# Presentation: 591442

## EXAFS in theory: an experimentalist’s guide to what it is and how it works 

- Presented at the SSRL School on Synchrotron X-ray Absorption Spectroscopy, May 20, 2008

***Corwin H. Booth***
**Chemical Sciences Division**
**Glenn T. Seaborg Center**
**Lawrence Berkeley National Laboratory**

## Topics

**Overview**
**Theory**
**Simple “heuristic” derivation**
**A real derivation (just for show)**
**polarization: oriented vs spherically averaged**
**Experiment: Corrections and Other Problems**
**Dead time (won’t cover)**
**Self-absorption**
**Sample issues (size effect, thickness effect, glitches) (won’t cover)**
**Energy resolution (won’t cover)**
**Data Analysis**
**Fitting procedures (Wednesday)**
**Fourier concepts**
**Systematic errors**
**“****Random” errors**
**F-tests**

## X-ray absorption spectroscopy (XAS) experimental setup

- “white” x-rays from synchrotron

- double-crystal monochromator

- collimating slits

- ionization detectors

- I0

- I1

- I2

- beam-stop

- LHe cryostat

- sample

- reference sample

-  sample absorption is given by
-      ** ******** ****t = log****e****(I****0****/I****1****)**
-  reference absorption is
-     ** ********REF ****t = log****e****(I****1****/I****2****)**
-  NOTE: because we are always taking relative-change ratios, *detector gains don’t matter!*

## X-ray absorption spectroscopy

**Main features are ****single-electron excitations.**
**Away from edges, energy dependence fits a power law: ********A*****E*****-3****+B*****E*****-4**** (Victoreen).**
**Threshold energies *****E*****0****~*****Z*****2****, absorption coefficient ********~*****Z*****4****.**

- From McMaster Tables

## X-ray absorption fine-structure (XAFS) spectroscopy

**occupied states**

- 2p

- filled 3d

- continuum

- EF

- 

- core hole

- unoccupied
- states

**pre-edge**

**“**_**edge region**_**”:  x-ray absorption near-edge structure (XANES)     near-edge x-ray absorption fine-structure (NEXAFS)**

_**EXAFS region**_**: extended x-ray absorption fine-structure**

- ~E0

***E*****0****: photoelectron threshold energy**

- e- :  l=±1

**“****pre-edge” subtraction**

**“****post-edge” subtraction**

**determine *****E*****0**

## How to read an XAFS spectrum

**Do NOT read this strictly as a radial-distribution function! Must do detailed FITS!**

**How to read an XAFS spectrum**

## “Heuristic” derivation

**In quantum mechanics, absorption is given by “Fermi’s Golden Rule”:**

**Note, this is the same as saying this is the change in the absorption *****per photoelectron***

## How is final state wave function modulated?

**Assume photoelectron reaches the continuum within dipole approximation:**

## How is final state wave function modulated?

**Assume photoelectron reaches the continuum within dipole approximation:**

## How is final state wave function modulated?

**Assume photoelectron reaches the continuum within dipole approximation:**

**central atom phase shift ********c****(k)**

## How is final state wave function modulated?

**How is final state wave function *****modulated*****?**

**Assume photoelectron reaches the continuum within dipole approximation:**

**electronic mean-free path ********(k)**

**central atom phase shift ********c****(k)**

## How is final state wave function modulated?

**Assume photoelectron reaches the continuum within dipole approximation:**

**central atom phase shift ********c****(k)**

**electronic mean-free path ********(k)**

**complex backscattering probability f(********,k)**

## How is final state wave function modulated?

**Assume photoelectron reaches the continuum within dipole approximation:**

**central atom phase shift ********c****(k)**

**electronic mean-free path ********(k)**

**complex backscattering probability f(********,k)**

**complex=magnitude and phase: **
**backscattering atom phase shift ********a****(k)**

## How is final state wave function modulated?

**How is final state wave function *****modulated*****?**

**Assume photoelectron reaches the continuum within dipole approximation:**

**central atom phase shift ********c****(k)**

**electronic mean-free path ********(k)**

**complex backscattering probability *****k*****f(********,*****k*****)**

**complex=magnitude and phase: **
**backscattering atom phase shift ********a****(k)**

**final interference modulation per point atom!**

## Other factors

**Other factors**

**Allow for multiple atoms *****N******i***** in a shell *****i***** and a distribution function function of bondlengths within the shell g(*****r*****)**

