Lucidi Duino 11 - ELETTRA
Transcript of Lucidi Duino 11 - ELETTRA
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Paolo Fornasini Dipartimento di Fisica Università di Trento
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Basic attenuation mechanisms for X-rays Paolo Fornasini
Univ. Trento
outgoing beam
€
hν
€
hν '
€
hν
€
hνF
€
E
€
EA
Elastic scattering
Inelastic scattering
Photo-electrons
Auger electrons
Fluorescence Absorption
Scattering
Beam attenuation Incoming beam
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X-RAYS and X-ray techniques Paolo Fornasini
Univ. Trento
X-rays
10-12 10-10 10-8 10-6 10-4 10-2 1 Å 1 µm
λ (m)
U.V. I.R.
E[keV] = 12.4λ[Å]
Scattering
Bragg angle
Inte
nsity
λ = 2dsinθ
Spectroscopy
Abso
rptio
nEnergy
Imaging
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Structural techniques
Scattering
Paolo Fornasini
Univ. Trento
X-rays
Neutrons
Spectroscopy
Abso
rptio
n
Energy
XAFS
Long-range order Short-range order
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• X-rays absorption – phenomenology • X-rays absorption - theory • EXAFS: theoretical background • EXAFS experiments • EXAFS: data analysis, examples
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X-rays absorption - phenomenology
X-rays, 1897
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Attenuation of X-rays Paolo Fornasini
Univ. Trento
source monochromator
x
sample
detectors
Φ0 Φ
Φ = Φ0 exp −µ ω( ) x[ ]
µ ω( ) =1xlnΦ0
Φ
Exponential attenuation
Attenuation coefficient
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Atomic cross sections Paolo Fornasini
Univ. Trento
10-15
10-10
10-5
10-2 100 102 104 106
Cro
ss s
ectio
n (
Å2 )
E (keV)
Photoelectric absorption
Coherent sc.
Incoherent sc.
Pair prod.nucl. field
elec. field
32 - Ge
€
µ ω( ) =NaρA
µa ω( )
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Photo-electric absorption Paolo Fornasini
Univ. Trento
100
101
102
103
104
1 10 100
µ/ρ (cm2/g)
60-Nd, 65-Tb, 70-Yb
hω (keV)
K
L1-3
M1-5
100
102
104
106
1 10 100
µ/ρ (cm2/g)
Photon energy (keV)
τ
σc
σi
32 - Ge
Edges
€
τ ω( )∝ Z 4
ω( )3
+
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Excitation and ionization
-10000
-8000
-6000
-4000
-2000
0
Ep (eV)
- 13,6
- 3,4
0 Ep (eV)
1- Hydrogen 29 - Copper
Paolo Fornasini
Univ. Trento
unoccupied free-levels
unoccupied free-levels
unoccupied bound-levels
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Absorption edges
100
101
102
103
104
1 10 100
µ/ρ (cm2/g)
60-Nd, 65-Tb, 70-Yb
hω (keV)
K
L1-3
M1-5
Z > 9 Z > 29
L1 2s
L2 2p1/2 L3 2p3/2
K 1s
M1 3s M3 3p3/2
M2 3p1/2 M4 3d3/2 M5 3d5/2
Paolo Fornasini
Univ. Trento
1
10
100
0 20 40 60 80 100
Bind
ing
ener
gy (k
eV)
Z
K
L3
M5
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Physical Review 16, 202 (1920)
Absorption edge fine structure
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Atomic gases: edge fine structure
-6 -4 -2 0 2
µ (a
.u.)
E - Eb (eV)
Ar Eb = 3205.9 eV
Paolo Fornasini
Univ. Trento
Core level Unoccupied bound levels (Rydberg levels)
L. G. Parrat, Phys. Rev. 56, 295 (1939)
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Atomic gases: smooth absorption coefficient
14.2 14.5 14.8 15.1Photon energy (keV)
µ (a
.u.)
Kr
Paolo Fornasini
Univ. Trento
Core level Unoccupied free levels (continuum)
Edge Extended region
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Molecular gases: Fine structure Paolo
Fornasini Univ. Trento
Abso
rptio
n In
tens
ity (
arb
. Uni
ts)
K-shell - gas-phase N2
x10
400 405 410 415 420 Photon energy (eV)
Core level Unoccupied bound levels Unoccupied free levels
C.T. Chen and F. Sette, Phys. Rev. A 40 (1989)
Vibrational levels
Rydberg series
Double excit.
Shape reson.
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Condensed systems: Fine structure
11 11.5 12Photon energy (keV)
Ge
µ (a
.u.)
Paolo Fornasini
Univ. Trento
Core level Unoccupied bound levels Unoccupied free levels
Edge
Extended region
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XAFS: X-ray Absorption Fine Structure
X-ray Absorption Near Edge StructureNear Edge X-ray Absorption Fine Structure
Paolo Fornasini
Univ. Trento
11 11.5 12Photon energy (keV)
XANESNEXAFS
EXAFS
Edge
Extended X-ray AbsorptionFine Structure
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XAFS: edge and pre-edge
5 4.98 4.96 5.02 5.04 Photon energy (keV)
Edge Pre-edge
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Lectures on XAFS
EXAFS • introduction • basic theory • experiments • data analysis
P. Fornasini
XANES phenomenological approach
C. Meneghini
XAFS multiple scattering approach
M. Benfatto
Applications
XAFS & Materials science F. Boscherini
SR & Earth science S. Quartieri
SR & Cultural heritage S. Quartieri
SR & Chemistry A. Martorana
SR & Environmental science P. Lattanzi
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X-rays absorption - theory
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X-ray attenuation
• Linear attenuation coefficient
• Mass attenuation coefficient
µρ=Na
Aµ a
Elements:
µρ
tot
= xµρ
P
APM
+ yµρ
Q
AQM
+ …
Ai = atomic weights, M = molecular weight
Chemical compounds PxQy...
x
Φ0 Φ
Energy density u = ε0E02
2=ε0ω
2A02
2
µ ω( ) = −1ududx
=1xlnΦ0
Φ=NaρA
µ a ω( )
Na = Avogadro numberA = atomic weightr = mass density
µa = atomic cross section
Paolo Fornasini
Univ. Trento
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Absorption coefficient
Pif = ?
Paolo Fornasini
Univ. Trento
Absorption coefficient
Atomic density
Vector potential amplitude
Transition probability per unit time
i = initial ground state f = final excited states
€
E f − Ei = ω
€
µ ω( ) =2
ε0ωA02 n
dPifdtf
∑
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Radiation-matter interaction
Pif = ?
