Basics of EXAFS Data Processing 2009 - X-ray Absorption€¦ · Microsoft PowerPoint - Basics of...

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

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Basics of EXAFS Processing

Shelly Kelly

© 2009 UOP LLC. All rights reserved.

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X-ray-Absorption Fine Structure

• Attenuation of x-rays It= I0e-µ(E)·x

• Absorption coefficient µ(E) ∝ If/I0

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X-ray-Absorption Fine Structure

R0

Photoelectron

ScatteredPhotoelectron

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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 http://www.xafs.org/Common_Mistakes

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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

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χ(E) = µ(E) - µ0(E)∆µ(E)

Definition of EXAFS

Measured Absorption coefficient

Evaluated at the Edge step (E0)

~ µ(E) - µ0(E)∆µ(E0)

Bkg: Absorption coefficient without contribution from neighboring atoms (Calculated)

Normalized oscillatory part of absorption coefficient

=>

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• 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

Absorption coefficient

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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 E0

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Athena normalization parameters

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k2 = 2 me (E – E0)ħ

Energy to wave number

E0

Must be somewhere on the edge

Mass of the electron

Plank’s constant

Edge Energy

~ ∆E3.81

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Athena edge energy E0

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• 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

Fourier Transform is a frequency filter

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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

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• 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.

Fourier Transform is a frequency filter

Regularly spaced rippleIndicates a problem

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Fourier Transform

Multiplying the discrete sine wave by a window that gradually increases the amplitude of the data smoothes the FT of the data.

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Fourier Transform Windows

dk

kmin

kmax

dk WelchParzen

Sine

Kaiser-Bessel

Hanning

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Athena plotting in R-space

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Parts of the Fourier transform

• The Magnitude of the Fourier transform does not contain as much information as the Real or Imaginary parts of the FT.

0 2 4 6 8-0.8

-0.4

0.0

0.4

0.8

0 2 4 6 8 10 12

-0.8

-0.4

0.0

0.4

0.8

0 2 4 6 8-0.8

-0.4

0.0

0.4

0 2 4 6 80.0

0.2

0.4

0.6

0.8

0 2 4 6 8 10 12-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 2 4 6 8-0.8

-0.4

0.0

0.4

0.8R

e(FT

(χ(k

)·k3 ))

(Å-4)

R (Å)k (Å-1)

χ(k)

·k2 (Å

-2)

R (Å)

Im(F

T(χ(

k) k

2 )) (Å

-3)

|FT(χ(

k) k

2 )| (Å

-3)

R (Å) k (Å-1)

|χ(q

)| an

d χ(

k)k2 (Å

-2)

Parts

of F

T(χ(

k) k

2 ) (Å-3

)

R (Å)

Real part of FT χ(k) = Re [χ(R)]

magnitude of FT χ(k) = |χ(R)|

Imaginary part of FT χ(k) = Im [χ(R)]

χ(k) data and FT window

back FT of χ(R) = χ(q)

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Backward Fourier transform

• 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)

0 1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

1.2

|FT(χ(

k) k

2 )| (Å

-3)

R (Å)0 2 4 6 8 10 12 14

-0.75

0.00

0.75

1.50

2.25

3.00

k (Å-1)

Re[χ(

q)] a

nd χ

(k)k

2 (Å-2)

4321

4321

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How to Choose Minimum K of FT

• Choose Kmin in the region where the background doesn’t change rapidly.- Often around 2 to 4 Å-1

- Vary E0 and plot the resulting χ spectra with low k-weight to determine the best value.

0 2 4 6 8 10 12

-0.4

-0.2

0.0

0.2

0.4

17150 17200 172500.00.10.20.30.40.50.60.70.80.91.0

k (Å-1)

χ(k)

·k (Å

-1)

µ(E

)·x

Incident x-ray energy (eV)

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Choosing Maximum K-range

• 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

0 2 4 6

-4

-2

0

2

4

0 2 4 6

-4

-2

0

2

4

0 2 4 6 8 10 12-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15 kmax = 8 Å-1

kmax = 9 Å-1

kmax = 10 Å-1

R (Å)R

e [F

T(χ(

k)·k

1 )] (Å

-2)

kmax = 10 Å-1

kmax = 11 Å-1

kmax = 12 Å-1

Re

[FT(χ(

k)·k

1 )] (Å

-2)

R (Å)

χ(k

)·k2 (Å

-2)

k (Å-1)

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Effect of K-weight on FT

• 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

0 2 4 60

5

10

15

0 2 4 6-10

-5

0

5

10

15

0 2 4 6 8 10 12 14-20

-10

0

10

20

R (Å)

|FT(χ(

k) k

x )| (Å

-x-1)

Re

(FT(χ(

k) k

x )) (Å

-x-1)

R (Å) k (Å-1)

χ(k).k

kw (Å

-1-k

w)

Kw=1Kw=2Kw=3

OP

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Fourier transform parameters in Athena

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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-valuesFT is a frequency filter – use it to separate the data from the background!

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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 3rd 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.

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Rbkg value in Athena

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How to choose Rbkg value

An example where background distorts the first shell peak.Rbkg should be about half the R value for the first peak.

Rbkg= 2.2Rbkg= 1.0

A Hint that Rbkg may be too large.Data should be smooth, not pinched!

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FT and Background function

Rbkg= 1.0 Rbkg= 0.1

An example where long wavelength oscillations appear as (false) peak in the FT

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Frequency of Background function

Constrain background so that it cannot contain wavelengths that are part of the data.

Use information theory, number of knots = 2 Rbkg ∆k / π8 knots in bkg using Rbkg=1.0 and ∆k = 14.0

Background may contain only longer wavelengths. Therefore knots are not constrained.

Data contains this and shorter wavelengths

Bkg contains this and longer wavelengths

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The EXAFS Equation

(NiS02)Fi(k) exp(i(2kRi + ϕi(k)) exp(-2σi

2k2) exp(-2Ri/λ(k))kRi

2χi(k) = Im( )Ri = R0 + ∆R

k2 = 2 me(E-E0)/ ħ

Theoretically calculated valuesFi(k) effective scattering amplitudeϕi(k) effective scattering phase shiftλ(k) mean free pathStarting valuesR0 initial path length

Parameters often determined from a fit to dataNi degeneracy of pathS0

2 passive electron reduction factorσi

2 mean squared displacement of half-path length

E0 energy shift∆R change in half-path length

χ(k) = Σi χi(k)

with

R0

Photoelectron

ScatteredPhotoelectron

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More Information

• www.xafs.org

• 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 367.