Paolo Fornasini Dipartimento di Fisica Università di Trento

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

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

Structural techniques

Scattering

Paolo Fornasini

Univ. Trento

X-rays

Neutrons

Spectroscopy

Abso

rptio

n

Energy

XAFS

Long-range order Short-range order

•  X-rays absorption – phenomenology •  X-rays absorption - theory •  EXAFS: theoretical background •  EXAFS experiments •  EXAFS: data analysis, examples

X-rays absorption - phenomenology

X-rays, 1897

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

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( )

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

+

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

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

Physical Review 16, 202 (1920)

Absorption edge fine structure

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)

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

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.

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

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

XAFS: edge and pre-edge

5 4.98 4.96 5.02 5.04 Photon energy (keV)

Edge Pre-edge

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

X-rays absorption - theory

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

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

∑

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

Perturbation approach

Paolo Fornasini

Univ. Trento

System = atom (quantum treatment)

Weak perturbation = electromagnetic field (classical treatment)

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

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

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

''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

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

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

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

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

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 −…

€

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

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

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

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

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

EXAFS: theoretical background

€

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

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

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

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

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

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

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

€

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

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

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)

Many-atomic systems Paolo Fornasini

Univ. Trento

r

Radial Distribution Functions

r

Coordination shells

Crystals Amorphous systems

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( )[ ]

Single and multiple scattering

SS = Single scattering MS = Multiple scattering

Scattering paths

Paolo Fornasini

Univ. Trento

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

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

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

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( )

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

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

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

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

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

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

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( ) ?

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

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

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

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

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

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 ?

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

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

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

EXAFS experiments

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

XAFS: experimental

♠ Monochromators and mirrors

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

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

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

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

Crystals detuning Paolo Fornasini

Univ. Trento

(111) (333)

Silicon crystal (EB = 10 keV)

E-EB (eV)

3.5 arcsec

0 1 2

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

XAFS: experimental

♠ Detection schemes

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

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

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

€

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

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( )[ ]

Fluorescence signal

I0 If

Background signals

Elastic scattering Compton scattering Other fluorescences

Energy selective detectors Filters + Soller slits

Paolo Fornasini

Univ. Trento

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

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

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

XAFS: experimental

♠ Alternative layouts

Dispersive XAFS (a) Paolo Fornasini

Univ. Trento

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

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

EXAFS: data analysis, examples

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.

EXAFS data analysis

♠ Extraction of EXAFS signal

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'

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

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

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

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

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.

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)

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

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

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

EXAFS data analysis

♠ Fourier transform

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

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)

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

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

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

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

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

EXAFS data analysis

♠ First shell analysis

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

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

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

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

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

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

EXAFS data analysis

♠ 1st shell phase and amplitude analysis

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

“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

“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

“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 ?

“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

EXAFS data analysis

♠ Outer shells analysis

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

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 !!

r

k

Δk Δr

€

N ind =2 Δk Δrπ

+ 1

Maximum number of independent parameters

Analysis - Independent parameters Paolo Fornasini

Univ. Trento

Correlation ofparameters

EXAFS data analysis

♠ Interpretation of results

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

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

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

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

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

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)

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

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

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

The end

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/