Initial atomic state
Stationary ground state Ψi
Final atomic state
Stationary excited state Ψf
Interaction
Ψ t( )
Paolo Fornasini
Univ. Trento
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Perturbation approach
Paolo Fornasini
Univ. Trento
System = atom (quantum treatment)
Weak perturbation = electromagnetic field (classical treatment)
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Hamiltonian for atom in e.m. field Paolo Fornasini
Univ. Trento
∇ ⋅ A =0
Φ = 0 J = 0
Radiation gauge
E = ∂
A /∂t
B = ∇ × A
∇ ⋅ A = A ⋅ ∇
Vector potential
€
H =12m
p j − q A r j ,t( )[ ]
2−
qm s j ⋅ B r j ,t( )
j
∑ + V r 1… r N( )
=p j2
2m+ V r 1…
r N( )
j∑ +
em p j ⋅ A r j ,t( ) +
em s j ⋅ B r j ,t( ) +
e2
2mA2 r j ,t( )
j∑
€
H0
€
HI
Unperturbed Interaction
e-e and e-p interact.
Sum over electrons Electron
spin
€
q = −e < 0
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Interaction Hamiltonian
Sum over electrons
€
A = Re A0 ˆ η ei(
k ⋅ r −ωt )[ ]
Paolo Fornasini
Univ. Trento
€
HI =em p j ⋅ A r j ,t( ) +
em s j ⋅ B r j ,t( ) +
e2
2mA2 r j ,t( )
j∑
Relevant terms
Time dependence:
sinusoidal perturbation frequency ω
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Transition to continuum states Paolo Fornasini
Univ. Trento Time dependent perturbation theory
€
Ψi
€
Ψf
Paolo Fornasini
Univ. Trento
€
dE f
Density of states
€
dPifdE f
= ρ(E f ) Ψf HI Ψi
2t δ E f − Ei − ω( ) 2π
Probability density
Time Energy conservation
1st-order perturbation
Transition to continuum states
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''Golden rule''
€
HI =em p j ⋅ A rj ,t( )
j∑
€
wif =πe2
m2 Ao2
Ψi ei k ⋅ r j ˆ η ⋅ ∇ jj∑ Ψf
2ρ E f( ) δ E f − Ei − ω( )
€
A = Re A0 ˆ η ei
k ⋅ r [ ]
Paolo Fornasini
Univ. Trento
€
wif =dPifdE f dt
= ρ(E f ) Ψf HI Ψi
2δ E f − Ei − ω( ) 2π
Probability density per unit time
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Golden rule
Initial atomic state
Stationary ground state Ψi
Final atomic state
Stationary excited state ΨfInteraction
with e.m. field
HI
Time-dependent perturbation theory (1st-order)
matrix element density of final states
€
wif ∝ Ψi HI Ψf
2ρ E f( )
Paolo Fornasini
Univ. Trento
€
E f − Ei = ω
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One-electron approximation
µtot ω( ) = µ el ω( ) + µinel ω( )
• 1 core electron excited • N-1 passive electrons relaxed
• 1 core electron excited • Other electrons excited
EXAFS coherent signal
€
µel ω( )∝ ΨiN−1ψ i ei
k ⋅ r ˆ η ⋅ p ψ fΨf
N−12ρ ε f( )
Paolo Fornasini
Univ. Trento
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Electric dipole approximation
€
µel ω( ) ∝ ΨiN−1ψ i ˆ η ⋅
r ψ fΨf
N−12
€
ei k ⋅ r = 1+ i
k ⋅ r −… ≈ 1
€
HI ∝ ei k ⋅ r ˆ η ⋅ p ≈ ˆ η ⋅
p = ω 2 ˆ η ⋅ r
€
Δ = ± 1 Δs = 0Δj = 0,±1, Δm = 0,± 1
Dipole selection rules:
Paolo Fornasini
Univ. Trento
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Validity of dipole approximation Paolo Fornasini
Univ. Trento
€
ei k ⋅ r = 1+ i
k ⋅ r −… ≈ 1
€
λ >> 2πr ⇔ kr <<1
0.0
0.1
0.2
0.3
10 30 50 70
K edgekr
Z€
r ≈ a0Z
a0 = 0.53Å( )
€
λ ≈1Z 2
1s orbital - K edge
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Beyond the electric dipole approximation Paolo
Fornasini Univ. Trento
€
HI =em p j ⋅ A r j ,t( ) +
em s j ⋅ B r j ,t( ) +
e2
2mA2 r j ,t( )
For one electron:
Electric dipole
Electric quadrupole Magnetic dipole
Spin magnetic moment
Orbital magnetic moment
2-photon processes
€
ei k ⋅ r = 1+ i
k ⋅ r −…
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€
µel ω( )∝ ψi ˆ η ⋅ r ψ f
2ρ ε f( ) Ψi
N−1 ΨfN−1 2
Sudden approximation
ΨN −1ψ = Ψ N −1 ψ
1 active electron N-1 passive electrons
S02 ≈ 0.6 ÷ 0.9
Structuralinformation
No interaction between photoelectron and passive electrons
Paolo Fornasini
Univ. Trento
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The final state
€
ψ f
Molecular orbitals theories
Band theories
Multiple scattering approach
Single scattering approximation
Fine structure
€
ψ f
11 11.5 12Photon energy (keV)
Ge
µ (a
.u.)
Basic EXAFS mechanism
at the core site
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EXAFS: the mechanism
Superpositionat the core site.
Photo-electron propagation.
Photo-electron back-scattering.
B
A
X-ray photon absorption.Photo-electron emission.
