Post on 14-Aug-2020
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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Sakura Pascarelli European Synchrotron Radiation Facility, Grenoble, France
An introduction to
X-ray Absorption Spectroscopy
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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Outline
X-ray Absorption Spectroscopy
X-ray Absorption Fine Structure (EXAFS and XANES)
Major historical EXAFS breakthroughs
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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Outline
X-ray Absorption Spectroscopy
X-ray Absorption Fine Structure (EXAFS and XANES)
Major historical EXAFS breakthroughs
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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Main X-ray based techniques
elastic diffusion microscopic geometric structure
diffraction (crystalline solids)
scattering (amorphous solids, liquids)
Two fundamental X-ray-matter interactions:
photoelectric absorption
scattering (elastic, inelastic)
spectroscopy electronic structure, local structure of matter
absorption
emission
inelastic scattering
Two families of experimental techniques:
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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The Absorption Coefficient m
I0 I
t
I = I0 exp[-mt]
linear absorption coefficient
mt = ln [ I0 / I ]
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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1. Measure I0 and I as a function of EX
polychromatic X-rays
monochromatic X-rays
synchrotron source monochromator
incident flux monitor
transmitted flux monitor
sample I0 I
t
2. Calculate: m t = ln [I0/I ]
The Absorption Coefficient m
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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m has sudden jumps (absorption edges) which occur at energies
characteristic of the element.
m/r
[bar
ns/a
tom
]
3
4
EA
Zrm
m depends strongly on X-ray energy E and atomic number
Z, and on the density r and atomic mass A
The Absorption Coefficient m
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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Photoelectric absorption dominates the absorption
coefficient in this energy range
total absorption coefficient
photoelectric absorption
elastic scattering
inelastic scattering
E(eV)
Germanium
m [
cm-1]
102 103 104 105
106
104
102
100
10-2
10-4
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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Photoelectric Absorption
X-rays (light with wavelength 0.06 ≤ l ≤ 12 Å or energy 1 ≤ E ≤ 200 keV) are
absorbed by all matter through the photoelectric effect:
An x-ray is absorbed by an atom when the energy of the x-ray is transferred to a core-level electron (K, L, or M shell) which is ejected from the atom.
The atom is left in an excited state with an empty electronic level (a core hole).
Any excess energy from the x-ray is given to the ejected photoelectron.
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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De-excitation: Fluorescence and Auger Effect When x-rays are absorbed by the photoelectric effect, the excited core-hole
will relax back to a “ground state” of the atom. A higher level core electron
drops into the core hole, and a fluorescent x-ray or Auger electron is emitted.
X-ray fluorescence and Auger emission occur at discrete energies characteristic of the absorbing atom, and can be used to identify the absorbing atom.
Auger Effect:
An electron is promoted to the continuum from another core-level.
X-ray Fluorescence:
An x-ray with energy = the difference of the core-levels is emitted.
Ka : LK, Kb : M K 1s
2p
3p
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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XAS measurements
XAS measures the energy dependence of the x-ray absorption coefficient μ(E) at and above the absorption edge of a selected element. μ(E) can be measured in several ways: Transmission: The absorption is measured directly by measuring what is transmitted through the sample:
I = I0 e −μ (E)t
μ(E) t = − ln (I/I0) Fluorescence: The re-filling the deep core hole is detected. Typically the fluorescent x-ray is measured. μ(E) ~ IF / I0
synchrotron source
monochromator
sample
I0
IF
I
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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Outline
X-ray Absorption Spectroscopy
X-ray Absorption Fine Structure (EXAFS and XANES)
Major historical EXAFS breakthroughs
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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What is XAFS ? X-ray Absorption Fine Structure: oscillatory variation of the X-ray
absorption as a function of photon energy beyond an absorption edge.
As K-edge in InAsP
0.80
1.00
1.20
1.40
1.60
1.80
2.00
2.20
2.40
12000 12400 12800 13200
Ab
sorp
tio
n
E (eV)E(eV)
Abso
rpti
on c
oeff
icie
nt m
Proximity of neighboring atoms strongly modulates the absorption coefficient
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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EXAFS and XANES XAFS is often broken into 2 regimes: XANES X-ray Absorption Near-Edge Spectroscopy EXAFS Extended X-ray Absorption Fine-Structure which contain related, but slightly different information about an element’s local coordination and chemical state.
