EXAFS in theory: an experimentalist’s guide to what it is and how it works Presented at the SSRL School on Synchrotron X-ray Spectroscopy Techniques in

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

Presented at the SSRL School on Synchrotron X-ray Spectroscopy Techniques in Environmental and Materials Sciences: Theory and Application, June 2-5, 2009

Corwin H. BoothChemical Sciences Division

Glenn T. Seaborg CenterLawrence Berkeley National

Laboratory

Advanced EXAFS analysis and considerations: What’s under the hood

Presented at the Synchrotron X-Ray Absorption Spectroscopy Summer School , June 28-July 1, 2011

Corwin H. BoothChemical Sciences Division

Glenn T. Seaborg CenterLawrence Berkeley National

Laboratory

Topics

1. Overview2. Theory

A. Simple “heuristic” derivationB. polarization: oriented vs spherically averaged (won’t cover)C. Thermal effects

3. Experiment: Corrections and Other ProblemsA. Sample issues (size effect, thickness effect, glitches) (won’t cover)B. Fluorescence mode: Dead time and Self-absorption (won’t cover)C. Energy resolution (won’t cover)

4. Data AnalysisA. Correlations between parameters (e.g. mean-free path)B. Fitting procedures (After this talk)C. Fourier conceptsD. Systematic errors (won’t cover)E. “Random” errors (won’t cover)F. F-tests

X-ray absorption spectroscopy (XAS) experimental setup

“white” x-rays from

synchrotron

double-crystal monochromator

collimating slits

ionization detectors

I0I1

I2

beam-stop

LHe cryostatsample

reference sample

• sample absorption is given by

t = loge(I0/I1)

• reference absorption is

REF t = loge(I1/I2)

• sample absorption in fluorescence

IF/I0

•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: AE-3+BE-4 (Victoreen).

• Threshold energies E0~Z2, absorption coefficient ~Z4.

1 10 100

0.01

0.1

1

10

M

LIII

, LII, L

I

K

Xenon

(cm

-1)

E (keV)From McMaster Tables 1s

filled 3d

continuum

EF

core hole

unoccupiedstates

X-ray absorption fine-structure (XAFS) spectroscopy

occupied states

2p

filled 3d

continuum

EF

core hole

unoccupiedstates

16800 17000 17200 17400 17600 17800 18000 182001.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

log(

I 1/I0)

E (eV)

U L3 edge

UPdCu4

pre

-ed

ge

“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

E0: photoelectron threshold energy

e- : ℓ=±1

Data reduction: (E)(k,r)

17000 17200 17400 17600 17800 180000.0

0.2

0.4

0.6

0.8

1.0

1.2

a

E(eV)

a=-

pre

0

17000 17200 17400 17600 17800 180001.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

t+

cons

t

E(eV)

data Victoreen-style background

17100 17200 17300

0.0

0.7

1.4

a

E(eV)

data 10*derivative

“pre-edge” subtraction

“post-edge” subtraction

determine E0

• Peak width depends on back-scattering amplitude F(k,r) , the Fourier transform (FT) range, and the distribution width of g(r), a.k.a. the Debye-Waller s.

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

1 2 3 4

-40

-20

0

20

40

16 U-Cu4 U-Pd12 U-Cu

r (Å)

FT o

f k3 (

k)

U LIII

edge

data fit

Envelope = magnitude

[Re2+Im2]1/2

Real part of the complex

transformfunctionon distributi-pair radial a is

)2sin(),()()(

g

drkrrkFrgNki

cii

How to read an XAFS spectrum

3.06 Å

2.93 Å

“Heuristic” derivation

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

2ˆˆ~ irf

fff 0

.....