Modulation ofabsorption coefficient -0.4
0.0
0.4
0 5 10 15 20
k χ(k)
Photo-electron wavenumber (Å-1)
Paolo Fornasini
Univ. Trento
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De-excitation mechanisms
Radiative: fluorescence
Non-radiative: Auger
X e-
X
X e- e-
X
A
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100
η
Z
ηK
ηL
ηM
Fluorescence yield
η =X
X + A
Paolo Fornasini
Univ. Trento
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Core-hole lifetime
0
5
10
15
20
20 40 60 80
Γh (eV)
Z
K edges
L3 edges
Lifetime of the excited stateτh~10-16 -10-15 s
Energy width of the excited state
Γh
τh ≈1/Γh
Contribution tophoto-electron life-time
τh
Energy resolutionof XAFS spectra
Γh
Paolo Fornasini
Univ. Trento
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EXAFS: theoretical background
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€
2m2 hν − Eb( ) =
2m2 ε = k =
2πλ
Photon → photo-electron
0
1
2
11 11.5 12hν (keV)
µx
Photon energy Photo-electron wavenumber
0 5 10 15k (Å-1)
χ (k)
Paolo Fornasini
Univ. Trento
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Angular emission of photo-electron Paolo Fornasini
Univ. Trento
Photoelectron emission
Photon polarisation
€
θ
asimmetry parameter
€
N(θ)∝1+β23cos2θ −1( )
€
θ
€
ˆ η
€
N(θ) ∝ 3cos2θ = 3 ˆ η ⋅ ˆ r 2€
β = 2Emission from s orbitals
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Photo-electron parameters
0
5
10
15
20
0 500 1000 1500
k (Å
-1)
εf (eV)0
1
2
3
0 500 1000 1500
λ (Å
)
εf (eV)
Wave-number Wave-length
Energy
Paolo Fornasini
Univ. Trento
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EXAFS normalisation
€
µ ω( )∝ ψ i ˆ η ⋅ r ψ f
2
µ0
hν
€
χ(k) =µ − µ 0
µ 0
hν
µ
€
µ0 ω( )∝ ψ i ˆ η ⋅ r ψ f
0 2
Paolo Fornasini
Univ. Trento
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The EXAFS function (a)
-0.06
-0.03
0
0.03
0.06
0 4 8 12 16k (Å-1)
χ (k)
€
µ0 ω( )∝ ψ i ˆ η ⋅ r ψ f
0 2
µ ω( )∝ ψ i ˆ η ⋅ r ψ f
2 ?
€
ψ f = ψ f0 + δψ f
(weak interaction) €
χ(k) =µ − µ 0
µ 0
Paolo Fornasini
Univ. Trento
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The EXAFS function (b)
-0.06
-0.03
0
0.03
0.06
0 4 8 12 16k (Å-1)
χ (k)
?
€
χ(k) =µ − µ 0
µ 0
Quantum states → wavefunctions
€
χ k( ) =2Re d r ψi ˆ η ⋅
r ψ f0*( )∫ ψi
* ˆ η ⋅ r δψ f( )
d r ψi* ˆ η ⋅ r ψ f
0∫2
Core orbital = source & detector
Paolo Fornasini
Univ. Trento
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EXAFS: Two-atomic system (a)
• Scattering theory in plane-wave approximation
• Muffin tin potential
Paolo Fornasini
Univ. Trento
R
II I III
Absorber Scatterer
€
δψ f ∝ψ f0 i eiδ
exp ikR( )2kR
f k,π( )exp ikR( )
Reiδ
At the absorber core site
Propagators
Interaction
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€
δψ f ∝ψ f0 i eiδ
exp ikR( )2kR
f k,π( )exp ikR( )
Reiδ
EXAFS: Two-atomic system (b)
€
χ k( ) = 3 ˆ η ⋅ ˆ R 2 1
kR2 Im f k,π( ) exp 2iδ1( ) exp 2ikR( )[ ]
spherical wave attenuation
back & forth path
central atom phase-shift
back-scattering amplitude
polarisation
Paolo Fornasini
Univ. Trento
€
χ k( ) =2Re d r ψi ˆ η ⋅
r ψ f0*( )∫ ψi
* ˆ η ⋅ r δψ f( )
d r ψi* ˆ η ⋅ r ψ f
0∫2
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Basic interference effect
€
χ k( ) = 3 ˆ η ⋅ R 1
kR2 Im f k,π( ) exp 2iδ1( ) exp 2ikR( )[ ]
€
χ k( ) = 3 ˆ η ⋅ ˆ R 2 1
kR2 f k,π( ) sin 2kR + φ k( )[ ]€
f k,π( ) e2iδ = f k,π( ) eiφ
0 5 10 15 20k (Å-1)
R = 2 Å
R = 4 Å
R
EXAFS frequency
Inter-atomic distance
Paolo Fornasini
Univ. Trento
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Amplitudes and phase-shifts
[Calculated by FEFF 6.01] Z dependence
Central-atom Back-scattering
Amplitude Phase-shift Phase-shift
Paolo Fornasini
Univ. Trento
0.0
0.2
0.4
0.6
0.8
0 5 10 15 20
Back-scattering amplitude(a.u.)
k (Å-1)
6 - C
32 - Ge
78 - Pt
-25
-20
-15
-10
-5
0
0 5 10 15 20k (Å-1)
32 - Ge
6 - C
78 - Pt
Back-scattering phase-shift(rad)
-2
0
2
4
6
0 5 10 15 20k
32 - Ge
78 - Pt
6 - C
Central atom phase-shift
k (Å-1)
(rad)
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Many-atomic systems Paolo Fornasini
Univ. Trento
r
Radial Distribution Functions
r
Coordination shells
Crystals Amorphous systems
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Coordination shells
Coordination shells
Coordination number
€
χ k( ) =1k
Nsshell∑ Im fs k,π( ) e2iδ1 1
Rs2 exp 2ikRs( )
€
3 ˆ η ⋅ ˆ R 2
=1Isotropic samples:
Paolo Fornasini
Univ. Trento
€
χ k( ) = 3 ˆ η ⋅ R 1
kR2 Im f k,π( ) e2iδ1 exp 2ikR( )[ ]
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Single and multiple scattering
SS = Single scattering MS = Multiple scattering
Scattering paths
Paolo Fornasini
Univ. Trento
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Multiple scattering series
Multiple Scattering ( ) ( ) ( ) ( ) ( )[ ]…++++= kkkkk 4320 1 χχχµµ
Single Scattering χ k( ) =µ − µ0µ0
( ) ( ) ( )[ ]kkk χµµ += 10
Paolo Fornasini
Univ. Trento
0
1
2
11 11.5 12Photon energy hν (keV)
µxµ0x
Es
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Multiple scattering paths
χ2
χ3
χ4
g4 g3 g2
gn = n-body correlation function
Multiple Scattering
€
µ k( ) = µ0 k( ) 1+ χ2 k( ) + χ3 k( ) + χ4 k( ) +…[ ]
Single Scattering χ k( ) =µ − µ0µ0
€
µ k( ) = µ0 k( ) 1+ χ k( )[ ]
Contribution from all n-order paths χn k( )
Paolo Fornasini
Univ. Trento
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Multiple scattering contributions
µ k( ) = µ0 k( ) 1 + χ2 k( ) + χ3 k( ) +…χ n k( ) +…[ ]
Photo-electron path Set of coordinates
Functions of potential
χn k( ) = A k, r { }( ) sin kRp +φ k, r { }( )[ ] Neglecting
thermal disorder !
