Practical approach to EXAFS Data AnalysisPractical approach to EXAFS Data Analysisby Alexei Kuzmin
Institute of Solid State Physics, University of Latvia
Kengaraga street 8, LV-1063 Riga, LatviaE-mail: [email protected]
2A.Kuzmin, October 2011.
http://www.esrf.eu/UsersAndScience/Experiments/TBS/SciSoft/Links/xafs
Software for EXAFS analysis & simulationsSoftware for EXAFS analysis & simulations
• IFEFFIT (Athena/Artemis/Feffit)
• EXTRA
• FDMNES
3A.Kuzmin, October 2011.
http://www.dragon.lv/eda/
EEXAFS XAFS DData ata AAnalysis Software Packagenalysis Software Package
EDAFORM: converts experimental data into the EDA's file format (ASCII, 2 columns).EDAXANES: extracts the XANES part and calculates its first and second derivatives.EDAEES: extracts the EXAFS part using improved algorithm for the atomic-like ("zero-line") background
removal.EDAFT: performs Fourier filtering procedure with or without amplitude/phase correction and with the
rectangular, Gaussian, Kaiser-Bessel, Hamming and Norton-Beer F3 window functions.EDAFIT: a non-linear least-squares fitting code, based on a high speed algorithm without matrix inversion.
It uses the multi-shell Gaussian/cumulant model within single-scattering approximation and allows simultaneous analysis up to 20 shells with 8 fitting parameters (Ni So
2, Ri, σi2, ΔE-0i, C3i, C4i, C5i, C6i)
in each. The range of values for any fitting parameter can be limited by boundaries or fixed to a constant value. The covariance and correlation matrices can be also calculated.
EDARDF: a hybrid regularization/least-squares-fitting code allowing to determine model-independent radial-distribution-function (RDF) in the first coordination shell for a compound with arbitrary degree of disorder.
FTEST: performs analysis of variance of the fit results based on the Fisher's F0.95-test.EDAPLOT: general-purpose program for plotting, comparison and mathematical calculations frequently used
in the EXAFS analysis (more than 20 functions !!!).EDAFEFF: extracts the scattering amplitude and phase functions from FEFF****.dat files for use with EDAFIT
or EDARDF codes (works under Windows).
4A.Kuzmin, October 2011.
General scheme of the EDA packageGeneral scheme of the EDA package
XANESEDAFORM
EDAEES
EDAFT
EDAXANES
EDAFIT
FEFF+EDAFEFF or
EDAFT+EDAPLOTEDARDF
EDAPLOT
FTEST
EDAFT+EDAPLOT
5A.Kuzmin, October 2011.
EDAEES: EDAEES: EExtraction of xtraction of EEXAFS XAFS SSignalignal
STEP 1: STEP 2:)()()()(
)(0
0exp
EEEE
E Ib
μμμμ
χ−−
=
)()()()( 0000 kkEE IIIIII μμμμ ++=
)()(0 EPE nI =μ
)()(0 kPk mII =μ
),()( 30 pkSkIII =μ
Pn – polynomial of n-order
S3 – smoothing cubic spline
n = 2,...,43)(EBAEb −=μ
( )02
e )/2( EEmk −= h
6A.Kuzmin, October 2011.
STEP 3: STEP 4:
EDAEES: EDAEES: EExtraction of xtraction of EEXAFS XAFS SSignalignal
)()()( 0 EEk II μμμ −= )()()( 0 kkk IIIII μμμ −=
)()(0 kPk mII =μ
m = 1,...,7
),()( 30 pkSkIII =μ
p ≥ 0 – smoothing spline parameter
7A.Kuzmin, October 2011.
)()()()( 0000 kkEE IIIIII μμμμ ++=
EDAEES: EDAEES: EExtraction of xtraction of EEXAFS XAFS SSignalignal
STEP 5:
)()()()(
)(0
0exp
EEEE
E Ib
μμμμ
χ−−
=
Influence of zero-line on the EXAFS.
8A.Kuzmin, October 2011.
EDAEES: EDAEES: EExtraction of xtraction of EEXAFS XAFS SSignalignal
Influence of zero-line removal on the FT of the EXAFS.
9A.Kuzmin, October 2011.
Fourier transform (FT)Fourier transform (FT)
W(k)Window function W(k):
A~1-2
10A.Kuzmin, October 2011.
BackBack--Fourier transform (BFT)Fourier transform (BFT)
AMPL(χ(k))
Rmin Rmax
11A.Kuzmin, October 2011.
