Transcript analisi exafs

Interaction X-rays - Matter
Pair production
h > 1M eV
Photoelectric absorption
h
h
MATTER
Transmission
X-rays
Scattering
h'  h
Decay processes
Compton
h
Thomson
h f
Fluorescence
Auger electrons
Primary competing processes and some radiative and non-radiative decay processes
Cross section (barns/atom)
X-ray attenuation: atomic cross section:
Photoelectric absorption
103
Thomson
Observed data
Electron positron
pairs
1
Compton
s
(Barns/atom)
10 eV 1 KeV
104
sample
Cu Z=29
106
Li Z=3
104
Photonuclear
absorption
1 MeV
Energy
Ge Z=32
1 GeV
104
102
102
102
100
100
100
100
102 104
100 102 104
Energy (KeV)
h
h
Gd Z=64
100 102 104
Photoelectric absorption coefficient
The total photoelectric absorption coefficient  consists of the sum of the coefficients
for the single shells.
 = K + L1 + L2 + L3 + M1 + . . .
Z=57 Lanthanum

-- K
-- K-K - L1
 - KKL1- L1- L2
 - - -
 - KKL1- L2L1- L2 - L3
10000
L3, L2, L1
edges
1000
2
2
(cm
/g)
M as s A bs orption Coeffic ient
/ g
]m
[c

100
 - K - L1 - L2 - L3
K edge
10
1
0.1
Z=57 Lanthanum
0.01
0.001
1
10
100
E nergy [k eV ]
When the photons reach an energy equal to an absorption edge,  increases abruptly
because the photons are suddenly able to remove some more electrons.
Between two absorption edges  decreases with the photon energy approximately
following Bragg-Pierce law
Cross section (cm2g)
X-ray absorption spectroscopy
10
6
L3, L2, L1 edges
Ge
K edge
10
4
Thomson
102
Compton
10
0
1s  p
L1
2s  p
L2
2p1/2  s, d
L3
2p3/2  s, d
Z dependence 
0
1
K
10
Photon energy (keV)
100
atomic selectivity
x

Photon flux
h
h
 x
  0 e -  af
sample
X-ray energy range 1 - 30 KeV
af
E 
1 
ln
x 0
EXAFS analysis
Detectors
0 
Sample
Double - crystal
monochromator
Storage ring
I0
I
bg b g
ln I 0 I  ln 0   C   t x  C
 0 , , photon fluxes; x sample thickness, C depends on the detectors efficiency.
Ln ( I / I0 )
1.0
0.0
x
-1.0
n x
-2.0
8800
9200
9600
E (eV)
10000
XAFS measurements
SAMPLE
Incident X-rays
Transmitted X-rays
h
Visible light
XEOL
h
e-
Fluorescence X-rays
TEY
XAFS spectrum

X-ray energy
Which method for which application?
hfluor .
h
Ie-
Always transmission, if possible
Most accurate method, best overall S/N
counting statistics of about 10-4 from
beamlines with more than 108 photons/s)
h
e-
The most important criterion:
The best signal to noise ratio for the
element of interest
h lu min.
Fluorescence for very diluted samples
A 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.
XEOL X-ray excited optical luminescence
VIS/UV detection from luminescent samples
Measurement sensitivity in transmission mode
Sample = Species (A + B)
A = solute, B = Solvent
I  I 0 e- x  I 0 e- A B x
By changing the solute absorption coefficient
For I 
1
I0
10
Statistically:
so that
x  2
and
I  I and
I
I
 0
I
e
x
I
 I
I
2 I0  A
A


 A
e

Assuming that the solute X-ray cross section is almost equal to that of the solvent:
sA sB
 A 2 I 0 nA 2 I 0


R
 A
e n
e
 A  s A nA    s A nA  s B nB  s A n
 A
A
is proportional to
1
R
R= Dilution Ratio
Evaluation of the absorption coefficient
2
n = atomic density
 
n  if
2
A0 = vector potential amplitude
0  A0 f
af
Transition probability if
INITIAL STATE
INTERACTION
i
FINAL STATE
f
Core hole +
excitation or ionization
Ground state
GOLDEN RULE (weak interaction, time dependent perturbation theory (1st order))
if 
2
 i Hi  f

