Download - Introduction to EXAFS I
Introduction to EXAFS I
Outline• Brief history• Why EXAFS? Some general differences
between local and average structure• What information can one extract?• Experimental set-up and requirements• EXAFS equation• Data reduction • Data analysis
F. Bridges Chalmers 2011
F. Bridges Physics Dept. UCSC,
MC2 Chalmers
Scott MedlingMichael KozinaBrad CarCarley CorradoJin Zhang Yu (Justin) JiangLisa DownwardJohn J. Neumeier T. TysonCollin BroholmSatoru Nakatsuji
Brief History of EXAFS(Extended X-ray Absorption Fine Structure)
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• Every atom has well defined absorption steps.
• For a solid, above each step there is structure in the absorption as a function of energy; known since ~ early ’20s-’30s Kossel , Kronig.
• Many explanations proposed – some were multi-electron one was single electron (Kronig).
• Stern and Sayers (Stern’s student) developed a useful model to use the oscillations to understand structure ~ 1970-71 – used Lytle’s data.
• Argument as to who first suggested using Fourier Transforms will never be resolved.
1 10 1000.01
0.1
1
10
M
LIII, LII, LI
K
Xenon
(cm
-1 )
E (keV)
What information does EXAFS provide?
• Can measure bond lengths and further pair distances. Usually for bond lengths, absolute distances to 0.01Å; relative change even smaller.
• Types of neighbors – but need change in Z (4 or more)• Coordination numbers – generally better than 20%; sometimes to
10%. Limited by correlations between parameters. • Disorder/Distortions – static, thermal phonons, lattice distortions
(Jahn-Teller), polarons etc F. Bridges Chalmers 2011
• Local structure about a selected atom type; determined by X-ray energy used. (Moseley’s Law for fluorescence)
• E.g. for Cu use X-ray energies near 9000 eV; Rb, 15,200 eV; Ag, 25,500eV
Bunker
Types of materials/Samples• Solids – usually complex; often distorted• Nanoparticles; small – and doped• Thin films (nano-sized grains)• Amorphous materials• Liquids (and gases); solutions• Need samples to be homogeneous and uniform
in thickness. Uniform thickness – no pinholes, tapers etc., no concentration gradients. (Address in later lecture) Choose appropriate thickness for edge energy. Step height <1.
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Experimental set-up for X-ray spectroscopy (solids)
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Top View
Side View
Source
Source
Typical set-up for transmission
µt = ln Io/I1
µReft = ln I1/I2
Set-up for fluorescence detection.For low concentrationsµ ~ If/Io
Need very linear detectors and amps. Because use ratios, gains not important, but need dynamic range.
Typical set-up (CH Booth)
“white” x-rays from
synchrotron
double-crystal monochromator
collimating slits
ionization detectors
I0 I1 I2
beam-stop
LHe cryostatsample
reference sample
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Windows in detectors – thin Kapton – often aluminized; 50 –75 µm .
• Choose gas in detectors for energy range (He, N2, Ar, Kr, etc.) Can use mixtures to optimize. Not too much absorption in ionization detectors.
• Choose edge/energy; check no other edges overlap
• Prepare uniform sample
Leave gap between Io and sample – and I1 and sample.Keep reference sample away from I1
Reduce higher harmonics(2dsinθ = nλ)
X-ray Absorption SpectroscopyExtended X-ray Absorption Fine structure (EXAFS)
• Main features are single-electron excitations.• Away from edges, energy dependence fits a power law: AE-3+BE-4 (Victoreen).• Threshold energies E0~Z2, absorption coefficient ~Z4.• Core-hole lifetime ~ 10-15 sec sets response time – a very fast probe – but data
collection slow.
1 10 1000.01
0.1
1
10
M
LIII, LII, LI
K
Xenon
(cm
-1 )
E (keV) From McMaster Tables 1s
filled 3d
continuum
EF
core hole
unoccupiedstates
K-edge
From G. Bunker
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EXAFS /XANES overview
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-2.4
-2.1
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-1.2 raw data pre-edge fit
a)
Abs
orpt
ion
(a.u
.)
E (eV)7600 7800 8000 8200 8400
0.0
0.3
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1.2
1.5
1.8 normalized data 7-knots spline
b)
Abs
orpt
ion
(a.u
.)
