Practical approach to EXAFS Data Analysis

Transcription

1 Practical approach to EXAFS Data Analysis by Alexei Kuzmin Institute of Solid State Physics, University of Latvia Kengaraga street 8, LV-1063 Riga, Latvia

2 Software for EXAFS analysis & simulations IFEFFIT (Athena/Artemis/Feffit) EXTRA FDMNES 2

3 EXAFS Data Analysis Software Package EDAFORM: EDAXANES: EDAEES: EDAFT: EDAFIT: EDARDF: FTEST: EDAPLOT: EDAFEFF: converts experimental data into the EDA's file format (ASCII, 2 columns). extracts the XANES part and calculates its first and second derivatives. extracts the EXAFS part using improved algorithm for the atomic-like ("zero-line") background removal. performs Fourier filtering procedure with or without amplitude/phase correction and with the rectangular, Gaussian, Kaiser-Bessel, Hamming and Norton-Beer F3 window functions. 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 (N i S o2, R i, σ i2, ΔE- 0i, C 3i, C 4i, C 5i, C 6i ) 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. a hybrid regularization/least-squares-fitting code allowing to determine model-independent radialdistribution-function (RDF) in the first coordination shell for a compound with arbitrary degree of disorder. performs analysis of variance of the fit results based on the Fisher's F test. general-purpose program for plotting, comparison and mathematical calculations frequently used in the EXAFS analysis (more than 20 functions!!!). extracts the scattering amplitude and phase functions from FEFF****.dat files for use with EDAFIT or EDARDF codes (works under Windows). 3

4 General scheme of the EDA package EDAXANES XANES EDAFORM EDAEES EDAFT EDAPLOT FTEST EDAFIT EDAFT+EDAPLOT FEFF+EDAFEFF or EDAFT+EDAPLOT EDARDF 4

5 STEP 1: EDAEES: Extraction of EXAFS Signal μ χ( E) = exp ( E) μ ( E) μ ( E) I μ ( E) 0 b 0 STEP 2: μ ( E) b = A B E 3 I II III μ 0 ( E) = μ0 ( E) + μ0 ( k) + μ0 ( k) 2 e / ) ( E ) k = ( 2m h E 0 I μ0 ( E) = Pn ( E) II μ0 ( k) = Pm ( k) III μ0 ( k) = S3( k, p) n = 2,...,4 P n polynomial of n-order S 3 smoothing cubic spline 5

6 EDAEES: Extraction of EXAFS Signal STEP 3: STEP 4: I I μ ( k) = μ( E) μ0 ( E) II μ0 ( k) = Pm ( k) m = 1,...,7 II I II μ ( k) = μ ( k) μ0 ( k) III μ0 ( k) = S3( k, p) p 0 smoothing spline parameter 6

7 EDAEES: Extraction of EXAFS Signal STEP 5: I II III μ 0 ( E) = μ0 ( E) + μ0 ( k) + μ0 ( k) Influence of zero-line on the EXAFS. μ χ( E) = exp ( E) μ ( E) μ ( E) I μ ( E) 0 b 0 7

8 EDAEES: Extraction of EXAFS Signal Influence of zero-line removal on the FT of the EXAFS. 8

9 Fourier transform (FT) Window function W(k): W(k) A~1-2 9

10 Back-Fourier transform (BFT) AMPL(χ(k)) R min R max 10

11 Influence of backscattering amplitude and phase on EXAFS and FT N.B. The RDF G(R) is the same for each atoms pair! Backscattering amplitude f(π,k,r) O Ni W Wavenumber k (Å -1 ) EXAFS χ(k)k 2 (Å -2 ) Model: N=1, R=2 Å, σ 2 =0 Å 2 Ni-O Ni-Ni Ni-W Wavenumber k (Å -1 ) Total backscattering phase φ(π,k,r)=φ(π,k,r)+δ c l(k) Wavenumber k (Å -1 ) Ni-O Ni-Ni Ni-W FT χ(k)k 2 (Å -3 ) Model: N=1, R=2 Å, σ 2 =0 Å Distance R (Å) Ni-O Ni-Ni Ni-W FEFF8 simulations for the Ni K-edge 11

12 Simulation of the EXAFS signal The EXAFS signal χ(k) can be described by the following equations depending on the way how the RDF G(R) is described. 1. The parametrized multi-component Gaussian/cumulant model (EDAFIT): 2. The general RDF model (EDARDF): R max 2 G ( R ) χ model ( k ) = S 0 f ( π, k, R ) sin( 2kR + φ ( π, k, 2 kr R min R )) dr 3. The splice model (EDAPLOT + EDAFT): Reconstructed from cumulant Experimental =0 12

13 Where to get amplitude f(π,k,r) ) and phase φ(π,k,r)? Ab initio theory: FEFF8, Crystal structure parameters atoms.inp feff.inp feff****.dat ATOM FEFF8 EDAFEFF amp****.dat pha****.dat Absolute amplitude and phase Reference compound: a crystal with well known structure Experimental EXAFS χ ref (k) EDAFT: FT+BFT AMPL(k) PHASE(k) EDAPLOT Crystal structure parameters: N, R Relative amplitude and phase 13

