1. Basics of EXAFS data analysis
Shelly Kelly
Argonne National Laboratory, Argonne, IL
Hello, I will be talking on the basics of EXAFS data analysis.
First I would like to give a quick overview of the analysis procedure, and then we will go though the details
This way the big picture will not get lost among the details.
I will show how to set the parameters using Athena and highlight several mistakes commonly made.

2. Investigation of the interactions of U Species at the Bacteria-Geosurface Interface
Bacterium ? ? U ? ?
By way of introduction I wanted to take a minute to tell you about our Molecular environmental science group at Argonne national laboratory. We are interested in studing the interactions of contaminants (for example U) with Fe and Mn oxides and bacteria. Today I will be using some data taken from a system consisting of GreenRust and 1% Cu. GreenRust is an Fe2+/Fe3+ oxyhydroxide. It is formed in anoxic conditions as a byproduct of microbial activity and is what gives wet soil its greenish ting. GR is know to reduce several contaminants for example carbon tetrachloride. The rate of reduction is enhanced dramatically with the addiction of Cu2+. The data that I Therefore we added Cu2+ to a GR suspension to determine it’s role in the reduction process.
O O O O
Fe,Mn Fe,Mn Fe, Mn

3. X-ray-Absorption Fine Structure
Attenuation of x-rays
It= I0e-(E)·x
Absorption coefficient
(E)  If/I0
Here is a cartoon of a typical XAFS experimental setup. All of our measurements were made at the Advanced Photon Source, which provides high energy x-rays. We use a monochromator to select a specific x-ray energy. The x-rays are then spacially defined using some slits. The incident and fluorescent x-ray intensity is monitored using and Ion chamber we can also monitor the transmitted x-ray intensity.
This simple relationship describes the final x-ray intensity which is proportional to the incident x-ray intensity decreased exponentially by the thickness of the sample and its absorption coefficient mu(E). This absorption coefficient is just the probability for an x-ray to be absorbed and is a function of the x-ray energy. If we tune the incident x-ray energy, using the monochromator, to match the energy required to excite a core hole electron from the Cu atoms then the probability for the absorption is greatly increased corresponding to and absorption edge. As the energy of the incident x-ray is increased we find oscillations in the absorption coefficient. For very dilute samples we measure the fluorescent x-ray intensity which is the energy released when the core-hole is filled with an electron.

4. X-ray-Absorption Fine Structure
Photoelectron R0
Scattered
Photoelectron
XAFS is defined as the normalized oscillatory part of the absorption coefficient. Here I show the XAFS signal for 1% Cu2+ added to a greenrust suspension. On the right I show a cartoon of the absorption process. XAFS is a direct consequence of the wave nature of the photoelectron.. This central circle represents a Cu atom which has absorbed an x-ray and excited an core electron leaving behind a core hole. The photoelectron propagates away from the Cu atom as a wave where the crests are represented by the solid circles. As it propagates away it is scattered from the neighboring atoms. The crests of the scatter wave is shown by the dashed circles. The interference between these two waves at the core-hole atom creates the oscillations in the absorption coefficient. The interference depends on the distance R and the wave length of the photoelectron which we change as we increase the incident x-ray energy. Therefore each shell of atoms about the copper atom contributes a sine wave with a frequency related to the distance from the cu and an amplitude that depends on the type and number of atoms.

5. Fourier Transform of (k)
Similar to an atomic radial distribution function
Distance
Number
Type
Structural disorder
Here I show the Fourier transform of the Cu GR sample. Each peak corresponds to shell of atoms about the copper atoms. Here is a cartoon of two dimensional copper metal. Which shows three shells of atoms about each copper. Each corresponding to a peak in the Fourier transform of the data. Therefore the Fourier transform of the data is similar to an atomic radial distribution function the height of the peaks depends on the type and number of atoms at a particular distance and the width depends on the structural disorder of the atoms at that distance.

6. Outline Definition of EXAFS Fourier Transform (FT) of (k)
Energy to wave number
Edge Step
Fourier Transform (FT) of (k)
FT of sine wave is a delta function
FT of a discrete data set
Different parts of a FT and backward FT
FT windows and sills
Information content
Autobk method for constructing the bkg
FT and background (bkg) function
Wavelength of bkg
Fit the bkg
EXAFS Equation
Here is an outline of the rest of the talk. First we will look at the definition of exafs, including how the edge step is defined and the conversion of energy to wave number. Then we will look a FT. The different parts and windows and sills and finally information content. Then I will discuss the Autobk method for construction the background function. Finally I’ll go over the exafs equation which is used to generate the theory used to model the exafs data and extract structural parameters.