As K-edge in InAsP
0.80
1.00
1.20
1.40
1.60
1.80
2.00
2.20
2.40
12000 12400 12800 13200
Abso
rpti
on
E (eV)
Abso
rpti
on c
oeff
icie
nt m
XANES EXAFS
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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0.80
1.00
1.20
1.40
1.60
1.80
2.00
2.20
2.40
12000 12400 12800 13200
Abso
rpti
on
E (eV)
Abso
rpti
on c
oeff
icie
nt m
EXAFS and XANES XANES: transitions to unfilled bound states, nearly bound states, continuum local site symmetry, charge state, orbital occupancy
XANES
EXAFS
EXAFS: 50 - 1000 eV after edge due to transitions to continuum local structure (bond distance, number, type of neighbors….)
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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EXAFS qualitatively
isolated atom
E
e-
condensed matter
e-
R
l
m
k
m
pEEEkin
22
222
0
=== l = 2 p/k
The kinetic energy of the ejected photoelectron Ekin is:
E0
Ekin
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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Where do the oscillations come from ?
1. As E is scanned above E0, Ekin is varied, and consequently k and l.
2. The outgoing and backscattered parts of the wave interfere either constructively or destructively, depending on the ratio between l
and R.
3. It is the interference between outgoing and incoming waves that gives rise to the sinusoidal variation of m(E)
Due to a quantistic effect, the autointerference of photoelectron wave
modifies the absorption coefficient value:
frequency ~ distance from neighbors
amplitude ~ number and type of neighbors
e-
R
l
m
k
m
pEEEkin
22
222
0
===
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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Kr gas Rh metal
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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Absorption coefficient
2ˆ)(
f
fIi HEm
photoelectron
core hole
1
2 3 4
continuum
1s electron
1
2 3 4
continuum
in principle, all electrons are involved multi body process
single electron
211 ˆ
f
f
N
fi
N
i r
j
j
j
j
rAA
ˆˆ00
dipole
2
ˆˆ f
fi r
2112
0 f
NN
fiSsudden
22
0ˆ
f
fi rS
j
j
jI rAH
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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Consequences of dipole approximation: Selection rules
Dl = ± 1 Ds = 0 Dj = ± 1 Dm = 0
For 1-electron transitions: edge initial state final state K, L1 s (l=0) p (l=1)
L2, L3 p (l=1) s (l=0), d (l=2)
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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Absorption coefficient
2
ˆ)( f
fi rE m
if EE = pf =
photoelectron
core hole
1
2 3 4
continuum
si 1=iE
|f> very complicated final state strongly influenced by environment
|i> relatively easy ground state of atom; i.e. 1s e- wavefunction
1s electron
1
2 3 4
continuum
Approx: dipole + single electron + sudden
: photon polarization r: electron position
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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outgoing spherical wave originating from the absorbing atom
0
ff =
Isolated atom: atomic absorption coefficient
photoelectron free to travel away undisturbed
20
0ˆ
fi r m
20*
0ˆ rrrrd fi
m
overlap integral of initial and final state wavefunctions: monotonically decreases as function of E
e-
h
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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Non-isolated atom
sum of the outgoing and all the incoming waves, one per each neighboring atom.
fff = 0
20ˆffi r m
20* ˆ rrrrrd ffi
m
2*
**020*
ˆ
ˆˆRe2ˆ
rrrrd
rrrrrrrdrrrrd
fi
fififi
m
EXAFS
e-
h
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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Origin of EXAFS
The region where i 0 represents simultaneously
the source and the detector for the photoelectron that probes the local structure around the absorber atom
c : fractional change in m introduced by the neighbors
cmm = 10
Interference between outgoing wavefunction and backscattered wavelets
Dominant contribution to integral comes from spatial region close to absorber atom nucleus, where the core orbital wavefunction i ≠ 0.
=
20*
**0
ˆ
ˆˆRe2
rrrrd
rrrrrrrdk
fi
fifi
c (1)
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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The EXAFS signal c
We’re interested in the energy dependent oscillations in μ (E), as these will tell us
something about the neighboring atoms, so we define the EXAFS as:
00
0
E
EEE
m
mmc
D
=
We subtract off the smooth “ bare atom” background μ0(E), and divide by the
“edge step” D μ0(E0), to give the oscillations normalized to 1 absorption event.