2
of the final states compatible
bg bE giswiththe thedensity
energy conservation principle
 Ef
f
 i ,  f atomic initial and final states
Interaction Hamiltonian
  
e
e2
H i  i   A rj  j A 2

m j
2m j
ch
if 
e
2
A
i
0
2
m
2
e

i k  rj
2

 j f

bg
 Ef
j

j = electrons;   polarization unit vector; k = radiation wavevector
An equivalent expression often used is:
if 
e
2
 2 A0
2
2
m
Electric dipole approximation
i

2
   rj  f
di
 Ef
j
e

i k rj

 1  i k rj - ....  1
 2
This approximation is valid if: k  rj  1
  rj
i.e. for the K edges for energies up to 25 - 30 K eV and for the L edges
Selection rules:
l  1
R
||s  0
S
||j  1, 0
Tm  0
1s  p
2 p  s, d
The direct and inverse problem
Direct problem: given:  i ,  f , E i , E f , 
if


Inverse problem
In practice one is interested to the inverse problem.
Given: , experimentally determined, one wants to know from XAFS, the
final state  f
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.
 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
One electron approximation
af
af
af
 TOT    elastic    anelastic 
Elastic transitions:
1 core electron excited
N-1 passive electrons (relaxed)
Anelastic transitions:
1 core electron excited
Other electrons excited; shake up and shake off
High photoelectron energy  sudden approximation
 TOT

  rj  f
  i 
af
Local structure 
2
bg
N-1  N-1 
N -1
i
 Ef  
N -1
f

2
1 active electron
N-1 electrons
 TOT() if So2 = 1
= So2  0.7  

R
S
T
f
f
Evaluation of the final state!
Cross section (cm2g)
X-ray Absorption Fine Structure (XAFS)
10
6
L3, L2, L1 edges
Ge
K edge
104
Thomson
102
Compton
100
1
10
Photon energy (keV)
100
x
1.2
XAFS
=
XANES + EXAFS
a-Ge
0.8
EXAFS
XANES
0.4
XANES = X-ray absorption
Near Edge Structure
EXAFS = Extended X-ray
Absorption Fine Structure
0.0
11
11.5
12
X-ray photon energy (keV)
XANES: pre edge structure
E
Fermi Golden rule
 0
f >
 0
i >
E
 a
af
f >
 f
i >
b g
if   Mif  0 - 
if 
2
f Hi i
2

z
2 b
   2
f Hi i
 a
af
 b

b g
2
2
 0 - 
b gaf
  f -    d
af
   0
2
if   2 f Hi i

af
E
2
b g
  f -  0
 0

Arctangent
curve
}
Inflection
point
E

 0
i >
E
 0
E
Experimental K- edge of metallic Iridium and its simulation
obtained by the superposition of an arctang function and a Lorentian

2
Ir metal (Experimental signal)
1.5
1
Theoretical edge
0.5
Arctangent
Lorentian
0
1160
1160
1160
Energy
1160
1160
XANES
WL
x
Ar K
1snp, n3
Single particle binding energy (eV)
X-ray edges
s pd
100
EXAFS
Arctangent
curve
50
Visible
UV
XANES
0.58 eV
0
EDGE
2p
2
4
x
6
Energy eV
8
1s2p
Ne K
2s
1s
4p
5p
866
868
Energy eV
870
eV
K-shell photoabsorption of N2 molecule
C.T. Chen and F. Sette, Phys. Rev. A 40 (1989)
N 1s1g*
(a)
(c)
N 1s 1g*
N 1s 
Rydberg series
Shape
resonance
Double
excitations
x10
400
405
410
415
Photon energy (eV)
Absorption Intensity (arb. Units)
Absorption Intensity (arb.units)
K-shell photoabsorption of gas-phase N2
Double excitations
400
401
402
414
415
3
(b)
N 1sRydberg series
4
5
6
7
10
8
1
9
3
11
12 13
2
420
406
407
408
409
Photon Energy (eV)
410
(a) 8 vibrational levels observed in the absorpttion spectrum: N 1s1g*
(b) N 1sRydberg series
(c) Double excitations in the N2 spectrum Rydberg associated to the N 1s1g* transition.
XANES
Chemical information: oxidation state

