Transmission MATTER Scattering Compton Thomson Photoelectric absorption Pair production > 1M eV...
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Transcript of Transmission MATTER Scattering Compton Thomson Photoelectric absorption Pair production > 1M eV...
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