Paolo Fornasini
Univ. Trento
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Intrinsic losses
Attenuation factor
µtot ω( ) = µ el ω( ) + µinel ω( )
hν
µ
€
χexp k( ) =µ −µ0
µ0
< χ th k( )
From experiment One-electron theory
€
S02 = Ψi
N−1 ΨfN−1 2
≈ 0.6 ÷ 0.9
Paolo Fornasini
Univ. Trento
€
χexp k( ) = S02 χ th k( )
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Photo-electron mean-free-path
0
10
20
30
40
0 5 10 15 20
λ (Å
)
k (Å-1)
λe
λh
Core-hole lifetime τh → λh
Photo-electron lifetime τe ⇔ λe
€
exp − 2Rλ k( )
€
1λ
=1λh
+1λe
Paolo Fornasini
Univ. Trento
Attenuation factor
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Disorder effects ?
EXAFS and inelastic effects
Atoms frozen in equilibrium positions !
€
χ k( ) =S02
kNs
shell∑ Im fs k,π( ) e2iδ1 e
−2Rs /λ (k )
Rs2 exp 2ikRs( )
Intrinsicinelastic effects
0 5 10 15k (Å-1)
χ (k)
Photo-electronmean-free-path
Paolo Fornasini
Univ. Trento
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Local disorder
Distance Distribution of distances
Structural disorder - examples Thermal disorder
r
τvib ≈ 10-12 sτexafs ≈ 10-15 s
+• Sites disorder
+
r• Free-volume models
• Distorted coordination shells
Paolo Fornasini
Univ. Trento
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Thermal disorder in crystals
Separability of coordination shells ?
Paolo Fornasini
Univ. Trento
0
20
40
60
0 2 4 6 8r (Å)
Ge, 300 K
0
20
40
60
0 2 4 6 8r (Å)
Ge, 1200 K
Simulated distributions for c-Ge
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Distributions of distances
Thermal + structural disorder
⇒ distribution of distances
r
Real distribution ρ(r)
Effective distributionP(r,λ)
€
χ k( ) =S02
kNs
shell∑ Im fs k,π( ) e2iδ1 ρs (r)
0
∞
∫ e−2rs /λ (k )
rs2 e2ikrs dr
Paolo Fornasini
Univ. Trento
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Real and effective distributions
0
20
40
60
0 2 4 6 8r (Å)
Ge, 300 K
0
20
40
60
0 2 4 6 8r (Å)
Ge, 1200 K
0
1
2
3
4
0 2 4 6 8r (Å)
Ge, 300 K
0
1
2
3
4
0 2 4 6 8r (Å)
Ge, 1200 K
Paolo Fornasini
Univ. Trento
€
ρ r( )
€
ρ r( )r2
€
P r,λ( ) =ρ r( )r2
e−2r /λ
(λ = 8Å)
EXAFS = short-range probe
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The inversion problem Paolo
Fornasini Univ. Trento
€
χ k( ) =S02
kNs
shell∑ Im fs k,π( ) e2iδ1 ρs (r)
0
∞
∫ e−2rs /λ (k )
rs2 e2ikrs dr
B
k
EXAFS
Characteristic function
χ k( ) ⇒ ρ r( ) ?
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Structural models and fitting procedure
χ(k)
k
ExperimentalEXAFS
Prametrisedstructural model Change
parameters
Compare with experiment
ρ (r)
r
Distributionof distances
kχ(k)∝ ρ(r) e−2r /λ
r 20
∞
∫ e2ikrdr
Paolo Fornasini
Univ. Trento
Initial guessparameters
Calculate EXAFS
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The simplest model: gaussian approximation Paolo Fornasini
Univ. Trento
r
€
r€
σ€
P(r,λ) =1
σ 2πexp
r − r( )2
2σ 2
€
C2 = σ 2 = r − r( )2
Distribution width (EXAFS Debye-Waller factor)
Average distance
€
C1 = r eff = r real −2σ 2
r1−
rλ
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Gaussian parametrization of EXAFS (one shell)
Coordination number
Debye-Waller
Average distance
€
k χ k( ) =S02 e−2C1 /λ
C12 f (k,π ) N exp −2k2σ 2[ ] sin 2kC1 + φ(k)[ ]
Inelasticterms
Totalphase-shift
Back-scatteringamplitude
• Theory (interaction potentials + scattering theory) • Experiment (reference samples)
Approx.: Single Scattering Plane waves
N
σ2
C1
Paolo Fornasini
Univ. Trento
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Including weak asymmetry Paolo Fornasini
Univ. Trento
r
Asymmetric distribution
€
r€
σ
€
C3 €
C2 = σ 2 = r − r( )2
€
C1 = r eff = r real −2σ 2
r1−
rλ
€
C3 = r − r( )3
Third cumulant Asymmetry parameter
Better for first shell
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EXAFS including asymmetry (one shell)
Coordination number
Debye-Waller
Average distance and asymmetry
€
k χ k( ) =S02 e−2C1 /λ
C12 f (k,π ) N exp −2k2σ 2[ ] sin 2kC1 − 43 k
3C3 + φ(k)
Inelasticterms
Totalphase-shift
Back-scatteringamplitude
• Theory (interaction potentials + scattering theory) • Experiment (reference samples)
Approx.: Single Scattering Plane waves
N
σ2
C1 C3
Paolo Fornasini
Univ. Trento
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XAFS and other techniques Paolo
Fornasini Univ. Trento
Photo-ionization
XPS - X-ray photo-electron spectroscopy
XAFS – X-ray absorption fine structure
e-
e-
Info on bound electronic states
Info on • free electronic states • local structure
XAFS = structural probe – Comparison with diffraction ?
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Bragg diffraction .vs. EXAFS
k0 k1
Bragg diffraction EXAFS
Structural probe X-ray or neutron plane waves photo-electron spherical wave
Paolo Fornasini
Univ. Trento
• long-range sensitivity • atomic positions • atomic thermal factors
• short-range sensitivity • inter-atomic distances • relative displacements
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EXAFS: a structural probe
Frequency
Inter-atomic distance
Amplitude
Coordinationnumber
Damping
Disorder
Selectivity of atomic species Insensitivity to long-range order
Paolo Fornasini
Univ. Trento
0 5 10 15k (Å-1)
R = 2 Å
R = 4 Å
0 5 10 15k (Å-1)
N = 4
N = 2
0 5 10 15k (Å-1)
σ2 = 0.01 Å2
σ2 = 0.005 Å2
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EXAFS applications
Non-crystalline materials
mono-atomic
many-atomic
• Inorganic heterogeneos catalysts • Metallo-proteins • Impurities in semiconductors • Luminescent atoms
Active sites embedded in a matrix
Local properties different from
average properties
• Crystalline ternary random alloys • Lattice dynamics studies • Negative thermal expansion
Paolo Fornasini
Univ. Trento
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EXAFS experiments
![Page 74: Lucidi Duino 11 - ELETTRA](https://reader031.fdocuments.net/reader031/viewer/2022012215/61df9d9d47f5382d1c6bee08/html5/thumbnails/74.jpg)
XAFS: experimental layout
• storage ring • beamlines
• monochromators • mirrors
• sample holder • detectors
Sample conditioning: cryostat ovenreactor manipulators
Detection: transmissionfluorescenceelectron yield.......
sample holder
storage ring monochromator
mirror
detectors Optical apparatus Source Experim. apparatus
Alternative layouts • dispersive EXAFS • refl-EXAFS......