Influence of backscattering amplitude and phaseInfluence of backscattering amplitude and phaseon EXAFS and FTon EXAFS and FT
0 2 4 6 8 10 12 14 16 18 200.0
0.2
0.4
0.6
Ni
W
Back
scat
terin
g am
plitu
def(π
,k,R
)
Wavenumber k (Å-1)
O
0 2 4 6 8 10 12 14 16 18 20-20
-15
-10
-5
0
Tota
l bac
ksca
tterin
g ph
ase
φ(π,
k,R
)=Φ
(π,k
,R)+δ cl (k
)
Wavenumber k (Å-1)
Ni-ONi-NiNi-W
0 2 4 6 8 10 12 14 16 18 20-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0Model:N=1, R=2 Å, σ2=0 Å2
EXAF
S χ(
k)k2 (Å
-2)
Wavenumber k (Å-1)
Ni-ONi-NiNi-W
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0-4
-3
-2
-1
0
1
2
3
4
Model:N=1, R=2 Å, σ2=0 Å2
FT χ
(k)k
2 (Å-3)
Distance R (Å)
Ni-ONi-NiNi-W
FEFF8 simulations for the Ni K-edge
N.B. The RDF G(R) is the same for each atoms pair !
12A.Kuzmin, October 2011.
The EXAFS signal χ(k) can be described by the following equations depending on the way how theRDF G(R) is described.
Simulation of the EXAFS signalSimulation of the EXAFS signal
1. The parametrized multi-component Gaussian/cumulant model (EDAFIT):
2. The general RDF model (EDARDF):
3. The splice model (EDAPLOT + EDAFT):
dRRkkRRkfkR
RGSkR
R
)),,(2sin(),,()()(max
min
220model πφπχ += ∫
=0Reconstructed from cumulant
Experimental
13A.Kuzmin, October 2011.
Where to get amplitude Where to get amplitude ff((ππ,,k,Rk,R) and phase ) and phase φφ((ππ,,k,Rk,R) ) ??
• Ab initio theory: FEFF8, …
• Reference compound: a crystal with well known structure
ATOMCrystal
structure parameters
FEFF8 EDAFEFFatoms.inp feff.inp feff****.dat amp****.dat
pha****.dat
“Absolute” amplitude and phase
Crystal structure parameters: N, R
Experimental EXAFSχref(k)
EDAFT: FT+BFT EDAPLOTAMPL(k)
PHASE(k)
Relativeamplitude and phase
14A.Kuzmin, October 2011.
Gaussian/Gaussian/cumulantcumulant parametrizationparametrizationof the EXAFS signalof the EXAFS signal
( ) ( ) ( )( ) ( ) ⎟
⎠⎞
⎜⎝⎛ −+++−×
⎟⎟⎠
⎞⎜⎜⎝
⎛−−+−= ∑
=
πδπφ
λσπχ
lkRkkCkCkR
kRkCkCkRkf
kRNSk
lciliii
M
i
iiiiil
i
i
2,,154
342sin
2exp)454
322exp(,,
55
33
1
66
44
222
20
Correlation between parametersin the amplitude of EXAFS function: S0
2N, σ2, C4, C6
in the phase of EXAFS function: R, C3, C5, ΔE0
Example: ΔE0 = 1 eV ⇒ ΔR ≈ 0.005 Å ΔN = 0.5 ⇒ Δσ2 ≈ 0.001 Å2
ΔE0 = 5 eV ⇒ ΔR ≈ 0.025 Å ΔN = 1.0 ⇒ Δσ2 ≈ 0.002 Å2
ΔE0 = 10 eV ⇒ ΔR ≈ 0.048 Å
02
e )/2( Emk Δ= h
1.Fix E02.Fix N
15A.Kuzmin, October 2011.
Relationship between the Relationship between the kk and and RR spacespaceNyquistNyquist theoremtheorem
From the properties of Fourier transform:if χ(k) is given in the k-space from kmin=0 to kmax with a step dk, then G(R) will be given in the R-space from Rmin=0 to Rmax≈π/2dk withthe spatial resolution δR ≈ 1/2kmax.
For example: δR = 0.03 Å for kmax = 16 Å-1.
The total number of parameters Mmax used in the model must be less thanthat given by the Nyquist theorem:
For a single shell, Δk = kmax - kmin = 15 Å-1 and ΔR = Rmax - Rmin = 1 Å, then Mmax ≈ 11.5.
E.O. Brigham, The Fast Fourier Transform (Prentice Hall, Englewood Cliffs, New Jersey, 1974).E.A. Stern, Phys. Rev. B 48 (1993) 9825.
16A.Kuzmin, October 2011.
The FTEST program allows one to perform the analysis of variance of theresults of the multi-shell fit using the Fisher’s F0.95 test.
• The experimental EXAFS χexp(k) is given from kmin to kmax and corresponds to the range of the back-Fourier transform ΔR. • It was fitted by two models χ1(k) and χ2(k) having the number of fitting parameters M1 and M2.
FisherFisher’’ss FF0.950.95 testtest
D.J. Hudson, Statistics Lectures on Elementary Statistics and Probability (CERN, Geneva, 1964)
The variance is( ) ( )∑
=
−−
=N
iieli kk
MMNMD
1
2modexp
fitmax
max )()( χχ
According to the Fisher’s F0.95 test (95% probability), the second model shouldbe accepted when
95.02fit
1fit FDD
>
17A.Kuzmin, October 2011.
Analysis of the experimental Re L3-edge EXAFS χexp(k) from the first coordination shell (Re-O1) in ReO3.
