Transcript XAFS_course2012_2_Theory

Introduction to X-ray Absorption Spectroscopy:
Introduction to XAFS Theory
K. Klementiev, Alba synchrotron - CELLS
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Introduction
In the present lecture:
• Channels of interaction between x-rays and matter
• Discussion on Fermi’s Golden Rule
• General steps in derivation of EXAFS formula, early formulation
• Modern formulation
• Qualitative picture of XANES
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Interaction cross sections
data from physics.nist.gov/PhysRefData/Xcom/Text/XCOM.html
6
6
1
0
6
1
0
S
(
Z
=
1
6
)
1
1
0
n
to
p
m
o
C
4
1
0
5
1
0
p
h
o
t
o
n
e
n
e
r
g
y
(
e
V
)
1
3
6
1
0
1
0
R
a
y
le
ig
h
t
1
3
1
0
2
1
0
3
1
0
ffe c
1
1
0
to - e
R
a
y
le
ig
h
R
a
y
le
ig
h
ect
3
1
0
4
1
0
o -e ff
2
1
0
4
1
0
ect
3
1
0
P
b
(
Z
=
8
2
)
5
1
0
phot
o -e ff
4
1
0
C
u
(
Z
=
2
9
)
5
1
0
pho
5
1
0
phot
cross section (barn/atom)
1
0
2
1
0
1
1
0
n
to
p
m
o
C
4
1
0
5
1
0
1
6
3
1
0
1
0
p
h
o
t
o
n
e
n
e
r
g
y
(
e
V
)
n
to
p
m
o
C
4
1
0
5
1
0
6
1
0
p
h
o
t
o
n
e
n
e
r
g
y
(
e
V
)
• Two principal channels: absorption and scattering. The cross sections are Z- and energydependent.
• Photoelectric process is the most probable in the synchrotron energy range
(the range of the ALBA-CLÆSS beamline is marked by green).
• Electron-positron pair production and photonuclear absorption occur at E>1MeV (not shown).
• The shown cross-sections are for a single atom. The collective effects, like Bragg peaks, can
be more intense.
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Fermi’s Golden Rule
Fermi’s Golden Rule
in one-electron approximation:
E f EF
 (E ) 

2
f H int i
 (E  E f  Ei )
f
• i is an initial deep core state (e.g. |1s = 2Z3/2e-Zr/√4): strongly localized.
• f  is an unoccupied state in the presence of a core hole [a collective response of the
other electrons which is effectively described as a single particle of a positive charge called 'hole'],
• Hint is the electron transition operator:
• Hint = p·A(r);
• The photon is taken to be a classical wave: A(r) = eA0eik·r:
• For deep-core excitations eik·r ≈ 1 (dipole approximation) because
r is small due to the strong localization of the initial state
• The next term +ik·r (quadrupole approximation) is ~Z/(2·137) times
weaker, and for heavy elements like Pb, Au, Pt is not negligible
(but anyway is normally neglected)
• equivalent representations: momentum form p·e and position form (ħ/m)r·e.
For example, consider a photon propagating along z with e||x and its K-absorption:
Then r·e = x = r sin cos  Y1±1() and (E)  |f Y1±1 Y00d.
Hence for K absorption the final states f  can only be of Y1±1 (i.e. p) symmetry
(in general, l=±1).
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Fermi’s Golden Rule. Summary
• Photo-electric absorption is the main process in the x-ray range of photon
energies (apart from coherent effects).
• XAS is element specific because the photon energy is tuned to a specific
absorption edge. All elements can be selected, there are no
spectroscopically silent ones.
• Due to the selection rules the empty states can be selected (via selecting
the absorption edge). A common error: angular momenta are always about
the origin! A p orbital on a neighbor is not a p orbital with respect to the
central atom!
• XAS is sensitive to the filling of the final state bands (because we sum over
the final states) and thus sensitive to valence.
• The scalar product r·e means sensitivity to anisotropy in respect to photon
polarization. Oriented samples provide more information on low symmetry
sites.
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Simple derivation of EXAFS
General steps in wave function approach (early EXAFS history):
• The final states are perturbed by neighboring atoms: f= f0+f,
where f0 is a pure atomic state and is found as spherical Hankel function with a
phase shift found from its asymptotic behavior at
infinity.
E

E
2
•  is now factored: 0(1+), where 
(
E
)

f
H
i
(
E

E

E
)