ˆˆˆˆˆˆ

ˆˆˆˆ~

0

2

0

2

0

2

tohcc

irffriirf

irffirf

00

0

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:

2)ˆˆ( r



kr

er

ikr

2)ˆˆ(



central atom phase shift c(k)

kr

er

ikr

2)ˆˆ(

kr

er

kiikr c )(2)ˆˆ(



electronic mean-free path (k)


kr

er

ikr

2)ˆˆ(

kr

er

kiikr c )(2)ˆˆ(

)(/)(

2)ˆˆ( kRkiikR

ekR

er

c





complex backscattering probability f(,k)

kr

er

ikr

2)ˆˆ(

kr

er

kiikr c )(2)ˆˆ(

)(/)(

2)ˆˆ( kRkiikR

ekR

er

c

)(/)(

2 ),()ˆˆ( kRkiikR

ekkfkR

er

c





complex backscattering probability f(,k)

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

kr

er

ikr

2)ˆˆ(

kr

er

kiikr c )(2)ˆˆ(

)(/)(

2)ˆˆ( kRkiikR

ekR

er

c

)(/)(

2 ),()ˆˆ( kRkiikR

ekkfkR

er

c

kR

eekfk

kR

er

kikirRikkR

kiikR acc )()()()(/

)(2 ),()ˆˆ(



kr

er

ikr

2)ˆˆ(

kr

er

kiikr c )(2)ˆˆ(

)(/)(

2)ˆˆ( kRkiikR

ekR

er

c

)(/)(

2 ),()ˆˆ( kRkiikR

ekkfkR

er

c

kR

eekfk

kR

er

kikirRikkR

kiikR acc )()()()(/

)(2 ),()ˆˆ(

)(/22

)()(222 ),()ˆˆIm( kR

kikikRi

ekfkR

er

ac



complex backscattering probability kf(,k)

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

final interference modulation per point atom!

Assumed both harmonic potential AND k<<1: problem at high k and/or (good to

k of about 1)

Requires curved wave scattering, has r-dependence, use full curved wave theory:

FEFF

Other factors

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

rdkr

kkkrrgekfrNSk

i

ackri

2)(/222

0

)()(22sin)(),()ˆˆ()(

2

2

2

)(

2

1)(

iRr

erg

i

ackkri kr

kkkreekfrNSk

22)(/222

0

)()(22sin),()ˆˆ()(

22

where and S02 is an inelastic loss factor

Some words about Debye-Waller factors

rdkr

kkkrrgekfrNSk

i

ici

kriii

2)(/222

0

)()(22sin)(),()ˆˆ()(

2

2

2

)(

2

1)(

irr

i erg

• Important: EXAFS measures MSD differences in position (in contrast to diffraction!!)

jiiiij

jiii

jijijiij

UUUUσ

PPPP

PPPP)PP(σ

2

2

2

222

22

2222

• Harmonic approximation: Gaussian

• Ui2 are the position mean-squared

displacements (MSDs) from diffraction

Pi

Pj (non-Gaussian is advanced topic: “cumulant expansion”)

Lattice vibrations and Debye-Waller factors

2

222

22

22

with

2

1

2

12

1

2

1

r

r

r

m

xx

xvmE

xvmE

• isolated atom pair, spring constant

mr=(1/mi+1/mj)-1

T

TkE

TTkE

nE

B

B

lowat

high at 2

1

2

2

Classically… Quantumly…

)2coth)(

22

Tk

d

m Bj

rj

Some words about Debye-Waller factors

• The general formula for the variance of a lattice vibration is:

where j() is the projected density of modes with vibrational frequency

Poiarkova and Rehr, Phys. Rev. B 59, 948 (1999).

• Einstein model: single frequency

• correlated-Debye model: quadratic and linear dispersion cD = c kcD

A “zero-disorder” example: YbCu4X

0 1 2 3 4-60

-40

-20

0

20

40

60 F

T o

f k3

(k)

r (Å)

Ag K edge YbCu4Ag

Pd K edge UCu4Pd

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

X-Cu

2 (Å

2 )

data fit X Tl In Cd Ag

0 50 100 150 200 250 3000.000

0.002

0.004

0.006

0.008

0.010

X-Yb

T (K)

2 (Å

2 )

X S02

cD(K) static2(Å)

X/Cu inter-

changeCu 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)%

AB2(T)=static

2+F(AB,cD)

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

XAFS of uranium-bacterial samples

0 1 2 3 4-15

-10

-5

0

5

10

15

U LIII

edge

data at 298 K fit

FT

of

k3 (k

)

r (Å)

Bacteria

• XAFS can tell us whether uranium and phosphate form a complex

• U-Oax are very stiff cD~1000 K

• U-Oax-Oax also stiff! Won’t change with temperature

• U-P much looser cD~300 K: Enhance at low T w.r.t. U-Oax-Oax!

