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Interaction of X-raysInteraction of X-rayswith matterwith matter
Paolo FornasiniDepartment of PhysicsUniversity of Trento, Italy
OverviewOverview
Attenuation of x-rays
Classical Thomson scattering
Basic interference phenomenon
Scattering: deviations from classical behaviour
Photoelectric absorption
Attenuation of X-raysPaolo
FornasiniUniv. Trento
source monochromator
x
sample
detectors
Φ0 Φ
Φ = Φ0 exp −µ ω( ) x[ ]
µ ω( ) =1xlnΦ0
Φ
Exponential attenuation
Attenuation coefficient
Interaction of x-rays with matter PaoloFornasini
Univ. Trento
Incoming beam(monochromatic)
€
hν
€
hν '
€
hν
€
hνF
€
E
€
EA
Elastic scattering
Inelastic scattering
Photo-electrons
Augerelectrons
FluorescencePhoto-electric
absorption
Scattering
Beam attenuation
Structural techniques
Scattering
PaoloFornasini
Univ. Trento
X-rays
Neutrons
Spectroscopy
Abso
rptio
n
Energy
XAFS
Long-range order(crystals)
Short-range order(amorphous solids)
Scattering from a free chargeclassical treatment (Thomson scattering)
Charge acceleration
€
r E in t( ) =
r E 0 cos(ωt)
€
r a t( ) =qm
r E 0 cos(ωt)
Incoming electric field Charge acceleration
PaoloFornasini
Univ. Trento
Ein a
Magnetic field effects are negligible
Electrons .vs. protons
€
r a t( ) =qm
r E 0 cos(ωt)
PaoloFornasini
Univ. Trento
Ein -e
a +e
Ein a
€
q = −e
€
q = +e
€
Mm≈ 1836
€
em≈ 1.7×1011 C/kg
€
eM
≈ 0.96×108 C/kg
Electron Proton
π phase-shift
Dipole emission of radiation
€
r E out
r r ,t( ) =e r a ⊥ t '( )4πε0 r c2
Accelerating charge ⇒ electromagnetic field
PaoloFornasini
Univ. Trento
Ein -e
a
Eoutr
a⊥
a||
a
Ein -eEout
r
a⊥a||
•charge velocity:•charge distribution:•observer distance:
v << cd << λr >> λ
Dipole approximation:
Electron:
€
q = −e
€
r a t( ) = −em
r E 0 cos(ωt)
Outgoing electric field PaoloFornasini
Univ. Trento
€
Eoutr r ,t( ) = −
re
rE0 cos ωt '( )
π polarisation
€
Eoutr r ,t( ) = −
re
rE0 cos ωt '( ) cos 2θ( )
EoutEin
e-
€
2θ
σ polarisation
€
re =e2
4πε0c2m
Thomson scattering length
€
2θEin
Eout
e-
Classical electron radius
or
Thomson scattering length PaoloFornasini
Univ. Trento
= Classical electron radius
€
U ≈14πε0
e2
re= mc2
€
re =e2
4πε0c2m
= 2.8x10-15m = 2.8x10-5Å
Energy of a uniformly charged:
€
U =12
14πε0
q2
re
€
U =35
14πε0
q2
re
spherical shell
spherical volume
Electron
Relativisticrest energy
Radiated power
€
I0 =12ε0E0
2c W/m2[ ]
Incoming power flux
€
P θ( )dθ = re2 1+ cos2 2θ( )
2
I0 dθ
Emitted power
Polarisation factor
PaoloFornasini
Univ. Trento
Un-polarized beam
Independent from radiation wavelength
e-
€
2θ
Polarisation factor PaoloFornasini
Univ. Trento
π polarisation
EoutEin
e-
€
2θ
σ polarisation
Ein
Eout
e-
€
12
+cos2 2θ( )
2
Un-polarised beam
Laboratory x-ray sources
Total electron cross-section PaoloFornasini
Univ. Trento
Electron cross-section
€
σ e = 6.66x10-29m2
e-
Independent from radiation wavelength