Paolo Fornasini
Univ. Trento
![Page 75: Lucidi Duino 11 - ELETTRA](https://reader031.fdocuments.net/reader031/viewer/2022012215/61df9d9d47f5382d1c6bee08/html5/thumbnails/75.jpg)
XAFS: experimental
♠ Monochromators and mirrors
![Page 76: Lucidi Duino 11 - ELETTRA](https://reader031.fdocuments.net/reader031/viewer/2022012215/61df9d9d47f5382d1c6bee08/html5/thumbnails/76.jpg)
X-ray crystal monochromators
Si (111) 6.2708Si (220) 3.84Si (311) 3.28Si (331) 2.5Si (511) 2.08 Ge (111) 6.5328Ge (220) 4.0004
2d
Paolo Fornasini
Univ. Trento
- Forbidden ‘reflections’- Harmonics- Spurious reflections
dθ
Bragg law
€
2dhkl sinθ = nλ
Incidence angle ⇔ wavelength
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Crystal reflectivity Paolo Fornasini
Univ. Trento
0.4
1
0.8
0.6
0.2
-1 0 3 4 1 2 Δθ (arc sec)
Darwinwidth
Rocking curve (from dynamical theory of diffraction)
Higher order reflections have narrower rocking curves.
θB
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Energy resolution
Darwin width (Intrinsic resolution) Total angle ΔΘ
Beam divergence
= +
0
2
4
6
8
10 20 30E (keV)
Si (111)
ΔE (eV)
10 20 30
Si (220)
E (keV)
Darwin width
Core-levelwidth
€
ΔEE
=Δλλ
= ΔΘ cotgθB
Paolo Fornasini
Univ. Trento
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Two-crystal monochromators
“Channel-cut”
Mechanical simplicity Stability Harmonics Spurious reflections
Independent crystals
Detuning: harmonic reduction Possibility of focussing Mechanical complexity Instability
Paolo Fornasini
Univ. Trento
Horizontal output beam
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Crystals detuning Paolo Fornasini
Univ. Trento
(111) (333)
Silicon crystal (EB = 10 keV)
E-EB (eV)
3.5 arcsec
0 1 2
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X-ray mirrors
θ
€
θ c = 2δ ∝ λ ρ
harmonics rejection
grazing incidence θ ≈ 10-3 rad
n = 1−δ − iβ
δ ≈10−6 ÷ 10−5
for x - raysabsorption
Complex refractive index
Total external reflection : θ < θc
Beam collimation and
focalisation
Surface bending
Paolo Fornasini
Univ. Trento
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XAFS: experimental
♠ Detection schemes
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XAFS: direct transmission measurements
V ≈ 100 V/cm I ≈ 10-10 ÷ 10-8 A
Detectors: ionisation chambers
Paolo Fornasini
Univ. Trento
Sample: • Powders or thin films • Thickness ≈ 10 µm • No holes or inhomogeneities
Ι0 Ι1
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Direct transmission measurements Paolo Fornasini
Univ. Trento
Thick samples
Diluted samples
Surfaceinformation
Thin samples
Non-diluted samples
Homogeneous samples
Bulk information (not fom surface) from:
OK NO
High accuracy attainable
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Indirect detection methods Paolo Fornasini
Univ. Trento
• X-ray fluorescence
Detection of decay products
• Electrons
• Optical luminescence XEOL-PLY =
FLY = FLuorescence Yield
AEY = Auger Electron Yield
PEY = Partial Eletron Yield
TEY = Total Electron Yield
X-ray Ecxited Optical Luminescence Photo Luminescence Yield
Ι0
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€
I f zn( ) dz = I0 (ω) exp −µs ω( )znsinθi
η fµa ω( ) dz
sinθiexp −
µs ω f( )znsinθ f
Ω2π
XAFS: fluorescence detection (FLY)
I0 If
θi θf
Ω
zs zn
Absorption Fluorescence Absorption
Paolo Fornasini
Univ. Trento
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Fluorescence: total intensity
Sample of thickness zs θi = θf = 45°
Thin samples
€
1− exp A( ) ≈ 1−1− A = − A
I f ∝µa ω( )
Thick samples
€
1− exp A( ) ≈ 1
I f = I0 ω( )η fΩ4π
µa ω( )µs ω( ) + µs ω f( )
OK for diluted samples (< 1%)
Paolo Fornasini
Univ. Trento
€
I f = I0 ω( )η fΩ4π
µa ω( )µs ω( ) + µs ω f( )
1− exp A( ){ } A = − 2zs µs ω( ) + µs ω f( )[ ]
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Fluorescence signal
I0 If
Background signals
Elastic scattering Compton scattering Other fluorescences
Energy selective detectors Filters + Soller slits
Paolo Fornasini
Univ. Trento
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XAFS: electron detection (a) Paolo Fornasini
Univ. Trento
Auger electrons: • Fixed energy ⇒ atomic selectivity • Intensity ∝ µx
⇒ XAFS signal
Photo-electrons: • Energy varies with hν • Intensity∝ µx
⇒ XAFS signal
Ι0 hν
Electron mean free path: • adsorbates • thin layers
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Indirect processes and escape depth Paolo Fornasini
Univ. Trento
Photons penetration depth ~ 500 Å
Secondary electrons escape depth ~ 50 - 100 Å
Primary electrons mean free path ~ 5 - 10 Å
Adsorbate
Bulk
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XAFS: electron detection (b) Paolo Fornasini
Univ. Trento
AEY = Auger Electron Yield - narrow energy window - only direct Auger electrons - spurious structures from photoelectrons
PEY = Partial Eletron Yield - large energy window - Auger (direct + secondary) = XAFS signal - Photoel. (direct + secondary) = background
TEY = Total Electron Yield - all electrons collected
- Auger (direct + second.) = XAFS signal - Photoel. (direct + second.) = background
- XAFS from Auger and photoel.