**where                                   and *****S*****0****2**** is an inelastic loss factor**

## EXAFS equation derivation

**This “simple” version is from the Ph.D. thesis of Guoguong Li, UC Santa Cruz 1994, adapted from Teo, adapted from Lee 1974. See also, Ashley and Doniach 1975.**

**The absorption is therefore:**

**where:**

******-******

****

## derivation continued...

**Some, er, “simplifications”:**

## Rewrite I1, I2 and I3

**derivation continued...**

## derivation continued...

## Finishing derivation, beginning polarization

**Finishing derivation, beginning polarization**

**Notice ******** (angle w.r.t. polarization): can eliminate certain peaks!**

## L2 and L3 edges appear more complicated

**2p****1/2**** or 2p****3/2**** core hole and a mixed s and d final state**

- Heald and Stern 1977

## polarization vs. spherically averaged

**L****2**** and L****3**** mostly d final states (yeah!)**

- Stern 1974
- Heald and Stern 1977

## Corrections and Concerns

**“****Normal” EXAFS performed on powder samples in transmission**
** ****can tune the thickness**
**Want ********t~<1 and ********t < 3**
**We like stacking strips of scotch tape**
** ****can make a flat sample**
** ****diffraction off the sample not a problem**
**Working with oriented materials: single crystals, films**
** ****usually *****cannot***** get the perfect thickness: too thick**
**fluorescence mode data collection**
**self-absorption can be substantial!**
**dead-time of the detector**

## Fluorescence mode

- 1s

- filled 3d

- continuum

- EF

- 

- core hole

- unoccupied
- states

- 2p

**Fluorescing photon can be absorbed on the way out**
**Competing effects:**
**glancing angle, sample acts very thick, always get a photon, XAFS damped**
**normal-incidence: escaping photon depth fixed**

- L. Tröger , D. Arvanitis, K. Baberschke, H. Michaelis, U. Grimm, and E. Zschech, Phys. Rev. B **46**, 3283 (1992).

## The full correction

**With the above approximation, we can finally write the full correction:**

- where

- In the thick limit (*d*), this treatment gives:

- Booth and Bridges, Physica Scripta **T115**, 202 (2005)** **

- L. Tröger , D. Arvanitis, K. Baberschke, H. Michaelis, U. Grimm, and E. Zschech, Phys. Rev. B **46**, 3283 (1992).

## Correction applied to a 4.6 m Cu foil

- Data collected on BL 11-2 at SSRL in transmission and fluorescence using a 32-element Canberra germanium detector, corrected for dead time.

## Fitting the data to extract structural information

**Fit is to the standard EXAFS equation using either a theoretical calculation or an experimental measurement of F****eff**
**Typically, polarization is spherically averaged, doesn’t have to be**
**Typical fit parameters include: *****R******i*****, *****N******i*****, ***********i*****, *********E******0***
**Many codes are available for performing this fits:**
**EXAFSPAK**
**IFEFFIT**
**SIXPACK**
**ATHENA**
**GNXAS**
**RSXAP**

## FEFF: a curved-wave, multiple scattering EXAFS and XANES calculator

**The FEFF Project is lead by John Rehr and is very widely used and trusted**
**Calculates the complex scattering function F****eff****(k) and the mean-free path ******

-  TITLE   CaMnO3 from Poeppelmeier 1982
-  HOLE 1   1.0     Mn K edge  (  6.540 keV), s0^2=1.0
-  POTENTIALS
-  *   ipot   z  label
-        0   22   Mn
-        1    8   O
-        2   20   Ca
-        3   22   Mn
-  ATOMS
-    0.00000     0.00000     0.00000    0   Mn    0.00000
-    0.00000    -1.85615     0.00000    1   O(1)  1.85615
-    0.00000     1.85615     0.00000    1   O(1)  1.85615
-   -1.31250     0.00000     1.31250    1   O(2)  1.85616
-    1.31250     0.00000    -1.31250    1   O(2)  1.85616
-    1.31250     0.00000     1.31250    1   O(2)  1.85616
-   -1.31250     0.00000    -1.31250    1   O(2)  1.85616
-    0.00000     1.85615    -2.62500    2   Ca    3.21495
-   -2.62500     1.85615     0.00000    2   Ca    3.21495
-   -2.62500    -1.85615     0.00000    2   Ca    3.21495
-    0.00000     1.85615     2.62500    2   Ca    3.21495