E (eV)
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LaCoO3 r-space data
FFT
( k
(k) )
r (Å)
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LaCoO3 k-space data
k(k
)
k (Å-1)
)()()()(
0
0
EEEE
FT
4K
Dotted background constrained by Victoreen Equ. and the step height.
0512.0 EEk
XANESµt = ln Io/I1
Raw absorption data, pre-edge fit
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-2.4
-2.1
-1.8
-1.5
-1.2 raw data pre-edge fit
a)
Abs
orpt
ion
(a.u
.)
E (eV)
Co K-edge
• Pre-edge region –absorption from all other material in beam; includes other atom in sample.
• Varies as AE-3+BE-4 (Victoreen) -- tabulated in McMaster tables.
• Using measured step height and Victoreen equation below edge and well above edge, can extract background above edge.
• Keep step height below 1 (between 0.3 and 0.7)
• Need to also know absorption from rest of sample!!
Pre-edge subtracted data (Co K-edge LaCoO3)
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0.3
0.6
0.9
1.2
1.5
1.8 normalized data 7-knots spline
b)
Abs
orpt
ion
(a.u
.)
E (eV)
• Eo – we define as the half step energy.• Slope at high energy agrees with
Victoreen formula.• Errors in slope add (or subtract) to
σ2.• If slope varies from trace to trace
(e.g. in a temperature dependant study) get fluctuations in σ2.
• EXAFS function χ(E) – the oscillations on top of a background function µo – red line.
)()()()(
0
0
EEEE
0512.0 EEk
E-Eo =ħ2k2/2m
Co k-space data
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LaCoO3 k-space data
k(k
)
k (Å-1)
• Usually plot as knχ(k); here kχ(k). • Would like kmax as high as
possible – but limited by noise and time to collect data.
• Sum of sine waves of form sin(2kr + Φ(k))
• Take Fourier Transform to get an r-space plot.
i i
bcikkri kr
kkkreekfrNSk i2
2)(/2220
)]()(22sin[),()ˆˆ()(22
r-space (Co)
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LaCoO3 r-space data
FFT
( k
(k) )
r (Å)
Peaks in r-space correspond to different shells of neighbors
Usually can fit out to 4-6 Å depending on the structure.
EXAFS peaks shifted from real distances.
Co-O Co-LaCo-Co
More r-space (FT Spectra)Example: cubic ZnS:Cu,Mn
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• Fast oscillation – real part , R, of transform
• Imaginary part , I, not shown.• Envelope function ±√(R2
+ I2)• Fit to R and I • Peaks in EXAFS shifted in position by
well known amount - Δr (from phase shifts δc and δb in term
sin[2kr + 2δc(k) + δb(k)]
2δc(k) + δb(k) ≈ -2kΔr + f(k)RR
Zn-S; Mn-SZn-Zn,; Mn-Zn
0 5 10 15 20-5
0
5
10
Phas
e sh
ift (r
ad)
k(Å-1)
2c
a
Co-O, Co K edge
Bunker
Chalmers 2011
X-ray Absorption Spectroscopy
“I was brought up to look at the atom as a nice hard fellow, red or grey in colour according to taste.”
- Lord Rutherford
1s
filled 3d
continuumEF
core hole
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Time scale – 10-15 sec.
Simple model for absorption• Use Fermi’s golden rule µ ~ <f |(ε·r)2|i>• Final state f is modified by backscattering +
interference of outgoing and backscattered waves, i.e. f = fo + Δf ; dipole selection rules
• Can write µ = µo(1+χ); or χ = (µ-µo)/µo
µ, µo, and χ are all energy dependent
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i i
bcikkri kr
kkkreekfrNSk i2
2)(/2220
)]()(22sin[),()ˆˆ()(22
How is final state wave function modulated?
• Assume photoelectron reaches the continuum within dipole approximation:
2)ˆˆ( r
From CH Booth LBL
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How is final state wave function modulated?
• Assume photoelectron reaches the continuum within dipole approximation:
krer
ikr
2)ˆˆ(
F. Bridges Chalmers 2011
How is final state wave function modulated?
• Assume photoelectron reaches the continuum within dipole approximation:
central atom phase shift c(k)
krer
ikr
2)ˆˆ(
krer
kiikr c )(2)ˆˆ(
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How is final state wave function modulated?
• Assume photoelectron reaches the continuum within dipole approximation:
electronic mean-free path (k)
central atom phase shift c(k)
krer
ikr
2)ˆˆ(
krer
kiikr c )(2)ˆˆ(
)(/)(
2)ˆˆ( kRkiikR
ekR
erc
F. Bridges Chalmers 2011
How is final state wave function modulated?
