Basics of EXAFS Processing

Transcription

1 Basics of EXAFS Processing Shelly Kelly 2009 UOP LLC. All rights reserved.

2 X-ray-Absorption Fine Structure Attenuation of x-rays It= I0e-µ(E) x Absorption coefficient µ(e) If/I0 2 File Number

3 X-ray-Absorption Fine Structure Photoelectron R 0 Scattered Photoelectron 3 File Number

4 Fourier Transform of χ(k) Similar to an atomic radial distribution function - Distance - Number - Type - Structural disorder Fourier transform is not a radial distribution function - See 4 File Number

5 Outline Definition of EXAFS - Edge Step - Energy to wave number Fourier Transform (FT) of χ(k) - FT is a frequency filter - Different parts of a FT and backward FT - FT windows and sills - Determining Kmin and Kmax of FT IFEFFIT method for constructing the background function - FT and background (bkg) function - Wavelength of bkg EXAFS Equation 5 File Number

6 Definition of EXAFS => Normalized oscillatory part of absorption coefficient Measured Absorption coefficient χ(e) = µ(e) - µ 0 (E) µ(e) Bkg: Absorption coefficient without contribution from neighboring atoms (Calculated) ~ µ(e) - µ 0 (E) µ(e 0 ) Evaluated at the Edge step (E 0 ) 6 File Number

7 Absorption coefficient Pre-edge region 300 to 50 ev before the edge Edge region the rise in the absorption coefficient Post-edge region 50 to 1000 ev after the edge 7 File Number

8 Edge step Pre-edge line 200 to 50 ev before the edge Post-edge line 100 to 1000 ev after the edge Edge step the change in the absorption coefficient at the edge - Evaluated by taking the difference of the pre-edge and post-edge lines at E 0 8 File Number

9 Athena normalization parameters 9 File Number

10 Energy to wave number E 0 Must be somewhere on the edge Mass of the electron Edge Energy k 2 = 2 m e (E E 0 ) ħ Plank s constant ~ E File Number

11 Athena edge energy E0 11 File Number

12 Fourier Transform is a frequency filter FT of Sin(2Rk) is a peak at R=1 FT of infinite sine wave is a delta function Signal that is de-localized in k-space is localized in R-space FT is a frequency filter 12 File Number

13 Fourier Transform of a function that is: De-localized in k-space localized in R-space Localized in k-space de-localized in R-space 13 File Number

14 Fourier Transform is a frequency filter Regularly spaced ripple Indicates a problem The signal of a discrete sine wave is the sum of an infinite sine wave and a step function. FT of a discrete sine wave is a distorted peak. EXAFS data is a sum of discrete sine waves. Solution for finite data set is to multiply the data with a window. 14 File Number

15 Fourier Transform Multiplying the discrete sine wave by a window that gradually increases the amplitude of the data smoothes the FT of the data. 15 File Number

16 Fourier Transform Windows k min kmax dk dk Parzen Welch Sine Hanning Kaiser-Bessel 16 File Number

17 Athena plotting in R-space 17 File Number

18 Parts of the Fourier transform χ(k) k 2 (Å -2 ) FT(χ(k) k 2 ) (Å -3 ) χ(k) data and FT window k (Å -1 ) magnitude of FT χ(k) = χ(r) Parts of FT(χ(k) k 2 ) (Å -3 ) Re(FT(χ(k) k 3 )) (Å -4 ) R (Å) Real part of FT χ(k) = Re [χ(r)] R (Å) R (Å) R (Å) k (Å -1 ) Im(FT(χ(k) k 2 )) (Å -3 ) χ(q) and χ(k)k 2 (Å -2 ) Imaginary part of FT χ(k) = Im [χ(r)] back FT of χ(r) = χ(q) The Magnitude of the Fourier transform does not contain as much information as the Real or Imaginary parts of the FT. 18 File Number

19 Backward Fourier transform FT(χ(k) k 2 ) (Å -3 ) R (Å) Re[χ(q)] and χ(k)k 2 (Å -2 ) k (Å -1 ) Only the wavelengths that are contained in the back Fourier transform R range are present in the Re[chi(q)] spectra As a larger R range is included the back FT looks more like the original spectra (blue symbols) 19 File Number

