hTransmissionMATTER

hScattering

Compton

h h '

Thomson

Photoelectric absorption

Pair productionh > 1M eV

X-rays

Interaction X-rays - Matter

h

Decay processes

Fluorescence Auger electrons

h f

Primary competing processes and some radiative and non-radiative decay processes

Thomson Observed data

Electron positron

pairs

Compton

Photoelectric absorption

Photonuclear

absorptionCro

ss s

ecti

on (

barn

s/at

om)

1

103

106

10 eV 1 KeV 1 GeV1 MeV

Cu Z=29

Energy

Li Z=3 Ge Z=32 Gd Z=64

Energy (KeV)100 102 104104100 102 104

100

102

104

100 102

(

Bar

ns/a

tom

)

100

102

104

100

102

104

X-ray attenuation: atomic cross section: h h

sample

The total photoelectric absorption coefficient consists of the sum of the coefficientsfor the single shells.

= K + L1 + L2 + L3 + M1 + . . .

1 10 1000.001

0.01

0.1

1

10

100

1000

10000

Z=57 Lanthanum

-K

-K-L1

-K-L1-L2

-K-L1-L2-L3

Mas

s A

bsor

ptio

n C

oeffi

cien

t [c

m2 /

g ]

Energy [keV]

When the photons reach an energy equal to an absorption edge, increases abruptlybecause the photons are suddenly able to remove some more electrons.Between two absorption edges decreases with the photon energy approximatelyfollowing Bragg-Pierce law

Z=57 Lanthanum

LLL

LLL

L3, L2, L1 edges

K edge

(cm

2 /g)

Photoelectric absorption coefficient

100

102

104

106

1 10 100

Cro

ss s

ecti

on (

cm2 g

)

Photon energy (keV)

K edge

L3, L2, L1 edges Ge

Compton

Thomson

K 1s p

L1 2s p

L2 2p1/2 s, d

L3 2p3/2 s, d

Z dependence atomic selectivity

X-ray energy range 1 - 30 KeV

Ex

af1

0

ln

hh

sample

0

0 e x af

Photon flux

X-ray absorption spectroscopy

x

ln lnI I C x Ct0 0b g b g

photon fluxes; x sample thickness, C depends on the detectors efficiency.0 ,,

Storage ring Double - crystalmonochromator

Sample

Detectors

I0 I

0

EXAFS analysis

8800 9200 9600 10000

-2.0

-1.0

0.0

1.0

Ln

( I

/ I0 )

E (eV)

x

n x

XAFSXAFS measurementsmeasurements

X-ray energy

XAFS spectrum

Fluorescence X-rays TEY

SAMPLE

Incident X-rays Transmitted X-rays

Visible lightXEOL

h

h

e-

Always transmission, if possibleMost accurate method, best overall S/Ncounting statistics of about 10-4 from beamlines with more than 108 photons/s)

Always transmission, if possibleMost accurate method, best overall S/Ncounting statistics of about 10-4 from beamlines with more than 108 photons/s)

Which method for which application?

The most important criterion:The best signal to noise ratio for the element of interest

hh fluor .

e-h lu min.

h

Ie-

Fluorescence for very diluted samplesA specific signal reduces the large background (but maximum tolerable detector count-rate can result in very long measuring times).

Fluorescence for very diluted samplesA specific signal reduces the large background (but maximum tolerable detector count-rate can result in very long measuring times).

Total electron yield (TEY)for surface sensitivity and surface XAFS (adsorbates on surfaces) TEY for thick samples that cannot be made uniform.

Total electron yield (TEY)for surface sensitivity and surface XAFS (adsorbates on surfaces) TEY for thick samples that cannot be made uniform.