0 1 2 3 4

-14-12-10-8-6-4-202468

101214

FT

of

k3 (k

)

r (Å)

30 K 300 K

SalleiteU L

III edge

Fitting the data to extract structural information

• Fit is to the standard EXAFS equation using either a theoretical calculation or an experimental measurement of Feff

• Typically, polarization is spherically averaged, doesn’t have to be

• Typical fit parameters include: Ri, Ni, i, E0

• Many codes are available for performing these fits:

— EXAFSPAK

— IFEFFIT• SIXPACK

• ATHENA

— GNXAS

— RSXAP

rdkr

kkkrrgekfrNSk

i

ici

kriii

2)(/222

0

)()(22sin)(),()ˆˆ()(

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 Feff(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

0 5 10 15 20-5

0

5

10

Phas

e sh

ift (

rad)

k(Å-1)

2c

a

Co-O, Co K edge

Phase shifts: functions of k

• sin(2kr+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)

0 1 2 3 40

10

20

30

40

50

60

Mag

nitu

de o

f F

T o

f k3

(k)

r (Å)

"CoCaO3" R=1.85 Å

R=3.71 Å

Species identification: phase and magnitude signatures

0 5 10 15 20

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Mag

nitu

de o

f F

T o

f k3

(k)

r (Å)

Co-Mn Co-Co Co-Ba

0 1 2 3 4

-60

-50

-40

-30

-20

-10

0

10

20

30

40

50

60

FT

of

k3 (k

)

r (Å)0 5 10 15 20

-17

-16

-15

-14

-13

-12

-11

-10

-9

-8

-7

-6

-5

-4

-3

a

k (Å-1)

Co-O Co-C

• First example: same structure, first neighbor different, distance between Re and Ampmax 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, E0 generally is allowed to float (vary)• In theory, a single E0 is needed for a monovalent absorbing

species• Errors in E0 act like a phase shift and correlate to errors in

R!consider error in E0: ktrue=0.512[E-(E0+)]1/2

for small , k=k0-[(0.512)2/(2k0)] eg. at k=10Å-1 and =1 eV, r~0.013 Å

• This correlation is not a problem if kmax is reasonably large

• Correlation between N, S02 and is a much bigger problem!

1 2 3 4

-40

-20

0

20

40

16 U-Cu4 U-Pd12 U-Cu

r (Å)

FT o

f k3 (

k)

U LIII

edge

data fit

Information content in EXAFS

• k-space vs. r-space fitting are equivalent if each is 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 typically incorrect: must Fourier filter data over r-range

• All knowledge from spectral theory applies! Especially, discrete sampling Fourier theory…

Fourier concepts

• highest “frequency” rmax=(2k)-1 (Nyquist frequency)

eg. for sampling interval k=0.05 Å-1, rmax=31 Å

• for Ndata, discrete Fourier transform has Ndata, too! Therefore…

FT resolution is R=rmax/Ndata=/(2kmax), eg. kmax=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!!