## Phase shifts: functions of k

**Phase shifts: functions of *****k***

**sin(2*****kr*****+********tot****(*****k*****))): linear part of ********(*****k*****) will look like a shift in r slope is about -2x0.35 rad ****Å, so *****peak in r will be shifted by about 0.35 *****Å**
**Both central atom and backscattering atom phase shifts are important**
**Can cause CONFUSION: *****sometimes possible to fit the wrong atomic species at the wrong distance!***
**Luckily, different species have reasonably unique phase and scattering functions (next slide)**

**R=1.85 ****Å**

**R=3.71 ****Å**

## Species identification: phase and magnitude signatures

**First example: same structure, first neighbor different, distance between Re and Amp****max**** shifts**
**Note Ca (peak at 2.8 ****Å) ****and C have nearly the same profile**
**Magnitude signatures then take over**
**Rule of thumb is you can tell difference in species within ********Z~2, but maintain constant vigilance!**

## More phase stuff: r and E0 are correlated

**When fitting,********E****0**** generally is allowed to float (vary)**
**In theory, a single ********E****0**** is needed for a monovalent absorbing species**
**Errors in ********E****0**** act like a ****phase shift**** and correlate to errors in *****R*****!**
***	******	*****consider error ******** in *****E*****0****: *****k*****true****=0.512[*****E*****-(*****E*****0****+********)]****1/2**
**	****	****for small ********, *****k*****=*****k*****0****-[(0.512)****2****/(2*****k*****0****)]******** **
**	****	*****eg*****. at *****k*****=10****Å****-1**** and ********=1 eV, *********r*****~0.013 ****Å**
**This correlation is not a problem if *****k*****max**** is reasonably large**
**Correlation between N, S****0****2**** and ******** is a much bigger problem!**

## Information content in EXAFS

**Information content in EXAFS**

**k-space vs. r-space fitting are *****equivalent***** if done correctly!**

**r-range in k-space fits is determined by scattering shell with highest R**
**k-space direct comparisons with raw data (i.e. residual calculations) are incorrect: must Fourier filter data over r-range**
**All knowledge from spectral theory applies! Especially, discrete sampling Fourier theory...**

## Fourier concepts

**highest “frequency” *****r*****max****=********(2*********k*****)****-1**** (Nyquist frequency)**
**	****	*****eg*****. for sampling interval *********k*****=0.05 ****Å****-1****, *****r*****max****=31 ****Å**
**for *****N*****data****, discrete Fourier transform has *****N*****data****, too! Therefore...**
**	****  ****FT resolution is ********R****=*****r*****max****/*****N*****data****=********/(2*****k*****max****), eg. *****k*****max****=15 ****Å****-1****, ********R****=0.1 ****Å**
**This is the ultimate limit, corresponds to when a beat is observed in two sine wave ********R**** apart. IF YOU DON’T SEE A BEAT, DON’T RELY ON THIS EQUATION!!**

## More Fourier concepts

**Assuming *****N*****data**** are independent data points, and a fit range over k (and r!):**

**Fit degrees of freedom ********=*****N*****ind****-*****N*****fit**
**Generally should never have *****N*****fit****>=*****N*****ind**** (********<1)**
**But what does this mean? It means that**
**	****	****For every fit parameter exceeding *****N*****ind****, there is another linear combination ****	****of the same *****N*****fit**** parameters that produces EXACTLY the same fit function**

## Systematic errors: calculations are not perfect!

- Kvitky, Bridges and van Dorssen, Phys. Rev. B **64**, 214108 (2001).** **

## Systematic errors for nearest-neighbor shells are about 0.005 Å in R,     5% in N, 10% in  (Li, Bridges, Booth 1995)
Systematic error sources:
 sample problems (pin holes, glitches, etc.)
 correction errors: self-absorption, dead time, etc.
 backscattering amplitudes
 overfitting (too many peaks, strong correlations between parameters)
Random error sources:
 some sample problems (roughly, small sample and moving beam)
 low counts (dilute samples)