1
CuO
Cu2O
Y-Ba-Cu-O
0.5
KCuO2
Cu
0
8970
8990
E (eV)
Oxidation Numbers (formal valences)
I Cu2O
II CuO
III
KCuO2
Higher transitions energy are expected for higher valence states.
(J.B. Boyce et al. Phys. Rev. B 1987)
XANES: projected density of state
Total DOS (a.u.)
SbSI
Sb L1
Absorbance (a.u.)
E
2s  p
I L1
-5
5
10
15
E - E0 (eV)
I L3
Absorbance (a.u.)
X-rays
DOS
0
-5
0
Sb L3
5
10
E - E0 (eV)
15
20
Total DOS (a.u.)
0
2p  p, d
20
(G. Dalba et al., J.of Condensed Matter, 1983)
EXAFS: phenomenological interpretation
h
B
B
B
e-
f0 
A
A
B

i k r  A
e
2kr
B
k
b g
2m
E - E0
2
Autointerference phenomenon of
the outgoing photoelectron with its parts that are
backscattered by the neighboring atoms
k  
k  -  0 k 
  j A j k sen j k 
 0 k 
EXAFS: phenomenological interpretation
Atoms
x
e-
A
Kr

Outgoing wave:
e i k r  A
f0 
2kr
14.2
14.6
15.0
E (KeV)
Molecules
Positive
interference
Negative
interference
14200
4
14600
kr
3
A
B
A
B
Br2
x
-50
4
2
0
50
3

 r f
 E  i 
af
2
bg
 Ef
2
1
Br2
XANES
1
f  f0  f
f smoothly varying in the EXAFS region
13400
13800
Photon energy (eV)
Scattering phases of the photoelectron

A
B
A
B
B
A
B
A
x
A
B
R
e i k r  A
f0 
2kr
f  f0

1
af
f  f0 H k r e
i 1
i e i k r i 1
 f0
e
2kr
i e i k R i 1
f  i f0
e
2kR
e i k R  1
e i k aR - x f
f  i f0
f , k
2kR
R-x
af
f  i f0
af
f , k i b2 k R  2 1   g
e
2
2kR

  C i   r f0  f
2
XAFS formula



 C i   r f0  2 Re i   r f0 i   r f
af
k 
 - 0
0
k
b g
2m
E - E0
2
F
I
G
HJ
K
fa
, k fsinb
2 k R  af
kg

i   r f
f
 k  2 Re
 2 Re

i   r f0
f0
af
af
 k -
1
k R2
Approximations
1) Inelastic scattering effect: S 20
2) Thermal disorder:
e-k
2
s2
S20
-k2
 k f , k e
2
kR
af
Electron mean free path
af
s2
b
afg
e - 2 R  sin 2 k R   k
Standard EXAFS formula


EXAFS formula
For several coordination shells:
1 st
2 nd
3 rd
Coordination
shells
R
0
From ab-initio calculations or from reference compounds
S20
 k k
af
af
fs , k - k 2
s Ns R2 e
s
Coordination
number
s s2
e -2 Rs
Debye Waller
factor
s
b
afg
sin 2 k R s  s k
Interatomic distance
Multiple scattering
EXAFS: single scattering processes

2
R
k
E - E 0  40 eV
R
XANES: multiple scattering processes

2
R
k
E - E 0  10  40 eV


k    0 k  1    ln k 
 n 2


ln   Aln k, p n sin kRpn  k, p n   20l
pn

B
RAB
Single scattering
C
A
B
Double scattering
C
A
B
Double scattering
C
A
B
Triple scattering
C
A
Multiple scattering
Si K-edge in c-Si
Experiment
Absorption
8
7
6
5
4
3
2
1

k    0 k  1    ln k 
 n 2


0
50
100
Energy E-E F (eV)
150
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.