Un-polarized beam
€
I0 =12ε0E0
2c W/m2[ ]
Incoming power flux
€
P =83π re
2 I0
Total radiated power
Beyond classical treatment PaoloFornasini
Univ. Trento
Thomson scattering:
Probabilistic distribution of e- charge
Free electron ⇒ Inelastic scattering (Compton)
Electrons are bound in atoms
Interference
€
re << λ
Free electron
Elastic scattering
Interference
Scattering from many electrons PaoloFornasini
Univ. Trento
In
Out
Depending on radiation wavelength
Waves from different electrons Interference
Scattering vector PaoloFornasini
Univ. Trento
€
r k in =
2πλ
ˆ s inr k out =
2πλ
ˆ s out
€
r k in =
r k out =
2πλ
Elasting scattering
€
r k in
€
r k out
€
r K
€
θ
€
r K =
r k out −
r k in
r K = 4π sinθ
λ
Scattering vector
€
r k in
€
r k out
€
2θ
Wave-vectors
Incoming wave PaoloFornasini
Univ. Trento
€
Ein = E0 cos ω t −r k in ⋅
r r ( )
= Re E0ei ω t−
r k in ⋅
r r ( )
€
A0
π polarisation
€
r r
€
Ein
Complex notation
Input amplitude
Outgoing wave
€
Eout = Re E0ei ω t−
r k in ⋅
r r ( ) −re( ) e−ikout R'
R'
€
A
PaoloFornasini
Univ. Trento
π polarisation
€
Ein
€
r R '
€
r R
€
r r
€
P
€
A0
Scattered amplitude
Geometrical tricks
€
Eout = Re E0ei ω t−
r k in ⋅
r r ( ) −re( ) e−ikout R'
R'
PaoloFornasini
Univ. Trento
π polarisation
€
Ein
€
r R '
€
r R
€
r r
€
P
€
e−ikoutR'
R'≈
e−ikoutR
Reikoutr cos
r k out ⋅
r r ( ) =e−ikoutR
Rei
r k out ⋅
r r
Scattered amplitude
€
Eout = Re E0ei ω t−
r k in ⋅
r r ( ) −re( ) e−ikoutR'
R'
PaoloFornasini
Univ. Trento
π polarisation
€
Ein
€
r R '
€
r R
€
r r
€
P
Scattered amplitude
€
A(r K ) = −E0 re
e−ikoutR
Reiωt ei
r k out −
r k in( )⋅r r
€
eir K ⋅r r
Phase factor
Amplitude and intensity
€
A(r K ) = −E0 re
e−ikoutR
Reiωt ei
r K ⋅r r
€
I = A(r K )
2=
PaoloFornasini
Univ. Trento
€
Eout = Re Ar K ( ){ }
€
E02 re
2 1R2
1+ cos2 2θ( )2
(π polarisation)
€
E02 re
2 1R2
Cannot be measured !
We measure:
(π polarisation)
un-polarized beam
Basic interference effect (a) PaoloFornasini
Univ. Trento
€
r R €
P
1 electron (π polarisation)
€
A(r K ) = A1(
r K )+ A2(
r K )
= Ael eir K ⋅r r 1 + ei
r K ⋅r r 2[ ]
2 electrons (π polarisation)
€
A(r K ) = −E0 re
e−ikoutR
Reiωt ei
r K ⋅r r
= Ael eir K ⋅r r
Basic interference effect (b) PaoloFornasini
Univ. Trento
€
A(r K ) = Ael ei
r K ⋅r r i
i∑
n electrons (π polarisation)
€
A(r K ) = Ael ρe
r r ( ) eir K ⋅r r ∫ dV
Continuous charge distribution(ρ = number density)
€
r R
€
P
€
r R
€
P
Fourier transform PaoloFornasini
Univ. Trento
The scattered amplitude is the Fourier transform of the electron density
€
A(r K ) = Ael ρe
r r ( ) eir K ⋅r r ∫ dV
(number density)
Amplitude and intensity (b) PaoloFornasini
Univ. Trento
€
A(r K ) = Ael ρe
r r ( ) eir K ⋅r r ∫ dV
€
Ir K ( ) = A(
r K )
2= Ael
2 ρer r ( ) ei
r K ⋅r r ∫ dV
2
Polarisation factorfor unpolarised beam
Amplitude
Intensity
€
Ael2
= re2 E0
2 1R2
1+ cos2 2θ( )2
Is measured, butphase informationis lost !
Cannot be measured !