20 How to Choose Minimum K of FT µ(e) x Incident x-ray energy (ev) χ(k) k (Å -1 ) k (Å -1 ) Choose Kmin in the region where the background doesn t change rapidly. - Often around 2 to 4 Å -1 - Vary E 0 and plot the resulting χ spectra with low k-weight to determine the best value. 20 File Number

21 Choosing Maximum K-range χ(k) k 2 (Å -2 ) k (Å -1 ) Re [FT(χ(k) k 1 )] (Å -2 ) kmax = 8 Å -1 kmax = 9 Å -1 kmax = 10 Å R (Å) Re [FT(χ(k) k 1 )] (Å -2 ) kmax = 10 Å -1 kmax = 11 Å -1 kmax = 12 Å R (Å) The FT should be smooth and free of ringing To choose Kmax make vary the kmax value and plot the data using the largest k-weight that will be used in modeling Look for ringing in the real or imaginary part of FT In the example above kmax of 10 or 11 Å -1 best 21 File Number

22 Effect of K-weight on FT FT(χ(k) k x ) (Å -x-1 ) O P R (Å) Re (FT(χ(k) k x )) (Å -x-1 ) Kw=1 Kw=2 Kw= R (Å) k (Å -1 ) χ(k).k kw (Å -1-kw ) These spectra have been k-weighted by 1, 2, and 3 and then rescaled so that the first peak in the FT are the same height The higher k-weight values give more importance to the data above 6 Å -1, this emphasizes the signal due to the P neighbor relative to the O in the first shell 22 File Number

23 Fourier transform parameters in Athena 23 File Number

24 Background function overview A good background function removes long wavelength oscillations from χ(k). Constrain background so that it cannot contain oscillations that are part of the data. Long wavelength oscillations in χ(k) will appear as peaks in FT at low R-values FT is a frequency filter use it to separate the data from the background! 24 File Number

25 Separating the background function from the data using Fourier transform Min. distance between knots defines minimum wavelength of background Rbkg Background function is made up of knots connected by 3 rd order splines. Distance between knots is limited restricting background from containing wavelengths that are part of the data. The number of knots are calculated from the value for Rbkg and the data range in k-space. 25 File Number

26 Rbkg value in Athena 26 File Number

27 How to choose Rbkg value R bkg = 2.2 R bkg = 1.0 A Hint that Rbkg may be too large. Data should be smooth, not pinched! An example where background distorts the first shell peak. R bkg should be about half the R value for the first peak. 27 File Number

28 FT and Background function R bkg = 1.0 R bkg = 0.1 An example where long wavelength oscillations appear as (false) peak in the FT 28 File Number

29 Frequency of Background function Data contains this and shorter wavelengths Bkg contains this and longer wavelengths Constrain background so that it cannot contain wavelengths that are part of the data. Use information theory, number of knots = 2 R bkg k/ π 8 knots in bkg using R bkg =1.0 and k = 14.0 Background may contain only longer wavelengths. Therefore knots are not constrained. 29 File Number

30 The EXAFS Equation Photoelectron χ(k) = Σ i χ i (k) with R 0 Scattered Photoelectron (N i S 02 )F i (k) exp(i(2kr i + ϕ i (k)) exp(-2σ i2 k 2 ) exp(-2r i /λ(k)) χ i (k) = Im( ) kr 2 i R i = R 0 + R k 2 = 2 m e (E-E 0 )/ ħ Theoretically calculated values F i (k) effective scattering amplitude ϕ i (k) effective scattering phase shift λ(k) mean free path Starting values R 0 initial path length Parameters often determined from a fit to data N i degeneracy of path S 02 passive electron reduction factor σ i2 mean squared displacement of half-path length E 0 energy shift R change in half-path length 30 File Number

31 More Information Kelly, S D, Hesterberg, D and Ravel, B. Analysis of soils and minerals using X-ray absorption spectroscopy. In Methods of soil analysis, Part 5 -Mineralogical methods; Ulery, A. L., Drees, L. R., Eds.; Soil Science Society of America: Madison, WI, USA, 2008; pp File Number