0.80
1.00
1.20
1.40
1.60
1.80
2.00
2.20
2.40
12000 12400 12800 13200
Ab
sorp
tio
n
E (eV)
m (
E)
E(eV)
Dm0
m0(E)
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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EXAFS : c (k)
2
2
0EEm
k
=
-0.10
-0.05
0.00
0.05
0.10
0 5 10 15 20
k (A-1
)
c (
k)
XAFS is an interference effect, and depends on the wave-nature of the
photoelectron.
It’s convenient to think of XAFS in terms of photoelectron wavenumber, k,
rather than x-ray energy
c (k) is often shown weighted by k2 or k3 to amplify the oscillations at high-k:
-3.00
-2.00
-1.00
0.00
1.00
2.00
3.00
0 5 10 15 20
k (A-1
)
k2 c
(k
)
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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A Fourier Transform of the EXAFS signal provides a photoelectron scattering profile as a function of the radial distance from the absorber.
Qualitative picture of local coordination in R space
The frequencies contained in the EXAFS signal depend on the distance between the absorbing atom and the neighboring atoms (i.e. the length of the scattering path).
R1(SS)
R2(SS), R3(SS), MS
R(A)
Am
plit
ude o
f F
T
-0.10
-0.05
0.00
0.05
0.10
0 5 10 15 20
k (A-1
)
c (
k)
c(k
)
k(A-1)
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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-4
-2
0
2
4
6
8
0 1 2 3 4 5 6 7 8
Structural determinations depend on the feasibility of resolving the data into
individual waves corresponding to the different types of neighbors (SS) and
bonding configurations (MS) around the absorbing atom.
As
In
As
P
As
As
InAsxP1-x
absorber As atom
0
2
4
6
8
0 1 2 3 4 5 6 7 8
R(A)
Am
plit
ude o
f F
T
As
In
As In
In As
P
In
As
As In
Quantitative structural determination
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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c : sum of damped waves
c (k) is the sum of contributions cj (k) from backscattered wavelets:
ksinkAk jjj c =
=j
j kk cc
Damping of the amplitude at large k, due to static and thermal disorder
222 k
jj ekfN
The larger the number of neighbors, the larger the signal kRk jj 2
The stronger the scattering amplitude, the larger the signal
Each shell contributes a sinusoidal signal which oscillates more rapidly the larger the distance
Each cj (k) can be approximated by a damped sine wave of the type:
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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InAsxP1-x
absorber As atom
k(A-1) k(A-1)
-0.2
-0.1
0
0.1
0.2
4 8 12 16 20
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
4 8 12 16 20
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
4 8 12 16 20
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
4 8 12 16 20
As
P
As
As
As
In
As
In
shape of the envelope of each wave indicative of nature of backscatterer atom:
Amplitudes
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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Frequencies: Single and Multiple Scattering paths
R1
SS g2(r) f = 2 R1
R1
MS g2(r)
f = 4 R1
MS g3(r)
f = R1 + R2 + R3 R1
R3
R2
MS g3(r) f = 2R1 + 2R3
R1
R3
The sum over paths in the EXAFS equation includes many shells of atoms (1st neighbor, 2nd neighbor, 3rd neighbor, . . . ), but can also include multiple scattering paths, in which the photoelectron scatters from more than one atom before returning to the central atom.
EXAFS can give information on the n-body distribution functions gn(r).
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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The EXAFS equation
To model the EXAFS, we use the EXAFS Equation
where f(k) and (k) are photoelectron scattering properties of the
neighboring atom. (The sum is over “shells” of similar neighboring atoms).
If we know these properties, we can determine:
R distance to neighboring atom.
N coordination number of neighboring atom.
2 mean-square disorder of neighbor distance.
The scattering amplitude f(k) and phase-shift (k) depend on atomic
number Z of the scattering atom, so we can also determine the species of
the neighboring atom.
]2[sin222
2
2
0 kkRekR
kfNSk jj
k
j
j
jjj c
=
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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The EXAFS equation: simple description
]2[sin2
kkRkR
kfk c =
With spherical wave for the propagating photoelectron: and a scattering atom at a distance r = R, we get:
where the neighboring atom gives the amplitude |f(k)| and phase-shift s(k) to the scattered photoelectron. Substituting into equation (1) and after some math we get:
|f(k)| eis(k)
for 1 scattering atom.