Structural info
Deviations from classical treatmenta) Electronic distribution
Electronic orbitals PaoloFornasini
Univ. Trento
Interference effects
€
r k in
€
r k out
€
r K
€
θ
€
r k in
€
r k out
€
2θ
€
r K = 4π sinθ
λ
€
A(r K ) = Ael ρe
r r ( ) eir K ⋅r r ∫ dV
PaoloFornasini
Univ. Trento
Electroncloud
Interference depends on wavelength
0.0
0.2
0.4
0.6
0.8
1.0
0 20 40 60 80
λ = 0.715 Å (Mo Kα)Z=1, hydrogen
θ (deg)
fe
0.0
0.2
0.4
0.6
0.8
1.0
0 20 40 60 80
fe
θ (deg)
λ = 1.542 Å (Cu Kα)Z=1, hydrogen
PaoloFornasini
Univ. Trento
€
2θ
€
2θ
€
2θ
Electron scattering factor PaoloFornasini
Univ. Trento
€
r K
€
r k in
€
r k out
Electron cloud
€
A(r K ) = Ael ρe
r r ( ) eir K ⋅r r ∫ dV
€
Ae.u.(r K ) =
A(r K )
Ael= fe
r K ( )
€
Ir K ( ) = A(
r K )
2= Ael
2 fe
r K ( )
2
€
Ie.u.r K ( ) =
A(r K )
2
Ael2 = fe
r K ( )
2
Electronic units
€
fe
r K ( )
Hydrogen scattering factor
€
fe
r K ( ) = ρe
r r ( ) eir K ⋅r r dV∫
= 4π r2ρe r( )0
∞∫ sinKr
Krdr
€
r K
€
r k in
€
r k out
Electron cloud
for spherical symmetry
PaoloFornasini
Univ. Trento
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 5 10K = 4π sinθ/λ (Å-1)
fe (e.u.)
Z=1, hydrogen
point-like electron
Atomic scattering factor (a)
€
f0r K ( ) = fe,n
r K ( )
n∑
= ρe,nr r ( )ei
r K ⋅r r dV∫
n∑
= ρr r ( ) ei
r K ⋅r r dV∫
= 4π r2ρe r( )0
∞∫ sinKr
Krdr
PaoloFornasini
Univ. Trento
for spherical symmetry
€
Ae.u.r K ( ) = f0
r K ( )
Ie.u.r K ( ) = f0
r K ( )
2
sum over electrons
totalelectronicdensity
€
r K
€
r k in
€
r k out
Atomic scattering factor (b)
€
f0r K ( ) = ρ
r r ( ) eir K ⋅r r dV∫
PaoloFornasini
Univ. Trento
€
r K
€
r k in
€
r k out
0
20
40
60
80
0 5 10 15 20
f0 (e.u.)
G = 4π sinθ / λ (Å-1)
79 - Au
8 - O32 - Ge
Deviations from classical treatmentb) Compton effect
Compton experiment (1922) PaoloFornasini
Univ. Trento
Unmodified (elastic) Modified (inelastic)
Compton effect PaoloFornasini
Univ. Trento
€
λ'= λ +h
mc1− cosθ( )
= λ +λc 1− cosθ( )
€
E' keV[ ] =E keV[ ]
1+ 0.001957 E 1− cosθ( )
0.002426 Å(Compton wavelength)
Modified scattering - 1 electron PaoloFornasini
Univ. Trento
€
Ie,unmod + Ie,mod =1
fe2 + Ie,mod =1
€
Ie,mod =1− fe2
Hydrogen
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 5 10
Inte
nsity
(e.u
.)
K = 4π sinθ/λ (Å-1)
Total intensity
Thomson
Compton
Modified scattering - atoms PaoloFornasini
Univ. Trento
€
Imod = Ie,mod( )∑ n
= (1− fe,n2 )∑
= Z − fe,n2
n=1
Z
∑
uncoherent scattering(sum of intensities)
€
≤ Z
€
Iunmod = fe,nn=1
Z
∑2 coherent scattering
(sum of amplitudes)
€
≤ Z 2
Thomson
Compton
Thomson .vs. Compton
0
50
100
150
200
0 0.2 0.4 0.6 0.8 1
Inte
nsity
(e.u
.)
