A )  k  0513
.
E
1.10
Remove
spline
0.04
-0.02
800
E = E - Eo
c) Get (k) by removing
atomic contribution
0.10
600
Photon Energy (eV)
Photon Energy (eV)
X(k)
400
1.00
0.95
2
4
6
8
10 12 14 16 18
-1
k (A )
II. Data Analysis:
a) Separation of amplitude and phase
Window filter
FT
2.5
0.10
0.08
2.0
0.06
Magnitude
0.04
X(k)
0.02
0.00
-0.02
-0.04
1.5
1.0
0.5
-0.06
0.0
-0.08
2
4
6
8
10
12
14
16
0
18
2
4
-1
8
10
R (A)
k(A )
80
0.06
0.05
0.04
0.03
0.02
0.01
0.00
-0.01
-0.02
-0.03
-0.04
-0.05
-0.06
-0.07
A(k)
70
60
Phase(k)
First shell amplitude
6
(k)
50
40
30
20
10
2
2
4
6
8
10
-1
k (A )
12
14
16
18
4
6
8
10
-1
k (A )
12
14
16
18
Fourier Transform in EXAFS
 ( k )   A j ( k ) sin[(2krj  ( k )]
j
Consider the Fourier transform of EXAFS
FT[  ( k )]    ( k )e

j
 2 ikr j
 A ( k ) sin[2kr
j
j
dk
  j ( k )]e
 2 ik j r
dk
  FT[  j ( k )]
j
Consider the FT of one single shell
FT[  j ( k )]  FT{ A j ( k ) sin[2krj   j ( k )]}
amplitude
phase
In k space, A(k) is convoluted in the oscillatory
behavior of the phase so that
FT[  j ( k )]  FT{sin[2krj   j ( k )]},  ( k )    k  a j  2b j k
FT[  j ( k )] 
kmin
r = rj + b j
2.5
kmax
0.04
0.02
x(k)
for r >0
2( r j  r  b j )
FT
<=>
0.00
-0.02
-0.04
-0.06
2.0
1.5
Magnitude
0.06
sin(r j  r  b j )( k max  k min )
1.0
0.5
0.0
2
4
6
8
10
-1
k (A )
12
14
16
18
0
2
4
6
8
10
-1
k(A )
The r space peak position is shorter than the real rj (b ~ - 0.2 to -0.3 Å);
this is often referred to as the phase-shift uncorrected value
For multi-shells
FT[  ( k )]  FT  j  j ( k )
0.10
0.08
Ni fcc
2.5
2.0
0.06
Magnitude
Ni
0.04
X(k)
0.02
0.00
-0.02
-0.04
1.5
1.0
a
0.5
-0.06
0.0
-0.08
2
4
6
8
10
12
14
16
0
18
2
4
-1
8
10
R (A)
k(A )
FCC
6
BCC
Diamond
HCP
Shell
Atom
Distance
Atom
Distance
Atom
Distance
Atom
Distance
1st
12
1/2
8
3/2
4
3/4
6
1/3+b/4
2nd
6
1
6
1
12
2/2
6
1
3rd
24
6/2
12
2
12
11/4
6
4/3+b/4
4th
12
2
24
11/2
6
1
2
b
Distance in unit of a, b = (c/a)2
k n weighting and filtering
FT is a frequency filter for an infinite
sine wave the FT is a  function (localized)
EXAFS is not an infinite sine function but a sum of
discrete sine waves, FT yields a broadened, sometimes
distorted peak, k n weighting and filtering are often
used to improve the quality of the analysis
k n weighting ( n = 0 – 3)
k1, k2, k3
30
800
20
n
Magnitude k
n
10
x(k)k
k3, k2, k1
600
0
-10
400
200
-20
-30
0
0
2
4
6
8
10
-1
k (A )
12
14
16
18
0
2
4
6
R (A)
8
10
Filtering window
Discrete FT leads to leakage in R space (side lobes), proper
window selection can suppress side lobes and distortion
We often use a filter window in the back transform
Windows often used are
Rectangle
Hamming
Hanning etc.
Methods in EXAFS analysis
Phase difference and log ratio method:
Effective when we are interested in the difference in bond length
between two closely related systems such as bulk-nano structures,
metal ions in solution and metal oxides with different oxidation
states, Let
( k ')  2 k ' r   ( k ' )
be the phase of the unknown and the phase of the model
( k ) m  2 krm  m ( k )
compound is
And k’ is an adjustable parameter then
( k ')   m ( k )  2 k ' r  2 krm   ( k ')  m ( k )
Bt adjusting Eo of k’ until (k’) -m(k) = 0
The bond length difference can be obtained
from the slope of 
Alternatively, we can consider the difference in
adjacent maximum of the oscillations in k space
Foil
4.0nm
2.4nm
1.6nm
1
X(k)k2
kmax
S-Au
Au-Au
FT
1.6nm
0
2
4
6
0
back-transform
3
5
7
9
O
11
-1
k (A )
 ( k )  A( k , r )( k , r )
( k )  sin  ( k )
 sin[(2r   ) k   ]
13
15
difference between
adjacent maximum

