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)