A B R
I II III
Region I: amplitude
of outgoing wave
Region II: amplitude of
wave arriving on B
Region III: amplitude of backscattering on B
Region II: amplitude of ingoing
wave, backscattered from B
Df ~ 0f
e ikR
2kR
e ikR
kR [i eic ] eic ]
kkk cs 2=
kr
eikr
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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Development of the EXAFS equation
For N scattering atoms, and with a thermal and static Gaussian disorder of 2, giving the mean square disorder in R*, we have
]2[sin222
2
2
0 kkRekR
kfNSk k c =
A real system will have neighboring atoms at different distances and of different types. We add all these contributions to get a version of the EXAFS equation:
To obtain this formula we used a spherical wave for the photoelectron:
* EXAFS takes place on a time scale much shorter than that of atomic motion, so the measurement serves as an instantaneous snapshot of the atomic configuration
]2[sin222
2
2
0 kkRekR
kfNSk jj
k
j
j
jjj c
=
kr
eikr
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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The photoelectron mean free path But the photoelectron can also scatter inelastically*, and may not be able to get
back the absorbing atom. Also: The core-hole has a finite lifetime**, limiting how
far the photoelectron can go.
)(/ krikr
ekr
e l
Using a damped wave-function: where l(k) is the photo electron’s
mean free path (including core hole lifetime), the EXAFS equation becomes:
The mean free path l depends on k.
For the EXAFS k range, l < 25 Å.
The l and R-2 terms make EXAFS a local atomic probe.
* Electrons that have suffered inelastic losses will not have the proper wave vector to contribute
to the interference process.
]2[sin/22
2
2
0
22
kkReekR
kfNSk jj
kRk
j
j
jjj c l
=
** the photoelectron and core hole exist simultaneously
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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S02 : Amplitude Reduction Term
Another important Amplitude Reduction Term is due to the relaxation of all the
other electrons in the absorbing atom to the hole in the core level:
where fN-1 accounts for the relaxation of the other N-1 electrons relative to
these electrons in the unexcited atom: 0N-1 . Typically S0
2 is taken as a constant:
0.7 < S02 < 1.0
which is found for a given central atom, and simply multiplies the XAFS c.
Note that S02 is completely correlated with N.
This, and other experimental and theoretical issues, make EXAFS amplitudes (and therefore N) less precise than EXAFS phases (and therefore R).
Usually S02 is found from a “standard” (data from a sample with well-known
structure) and applied to a set of unknowns as a scale factor.
2112
0 f
NN
fiS
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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Scattering Amplitude and Phase-Shift: f (k) and (k)
The scattering amplitude f (k) and phase-shift (k) depend on atomic number.
The scattering amplitude f (k) peaks at different k values and extends to higher-k for heavier elements. For very heavy elements, there is structure in f (k).
The phase shift (k) shows sharp changes for very heavy elements.
These scattering functions can be accurately calculated (say with the programs FEFF, GNXAS, etc.), and used in the EXAFS modeling.
Z can usually be determined to within 5 or so. Fe and O can be distinguished, but Fe and Mn cannot be.
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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Calculating f (k) and (k)
These days, we can calculate f (k) and (k) easily using different software codes These programs take as input:
1. a list of atomic x,y,z coordinates for a physical structure 2. a selected central atom.
The result is a set of files containing the f (k), and (k) for a particular scattering “shell” or “scattering path” for that cluster of atoms. Many analysis programs use these files directly to model EXAFS data. A structure that is close to the expected structure can be used to generate a model, and used in the analysis programs to refine distances and coordination numbers.
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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Diffraction Methods (X-rays, Neutrons)
Crystalline materials with long-range ordering -> 3D picture of atomic coordinates
Materials with only short-range order (amorphous solid, liquid, or solution) -> 1D RDF containing interatomic distances due to all atomic pairs in the sample.
XAFS vs Diffraction Methods
XAFS 1D radial distribution function (centered at the absorber)
Higher sensitivity to local distortions (i.e. within the unit cell)
Charge state sensitivity (XANES)
Element selectivity
Structural information on the environment of each type of atom:
distance, number, kind, static and thermal disorder
3-body correlations
Investigation of matter in the solid (crystalline or amorphous), liquid, solution or gaseous state with same degree of accuracy.
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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EXAFS: typical applications
Local structure in non-crystalline matter
Local environment of an atomic impurity in a matrix of different atomic species
Study of systems whose local properties differ from the average properties
Detection of very small distortions of local structure
Element selectivity
Local structure sensitivity
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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0.80
1.00
1.20
1.40
1.60
1.80
2.00
2.20
2.40
12000 12400 12800 13200
Abso
rpti
on
E (eV)
Abso
rpti
on c
oeff
icie
nt m
EXAFS and XANES XANES: transitions to unfilled bound states, nearly bound states, continuum local site symmetry, charge state, orbital occupancy
XANES
EXAFS
EXAFS: 50 - 1000 eV after edge due to transitions to continuum local structure (bond distance, number, type of neighbors….)