XAFS of adsorbates
AEY - PEY - TEY
TEY
Bulk materials
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XAFS: experimental
♠ Alternative layouts
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Dispersive XAFS (a) Paolo Fornasini
Univ. Trento
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Dispersive XAFS (b)
S.R. incoming white beam
Curved crystal poly-chromator
€
2d sinθ = λ
θ1 θ2 θ3
Position-sensitive detector
E1
E2
E3
Paolo Fornasini
Univ. Trento
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Dispersive XAFS (c)
Critical in terms of temporal and spatial beam stability and sample presentation
Only trasmission mode X-ray beam not perfectly focussed through the sample No reference measurements during acquisition
No mechanical movements (no dead times) Simultaneous acquisition of all data points Acquisition time determined by acceptable statistics
Paolo Fornasini
Univ. Trento
OK for time-resolved measurements
NO accurate quantitative results
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EXAFS: data analysis, examples
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Analysis - Available software Paolo Fornasini
Univ. Trento
List of available software: XAFS Society web-site = http://xafs.org/Software
FEFF: ab initio MS calculations of EXAFS and XANES for clusters of atoms. The code yields scattering amplitudes and phases, as well as various other properties.
Athena: interactive graphical utility for processing EXAFS data. Artemis: interactive graphical utility for fitting EXAFS data using theoretical standards from FEFF and sophisticated data modelling.library.
IFEFFIT: interactive program for XAFS analysis.
GNXAS: EXAFS data analysis based on MS calculations and advanced fitting of raw experimental data. Main peculiarities: MS associated with 2, 3, and 4- atom configurations, multi-electron excitation, various model peaks for distribution functions.
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EXAFS data analysis
♠ Extraction of EXAFS signal
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Total absorption coefficient
source monochromator
x
sample detectors
Φ0 Φ
Paolo Fornasini
Univ. Trento
0
0.4
0.8
1.2
8800 9200 9600 10000E (eV)
CuCl (15 K)
edge Cu K
€
ln I0I
= lnΦ0
Φ+C' = µtotx+C'
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Edge absorption coefficient
-2
-1
0
1
11 11.5 12Photon energy (keV)
µtot
x
Ge, 10 K
Experimental signal
-2
-1
0
1
11 11.5 12
µtot
x
Photon energy (keV)
µnx
µx
Extrapolationof pre-edge behaviour
0
1
2
11 11.5 12Photon energy (keV)
µx
Edge absorption coefficient
Paolo Fornasini
Univ. Trento
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Photoelectron wavenumber Paolo Fornasini
Univ. Trento
Edgeenergy
€
k =2m2 hν −Es( )
Es E = hν, photon energy
Photoelectron wavenumber
?
hν k
Es = 1st maximum of 1st derivative Experimental convention
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Atomic absorption coefficient Paolo Fornasini
Univ. Trento
EXAFS function
Isolated atom
Embedded atom
€
µ0 ?
€
χ k( ) =µ −µ0
µ0
0
1
2
11 11.5 12Photon energy h ν (keV)
µxµ0x
Es
Ge
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Best-fitting polynomial spline Paolo Fornasini
Univ. Trento
1.4
1.6
1.8
0 500 1000 1500
Ge - 10 Kµx
E-Es (eV)
1.4
1.6
1.8
0 5 10 15 20
Ge - 10 Kµx
k (Å-1)
Polynomial spline - best fit
E space k space €
χ k( ) =µ −µ0
µ0
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Fit optimization Paolo Fornasini
Univ. Trento
1.4
1.6
1.8
0 5 10 15 20
µx
k (Å-1)
Fit A
0
4
8
12
16
0 1 2 3 4 5
Fit A
r (Å)
0
4
8
12
16Fit B
0 1 2 3 4 5r (Å)
1.4
1.6
1.8
0 5 10 15 20
µxFit B
k (Å-1)€
µ −µ0
Fourier transf.
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EXAFS signal Paolo Fornasini
Univ. Trento
€
χ k( ) =µ −µ0
µ1
-0.4
0
0.4
0 5 10 15 20
Ge - 10 K
k χ(k)
k (Å-1)
-0.04
0
0.04Ge -10 K
χ (k)
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EXAFS signals: examples
Amorphous Germanium
Crystalline Germanium
0 5 10 15k (Å-1)
0 5 10 15k (Å-1)
-0.5
0
0.5
0 5 10 15
k χ
(k)
(Å-1
)
k (Å-1)
T = 77 K T = 300 K T = 77 K
1 coord. shell Several coord. shells
Temperature effect
Paolo Fornasini
Univ. Trento
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Diffraction .vs. EXAFS - (b)
-0.4
0.0
0.4
0 4 8 12 16 20
k χ(
k)
c-Ge
0
5000
10000
15000
0 2 4 6 8
c-SiO2
k i(k
) [a.
u]
-2.0
0.0
2.0
0 5 10 15 20 25
a-SiO2
k i(k
)
G (1/A)0 5 10 15 20
-0.4
0.0
0.4 a-Ge
k χ(
k)
k (1/A)
Diffraction EXAFS G k
Paolo Fornasini
Univ. Trento
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Quantitative analysis of EXAFS Paolo Fornasini
Univ. Trento
0 5 10 15k (Å-1)
€
kχ(k)
0
0.5
-0.5
€
χ(k) = Ai (k)sinΦi (k)i∑
Sum over: • S.S. paths (coord. shells)
• M.S. paths
Input for each path: • backscattering amplitude • phaseshifts • inelastic terms Different analysis procedures
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EXAFS data analysis
♠ Fourier transform
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Data analysis - Fourier Transform k→r
-40
0
40
5 10 15 20
k3 χ(k) W(k)
k (Å-1)
0 2 4 6r (Å)
Imaginary part
Modulus
1st
2nd
3rd
Ge, 10 K
F(r) = χ k( ) kn W k( )kmin
kmax
∫ e2ikr dk
window weight
Peak's position and shape influenced by: - total phaseshifts - disorder - Fourier transform window
Paolo Fornasini
Univ. Trento
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1.5 2 2.5 3 3.5
σ = 0.05 Å
σ = 0.1 Å
1.5 2 2.5 3 3.5r(A)
Fourier Transform and distribution Paolo Fornasini
Univ. Trento
F.T.: k=2.5-16 K3, square w.
EXAFS simulation (Ge phases and amplit.)
-0.2
0
0.2
2 6 10 14
k χ(
k)k (Å-1)
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i Ni Ri (Å)1 8 2.482 6 2.863 12 4.054 24 4.755 8 4.966 6 5.73
0
0.2
0.4
0.6
0 1 2 3 4 5 6
Mod
ulus
of
F.T.
(a.
u.)
r (Å)
Fe (T = 300K)
24
812
668 • Peak shift
• Superposition of shells
1 2
3
26 - Iron: bcc structure Paolo
Fornasini Univ. Trento
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0
40
80
120
0 1 2 3 4 5 6
Mod
ulus
of F
.T.