7. Definition of EXAFS => (E) = (E) - 0(E) ~ (E) - 0(E) (E)
Normalized oscillatory part of absorption coefficient
Measured Absorption coefficient
Bkg: Absorption coefficient without contribution from neighboring atoms (Calculated)
XAFS signal is defined as the normalized oscillatory part of the absorption coefficient which is just the measured absorption coefficient (shown in blue) minus the absorption coefficient without the contribution from the neighboring atoms (shown in red) divided by the change in the absorption coefficient as a function of energy (which is the change from here to here). Instead of calculating the change in the absorption coefficient as a function of energy we approximate it by taking the single value at E0. This approximation is easily corrected during the modeling of the EXAFS data. Notice that I’ve changed the ascissa from energy to wave number.
Next lets take a closser look at the relationship between the absorption coefficient and the chi data, define e0 and the relationship between energy and wave number.
(E) = (E) - 0(E)
(E)
Evaluated at the Edge step (E0)
~ (E) - 0(E)
(E0)

8. Energy to wave number E0 k2 = 2 me (E – E0) ~ 3.81 E ħ
Must be somewhere on the edge
~ 3.81 E
Plank’s constant
Mass of the electron
Fermi Energy
This is an enlargement of the absorption coefficient. By looking at this enlargement we can see the relationship between the absorption coefficient and the chi data. The background in red defines the zero line the the chi data. Here the background is below the absorption coefficient so we get a positive contribution here to the chi data. Then the bkg is above the absorption coefficient and we get a negative contribution here and so on. Notice that the abscissa has changed from energy to wave number. This relationship here defines the wave number in terms of energy. The absorption edge is generated by the excitation of core hole electrons. So right at the edge the x-ray energy is just enough to excite a core hole electron…slightly above the edge we have enough energy the excite the core hole electron and the additional energy is given to the core hole electron as kinetic energy. This kinetic energy can be expressed in terms of wave number of the photo electron. Zero wave number is defined at the absorption edge in terms of E zero the Fermi Energy. E0 is user defined and must be somewhere on the edge.
k2 = 2 me (E – E0) ħ

9. Athena
This brings us to Athena the the GUI interface for Ifeffit. Here you will find all the parameters for manipulating EXAFS data. The main pannel is broken into 5 subgroups. The name of the current file, Background parameters, FT parameters, Backward transform parameters and plotting parameters. The right side lists the data files each file is called a group. Here are the plotting buttons.
We will be working through most of the parameters shown on this page. Starting with E0 which is shown here. The x after the value indicates that you can press this button and then pick the value from the graph.
Next we will be looking at the definition of the edge step which is related to the pre-edge and normalization ranges shown here.

10. Absorption coefficient
Pre-edge region 300 to 50 eV before the edge
Edge region the rise in the absorption coefficient
Normalization region 50 to 1000 eV after the edge
Back to our absorption coefficient, I need to define some terminalogy so that we can discuss how the edge step is evaluated. The absorption coefficient can be roughly divided into three regions. The Pre-edge region (here) from 300 to 50 eV before the edge. The Edge region approximately 50 eV before to 50 eV after the rise in the absorption coefficient and the Post-edge or Normalization region (here) from 50 to 1000 eV after the edge.

11. Edge step Pre-edge line 200 to 50 eV before the edge
Normalization 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 normalization lines at E0
The pre-edge line is defined by fitting a line to two points here in the pre-edge region and the normalization line is defined by fitting a line to two points in the post-edge region, the second line is off of this graph. The edge step is the change in the absorption coefficient at the edge and is evaluated by taking the difference of the pre-edge and Normalization lines at E0.

12. Athena
The edge step values is shown here and the pre-edge and normalization ranges are listed here. Notice that the pre-edge and normalization ranges are relative to E0 so that the pre-edge region contains negitive numbers because it is before the edge and the normalization range contains positive numbers because it is after the edge.
Next we will be discussing the Fourier Transfom parameters.

13. Fourier Transform FT of infinite sine wave is a delta function
The Fourier transform of an infinite sine wave is a delta function with a peak at the frequency of the sine wave.
Since we have a peak at 1 the sine wave has a frequency of 2kR. Unfortunately we can not collect an infinite data set.

14. Fourier Transform FT of discrete sine wave is a distorted peak
Localized features in k-space become unlocalized in R-space
And the FT of a sine wave from 0 to 14 inverse angstroms is a distorted peak.
Notice these additional peaks in the FT, they are due to the edges were the sine wave
Ends. These edges are localized in k-space and so they show up in R-space at all frequencies as
Truncation wiggles. These truncation wiggles will distort our signal so we
Want to minimize them.
A FT by definition is evaluated from neg infinity to pos. infinity and since our data
Only goes from 0 to 14 inverse angstroms we multiply the data by the window shown here that
Is zero every where except from 3 to 13 inverse angstroms in this region the window is one.