sinθ/λ
Thomson
Compton
14 - Silicon
PaoloFornasini
Univ. Trento
Scattering of X-rays from an electron
Free electron
Electron in atom
Thomson scattering
Elastic(“coherent”, “unmodified”)
Classical treatment
Tightly bound
€
ν '= νCompton scattering
Inelastic(“incoherent”, “modified”)
Quantum treatment
Loosely bound
€
ν '≠ν
PaoloFornasini
Univ. Trento
Deviations from classical treatmentc) Electron bonding
Effect of electron bonding (a)Paolo
FornasiniUniv. Trento
€
d2r r dt2
= −em
r E 0e
iωt
Free electron
€
d2r r dt2
+ γdr r dt
+ ω02r r = −
em
r E 0e
iωt
Bound electronComplex notation
Damped oscillator model
Dampingforce
Elasticforce
E fieldE field
Model of bound electron
€
R
€
x
PaoloFornasini
Univ. Trento
Spherical distribution of negative charge centered on the positive nucleus
Displacement x
€
F =1
4πε0Qqxx2
≈1
4πε0Q2
x2x3
R3
€
F ∝ x
Restoring force
Effect of electron bonding (b) PaoloFornasini
Univ. Trento
€
d2r r dt 2
= −em
r E 0e
iωt
Free electron
€
d2r r dt 2
+γdr r dt
+ω02r r = − e
mr E 0e
iωt
Bound electron:
€
r r 0 =er E 0m
1ω2
€
r r 0 =er E 0m
1ω2 −ω0
2 − iγω
Oscillating solution Complex notation
€
r r (t) =r r 0 eiωt
Amplitudeof
oscillation
Effect of electron bonding (c)
Free electron Bound electron:
€
A(r K ) = Ael fe(
r K )
€
Ar K ( ) = Ael fe(
r K ) ω2
ω2 −ω02 − iγω
€
r r 0 =er E 0m
1ω2
€
r r 0 =er E 0m
1ω2 −ω0
2 − iγω
PaoloFornasini
Univ. Trento
Amplitudeof
oscillation
Amplitudeof
scattering
spatialdistribution
Complex notation
Anomalous scattering factor
Scattering factor
“Anomalous” terms (1 electron)
€
Ar K ( ) = Efree
out fe(r K ) ω2
ω2 −ω02 − iγω
= Efreeout fe(
r K )
ω2 ω2 −ω02( )
ω2 −ω02( )2 +γ 2ω2
+ i fe(r K ) γω3
ω2 −ω02( )2 +γ 2ω2
= Efreeout fe(
r K ) + fe(
r K )
ω02 ω2 −ω0
2( )−γω2
ω2 −ω02( )2 +γ 2ω2
+ i fe(r K ) γω3
ω2 −ω02( )2 +γ 2ω2
PaoloFornasini
Univ. Trento
“Anomalous” terms
€
Ar K ( ) = Ael fe(
r K ) + f 'e + ife"[ ]
“Anomalous” scattering factor (atoms) PaoloFornasini
Univ. Trento
anomalousterms
(~ indep. from K)
(for forward scattering)
Real part
Imaginarypart
€
fr K ( ) = f0
r K ( ) + f1 + i f2
Absorption
-10
0
10
2030
40
f0 + f
1
f0
32 - Ge
0
5
10
15
0 10 20 30E (keV)
f2
32 - Ge
Photo-electric absorptiona) Phenomenology
Attenuation of X-raysPaolo
FornasiniUniv. 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 PaoloFornasini
Univ. Trento
10-15
10-10
10-5
10-2 100 102 104 106
Cros
s se
ctio
n (
Å2 )
E (keV)
Photoelectric absorption
Coherent sc.
Incoherent sc.
Pair prod.nucl. field
elec. field
32 - Ge
€
µ ω( ) =NaρA
µa ω( )
Excitation and ionization
0 Un-occupiedbound levels
Un-occupiedfree levels
Energy
Excitation
Ionization
X-rays
Photoelectric absorption (a)
-10000
-8000
-6000
-4000
-2000
0
Ep (eV)
- 13,6
- 3,4
0Ep (eV)
Hydrogen Copper
PaoloFornasini
Univ. Trento
source monochromator
x
sample
detectors
Φ0 Φ
•Exponential attenuation
•Attenuation coefficient
Φ = Φ0 exp −µ ω( ) x[ ]
µ ω( ) =1xlnΦ0
Φ
101
102
103
104
105
106
1 10 100
µ (ω)
Photon energy hω (keV)
32 - Ge
Edges
µ ω( )∝
Z 4
hω( )3
+
X-ray absorption spectroscopy PaoloFornasini
Univ. Trento
X-ray 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
1
10
100
0 20 40 60 80 100
Bind
ing
ener
gy (k
eV)
Z
K
L3
M5
Z > 9
Z > 29
L1 2s
L2 2p1/2
L3 2p3/2
K 1s
M1 3s
M2 3p1/2
M3 3p3/2
M4 3d3/2
M5 3d5/2. . . . . . . . .