14 χ Gaussian/cumulant parametrization of the EXAFS signal M N = 0 2 l i= 1 kri sin 2kR 2 i ( k) S f ( π, k, R ) i i 4 3 C 2 exp( 2σ k 3i k i C 2 5i k C + φ l k C k )exp 2R i 6i λ( k) l ( π, k, R ) + 2δ ( k) lπ 4 i 6 c i k = ( 2m h ΔE 2 e / ) 0 Correlation between parameters in the amplitude of EXAFS function: S 02 N, σ 2, C 4, C 6 in the phase of EXAFS function: R, C 3, C 5, ΔE 0 Example: ΔE 0 = 1 ev ΔR Å ΔN = 0.5 Δσ Å 2 ΔE 0 = 5 ev ΔR Å ΔN = 1.0 Δσ Å 2 ΔE 0 = 10 ev ΔR Å 1.Fix E 0 2.Fix N 14

15 Relationship between the k and R space Nyquist theorem From the properties of Fourier transform: if χ(k) is given in the k-space from k min =0 to k max with a step dk, then G(R) will be given in the R-space from R min =0 to R max π/2dk with the spatial resolution δr 1/2k max. For example: δr =0.03 Å for k max = 16 Å -1. The total number of parameters M max used in the model must be less than that given by the Nyquist theorem: For a single shell, Δk = k max -k min = 15 Å -1 and ΔR = R max -R min = 1 Å, then M max E.O. Brigham, The Fast Fourier Transform (Prentice Hall, Englewood Cliffs, New Jersey, 1974). E.A. Stern, Phys. Rev. B 48 (1993)

16 Fisher s F 0.95 test The FTEST program allows one to perform the analysis of variance of the results of the multi-shell fit using the Fisher s F 0.95 test. The experimental EXAFS χ exp (k) is given from k min to k max and corresponds to the range of the back-fourier transform ΔR. It was fitted by two models χ 1 (k) and χ 2 (k) having the number of fitting parameters M 1 and M 2. The variance is D = N M ( M M ) max fit N ( χexp( k i ) χmodel ( ki ) max ) i= 1 2 According to the Fisher s F 0.95 test (95% probability), the second model should be accepted when D fit1 > D fit2 F 0.95 D.J. Hudson, Statistics Lectures on Elementary Statistics and Probability (CERN, Geneva, 1964) 16

17 Example of Fisher s F 0.95 test Analysis of the experimental Re L 3 -edge EXAFS χ exp (k) from the first coordination shell (Re-O 1 ) in ReO 3. Back-FT in the interval from R min =0.7 Å to R max =2.1 Å: ΔR=1.4 Å. Best-fit of χ (k) in the interval from k min = 1.5 Å -1 to k max = 15 Å -1. Model1 χ 1 (k,n,r,σ 2,ΔE 0 ): the number of fitting parameters M 1 =4. Model2 χ 2 (k,n,r,σ 2,ΔE 0,C 3,C 4,C 5,C 6 ): the number of fitting parameters M 2 =8. The variance is D = N M ( M M ) max fit N ( χexp( k i ) χmodel ( ki ) max ) i= 1 2 D fit1 > D fit2 F 0.95 D 1 = and D 2 = , so D 1 /D 2 =0.63 < F 0.95 =4.1 According to the Fisher s F 0.95 test, the second model should not be accepted! 17

18 Amplitude ratio and phase difference analysis within the Gaussian/cumulant approximation 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) :

19 EXAFS Data Analysis: how to do? Change E 0 X-ray Absorption Energy E (ev) EDAEES EXAFS χ(k)k 2 (Å -2 ) Change E Wavenumber k (Å -1 ) EDAFT FT χ(k)k 2 (Å -3 ) Compare χ(k) phase shell Distance R (Å) EDAFIT EXAFS χ(k)k 2 (Å -2 ) Wavenumber k (Å -1 ) χ FEFF (k) Crystal structure model atoms.inp feff.inp feff****.dat ATOM FEFF8 EDAFEFF amp****.dat pha****.dat 19

20 Examples of EXAFS data analysis by different approaches 20

21 Gaussian, cumulant and RDF models experiment model RDF Gaussian-model Cumulant-model RDF model Gaussian-model Cumulant-model RDF model A. Kuzmin, J. Physique IV (France) 7 (1997) C2-213-C

22 The RDFs derived by the splice and RDF techniques es for several k max values Dashed line splice method Solid line RDF method Circles experiment (model RDF) A. Kuzmin, J. Physique IV (France) 7 (1997) C2-213-C

23 The RDFs for the 1st shell in ReO 3, WO 3 and MoO 3 derived by the splice and RDF techniques es Dashed line experiment Solid line RDF method Dotted line splice method 23

24 Examples of best fits of the EXAFS signals in k-space EDAFIT (Gaussian model) EDARDF Re L 3 -edge in rhenium trioxide ReO 3 A. Kuzmin, J. Purans, G. Dalba, P. Fornasini, F. Rocca, J. Phys.: Condensed Matter 8 (1996)

25 Examples of best fits of the EXAFS signals in k-space by EDAFIT (Gaussian model) 1 st shell Ni-O 2 nd shell Ni-Ni Ni K-edge in polycrystalline and thin film nickel oxide NiO Solid lines experiment, dashed lines - model A. Kuzmin, J. Purans, A. Rodionov, J. Phys.: Condensed Matter 9 (1997)

26 Examples of best fits of the EXAFS signals in k-space EDAFIT (Gaussian model) EDARDF Ag K-edge in Ag 2 O-B 2 O 3 glasses Dashed lines experiment, solid lines - model A. Kuzmin, G. Dalba, P. Fornasini, F. Rocca, O. Šipr, Phys. Rev. B 73 (2006) :

27 Thank you for attention! Get more details at: and 27