XEOL X-ray excited optical luminescenceVIS/UV detection from luminescent samples

XEOL X-ray excited optical luminescenceVIS/UV detection from luminescent samples

Sample = Species (A + B)

Measurement sensitivity in transmission mode

xx BAeIeII 00

x

A = solute, B = Solvent

By changing the solute absorption coefficient

For 210

10 xII and

so thate

I

I

I 0

Statistically: II

IandII

A

A

A

e

I

02

Assuming that the solute X-ray cross section is almost equal to that of the solvent:

BA nnnn ABBAAAAA

Rtoalproportionis

A

A 1

Re

I

n

n

e

I A

A

A 00 22

R= Dilution Ratio

Evaluation of the absorption coefficient

af 2

0 02

A

n iff

n = atomic densityA0 = vector potential amplitude

Transition probability if

INITIAL STATE INTERACTION FINAL STATE

Ground state Core hole + excitation or ionization

i f

GOLDEN RULE (weak interaction, time dependent perturbation theory (1st order))

if i i f fH E2 2

bg E fbgis the density of the final states compatible with the energy conservation principle

i , atomic initial and final states f

Interaction Hamiltonian

H ie

mA r

e

mAi j j

j j

ch

22

2

if i

i k r

jj f f

e

mA e Ej

2

2 0

2

2

bgj = electrons; polarization unit vector; = radiation wavevector

k

Electric dipole approximation e i k ri k r

jj

1 1....

An equivalent expression often used is:

This approximation is valid if: k rj

21 rj

i.e. for the K edges for energies up to 25 - 30 K eV and for the L edges

Selection rules:

l

s s p

j p s d

m

RS||

T||

1

0 1

1 0 2

0

, ,

if i jj

f f

e

mA r E

2

22

0

2

2

d i

given E Ei f i f if: , , , ,

In practice one is interested to the inverse problem.

Given: experimentally determined, one wants to know from XAFS, the

final state

But also in this case one needs to know the initial and final states or, at least , to express in parametric form the interesting structural properties.

f

i is known because it is the fundamental state of the atom.

The calculation of f is complicated because the absorption process because:

• Many bodies interactions• The final state is influenced by the environment around the absorption atom

The direct and inverse problem

Direct problem:

Inverse problem

One electron approximation

TOT elastic anelasticaf af af

Elastic transitions:1 core electron excitedN-1 passive electrons (relaxed)

Anelastic transitions:1 core electron excitedOther electrons excited; shake up and shake off

High photoelectron energy sudden approximation N N 1 1

1 active electron N-1 electrons

TOT() if So2 = 1 = So

2 0.7

Local structuref

f

RST

Evaluation of the final state!

TOT i j f f iN

fNr Eaf bg 2 1 1 2

EXAFSXANES

x

X-ray photon energy (keV)

a-Ge

X-ray Absorption Fine Structure (XAFS)

100

102

104

106

1 10 100

Cro

ss s

ecti

on (

cm2 g

)

Photon energy (keV)

K edge

L3, L2, L1 edges Ge

Compton

Thomson

XAFS=

XANES + EXAFS

0.0

0.4

0.8

1.2

11 11.5 12

XANES = X-ray absorption Near Edge Structure

EXAFS = Extended X-ray Absorption Fine Structure

E

E

i >

E

f >

if if H i 2

2

2

0

b g

i >

E

af b gaf z22

2

f H i di f

a

b

Fermi Golden rule

if ifM af b g 0

i >

E

f >

f

af 0

if i ff H i

af b g 2

2

2

0

Arctangent curve

Inflectionpoint

XANES: pre edge structure

E

a

b

0

0

0

0

0

Arctangent

Theoretical edge

Lorentian

Ir metal (Experimental signal)

Experimental K- edge of metallic Iridium and its simulationobtained by the superposition of an arctang function and a Lorentian

2

Energy

1160 1160 1160 1160 1160

0

0.5

1

1.5

Sin

gle

part

icle

bin

ding

ene

rgy

(eV

)

1s

2s

2p

s p d

0

50

100EXAFS

XANES

EDGE

X-ray edges

VisibleUV

866 868 870 eV

Ne K1sp

p

p

2 4 6 8

1snpn

WL

Arctangentcurve

x

0.58 eV

Energy eV

Ar K

Energy eV

x

XANES

414 415

Double excitations

(c)

(c) Double excitations in the N2 spectrum Rydberg associated to the N 1s1g* transition.