0
0 int
f
i
f
and   f|(r·e)|i
• The scattering part f is found by expansion in spherical harmonics about the
origin with retaining the proper symmetry (e.g p for K-absorption).
Different derivations find f differently.
f
F
Assumptions:
• The central atom is accounted for by a phase shift, i.e. at the neighboring atoms its
potential is neglected. The criterion is k>> Z/a0. (E>>50 or even 100 eV)
• Spherically symmetric central potential. Will not work in asymmetric environment.
• The response of the environment to the absorption (core-hole potential) is weak.
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Early EXAFS expression
Finally, in the photoelectron momentum space, k = [2(E–EF)]½,
the  function was parameterized as [Stern, Sayers, Lytle]:




(
k
)


kf
(
k
)
[
N
exp(

r
)
/
r
]
exp(

k
/
2
)
sin[
2
kr

2
(
k
)]

j
j j
j
j
2
j
2
2
For each coordination shell j:
rj, Nj, 2j are the structural parameters (distance, coord. number and distance variance),
(k) is the phase shift due to (only!) the central atom,
f(k) is the global scattering factor.
Note:
There was no shell-attributed phase shift and the amplitude was global.
Nevertheless, first Fourier analysis was successfully applied to invert the
EXAFS equation: Stern, Sayers, Lytle, Phys.Rev.Lett. 1971
(beginning of modern EXAFS history).
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Modern EXAFS expression, FEFF
• Rewrite golden rule squared matrix element in terms of real-space Green’s function and
scattering operators [Ankudinov et al. PRB 58, 7565]:
1
(
E
)


Im
i
r
G
(
r
,
r
'
;
E
)

r
'
i
(
E

E
)
F
 

• Expand GF in terms of multiple scattering from distinct atoms
• Initial atomic potentials generated by integration of Dirac equation; modified atomic
potentials generated by overlapping (optional self-consistent field)
• Complex exchange correlation potential computed (gives mean free path)
• Scattering from atomic potentials described through k-dependent partial wave phase shifts
for different angular momenta l
• Radial wave function obtained by integration to calculate 0
• Unimportant scattering paths are filtered out
• feff for each path calculated and finally for the complex photoelectron momentum p:
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Modern EXAFS expression, FEFF
f(
p
)



(
p
)


(
p
)

S
Im
e
e
 
kR


2
0
2
2
eff

2
ikR

2
i
2
p

l

 2

For each path :
R, N, 2 are the structural parameters
(path half-length, coord. number and
distance variance),
Possible scattering paths:
multiple
(3-leg)
scattering
single
scattering
feff is the effective scattering amplitude
(is complex, thus also includes the phase),
S02 accounts for many-electron excitations.
There are still limitations in the modern (FEFF) theory:
• muffin-tin approximation is coarse. In the near-edge regime, the intra-atomic excitations
(shake-up, shake-off, resonances) do not have quantitative description.
• Many-body effects are not quantitatively understood.
• Self-energy is based on a simplistic electron-gas model (inaccurate mean free path)
See Rehr&Albers, Rev.Mod.Phys. 2000 for a big review
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M1–M5
L3 2p3/2
L2 2p1/2
L1 2s
K 1s
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EF
1) resonances
a) pre-edge
peak
EF
b) white
line
shifted downwards
due to core-hole
Ex-ray
continuum
E photoelectron
Qualitative Picture of XANES (difficulties)
2) “shake-up” and “shake-off”
3) “shake-down”
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Conclusions
• The fact that already the old derivation worked acceptably well tells us that when first
applied to a reference material and tuned, EXAFS can give reliable results.
Remember about references: this is an important logical step even when using
modern theory (yes, there are still some manually tweaked inputs)
• The modern EXAFS calculations (e.g. with FEFF) are quite reliable. The amplitude
factors still have some uncertainty due to inaccurate self-energy, simplified manybody effects and weak energy dependence of the neglected quadrupolar contribution.
The phases are more reliable.
• There are more and more XANES calculations appearing showing success also in the
near-edge region. To my point of view, quantitative agreement is still exception rather
than a rule. I haven’t seen a single example where a good agreement was got after a
fully automatic calculation, without good manipulation in inputs in these ‘ab-initio’
codes.
• The qualitative picture of XANES is well understood. XANES is mostly used for
fingerprint analysis (symmetric-asymmetric, oxidized-reduced) and for analysis of
mixtures.
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