Back-FT in the interval from Rmin=0.7 Å to Rmax=2.1 Å: ΔR=1.4 Å.Best-fit of χ (k) in the interval from kmin = 1.5 Å-1 to kmax = 15 Å-1.
Model1 χ1(k,N,R,σ2,ΔE0): the number of fitting parameters M1=4.Model2 χ2(k,N,R,σ2,ΔE0,C3,C4,C5,C6): the number of fitting parameters M2=8.
D1=4.02 ×10-4 and D2=6.37×10-4, so D1/D2=0.63 < F0.95 =4.1
Example of Example of FisherFisher’’ss FF0.950.95 testtest
The variance is ( ) ( )∑=
−−
=N
iieli kk
MMNMD
1
2modexp
fitmax
max )()( χχ
According to the Fisher’s F0.95 test, the second model should not be accepted !
95.02fit
1fit FDD
>
18A.Kuzmin, October 2011.
AmplitudeAmplitude rratioatio andand pphasehase ddiifffferenceerence aanalysnalysiisswithin the Gaussian/within the Gaussian/cumulantcumulant approximationapproximation
This method can be used to find relative variations of parameters in the EXAFS formula when the single shell EXAFS signal can be isolated.
P. Fornasini, S. a Beccara, G. Dalba, R. Grisenti, A. Sanson, M. Vaccari, F. Rocca, Phys. Rev. B 70 (2004) 174301:1-12.
19A.Kuzmin, October 2011.
EXAFS Data Analysis: how to do ?EXAFS Data Analysis: how to do ?
8100 8400 8700 9000 9300 96000.0
0.5
1.0
1.5
2.0
2.5
X-ra
y Ab
sorp
tion
Energy E (eV)0 2 4 6 8 10 12 14 16 18
-6
-4
-2
0
2
4
EXAF
S χ(
k)k2 (Å
-2)
Wavenumber k (Å-1)
EDAEES
ATOMCrystal
structure model
FEFF8 EDAFEFFatoms.inp feff.inp feff****.dat amp****.dat
pha****.dat
χFEFF(k)
Compare χ(k)
phase
Change E0
0 2 4 6
-4
-2
0
2
4
6
1 shell
FT χ
(k)k
2 (Å-3
)
Distance R (Å)2 4 6 8 10 12 14
-2
-1
0
1
2
EXAF
S χ(
k)k2 (Å
-2)
Wavenumber k (Å-1)
Change E0
EDAFT EDAFIT
20A.Kuzmin, October 2011.
Examples Examples of of
EXAFS data analysis EXAFS data analysis by by
different approachesdifferent approaches
21A.Kuzmin, October 2011.
Gaussian, Gaussian, cumulantcumulant and RDF modelsand RDF models
Gaussian-modelCumulant-modelRDF model
Gaussian-modelCumulant-modelRDF model
A. Kuzmin, J. Physique IV (France) 7 (1997) C2-213-C2-214.
••• “experiment”model RDF
22A.Kuzmin, October 2011.
TheThe RDFsRDFs derivedderived byby thethe splicesplice and RDFand RDF techniqutechniquesesfor several for several kkmaxmax valuesvalues
Dashed line – splice methodSolid line – RDF methodCircles – “experiment” (model RDF)
A. Kuzmin, J. Physique IV (France) 7 (1997) C2-213-C2-214.
23A.Kuzmin, October 2011.
TheThe RDFsRDFs forfor thethe 1st1st shellshell inin ReOReO33,, WOWO33 andand MoMoOO33derivedderived byby thethe splicesplice and RDFand RDF techniqutechniqueses
Dashed line – experiment Solid line – RDF methodDotted line – splice method
24A.Kuzmin, October 2011.
Examples of best fits of the EXAFS signals in Examples of best fits of the EXAFS signals in kk--spacespace
EDARDFEDAFIT (Gaussian model)
Re L3-edge in rhenium trioxide ReO3
A. Kuzmin, J. Purans, G. Dalba, P. Fornasini, F. Rocca, J. Phys.: Condensed Matter 8 (1996) 9083-9102.
25A.Kuzmin, October 2011.
Examples of best fits of the EXAFS signals in Examples of best fits of the EXAFS signals in kk--space space by EDAFIT (Gaussian model)by EDAFIT (Gaussian model)
Ni K-edge in polycrystalline and thin film nickel oxide NiOSolid lines – experiment, dashed lines - model
1st shell Ni-O 2nd shell Ni-Ni
A. Kuzmin, J. Purans, A. Rodionov, J. Phys.: Condensed Matter 9 (1997) 6979-6993.
26A.Kuzmin, October 2011.
Examples of best fits of the EXAFS signals in Examples of best fits of the EXAFS signals in kk--spacespace
EDARDFEDAFIT (Gaussian model)
Ag K-edge in Ag2O-B2O3 glassesDashed lines – experiment, solid lines - model
A. Kuzmin, G. Dalba, P. Fornasini, F. Rocca, O. Šipr, Phys. Rev. B 73 (2006) 174110:1-12.
27A.Kuzmin, October 2011.
Thank you for attention !Thank you for attention !
Get more details at:http://www.dragon.lv/eda
and http://www.dragon.lv/exafs