**Systematic error sources:**
** ****sample problems (pin holes, glitches, etc.)**
** ****correction errors: self-absorption, dead time, etc.**
** ****backscattering amplitudes**
** ****overfitting (too many peaks, strong correlations between parameters)**
**Random error sources:**
** ****some sample problems (roughly, small sample and moving beam)**
** ****low counts (dilute samples)**

**Systematic vs. Random error**

## Systematic vs. Random error

**Statistics:**

## Error analysis options

**Use error analysis in fitting code (generally from the covariance matrix)**
** ****Always requires assumptions**
**a single error at all r or k is assumed**
**systematic errors ignored**
**can be useful in conjunction with other methods**
**Collect several scans, make individual fits to each scan, calculate standard deviation in parameters p****i**
** ****Fewer assumptions**
**random errors treated correctly as long as no nearby minima in ********2****(p****i****) exist**
**systematic errors lumped into an unaccounted shift in <p****i****>**
**Best method(?): Monte Carlo**

**EXAFS as a technique is not count-rate limited: It is limited by the accuracy of the backscattering functions**
**This does NOT mean that you should ignore the quality of the fit!**
**DO a Chi2 test, observe whether Chi2=degrees of freedom**
** ****one limit: random noise is large, and you have a statistically sound fit**
** ****other limit: random noise is small, and you will then know how large the problem with the fit is**

## not so Advanced Topic: F-test

**F-test, commonly used in crystallography to test one fitting model versus another**
**	****	****	****	****F=(********1****2****/********1****)/(********0****2****/********0****)********0****/********1********R****1****2****/R****0****2**
**	****	****	****	*****(if errors approximately cancel)***
**	****	****alternatively: ****	****F=[(R****1****2-****R****0****2****)/(********1****-********0****)]/(R****0****2****/********0****)**
**Like ********2**** , F-function is tabulated, is given by incomplete beta function**
**Advantages of a ********2****-type test:**
**don’t need to know the errors!**

## F-test how-to

## Example and limitations

**consider 4 different samples with various amount of species TcS****x****: Are they interconnected?**
** ********r****=4.5-1 ****Å****	********k****=13.3-2 ****Å****-1****, *****n*****=26.8**
** ****model 0 has Tc neighbors and *****m*****=14 parameters, *****R*****0****=0.078 to 0.096**
**model 1 has only S neighbors and *****m*****=10, *****R*****0****=0.088 to 0.11**
**dimension of the hypothesis *****b*****=14-10=4**
**each data set, ******** between 35 and 82%, all together 99.9%**
**Effect of systematic error: increases *****R*****0**** and *****R*****1**** same amount**
**This will decrease the % improvement, making it harder to pass the F-test (right direction!)**
**Failure mode: fitting a peak due to systematic errors in *****F*****eff**

## Finishing up

**Never report two bond lengths that break the rule**
**Break Stern’s rule only with extreme caution**
**Pay attention to the statistics**

## Further reading

**Overviews:**
**B. K. Teo, “EXAFS: Basic Principles and Data Analysis” (Springer, New York, 1986).**
**Hayes and Boyce, *****Solid State Physics***** 37, 173 (192).**
**Historically important: **
**Sayers, Stern, Lytle, *****Phys. Rev. Lett*****. 71, 1204 (1971).**
**History**
**Lytle, *****J. Synch. Rad.***** 6, 123 (1999). (http://www.exafsco.com/techpapers/index.html)**
**Stumm von Bordwehr, Ann. Phys. Fr. 14, 377 (1989).**
**Theory papers of note:**
**Lee, *****Phys. Rev. B *****13, 5261 (1976).**
**Rehr and Albers, *****Rev. Mod. Phys.***** 72, 621 (2000).**
**Useful links**
**xafs.org (especially see Tutorials section)**
**http://www.i-x-s.org/ (International XAS society)**
**http://www.csrri.iit.edu/periodic-table.html (absorption calculator)**