PaoloFornasini
Univ. Trento
Fine Structure: Atoms
-6 -4 -2 0 2
µ (a
.u.)
E - Eb (eV)
Ar Eb = 3205.9 eVhν < binding energy
hν > binding energy
Core electron
continuum
Core electron
unoccupied levels
⇒ Edge Fine Structure
⇒ Smooth µ 14.2 14.5 14.8 15.1Photon energy (keV)
µ (a
.u.)
Kr
PaoloFornasini
Univ. Trento
Fine Structure: Molecules and Condensed systems
11 11.5 12Photon energy (keV)
Ge
µ (a
.u.)
hν ≈ binding energy
Core electron
unoccupied levels
⇒ Edge Fine Structure
hν > binding energy
Core electroncontinuum Outgoing wave-function
Incoming wave-function
Back-scatteringfrom
neighbouring atoms
INTERFERENCEExtended
FineStructure
PaoloFornasini
Univ. Trento
XAFS: X-ray Absorption Fine Structure
X-ray Absorption Near Edge StructureNear Edge X-ray Absorption Fine Structure
Electronic transitionsPhoto-electron multiple scattering
PaoloFornasini
Univ. Trento
11 11.5 12Photon energy (keV)
XANESNEXAFS
EXAFS
Edge
Extended X-ray AbsorptionFine Structure
Mainly photo-electron single scattering
Photo-electric absorptionb) Theory
Transition probability Wif
Wif = ?
n = atomic densityA0 = vector potential amplit.
Initial atomic state
Stationary ground state
Ψi
Final atomic state
Stationary excited state
Ψf
Interaction
Ψ t( )
€
µ ω( ) =2h
ε0ωA02 n Wif
f∑
PaoloFornasini
Univ. Trento
Approximations
Time-dependent perturbation theory1st order
Initial state Final state
Weak interaction
Electric dipole approximation
One electron
Absorption coefficient
€
µel ω( ) ∝ ψi ˆ η ⋅r r ψ f
2δ Ei + hω −E f( ) Ψi
N−1 Ψ fN−1 2
1 active electron passive electrons
Energy conservation Superposition integral
€
ψi ˆ η ⋅r r ψ f
2
Initialstate
Finalstate
X-raypolarization
Electronposition
€
ψi* r r ( )∫ ˆ η ⋅
r r ψ fr r ( ) dr r
2
Electric dipole approximation
€
µel ω( ) ∝ ΨiN−1ψ i ˆ η ⋅
r r ψ fΨf
N−12
€
eir k ⋅
r r = 1+ i
r k ⋅
r r −K ≈ 1
€
HI ∝ eir k ⋅
r r ˆ η ⋅
r p ≈ ˆ η ⋅
r p = ω 2 ˆ η ⋅
r r
€
Δl = ± 1 Δs = 0Δj = ±1, 0 Δm = 0
Dipole selection rules:
PaoloFornasini
Univ. Trento
De-excitation mechanisms
Radiative: fluorescence
Non-radiative: Auger
Xe-
X
Xe- 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
PaoloFornasini
Univ. Trento
Core-hole lifetime
0
5
10
15
20
20 40 60 80
Γh (eV)
Z
K edges
L3 edges
Lifetimeof the excited stateτh~10-16 -10-15 s
Energy widthof the excited state
Γh
τh ≈1/Γh
Contribution tophoto-electron life-time
τh
Energy resolutionof XAFS spectra
Γh
PaoloFornasini
Univ. Trento
Summary
Attenuation: scattering and absorption Thomson scattering from a free electron (classical) Limits of the classical treatment Basic interference phenomenon Electronic and atomic structure factors Comtpon effect Anomalous (or ‘resonant’) scattering Absorption coefficient and absorption edges Fine structure at absorption edges Introduction to theory of photoelectric absorption
Dolomite mountains at sunset (near Trento, Italy)