(a)

400 401 402

N 1s1g*

(a) 8 vibrational levels observed in the absorpttion spectrum: N 1s1g*

Abs

orpt

ion

Inte

nsit

y (a

rb. U

nits

)406 407 408 409 410

Photon Energy (eV)

(b)N 1sRydberg series

1

3

2

4

3

56

78

9

10

1112 13

(b) N 1sRydberg series

K-shell photoabsorption of N2 moleculeC.T. Chen and F. Sette, Phys. Rev. A 40 (1989)

K-shell photoabsorption of gas-phase N2

Abs

orpt

ion

Inte

nsity

(ar

b.un

its)

N 1s 1g*

N 1s Rydberg series

Doubleexcitations

Shaperesonance

x10

400 405 410 415 420Photon energy (eV)

XANESChemical information: oxidation state

Oxidation Numbers (formal valences)I Cu2OII CuO

III KCuO2

Higher transitions energy are expected for higher valence states.

KCuO2

CuOCu2O

Cu

E (eV)

8970 89900

0.5

1

Y-Ba-Cu-O

(J.B. Boyce et al. Phys. Rev. B 1987)

E

X-rays

0

DOS

XANES: projected density of state

SbSISb L1

I L1

Abs

orba

nce

(a.u

.)

E - E0 (eV)

Tot

al D

OS

(a.

u.)

-5 0 5 1510 20

-5 0 5 1510 20E - E0 (eV)

Tot

al D

OS

(a.

u.)

Abs

orba

nce

(a.u

.) Sb L3

I L3

2s p

2p p, d

(G. Dalba et al., J.of Condensed Matter, 1983)

he-

A

B

fe

k r

i k r A

0 2

A

B B

B B

EXAFS: phenomenological interpretation

Autointerference phenomenon ofthe outgoing photoelectron with its parts that are

backscattered by the neighboring atoms

km

E E 2

2 0b g

j jj0

0 ksenkAk

kkk

Kr

x

14.2 15.014.6 E (KeV)

Atoms

Molecules

Positiveinterference

Negativeinterference

f f f 0

e-

Outgoing wave: fe

k r

i k r A

0 2

A

A BBA

EXAFS: phenomenological interpretation

0 50-50

1

2

3

4

Br2

XANES

Photon energy (eV)13400 13800

1

2

3

4

x

14200 14600

Br2

kr

E i r f E faf bg 2

f smoothly varying in the EXAFS region

Scattering phases of the photoelectron

BAf f 0

BAf f H k r e f

i e

k rei

i k ri

0 1 01 1

2af

BA

BAx

BA

R

f i fi e

k Re

i k Ri 0 2

1

f i fe

k Rf k

e

R x

i k R i k R x

0

1

2,a f a f

f i f

f k

k Re i k R

0 2

2 2

21

,a f b g

fe

k r

i k r A

0 2

C i r f f C i r f i r f i r f Re

0

2

0 02

kaf 0

0

km

E E 2

2 0b g

k

i r f

i r f

f

faf

FHGIKJ2 2

0 0

Re

Re

kk R

f k k R kaf a f afb g 1

22

, sin

1) Inelastic scattering effect: S02 Electron mean free path

2) Thermal disorder: e k 2 2

kS

k Rf k e e k R kk Raf a f afb g 0

2

222 2

2, sin

Standard EXAFS formula

XAFS formula

Approximations

EXAFS formula

For several coordination shells:

kS

kN

f k

Re e k R ks

s

ss

k Rs s

s s saf a f afb g 02

222 2

2,

sin

Interatomic distanceCoordinationnumber

Debye Wallerfactor

From ab-initio calculations or from reference compounds

0

1 st

2 nd

3 rd

Coordination shells

R

Multiple scattering

XANES: multiple scattering processes

2

kR

EXAFS: single scattering processes

2

kR

E E eV 0 10 40

E E eV 0 40

2n

ln0 k1kk

R

n

np

0lnpn

ln

ln 2p,kkRsinp,kA

B

A

Single scattering

Double scattering

Double scattering

Triple scattering

RAB

A

B

C

A

B

C

A

B

C

C

0 50 100 150

8

Experiment

Si K-edge in c-Si

765

4321

A

bs

orp

tio

n

Energy E-EF (eV)

Comparison between the experimental Si K-edge XAFS for c-Si and calculations carried out with the FEFF code. The total multiple scattering in the first 8 shells around the absorption atom have been considered. 0 is the atomic absorption coefficient.

2n

ln0 k1kk

Multiple scattering