15. Fourier Transform
Multiplying the sine wave by a window that gradually increases the amplitude of the sine wave smoothes the FT of discrete sine wave is a distorted peak
Using a sine wave instead of a square wave for the window smoothes the FT and minimizes the truncation ripples.

16. Fourier Transform parts
Magnitude
Real part
The Fourier transform of the chi(k) data results in a real and imaginary parts. And from these parts the magnitude
Of the Fourier transform is defined. We can back Fourier transform the magnitude of the FT which is shown here
In Red and is called chi(q). I’ve plotted the Back FT on top of the chi(k) data to show that we have Fourier filtered the
High frequency noise from the data and you can see the effect of the window on the data. Which is shown here by the
Decreased amplitude.
Imaginary part
Back FT

17. Fourier Transform Windows
kmin
kmax dk dk
Welch
Parzen
Sine
Hanning
On this transparency I show 5 of the available FT windows in Athena. The window is in general defined by a Kmin and Kmax values
Which are at the center of the window sill. The sill is defined as the part of the window that goes from zero to one. And the width of the sill is defined as delta k. It is best to choose a node for the kmin and kmax values and the window sill is usually between 1 and 2 inverse angstroms.
Kaiser-Bessel

18. Fourier Transform window sill
dk=0
dk=2.0 Å-1
Here again I show the effect of the window sill on the FT of the data. The data in red has truncation wiggles that effect the amplitude of the data. Where as the data in blue is smoother.
A small sill can distort FT

19. Athena
Here is the k-range which are the kmin and kmax values, here is dk which is the window sill and the window type option.
And here are the same parameters for the backward FT.

20. Information content FT k-range = 2-8 Å-1
The amount of information in the data depends on the k-range and the R-range
Next we are going to get an idea how information content is communicated through a FT. On the left side I show the back FT of some chi(k) data with a very limited data range from 2 to 8 inverse angstroms. On the right side I show the magnitude of the FT. From both of these picture we and see that we have one broad frequency in the chi data. But if we increase the chi data range we will see the signal in the FT become more defined indicating that there is more information. Next I’m going to quickly increase the chi data range.

21. Information content FT k-range = 2-10 Å-1
The amount of information in the data depends on the k-range and the R-range

22. Information content FT k-range = 2-12 Å-1
The amount of information in the data depends on the k-range and the R-range

23. Information content FT k-range = 2-16 Å-1 (k) = sin(2k) + sin(3k)
Here is the data from 2 to 16 inverse angstroms and here is a comparison of the magnitude of the FT of the data using the different chi data ranges. This chi data is simply a sum of two sine waves. From this illustration we can see that the amount of information in the FT of
Some data depends on the k-range. And, For example if we use only the first peak here we have half the information as compared to using both peaks. And we also see that the amount of information or the number of independent points goes like the 2 delta K deltaR divided by pi.
Number of
independent ~ R k
points 

24. Background function overview
A good background function removes long wavelength oscillations from (k).
Long wavelength oscillations in (k) will appear as peaks in FT at less than half the R-value for the first peak.
Constrain background so that it cannot contain wavelengths that are part of the data.
Now we are ready to look at how autobk creates the background function which is shown in Red here on top of the absorption data.
A good background function removes long wavelength oscillations from chi(k). Long wavelength oscillations will appear as peaks in FT at very low R-values. And a good background is contrained from containing wavelengths that are a part of the data. Next we will look at these requirements more closely.

25. FT and Background function
Rbkg= 1.0
Rbkg= 0.1
At the top I show the same absorption data with two different background functions in Red. The first has removed all long wavelengths in the chi(k) data that would appear in the FT at less than 1 angstrom. Hence the Rbkg value of The second has removed long wavelengths in the chi(k) data that would appear in the FT at less than 0.1 angstroms. The chi(k) data is shown here and you can see a very long wavelength in the blue data that has been removed in the red data. And the FT also shows the false peak in the FT of the blue data. So the value for RBK sets the limit to the shortest wavelengh that can be removed from the data by the background.
An example where long wavelength oscillations appear as (false) peak in the FT

26. FT and Background function
Rbkg= 2.2
Rbkg= 1.0
An example where background distorts the first shell peak.
Rbkg should be about half the R value for the first peak.
We can also increase the value for Rbkg so that I does interfere with the data. Here is an example with Rbkg = 1.0 and rbkg = With the rbkg value of 2.2 the background is actually adding signal to the data that is of the same wavelength as the data. See the increased amplitude in the chi(k) data here and here and also in the magnitude of the FT as the first shell peak is too large.
In general Large Rbkg values will show up as a pinched FT at low R-values.

