Transcript XAFS_course2012_3_DataAnalysis

Introduction to X-ray Absorption Spectroscopy:
Data Analysis
K. Klementiev, Alba synchrotron - CELLS
13-15.09.2012
K. Klementiev - Introduction to data analysis
1 of 19
Conventional EXAFS Analysis Steps
• Instrumental corrections (energy calibration, deglitching etc.)
• Subtraction of pre-edge background
• Conversion from energy dependence to wave number dependence (E → k)
• Construction of the post-edge background 0
• Normalization data to edge step
• kw-weighting
• Fourier transform, with optional back Fourier transform
• Determination of experimental errors in EXAFS signal
• Calculation of theoretical scattering amplitudes and phases (ATOMS, FEFF)
or extraction from reference experimental spectra
• Fitting data in k-space or r-space using theoretical or experimental
scattering amplitudes and phases
• Determination of fitting errors
The next slides show the above steps as screenshots of programs
ATOMS, FEFF, VIPER and XANES dactyloscope. See the corresponding manuals.
13-15.09.2012
K. Klementiev - Introduction to data analysis
2 of 19
Conventional EXAFS Analysis Steps
• Instrumental corrections: energy calibration
derivative of absorption
spectrum of a foil measured
simultaneously (15 spectra
are aligned in energy)
• Instrumental corrections: deglitching
13-15.09.2012
K. Klementiev - Introduction to data analysis
3 of 19
Conventional EXAFS Analysis Steps
• Subtraction of pre-edge background
pre-edge background
• Conversion from energy dependence
to wave number dependence:
k2
m
(
E

E
)/
e
0
Positioning of E0 at the
1st derivative maximum of 
13-15.09.2012
K. Klementiev - Introduction to data analysis
4 of 19
Conventional EXAFS Analysis Steps
• Construction of the post-edge background 0
1) through varied knots
These knots are varied in Y
to achieve no FT signal here
2) as smoothing spline
13-15.09.2012
K. Klementiev - Introduction to data analysis
5 of 19
Conventional EXAFS Analysis Steps
• Fourier transform, with optional back Fourier transform
13-15.09.2012
K. Klementiev - Introduction to data analysis
6 of 19
Conventional EXAFS Analysis Steps
• Determination of experimental errors in EXAFS signal
by FT noise at high r:
by uncertainty in 0 knots:
here the experimental
errors are determined
through high-r FT...
here the error bars (noise)
and  envelope (signal)
refer to the right Y-axis
… but be cautious with
end jumps in  function!
They contribute to FT also
in the high-r portion.
13-15.09.2012
K. Klementiev - Introduction to data analysis
7 of 19
Conventional EXAFS Analysis Steps
• Calculation of theoretical scattering amplitudes and phases (ATOMS, FEFF)
Atoms by Bruce Ravel. I use WebATOMS (Google it).
There is a searchable database of input data for ATOMS.
For example, this input was generated from the database:
Visit ATOMS and FEFF web-pages
for more information!
It generates input file for FEFF6 or FEFF8
(save it as feff.inp):
13-15.09.2012
K. Klementiev - Introduction to data analysis
8 of 19
Conventional EXAFS Analysis Steps
• Calculation of theoretical scattering amplitudes and phases (ATOMS, FEFF)
I normally use the following way:
1) create a new folder for the new compound
2) put feff.inp and feff.exe (temporarily) there
3) run FEFF
4) move feff.exe to its usual location
Remember to edit feff.inp beforehand!
change the last '0' to '3'
to get feffNNNN.dat files
Remove or comment out SCF, XANES, FMS,
LDOS: you do not need them for EXAFS.
Put the 2nd parameter of HOLE as less than 0.1;
then S02 will be calculated.
13-15.09.2012
Ready in a second…
K. Klementiev - Introduction to data analysis
9 of 19
Conventional EXAFS Analysis Steps
• Fitting data
An example of the simplest fitting in VIPER (see VIPER manual and examples for more)
13-15.09.2012
K. Klementiev - Introduction to data analysis
10 of 19
Conventional EXAFS Analysis Steps
• Fitting data. Continued
• What if the coordination number is too low (for a reference spectrum)?
Return to FEFF calculations. Edit the feff.inp file:
*
ixc [ Vr Vi ]
EXCHANGE 0 0 “some value here”
Repeat fitting with the new feff files. Typically, 1-2 iterations are enough to get
the right coordination number within the confidence limits.
• What to do with S02?
See VIPER manual for a discussion.
13-15.09.2012
K. Klementiev - Introduction to data analysis
11 of 19
Conventional EXAFS Analysis Steps
• Determination of fitting errors
The number of independent
points in the data:

 NindP
Degree of freedom for
P varied parameters:
The figure
of merit:
2

k

r
N

2
ind

2
exp
pts
N
data
mod
)
ind (
i
i


2
N
i
exp
pts
i
2
N

Postulated: the variate 2 obeys the 2-distribution law with
 degrees of freedom. This can be used to determine i.
In the simplest case (without a priori knowledge) the
posterior probability P(p|d)  exp(2/2).
The 2nd central moments give the fitting errors.
See the VIPER manual/tutorial for more explanations.
On one hand, this topic is quite broad. The fitting errors depend on how you treat the experimental
uncertainties. They may also include a priori knowledge. Its relative weight in comparison to the data can be
considered differently.
On the other hand, many EXAFS works neglect the fitting errors completely. They report some 'standard'
errors, like dr = 0.02 Å and dN = 10%, independently of data quality, signal strength or correlations.
How deeply you want to go into the error analysis depends on you and on the referee of your publication.
13-15.09.2012
K. Klementiev - Introduction to data analysis
12 of 19
Conventional XANES Analysis Steps
•
•
•
•
•
Instrumental corrections (energy calibration, deglitching etc.)
Subtraction of pre-edge background
Normalization data to edge step
Principal component analysis (PCA), target transformation (TT)
Fitting by a linear combination of reference spectra (if TT was successful)
XANES dactyloscope:
13-15.09.2012
K. Klementiev - Introduction to data analysis
13 of 19
Principal Component Analysis
PCA answers the question “How many independent spectra?”
The given here explanation of the PCA/TT methods is not standard but I believe is
more transparent. For standard derivation based on SVD see
[1a] S. R. Wasserman, J. Phys. IV France 7 (1997) C2-203, which followed [1b]
[1b] E. R. Malinowski. Anal. Chem. 49 (1977) 606
[2] T. Ressler, et al., Env. Sci. & Technol. 34 (2000) 950
d
1 ...d
N

D

 ... 



2) make DTD (NN matrix) and find its eigen-pairs i and ei. Always: 1  iN |eiei|
1) N spectra  data matrix
3) for M linearly independent data vectors, this sum can be truncated at i=M and still
1  iM |eiei|.
4) compare D with DiM |eiei|. If they coincide within noise then truncate further.
Finally, M gives the number of independent vectors.
5) alternatively, compare the cut-off part Di=M+1N |eiei| with noise. How strongly it may
deviate from noise? Different spectra have different noise; which noise then to take?
For the answers, see the manual of XANES dactyloscope.
6) Finally, either you know the noise and infer the number of independent spectra or,
inversely, you assume the number of independent spectra and estimate the noise.
13-15.09.2012
K. Klementiev - Introduction to data analysis
14 of 19
PCA: test example 1
This example has two independent spectra (thick red and blue) and a spectrum constructed as
the average of the two plus normal noise with nominal σ = 0.005 (thick magenta). The thin cyan
curves are the PCA-transformed curves. How many independent spectra we have within
this set?
One needs 2 PCs to reproduce all 3 spectra
If we had all 3 spectra independent, the noise level
had to be within these bounds with 95% probability.
13-15.09.2012
K. Klementiev - Introduction to data analysis
15 of 19
PCA: test example 2
This example has 10 spectra which are all artificially constructed from a single spectrum with
added 10 various realizations of normal noise of nominal σ = 0.005.
One needs 1 PC to reproduce all 10 spectra
We have our noise within
these bounds and thus we can
say that with 95% probability
there is 1 PC.
If we had our noise within these
bounds (~1!) then we could tell
that there is 0 PCs and all the
data are just noise.
13-15.09.2012
K. Klementiev - Introduction to data analysis
16 of 19
Target Transformation
With TT one can answer the question whether a particular spectrum
(material) represents a mixture of the basis spectra (materials).
When answered positively, one can try fitting by a linear combination
of the basis spectra.
1) Construct basis matrix B from N basis spectra. Make BTB (NN matrix) and
find its eigenvalues i and eigenvectors ei
2) If the basis spectra are independent then (BTB)-1 exists.
3) The matrix
N
B( B B) B  
T
1
T
i
1
i
B ei ei BT
is an orthogonal projector to the basis space (it is equal to its square, check this).
Hence, if a spectrum S is a linear combination of the basis spectra then
B(BTB)-1BTS = S and inversely.
4) In practice one checks if B(BTB)-1BTS coincides with S within noise. How strongly the
difference is allowed to deviate from noise? For the answer, see the manual of
XANES dactyloscope.
13-15.09.2012
K. Klementiev - Introduction to data analysis
17 of 19
Linear combination fitting: real example
1. Basis spectra: precursor (Pd I) and metallic Pd particles.
2. The spectrum “catalyst 2” shows coincidence with its target transformation.
3. Linear combination fitting of XANES of “catalyst 2” by the two basis spectra.
4. The found linear combination (the same coefficients) found for the XANES spectra
is then successfully applied to EXAFS (see the curves below in k- and r-space).
catalyst 2
x(Pd nano)+(1-x)Pd I
x=0.326(5)
0
K Pd
-1
4
6
8
10
-1
12
14
16
k (A )
2
2
|FT(·k )|
normalized absorption
1
·k
2
1
1
basis spectra:
Pd I
Pd nano
0
24300
24400
energy (eV)
13-15.09.2012
0
2
4
r (A)
6
0
Palladium Nanoparticles immobilized on
Mesoporous Silica Support – New Efficient
Catalysts for Aerobic Alcohol Oxidation in
Supercritical Carbon Dioxide
Z. Hou et al., J. of Catalysis 258 (2008) 315
K. Klementiev - Introduction to data analysis
18 of 19
Conclusion
• Both XANES and EXAFS require careful attention to the data quality. If
necessary, one has to perform instrumental correction: energy calibration
and/or deglitching.
• EXAFS requires preliminary calculations of the scattering amplitudes and
phases with checking them on reference spectra.
• EXAFS fitting typically requires careful attention to the error analysis.
• There are two main classes of applications of XANES:
1) Fingerprint analysis: presence/absence of pre-edge peaks, edge shift,
white line height (see the introductory lecture). This analysis doesn’t
require sophisticated methods.
2) Factor analysis (PCA, TT) and linear combination fitting. This analysis
requires some mathematics and statistics and careful attention to the
estimation of noise. This part has been considered in the present
lecture. The examples can be found among the example projects of
XANES dactyloscope program and in its manual.
In the practical session we will follow all the above analysis steps.
13-15.09.2012
K. Klementiev - Introduction to data analysis
19 of 19