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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XANES Interpretation
The EXAFS Equation breaks down at low-k, and the mean-free-path goes up.
This complicates XANES interpretation:
We do not have a simple equation for XANES.
XANES can be described qualitatively (and nearly quantitatively ) in terms of coordination chemistry regular, distorted octahedral, tetrahedral, . . . molecular orbitals p-d orbital hybridization, crystal-field theory, . . . band-structure the density of available electronic states multiple-scattering multiple bounces of the photoelectron These chemical and physical interpretations are all related, of course:
What electronic states can the photoelectron fill?
XANES calculations are becoming reasonably accurate and simple. These can help explain what bonding orbitals and/or structural characteristics give rise to certain spectral features. Quantitative XANES analysis using first-principles calculations are still rare, but becoming possible...
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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K-Edge XANES
1s
3d
Ground State
Excited States
Total electron energy
Main edge
Mn3+
Pre-edge
Continuum
1s
3d
p
1s
3d
4p
1s
3d
Mn: [Ar] 3d5 4s2
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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1s
3d
4p
Mn4+
1s
3d
Mn4+
1s
3d
4p
Mn3+
Chemical Shift
1s
3d
Ground State
Mn3+
Excited States
Mn2O3
MnO2
Total electron energy
Mn: [Ar] 3d5 4s2
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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Edge Shifts and Pre-edge Peaks in Fe oxides
The shift of the edge position can be used to determine the valence state. The heights and positions of pre-edge peaks can also be sometimes used to determine Fe3+/Fe2+ ratios.
XANES for Fe oxides and metal
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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XANES Analysis: Oxidation State
XANES can be used simply as a fingerprint of phases and oxidation state.
The Normalized XANES from several Fe compounds:
XANES analysis can be as simple as making linear combinations of “known” spectra to get compositional fraction of these components.
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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The XANES of Cr3+ and Cr6+ shows a dramatic dependence on oxidation state and coordination chemistry.
For ions with partially filled d shells, the p-d hybridization changes dramatically as
regular octahedra distort, and is very large for tetrahedral coordination. This gives a dramatic pre-edge peak – absorption to a localized electronic state.
XANES Analysis: Coordination Chemistry
Oh centrosymm only weak quadrupole
channel Td non-centrosymm
p-d mixing dipole channel
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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XANES: Conclusions
XANES is a much larger signal than EXAFS
XANES can be done at lower concentrations, and less-than-perfect sample
conditions.
XANES is easier to crudely interpret than EXAFS For many systems, the XANES analysis based on linear combinations of known spectra from “model compounds” is sufficient.
XANES is harder to fully interpret than EXAFS The exact physical and chemical interpretation of all spectral features is still difficult to do accurately, precisely, and reliably. This situation is improving, so stay tuned to the progress in XANES calculations . . . .
S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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Outline
X-ray Absorption Spectroscopy
X-ray Absorption Fine Structure (EXAFS and XANES)
Major historical EXAFS breakthroughs
Major historical EXAFS breakthroughs
• Atomic scale structure in solid solutions
• Lattice distortions around impurities in dilute alloys
• Structure of amorphous materials
50 S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
Solid solutions: Vegard’s law and the Virtual Crystal Approximation
ABx C1-x
aAB aAC
AC
ACAC aR4
30 =ABAB aR
4
30 =
a(x)
xaaa)x(a ACABAC Vegard’s Law:
AB
VCA: xaxRxR ACAB4
3==
51 S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
Atomic scale structure in solid solutions
• Atomic scale structure not well understood: XRD averages structure over distances that are large on the scale of a lattice constant.