27. 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 Rbkg k / 
9 knots in bkg using Rbkg=1.0 and k = 14.0
Background may contain only longer wavelengths. Therefore knots are not constrained.
This slide shows how the Rbkg value works and constrains the background so that it cannot interfere with the data. At the top I show the chi(k) data in blue in both graphs. On the left I show the first shell wavelength in red. The data contains this wavelength and shorter wavelengths. The shorter wavelengths are shown here and here as modulations to the first shell wavelength. On the right I show the chi(k) data again but this time with the shortest wavelength allowed for the background. So the BKG contains this and all longer wavelengths. Clearly this long wavelength of the bkg function cannot add or subtract any data from the short wavelenghts contained in the data.
The shortest wavelenght for the background is determined by the rbkg value. In this case rbkg = So this wavelength corresponds to a peak in the FT at 1 angstrom. Autobk calcuates the bkg by determining the number of knots which are the number of times the bkg function crosses zero. The knots are spaced equally in k-space and the bkg function is not allowed to cross zero between the knots. So that the background can contain longer wavelengths the knots of the bkg function are not held at zero.
I want to show an example of this in my next slide.

28. Fit the background function
11 knots in bkg =2 Rbkg k using Rbkg = 1.8 and k = 9.7 
Here is the chi data in red with rbkg = 0.1 so that the background has not been properly removed as shown by the low-R peak in the magnitude of the FT also in red. And here in blue is a background function that corresponds to an Rbkg value of 1.8 that actually contains 11 knots equally spaced. As you can see we only needed 3 knots to remove the background properly.
These graphs were created by simultaneously fitting the data and the bkg frunction. As you can see the fit shown in green can easily follow this poorly removed background data. And the curves in Blue show the contribution from the bkg. As a result of this fit you can check the correlations between the bkg parameters and the fit parameters. This is a measure of how much the bkg is effecting the results. And it can also be used to create pretty pictures where the theory follows the data at low r-values.
Knots are not fixed
shortest wave length constrained by Rbkg.
Not yet implemented in Artemis?

29. Athena
Here is the Rbkg parameter in athena and the spline region in both k-space and e-space.
This concludes my talk on the basics of EXAFS data analysis
But before I let you go I was asked to introduce the exafs equation which is used to build theoretical models of the exafs data.

30. The EXAFS Equation ) (k) = i i(k) i(k) = Im( withR0
Photoelectron
Scattered
(k) = i i(k)
with
(NiS02)Fi(k) exp(i(2kRi + i(k)) exp(-2i2k2) exp(-2Ri/(k))
kRi2
i(k) = Im( )
Ri = R0 + R
k2 = 2 me(E-E0)/ ħ
Parameters often determined from a fit to data
Ni degeneracy of path
S02 passive electron reduction factor
i2 mean squared displacement
E0 energy shift
R change in half-path length
Theoretically calculated values
Fi(k) effective scattering amplitude
i(k) effective scattering phase shift
(k) mean free path
R0 initial path length
One of my first slides showed this simple picture of the process that results in the fine structure of the absorption coefficient.
The blue circle represents an atom that has absorbed an x-ray and created a p.e. The p.e. travels as a wave were the crests of the wave are represented by the solid circles. The p.e. scatters from the neighboring atoms and is represented by the dashed circles. And the oscillations in the absorption coefficient are a result of the interference of the outgoing p.e. with the scattered p.e. at the absorbing atom. And the interface modulates as we increase the energy of the incident x-ray and hence decrease the wavelength of the p.e.
The EXAFS equation is the sum of the contributions from each path of the p.e., which is represented here as a sum over I.
The signal from each path can be written like this. The spherical wave die’s off as 1/R2. The amplitude depends on the scattering amplitude F(k). And the number of atoms or degeneracy of the path N. There is also the passive electron reduction factor which accounts for the screening of the rest of the electrons as they contract when the core hole electron is created. This imaginary exponent gives the sine 2kR term which results in a peak at the R-value for each shell. This additional phase shift is usually small and is caused in part by the charge of the neighboring atoms. Then we have a broadening term which accounts for the fact that all the atoms in the shell are not at exactly the same distance but there is some Gaussian distribution of distances about the equilibrium path length r. and finally there is this decreasing exponential due to the mean free path of the photoelectron in the sample.