0 5 10 15 20-0.2

-0.1

0.0

0.1

0.2

exp(

-2k2

2 )sin

(2kr

)

k

r=2.0, =0.1 r=2.1, =0.1 average

0 5 10 15 20

-1.0

-0.5

0.0

0.5

1.0

sin(

2kr)

k

sin(2k2) sin(2k2.1) average

More Fourier concepts: Independent data points

• Spectral theory indicates that each point in k-space affects every point in r-space. Therefore, assuming a fit range over k (and r!):

• Fit degrees of freedom =Nind-Nfit

• Generally should never have Nfit>=Nind (<1)• But what does this mean? It means that:

For Nfit exceeding Nind, there are other linear combinations of Nfit that produce EXACTLY the same fit function

result EXAFS rule" sStern'" ...22

1212

ind

maxmaxmax

1212

tohN

rkNr

kkrr

kr

rkk

k

kr

Stern, Phys. Rev. B 48, 9825 (1993).Booth and Hu, J. Phys.: Conf. Ser. 190, 012028(2009).

not so Advanced Topic: F-test

• F-test, commonly used in crystallography to test one fitting model versus another

F=(12/1)/(0

2/0)0/1R12/R0

2

(if errors approximately cancel)

alternatively: F=[(R12-R0

2)/(1-0)]/(R02/0)

• Like 2 , F-function is tabulated, is given by incomplete beta function• Advantages over a 2-type test:

—don’t need to know the errors!

0 10 20 30 400

50

100

150

200

Stern

=14.0

2-distribution with =14.1±0.1

Fre

qu

en

cy (2 )

2

5000 simulations

Statistical tests between models: the F-test

• F-tests have big advantages for data sets with poorly defined noise levels

— with systematic error, R1 and R0 increase by ~constant, reducing the ratio (right direction!)

020

121

/

/

F0

20

0120

21

/

)/()(

F

b

mn

R

RF

)(1

2

0

1

For tests between data with

~constant error models that share degrees of freedom, Hamilton test with b=1-0

i

fii yyR 2)(

function beta incompletean is ][ and level confidence theis where

]2

,2

[1)( 2,,

y,zI

bmnIFFP

x

mnb

F-test example testing resolution in EXAFS: knowledge of line shape huge advantage

• 6 Cu-Cu at 2.55 Å, 6 at 2.65 Å

• kmax=/(2R)=15.7 Å-1

• 1 peak (+cumulants) vs 2 peak fits

• Fits to 18 Å-1 all pass the F-test

• Sn=50 does not pass at 15 Å-1

• Sn=20 does not pass at 13 Å-1

• Sn=4 does not pass at ~11 Å-10 5 10 15 20-30

-20

-10

0

10

20

30

Sn=4.0

k3 (k

)

k (Å-1)

0 5 10 15 20-40

-30

-20

-10

0

10

20

30

40

Sn=20

k3 (k

)

k (Å-1)

0 5 10 15 20-50

-40

-30

-20

-10

0

10

20

30

40

50

Sn=50

k3 (k

)

k (Å-1)

Unfortunately, systematic errors dominate

Example without systematic error:kmax=13 Å-1,0=6.19, 1=7.19R0=0.31R1=0.79=0.999 passes F-test

Example with systematic error at R=2%:

kmax=13 Å-1,0=6.19, 1=7.19R0=0.31+2.0=2.31R1=0.79+2.0=2.79=0.86 does not pass

Finishing up

• Never report two bond lengths that break the resolution 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 (1982).— “X-Ray Absorption: Principles, Applications, Techniques of EXAFS, SEXAFS

and XANES”, ed. by Koningsberger and Prins (Wiley, New York, 1988).• 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); Booth and Hu, J. Phys.: Conf. Ser. 190, 012028(2009).

• 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

Further reading

• Correlated-Debye model:

— Good overview: Poiarkova and Rehr, Phys. Rev. B 59, 948 (1999).

— Beni and Platzman, Phys. Rev. B 14, 1514 (1976).

— Sevillano, Meuth, and Rehr, Phys. Rev. B 20, 4908 (1979).

• Correlated Einstein model

— Van Hung and Rehr, Phys. Rev. B 56, 43 (1997).

Acknowledgements

• Matt Newville (Argonne National Laboratory)

• Yung-Jin Hu (UC Berkeley, LBNL)

• Frank Bridges (UC Santa Cruz)

Documents

EXAFS in theory: an experimentalist’s guide to what it is and how it works Presented at the SSRL School on Synchrotron X-ray Spectroscopy Techniques in