• Calculations of the properties of solid solutions have often relied on simple approximations (i.e. VCA)
• VCA assumes that all atoms occupy average lattice positions defined by X-ray lattice constants AxB1-xC
• With the use of the VCA, properties of alloys may be calculated whether or not the alloys’s lattice costant varies linearly with composition between those of the end members (follows Vegard’s law)
52 S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
Atomic scale structure in solid solutions
• GaAs and InAs bonds change only by 0.04 Å in whole x range !! • contradicts underlying assumptions of VCA • important distortions within unit cell accommodated by bond angle changes
53 S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
Lattice distortions around impurities in dilute alloys
Mn
Ni Ni
Ni Ni
• systematic study of substitutional impurities in fcc and bcc metals • important shifts in first shell bond length detected
2 % Mn in Ni
Mn K edge
54 S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
Lattice distortions around impurities in dilute alloys
Mn shifts 12 Ni nearest neighbors outwards by:
0.023 ± 0.004 Å
(1 % of distance)
XAFS ck2 Fourier Transform of ck2
Fit of 1st shell
55 S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
Comparison to band structure calculations
Displacements has two contributions: 1. Valence difference between impurity and host:
- change of charge density in impurity cell - parabolic dependence (Z)
2. Magnetoelastic contribution Cr, Mn, Fe in Cu majority and minority bands are split:
- large magnetic moment - low DOS at Fermi level - low binding energy - increased interatomic distance
Cu matrix
XAS
Band structure calculations
N. Papanikolau et al., Phys. Rev. B 55, 4157 (1997)
56 S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
Structure of amorphous materials
Random network model: GeO2 tetrahedra connected by bridging Oxygen with deviations about bond angles such that long range periodicity destroyed Microcrystalline model: GeO2 composed of 15-20 Å crystallites – to explain long range fluctuations in RDF after removing first 3 peaks
From X-ray scattering experiments on glasses:
57 S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
EXAFS determines: • identical 1st shell coordination number (to within 2%) • increased disorder on Ge-Ge shell
microcrystalline model definitively ruled out.
c-GeO2
g-GeO2
58 S. Pascarelli – Joint ICTP-IAEA Workshop - Trieste, 2014
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More information: web links
International XAFS Society: http://ixs.iit.edu/ Tutorials and other Training Material: http://xafs.org/Tutorials Software Resources EXAFS: http://xafs.org/Software http://leonardo.phys.washington.edu/feff http://gnxas.unicam.it/
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More information: Books and Review Articles
Fundamentals of XAFS Introduction to XAFS: A Practical Guide to X-ray Absorption Fine Structure Spectroscopy, G. Bunker, Cambridge
University Press, 2012 X-ray Absorption: Principles, Applications, Techniques of EXAFS, SEXAFS, and XANES, in Chemical Analysis 92 D. C. Koningsberger and R. Prins, ed., John Wiley & Sons, 1988. Basic Principles and Applications of EXAFS, Chapter 10 in Handbook of Synchrotron Radiation, pp 995–1014. E. A. Stern and S. M. Heald, E. E. Koch, ed., North-Holland, 1983
FEFF Theoretical approaches to x-ray absorption fine structure
J. Rehr et al., Rev. Mod. Phys. 72, 621 - 654 (2000)
GNXAS
X-ray absorption spectroscopy and n-body distribution functions in condensed matter (I): theory of the GNXAS data-analysis method
A. Filipponi, A. Di Cicco and C. R. Natoli, Phys. Rev. B 52, 15122 (1995)
MXAN Geometrical fitting of experimental XANES spectra by a full multiple-scattering procedure M.Benfatto and S. Della Longa J. Synchr. Rad. 8 , 1087 (2001)
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How to measure the absorption coefficient m
tN
a
ta
t
a
ee)(I
)t(I
N)(Iln)t(Iln
dxt
NxI
dI
t
dxNxIdI
a m
==
=
=
=
0
0
00
at/cm2 cm2/at
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m is related to the atomic cross section:
• in general you find tabulated the mass absorption coefficient m/r:
• for a generic sample PxQy…..:
1
3
2=
== cm
cm
gr
mole
grmole
at
at
cm
A
N
t
N aaa rm
=
M
Ay
M
Ax Q
Q
P
Ptotr
m
r
m
r
m
=
=
gr
cm
mole
grmole
at
at
cm
A
Naa
22
r
m
tN ee)(I
)t(Ia m ==
0
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Recipe for calculating t for transmission XAS
tN ee)(I
)t(Ia m ==
0
1. Total absorption above the edge must not be too high: m above edge t = 2 5 I / I0 ~ 0.14 0.007
ideally m above edge t = 2-3
2. Contrast at edge must be as large as possible: [ m above edge - m below edge ] t > 0.1 ideally [ m above edge - m below edge ] t = 1
If absorber is very dilute, and matrix absorbs a lot, then this is not possible fluorescence detection
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How to calculate m
energy density u carried by X-ray beam is:
linear absorption coefficient m measures the energy density
reduction due to the interaction with the system of atoms:
22
20
220 AE
u oo ==
dx
du
u
1=m
=
=
=
=
fif
ph
ph
WnA
ndx
d
A
ndx
d
A
dx
du
A
200
200
20
20
20
20
2
2
2
2
m
m
m
m
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I. Lets consider the interaction between:
Monochromatic X-ray beam ( = 2pn + monoatomic sample
EM field (classic) + atom (quantistic)
(semi-classical description)
II. m ~ m photoelectric absorption for 1 < E < 50 keV
III. Qualitatively, interaction process is:
core hole 1
2 3
continuum or free energy level
1s electron 1
2 3
| i > , Ei | f >, Ef = Ei +
X-ray Absorption
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Transition probability: Golden Rule m depends on:
– atomic density n
– transition probability Wif of atom from | i > to | f >
to calculate Wif : time-dependent perturbation theory based on power series of EM field - atom interaction potential.