• Assume photoelectron reaches the continuum within dipole approximation:
central atom phase shift c(k)
electronic mean-free path (k)
complex backscattering probability f(,k)
krer
ikr
2)ˆˆ(
krer
kiikr c )(2)ˆˆ(
)(/)(
2)ˆˆ( kRkiikR
ekR
erc
)(/)(
2 ),()ˆˆ( kRkiikR
ekkfkR
erc
F. Bridges Chalmers 2011
How is final state wave function modulated?
• Assume photoelectron reaches the continuum within dipole approximation:
central atom phase shift c(k)
electronic mean-free path (k)
complex backscattering probability f(,k)
complex=magnitude and phase: backscattering atom phase shift b(k)
krer
ikr
2)ˆˆ(
krer
kiikr c )(2)ˆˆ(
)(/)(
2)ˆˆ( kRkiikR
ekR
erc
)(/)(
2 ),()ˆˆ( kRkiikR
ekkfkR
erc
kReekfk
kRer
kirRikkR
kiikR bc )()()(/
)(2 ),()ˆˆ(
F. Bridges Chalmers 2011
How is final state wave function modulated?
• Assume photoelectron reaches the continuum within dipole approximation:
krer
ikr
2)ˆˆ(
krer
kiikr c )(2)ˆˆ(
)(/)(
2)ˆˆ( kRkiikR
ekR
erc
)(/)(
2 ),()ˆˆ( kRkiikR
ekkfkR
erc
kReekfk
kRer
kirRikkR
kiikR bc )()()(/
)(2 ),()ˆˆ(
)(/22
)()(222 ),()ˆˆ(Im kR
kikikRi
ekfkR
erac
central atom phase shift c(k)
electronic mean-free path (k)
complex backscattering probability kf(,k)
complex=magnitude and phase: backscattering atom phase shift a(k)
final interference modulation per point atom!
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Assumes both harmonic potential AND kσ<<1: problem at high k and/or σ (good to kσ
of about 1)
Requires curved wave scattering, has r-dependence, use full curved wave theory:
FEFF
Other factors• Allow for multiple atoms Ni in a shell i and a distribution function
function of bondlengths within the shell g(r)
2
2
2)(
21)(
iRr
erg
i
bckkri kr
kkkreekfrNSk 22)(/222
0)()(22sin),()ˆˆ()(
22
where and S02 is an inelastic loss factor
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Contributions to 2 • Static distortions – distribution of pair distances
from strains, impurities, etc.• Thermal phonons – Einstein or Debye models.• Polarons – a distortion associated with a partially
localized charge.• An (unresolved) split peak – effective is ~ r/2
where r is the peak splitting.
• Atoms A and B displaced, can be static or dynamic.
• is the second moment of the pair -distribution function.
• Primarily sensitive to radial displacements
For uncorrelated mechanisms:
σ2total = σ2
static + σ2thermal + σ2
polarons +
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2
2
2)(
21)(
orr
erg
Definition of σ for Gaussian PDF
Fitting data I
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0 2 4 6 8-0.8-0.6-0.4-0.20.00.20.40.60.8
LaCoO3 r-space data
FFT
( k
(k) )
r (Å)
• Parameters for each shell: r, σ, N• Quantities |f(π,k)|e-2r/λ(k) and phases δc, δb calculated using program FEFF (Rehr); or
obtained from experimental function for that atom-pair.• Two global parameters So
2 and ΔEo required when using FEFF; determined for highest amplitude data, usually at low T. ΔEo is the shift between the experimentally defined edge position and edge position where k = 0 in theory. These parameters are usually negligible when using experimental functions
Many codes are available for performing these fits:EXCURVE98 (Diamond England)EXAFSPAK (G. George)IFEFFIT (M. Neville)
-SIXPACK (Sam Webb)-ATHENA (Bruce Ravel)
GNXAS (Italy)RSXAP (Booth, Bridges)
i i
bcikkri kr
kkkreekfrNSk i2
2)(/2220
)]()(22sin[),()ˆˆ()(22
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Fitting data IIQuandary: r-space or k-space fitting
LaCoO3 example
• Since FT is a linear operator, if you do fits correctly should get same answers.
• Need to define both an FT range and a range in r-space e.g. 4-14Å-1 and 1.2-2.0 Å.