(a.u
.)
r (Å)
Cu (T=4K )
12
6
12
24
i Ni Ri (Å)1 12 2.552 6 3.613 24 4.424 12 5.105 24 5.706 8 6.25
2
1
4
3
• Peak shift • Focussing effect
1 4
29 - Copper: fcc structure Paolo
Fornasini Univ. Trento
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0 5 10 15 20k (Å-1)
k2 χ(k
) (Å
-2)
0
4
-4
0
0
4
4
-4
4 K
250 K
500 K
0 1 2 3 4 5 60
40
80
120
Mod
. of F
.T. (
arb.
uni
ts)
r (Å)
4 K
500 K
250 K
Fourier transforms EXAFS signals
29-Cu: temperature effects Paolo
Fornasini Univ. Trento
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i Ni Ri (Å)1 4 a(√3)/4 2.452 12 a/√2 4.003 12 a(√11)/4 4.694 6 a 5.665 12 a(√19)/4 6.166 24 a(√6)/2 6.93
0
20
40
60
0 1 2 3 4 5 6 7
Mod
ulus
of F
.T. (
a.u.
)
r (Å)
Ge(10 K)
24
121212
64
32 - Germanium: diamond structure Paolo
Fornasini Univ. Trento
a = 5.66 Å
1
2
3
4
56
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Fourier transforms
EXAFS signals
32-Ge: crystalline and amorphous Paolo
Fornasini Univ. Trento
0
20
40
60
0 2 4 6
F(r)
(arb
.u.)
r (Å)
10 K300 K
c - Ge
0 2 4 6r (Å)
10 K300 K
a - Ge
0 5 10 15 20k (Å-1)
300 K
10 K a-Ge
-0.4
0.0
0.4
k χ(
k)
10 K c-Ge
-0.4
0.0
0.4
0 5 10 15 20
k χ (
k)
k (Å-1)
300 K
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EXAFS data analysis
♠ First shell analysis
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1st-shell Fourier back-transform Paolo Fornasini
Univ. Trento
0 1 2 3 4 5
Ge - 10 K
r (Å)
-0.02
0
0.02
2 6 10 14 18
k (Å-1)
k χ(k)
€
χ ' k( ) = 2 π( ) F r( )W ' r( )rmin
rmax∫ e−2ikr dr
- No peak superposition- No MultipleScattering - F.T. artifacts
First-shell contribution
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1st-shell distribution of distances Paolo Fornasini
Univ. Trento
r
Gaussian approximation
€
r€
σ
r
Asymmetric distribution
€
r€
σ
€
C3
€
σ 2 = r − r( )2
€
C3 = r − r( )3
Better for first shell
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EXAFS distance Paolo Fornasini
Univ. Trento
€
R
€
R
€
r
€
r = r b − r a
EXAFS, diffuse scattering
€
R = r b −
r a
Bragg diffraction, dilatometry
€
r > R
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Real and effective distributions Paolo Fornasini
Univ. Trento
r
Real distribution ρ(r)
Effective distributionP(r,λ)
€
ρs (r)e−2rs /λ (k )
rs2
€
r eff = r real −2σ 2
r1−
rλ
€
C1
€
r
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EXAFS for first shell
Coordination number
Debye-Waller
Average distance and asymmetry
€
k χ k( ) =S02 e−2C1 /λ
C12 f (k,π ) N exp −2k2σ 2[ ] sin 2kC1 − 43 k
3C3 + φ(k)
Inelasticterms
Totalphase-shift
Back-scatteringamplitude
• Theory (interaction potentials + scattering theory) • Experiment (reference samples)
Approx.: Single Scattering Plane waves
N
σ2
C1 C3
Paolo Fornasini
Univ. Trento
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Analysis - non-linear fitting method
Theoretical EXAFS
Experimental EXAFS
Theory
Paolo Fornasini
Univ. Trento
€
f k,π( ) , φ k( ), λ
€
e0, C1, C3
€
S02N, σ 2
€
Eb
€
Es
€
e0
Non-linear fit • k-space • r-space
• Correlation • Accuracy
• abs. values • rel. values
Check theory against standards
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EXAFS data analysis
♠ 1st shell phase and amplitude analysis
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Separate evaluation of phase and amplitude
Total phase (rad)
k
Amplitude
k
Paolo Fornasini
Univ. Trento
€
A k( ) =S02 e−2C1 /λ
C12 f (k,π ) N exp −2k2C2 +
24k 4C4 +…
?
€
Φ k( ) = 2kC1 −43k 3C3 + ...+φ k( )
?
From complex Fourier transform and back-transform
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“Ratio method” - phases Paolo Fornasini
Univ. Trento
€
Φs −Φm = 2k C1s −C1
m( )− 43 k3 C3
s −C3m( )
€
Φs −Φm
2k= C1
s −C1m( )− 43 k
2 C3s −C3
m( )
If suitable model compound available …
€
Φs −Φm
2k
€
k2
€
ΔC1
€
ΔC3 = 0
€
k2€
ΔC3 > 0
€
ΔC1
s = sample m = model
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“Ratio method” - amplitudes Paolo Fornasini
Univ. Trento
If suitable model compound available … s = sample m = model
€
ln As
Am≅ ln N
s
Nm − 2k2 σ s2 −σm
2( )
intercept Slope
€
ln As
Am
€
k2
€
ln Ns
Nm
€
Δσ 2
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“Ratio method” - results Paolo Fornasini
Univ. Trento
€
N s
Nm
€
ΔC1Δσ 2
ΔC3
Ratio of coordination numbers
Relative values :
→ Thermal expansion
Width
Asymmetry Absolute values ? Physical meaning ?
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“Ratio method” - OK when … Paolo
Fornasini Univ. Trento
€
χ k( ) = A k( ) sinΦ k( )• Only Single Scattering • Only one distance • Suitable reference model available
• First coordination shell, one distance • Same sample-model chemical environment
T or p-dep. Studies Amorphous .vs. crystalline samples
• 1st shell in bcc structure (2 distances) • Superposed outer shells • M.S. contributions
• 1st shell, different sample-model chemical environment
• Separated outer shells, weak M.S. Depending on sought accuracy
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EXAFS data analysis
♠ Outer shells analysis
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Analysis - Outer shells back-transform r→k
2 6 10 14 18
k (Å-1)2 6 10 14 18
k (Å-1)
χ' k( ) = 2 π( ) F r( ) W' r( )rmin
rmax
∫ e−2ikr dr
0 1 2 3 4 5 6 r (Å)
1st
2nd
3rd
Ge, 10 K
Paolo Fornasini
Univ. Trento
- Peak superposition- MultipleScattering - F.T. artifacts
Sometimes OK for D.W. factors
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Analysis - non-linear fitting of outer shells
Theoretical EXAFS
Experimental EXAFS
Theory
Paolo Fornasini
Univ. Trento
€
fpath k,π( ) , φpath k( ), λ
€
e0, C1, C3
€
S02N, σ 2
Non-linear fit • k-space • r-space
• Reduction of free param. • Relations between param. • Approximations
SS & MS path sorting
For every path !!