The interaction is in general WEAK
Can limit series to 1st order: Golden Rule
=f
ifWnA2
00
2
m
ffIiif EHW rp 2
ˆ2=
(2)
(1)
IH
fIi H ˆ Matrix element of HI between initial and final state
EM field - atom interaction hamiltonian operator
fEr Density of final states, compatible with energy conservation:
Ef = Ei +
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the interaction hamiltonian for photoelectric absorption (see Appendix 1) is (to 1st order):
the transition probability for photoelectric absorption of a monochromatic, polarized and collimated photon beam is [(3) into (2)]:
jj
jI rAm
eiH =
(3)
ffj
j
rki
iif EeAm
eW j r
p2
2
02
2
ˆ =
(4)
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Dipole approximation
further simplification:
transition probability in dipole approximation:
alternative and equivalent expression :
finally one gets [(5) into (1)]:
1
!21
2
=
jj
rki rkrkie j
ff
j
jiif EAm
eW r
p2
2
02
2
ˆ =
ff
j
jiif ErAe
W rp
2
2
0
22
ˆ =
(5)
(6)
12
jrkif
=f
ne
0
22
pm
ff
j
ji Er r
2
ˆ
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if |i > and |f > are known (if wavefunctions and energies can be calculated):
1) calculate Wif
2) calculate m
in practice, one is interested in inverse process:
1) measure m
2) extract EXAFS
3) obtain information on local structure through |f >
but, to obtain structural info, one still needs to calculate |i > and |f >
or at least be able to express their structural properties in parametric form
– |i > relatively easy
- ground state of atom
– |f > in general very complicated
- in principle, all electrons are involved -> multi body process
- final state strongly influenced by environment
=f
ne
0
22
pm
ff
j
ji Er r
2
ˆ
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Single electron approximation
large part of m due to “elastic” transitions:
– only 1 electron out of N modifies its state: leaves its deep core level
– all other N-1 “passive” electrons relax their orbitals to adapt to the new potential created by presence of core hole
remaining part of m due to “inelastic” transitions:
– primary excitation of core electron provokes successive excitations of other (external) electrons (shake up, shake off processes)
– excess energy distributed among all excited electrons
mmm inelel =
ff
N
fi
N
iel r rm 211 ˆ
1 Ni Slater determinant of “passive” electrons’ wavefunctions
fr ,, Wavefunction, position vector, final energy of “active” electron
where
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“Sudden” approximation and overlap factor
– if photoelectron energy is sufficiently high (E > few 10 eV above edge)
time to exit atom << relaxation time of passive electrons its state not influenced by passive electrons relaxation
ffiel Sr rm 2
0
2ˆ
where 21120
= Nf
NiS
– Allows to reduce interpretation of EXAFS to the calculation
of the final state of ONLY the photoelectron
(S02 ~ 0.7 - 0.9)
(7)
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The polarization dependent dipole operator
dipole operator in terms of spherical harmonics
dipole operator a = x, y , z X-ray prop direction q = +1, 0, -1 polarization states
(q photon angular momentum)
2
ˆ)( f
fi rE m
electron position vector
photon polarization
vectors
linear polarization
circular polarization with k // z
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The dipole selection rules
By looking at the non-zero matrix elements we get the dipole selection rules
where q is the X-ray angular momentum
2
ˆ)( f
fi rE m
matrix elements factor into spin, radial and angular parts