2
 0  k max 
k
( 2r   )
Find  from model compound
Amplitude log ratio
 N S f ( k , )e
j
(k )   
2
kR

j
j

2
0
2 R j / e ( k )
e
 2 k 2 2j

 sin[2 kR j   ( k )]


For two closely related samples, we assume that in A(k)
everything is the same except N, R and , then we have
2
A1 ( k )
N 1 R2
2
2
2
ln
 ln

2
k
(



2
1 )
2
A2 ( k )
N 2 R1
A1 ( k )
ln
A2 ( k )
N 1 R2
2
N 2 R1
2
2
2
2
slope    2   1
intercept
k2
Beating method
When atomic shells are close to each other, the beating of
the two waves produces minima in the total amplitude
A(k) and “kinks” in the the total phase. The position of
the beat nodes satisfies
2 k n R  1  2  n ,
n  1,3....
Since the phase for different shells is the same we have
2 k n R  n ,
n  1, 3,....
bcc Fe, the first and second shell is very close, R = 0.385Å
A(k)
(k)
FT
2k1R = ,
R = 0.385Å
filtered
back FT
2k3R = 3,
R = 0.381Å
2 k n R  n ,
n  1, 3,....
Curve fitting
Useful parameters can be obtained by
fitting the data to the following equation
 N S 2 f ( k , )e 2 R j / e ( k ) e 2 k  j
j 0
(k )   
kR 2j
j 

2
2

 sin[2 kR j   ( k )]


There are many codes to perform fitting analysis. Details
will not be discussed in this course. Typically, theoretical
amplitude and phase (FEFF) or experimental data of
standards are used for the fitting. More details can be
obtained from web site references.
Several useful sites can be linked from
http://publish.uwo.ca/~tsham/
Click on “link” then “spectroscopy”
Data analysis computer codes
Ban: Dos based, good for preliminary
examination of data
Athena: Window based contemporary
version (public domain, by Bruce
Ravel of NIST) (recommended)
Winxas: Commercial soft ware
EXAFS: Other Examples
(CH3)4Si
Identical backscattering atoms
Si-C bond
single sinusoidal
Curve in k space
FT
Single
bond
length
Two types of backscattering atoms
FT shows
two peaks
associated
Si-C and
Si-Si bonds
Beating of
two sine waves
is apparent
Si-Si
Si-C
Si
C
Electron exchange reactions (nuclear tunneling)
Fe*(H2O)63+ + Fe(H2O)62+=> Fe*(H2O)62+ + Fe(H2O)63+
Fe2+
Fe3+
stretching motion
e
The bigger
the separation equal radius
the shorter
the bond
Fe3+
Acc. Chem. Res. 19, 99(1986)
Fe2+
EXAFS of other electron exchange pairs
Qualitatively, we
can predict which
complex has a
longer bond from
the progressive
mismatching of
the oscillation
maximum

2
 0  k max 
k
( 2r   )
accurate R
kmax
From one max
to the next, the
phase changes
by 2
2
k max 
2r  
2
2r   
k max
2
2r '   '
k max
k’max
 1
1

2( r ' r )  (    )  2

'
 kmax
kmax

 1

1

r   '

 kmax


k
max