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r
k
Δk Δr
€
N ind =2 Δk Δrπ
+ 1
Maximum number of independent parameters
Analysis - Independent parameters Paolo Fornasini
Univ. Trento
Correlation ofparameters
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EXAFS data analysis
♠ Interpretation of results
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0
1
2
3
0 200 400 600
Cu - 1st shell
T(K)
Expa
nsio
n (1
0-2 Å
)
XRDEXAFS
Thermal expansion Paolo
Fornasini Univ. Trento
€
R
€
R
€
r
Bragg diffraction lattice expansion
EXAFS bond expansion
Complementarity EXAFS - XRD: Info on perpendicular vibrations
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Mean Square Relative Displacements
Δu|| = 0
€
r ≈ R0 +Δu⊥
2
2R0σ 2 ≈ Δu||
2 MSRD||
MSRD⊥
Mean values (harmonic approximation)
r R0 €
Δu⊥
€
Δu||
Paolo Fornasini
Univ. Trento
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Bond distances Paolo Fornasini
Univ. Trento
€
R
€
R
€
r
€
r = r b − r a
EXAFS, diffuse scattering
“True” bond length “True” bond expansion
€
R = r b −
r a
Bragg diffraction, dilatometry
“Apparent” bond length “Apparent” bond expansion
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EXAFS Debye-Waller factor
MSDMean Square Displacements
DCFDisplacement
Correlation Function
€
σ 2 ≈ MSRD = Δu||2 = ˆ R ⋅ u b −
u a( )[ ] 2
= ˆ R ⋅ u b( )2
+ ˆ R ⋅ u a( )2− 2 ˆ R ⋅ u b( ) ˆ R ⋅ u a( )
Paolo Fornasini
Univ. Trento
Thermal factors from Bragg diffraction:
€
U||b = ˆ R ⋅ u b( )2
= σ ||b( )2
€
U||a = ˆ R ⋅ u a( )2
= σ ||a( )2
€
σ ||b
€
σ ||a
r Δu
Δu||
R
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Debye-Waller factor – Debye model
0
1
2
0 100 200 300 400 500T (K)
10-2 Å2
Copper
1st shell
3rd shell4th shell2nd shell
MSD
Paolo Fornasini
Univ. Trento
Absolute values from fit to theoretical models
€
σ 2 =32ωD
3µω coth ω
2kBT0
ωD∫ 1−sin ωqDR( )ωqDR
dω
Debye correlated model (OK for metals)
€
σ 2 • increases with T • depends on the shell
θD = 315 K θM = 313 K
θ4 = 321 K θ3 = 322 K θ2 = 283 K θ1 = 328 K
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Correlation
0
1
2
0 100 200 300 400 500T (K)
10-2 Å2
Copper
1st shell
3rd shell4th shell2nd shell
MSD
Paolo Fornasini
Univ. Trento
Complementarity EXAFS - XRD: Info on vibrations correlation
Bragg diffraction absolute vibrations
EXAFS relative vibrations
(along the bond)
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Debye-Waller factor – Einstein model
0
0.01
0.02
0 200 400
(Å2)
T (K)
Germanium
1st
2nd
3rd
2 MSD
Non-Bravais crystals
€
Δu||2 =
2µωE
coth ωE
2kT
µ m1
m2
k
ω
θD = 354 K θM = 290 K
θ3 = 290 K θ2 = 299 K θ1 = 460 K
Debye Einstein
(THz)
ν3 = 3.95 ν2 = 4.21 ν1 = 7.55 €
ν =ω / 2π(eV/Å2)
k3 = 2.18 k2 = 2.48 k1 = 8.15 €
k = µω2
Paolo Fornasini
Univ. Trento
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Parallel and perpendicular MSRD
€
Δu⊥2
€
Δu||2
Paolo Fornasini
Univ. Trento
0
1
2
3
4
5
0 200 400 600
Cu 1st shell
MSR
D (
10-2
Å2 )
T (K)
||
⊥
0
1
2
3
4
5
0 200 400 600
Ge 1st shell
T (K)
||
⊥
€
Δu⊥2
Δu||22.7 6 Perpendicular-parallel
anisotropy
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First-shell distribution asymmetry
€
C3* T( ) ≈ − 2k3σ 0
4
k0z2 +10z+11− z( )2
Paolo Fornasini
Univ. Trento
0
1
2
3
4
5
0 200 400
C 3 (10-4
Å3 )
T (K)
Cu - 1st shell
r
€
r€
σ
€
C3K3 = -1.37 eV/A3
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The end
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Basic bibliography Paolo Fornasini
Univ. Trento
• G.S. Brown and S. Doniach: The principles of X-ray Absorption Spectroscopy, in Synchrotron Radiation research, ed. by E. Winick and S. Doniach, Plenum (New York, 1980) [A general introduction to X-Ray absorption]
• P.A. Lee, P.H. Citrin, P. Eisenberger, and B.M. Kincaid, Rev. Mod. Phys. 53, 769 (1981) [Review paper on EXAFS]
• T.M. Hayes and J. B. Boyce, Solid State Physics 27, 173 (1982) [Review paper on EXAFS]
• B.K. Teo: EXAFS, basic principles and data analysis, Springer (Berlin, 1986) [Introductory book on EXAFS]
• D.C. Koningsberger and R. Prins eds.: X-ray Absorption: principles and application techniques of EXAFS, SEXAFS and XANES, J. Wiley (New York, 1988) [Introductory book on XAFS]
• M. Benfatto, C.R. Natoli, and A. Filipponi, Phys. Rev. B 40, 926 (1989) [Paper on multiple scattering calculations]
• J. Stöhr: NEXAFS spectroscopy, Springer (Berlin, 1996) [Book on XANES] • J.J. Rehr and R.C. Albers, Rev. Mod. Phys. 72, 621 (2000) [Review paper on
EXAFS] • P. Fornasini, J. Phys.: Condens. Matter 13, 7859 (2001) [EXAFS and lattice
dynamics] • G. Bunker: Introduction to XAFS, Cambridge U.P. (2010) [Introductory book on XAFS]
XAFS Society home page: http://www.ixasportal.net/ixas/