• Straight forward in r-space for the Co-O peak. Here one fits a model to the real and imaginary parts of FT over a restricted range in r-space.
• k-space?? If fit the model for the Co-O peak to the full k-space data, poor fit.
• The other components (other peaks) act like noise or an oscillating background
• Need to go to r-space, and then back-transform the region of interest (Co-O peak) into k-space.
0 2 4 6 8 10 12 14-0.8-0.6-0.4-0.20.00.20.40.60.8
LaCoO3 k-space data
k(k
)
k (Å-1)
0 2 4 6 8-0.8-0.6-0.4-0.20.00.20.40.60.8
LaCoO3 r-space data
FFT
( k
(k) )
r (Å)
i i
bcikkri kr
kkkreekfrNSk i2
2)(/2220
)]()(22sin[),()ˆˆ()(22
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0 2 4 6 8-0.6
-0.4
-0.2
0.0
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0.620% bulk LSCO(SA)
FFT
( k
(k) )
r (Å)
Example of a fit – first peak;Co-O
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0.630% Bulk LSCOT = 4 K
FFT
( k
(k) )
r (Å)
Co
O
La/Sr
• Fit r-range, 1.1 – 1.6 Å; k-range 3.3 – 13 Å-
1.• Space group R-3c is used to calculate
theoretical Co-O function using FEFF code.• From the fit, we extract the width, σ, of the
PDF of the Co-O bond; σ = 0.059 Å • Its bond length (1.92 Å) agrees well with
diffraction results, ~ 0.01 Å.• Below 1 Å, often poor agreement – all
errors in background subtractions etc. , end up in this low r range.
• Region between 1.8 and 2.2 Å, interference between 1st and 2nd neighbors
4 K
300 KNearly cubic (trigonal),Co-O-Co 163-167°
Some caveats• Many places to develop systematic
errors - pre-edge subtraction and using multiple splines to obtain µo are two main areas for such errors. Errors in pre-edge subtraction can lead to error in σ.
• Using multiple splines is a type of filter. Want to extract any slow variation in background (part of µo) but don’t want to also filter out part of EXAFS. Termination of spline fit near edge is very important.
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7600 7800 8000 8200 84000.0
0.3
0.6
0.9
1.2
1.5
1.8 normalized data 7-knots spline
b)
Abs
orpt
ion
(a.u
.)
E (eV)
A spline is a cubic polynomial fit over a restricted range. For multiple splines, match value and slope where two splines join.
Further reading• Overviews:
– B. K. Teo, “EXAFS: Basic Principles and Data Analysis” (Springer, New York, 1986).
– Hayes and Boyce, Solid State Physics 37, 173 (1982).– “X-Ray Absorption: Principles, Applications, Techniques of EXAFS, SEXAFS and
XANES”, ed. by Koningsberger and Prins (Wiley, New York, 1988).• Historically important:
– Sayers, Stern, Lytle, Phys. Rev. Lett. 71, 1204 (1971).• History:
– Lytle, J. Synch. Rad. 6, 123 (1999). (http://www.exafsco.com/techpapers/index.html)
– Stumm von Bordwehr, Ann. Phys. Fr. 14, 377 (1989).• Theory papers of note:
– Lee, Phys. Rev. B 13, 5261 (1976).– Rehr and Albers, Rev. Mod. Phys. 72, 621 (2000).
• Useful links– xafs.org (especially see Tutorials section)– http://www.i-x-s.org/ (International XAS society)– http://www.csrri.iit.edu/periodic-table.html (absorption calculator)
More caveats II• Don’t go too low in k-space in choosing the FT range. Remember k =
0.512 (E-Eo)½; so for k = 3 Å-1 , E-Eo = 34.3 3V, and for k = 2 Å-1, E-Eo = 15.3 eV. XANES structure usually extends up to 20-30 eV above edge and sometimes higher, so dangerous to go below k = 3 Å-1. If not sure, do fits for various FT ranges- parameters should not change significantly. If large change in σ, say from kmin = 2.5 and 3 Å-1 then a problem.
• Strong correlations between N and σ. Don’t think of σ as a “throw-away” parameter, even when you are more interested in N and r. σ must be larger than zero-point motion value.
• kn weighting; depends on backscattering atom. Usually k2 or k3 make EXAFS spectra sharper – but be careful of noise at high k.
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A view of Monterey bay from above UCSC
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