Practical Approach to EXAFS Data Analysis

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


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).


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


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


STEP 3: STEP 4:


)()()( 0 EEk II μμμ −= )()()( 0 kkk IIIII μμμ −=

)()(0 kPk mII =μ

m = 1,...,7

),()( 30 pkSkIII =μ

p ≥ 0 – smoothing spline parameter


)()()()( 0000 kkEE IIIIII μμμμ ++=


STEP 5:

)()()()(

)(0

0exp

EEEE

E Ib

μμμμ

χ−−

=

Influence of zero-line on the EXAFS.



Influence of zero-line removal on the FT of the EXAFS.


Fourier transform (FT)Fourier transform (FT)

W(k)Window function W(k):

A~1-2


BackBack--Fourier transform (BFT)Fourier transform (BFT)

AMPL(χ(k))

Rmin Rmax


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 !


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


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


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


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.


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

>


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

>


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.


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


Examples Examples of of

EXAFS data analysis EXAFS data analysis by by

different approachesdifferent approaches


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


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.


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


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.


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.


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.


Thank you for attention !Thank you for attention !

Get more details at:http://www.dragon.lv/eda

and http://www.dragon.lv/exafs

Practical Approach to EXAFS Data Analysis

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