## Further reading

**Thickness effect: Stern and Kim, *****Phys. Rev. B***** 23, 3781 (1981).**
**Particle size effect: Lu and Stern, *****Nucl. Inst. Meth.***** 212, 475 (1983).**
**Glitches:**
**Bridges, Wang, Boyce, *****Nucl. Instr. Meth.***** *****A***** 307, 316 (1991); Bridges, Li, Wang, *****Nucl. Instr. Meth. A *****320, 548 (1992);Li, Bridges, Wang, *****Nucl. Instr. Meth. A***** 340, 420 (1994).**
**Number of independent data points: Stern, *****Phys. Rev. B***** 48, 9825 (1993).**
**Theory vs. experiment:**
**Li, Bridges and Booth, *****Phys. Rev. B***** 52, 6332 (1995).**
**Kvitky, Bridges, van Dorssen, *****Phys. Rev. B***** 64, 214108 (2001).**
**Polarized EXAFS:**
**Heald and Stern, *****Phys. Rev. B***** 16, 5549 (1977).**
**Booth and Bridges, *****Physica Scripta***** T115, 202 (2005). (Self-absorption)**
**Hamilton (F-)test:**
**Hamilton, *****Acta Cryst.***** 18, 502 (1965).**
**Downward, Booth, Lukens and Bridges, *****AIP Conf. Proc.***** 882, 129 (2007). http://lise.lbl.gov/chbooth/papers/Hamilton_XAFS13.pdf**

## A “zero-disorder” example: YbCu4X

| X | S02 | cD(K) |  | static2(Å) |  | X/Cu inter-change |
| --- | --- | --- | --- | --- | --- | --- |
|  |  | Cu | Yb | Cu | Yb |  |
| Tl | 0.89(5) | 230(5) | 230(5) | 0.0004(4) | 0.0005(5) | 4(1)% |
| In | 1.04(5) | 252(5) | 280(5) | 0.0009(4) | 0.0011(5) | 2(3)% |
| Cd | 0.98(5) | 240(5) | 255(5) | 0.0007(4) | 0.0010(5) | 5(5)% |
| Ag | 0.91(5) | 250(5) | 235(5) | 0.0008(4) | 0.0006(5) | 2(2)% |

*******AB*****2****(*****T*****)=********static****2****+*****F*****(*********AB*****,********cD****)**

- J. L. Lawrence et al., PRB 63, 054427 (2000).

## Interference of photoelectron waves

**Interference of outgoing and incoming part of photoelectron modulates absorption coefficient:**
_**Big advantage**_**: Atomic-species specific.**
***Disadvantages*****: very short range (<~5-6 Å), sensitive to multiple scattering, overlapping edges...**

*“**I was brought up to look at the atom as a nice hard fellow, red or grey in colour according  to taste.” *
*- Lord Rutherford*

## X-ray absorption spectroscopy (XAS) experimental setup

- “white” x-rays from synchrotron

- double-crystal monochromator

- collimating slits

- ionization detectors

- I0

- I1

- I2

- beam-stop

- LHe cryostat

- sample

- reference sample

-  sample absorption is given by
-      ** ******** ****t = log****e****(I****0****/I****1****)**
-  reference absorption is
-     ** ********REF ****t = log****e****(I****1****/I****2****)**
-  EXAFS ******(k)=[********(*****k*****)-********0****(*****k*****)]/********0****(*****k*****)**
- NOTE: because we are always taking relative-change ratios, *detector gains don’t matter!*
- Detuning:

**Diffracted flux**

**2******

## The Basic XAS Experiment

**The Basic XAS Experiment**

- SSRL BL 11-2

## Discovery of x-ray absorption fine structure

**First noticed before 1920**
**Many hair-brained (sort-of) explanations...**
**Closest by Kronig (Z. Phys. 70, 317, 1931; 75, 191,1932; 75, 468, 1932)**
**LRO (crystals) utilized gaps (actually a 2****nd****-order effect)**
**SRO (molecules) utilized backscattering photoelectrons**

- Coster and Veldkamp, Z. Phys. 70, 306 (1931).  See also  _X-Rays in Theory and Experiment_ (1935), by Compton and Allison, p. 663.

## The dawn of a new age...

- (*k*)  *N* sin(2*kr* + )

- Sayers, Stern and Lytle, *Phys. Rev. Lett*. **71**, 1204 (1971)

## How is final state wave function modulated?

**How is final state wave function *****modulated*****?**

**Want interference of outgoing and incoming wave:**

**This is the only part that modulates with k!**
**So we only worry about the backscattered wave.**

## EXAFS theory: Spherical waves

**The wave equation:**