exafs in theory: an experimentalist’s guide to what it is and how it works
DESCRIPTION
EXAFS in theory: an experimentalist’s guide to what it is and how it works. Corwin H. Booth Chemical Sciences Division Glenn T. Seaborg Center Lawrence Berkeley National Laboratory. Presented at the SSRL School on Synchrotron X-ray Absorption Spectroscopy, May 20, 2008. Topics. Overview - PowerPoint PPT PresentationTRANSCRIPT
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. BoothChemical Sciences Division
Glenn T. Seaborg CenterLawrence Berkeley National
Laboratory
Topics
1. Overview2. Theory
A. Simple “heuristic” derivationB. A real derivation (just for show)C. polarization: oriented vs spherically averaged
3. Experiment: Corrections and Other ProblemsA. Dead time (won’t cover)B. Self-absorptionC. Sample issues (size effect, thickness effect, glitches) (won’t cover)D. Energy resolution (won’t cover)
4. Data AnalysisA. Fitting procedures (Wednesday)B. Fourier conceptsC. Systematic errorsD. “Random” errorsE. 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)
• 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
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
Amplitude envelope
[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
How is final state wave function modulated?
• Assume photoelectron reaches the continuum within dipole approximation:
kr
er
ikr
2)ˆˆ(
How is final state wave function modulated?
• Assume photoelectron reaches the continuum within dipole approximation:
central atom phase shift c(k)
kr
er
ikr
2)ˆˆ(
kr
er
kiikr c )(2)ˆˆ(
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)
kr
er
ikr
2)ˆˆ(
kr
er
kiikr c )(2)ˆˆ(
)(/)(
2)ˆˆ( kRkiikR
ekR
er
c
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)
kr
er
ikr
2)ˆˆ(
kr
er
kiikr c )(2)ˆˆ(
)(/)(
2)ˆˆ( kRkiikR
ekR
er
c
)(/)(
2 ),()ˆˆ( kRkiikR
ekkfkR
er
c
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)
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(
How is final state wave function modulated?
• Assume photoelectron reaches the continuum within dipole approximation:
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
central atom phase shift c(k)
electronic mean-free path (k)
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
Assumed plane wave scattering, curved wave has r-dependence
solution: substitute Feff full curved wave theory
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
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.
222
)cos1( |ˆˆ),(),(
ˆˆ),(ˆˆ|)ˆ( re
r
kfkfre
r
kfkDkP ikr
cikr
)1
(]ˆˆ
)Re[(2)(ˆ)ˆ(4
1)(
33210 rODIe
r
rIIDkkdkPk ikr
The probability of emitting an electron in the direction is given by:k
The absorption is therefore:
kdkfr
I
kdr
kfkfekI
kdkfekI
kdkD
k
cikr
ikr
ˆ|),(|4
)ˆˆ(
ˆ),(),()ˆˆ(
4
1
ˆ),()ˆˆ(4
1
ˆ)ˆˆ(4
)(
22
3
2
cos1
20
where:
-
derivation continued…
• Some, er, “simplifications”:
)(cos|),(|)],0(Im[
12
2
12
1
12
2
12
1)()(
)1)(12(2
1)(
)(cos)(),(
)(cos)()1)(12(
),(),(3
4),(ˆ),(ˆ
2
,1
1
1 ,1
2
cos
,,
ddkfkf
ll
l
lldxxPxxP
elki
kf
Pkfkf
Pkrjle
YYk
llllll
il
lll
lll
lik
ml
lm
lmml
l
• Rewrite I1, I2 and I3
]),0(),([2
ˆˆ
])()()1[(2
ˆˆ
)]())(1([2
)(
12
)(2
2
ˆˆ
1for )sin(1
)( step,key for the now and
])()()())(1[(12
)(2
2
ˆˆ
sin),(cos4
ˆˆ
2
1
2
1
2
1
2
11
2
11
11
0
2
0
cos1
ikrl
ikr
l
ikrl
ikrl
l
il
ikril
ikril
ikril
ikr
l
ll
ll
ll
ll
ll
ikr
ekfekfkr
ri
ekfekfkr
ri
eeleelkri
i
l
kfr
krkrkr
krj
krjilkrjill
kfr
dkfedr
I
derivation continued…
ikri
ll
lcl
ikr
cikr
ekfer
ri
dxxPxilkir
kfer
dkfer
kfI
cl ),()1(
2
ˆˆ
)()1()12)(12(2
1
2
),(ˆˆ
sin),(coscos2
),(
2
1
1
02
),0(Im)ˆˆ(
2
2
3 kfkr
rI
derivation continued…
Finishing derivation, beginning polarization
• Notice (angle w.r.t. polarization): can eliminate certain peaks!
)],(Im[)ˆˆ(
3)( 22
2
2
kfekr
rDDk
cliikr
edge-K afor )](22sin[),(cos3
)],(Im[)ˆˆ(3
)(
2
2
222
2
0
0
kiikrkfkr
kfekr
r
k
cl
iikr cl
L2 and L3 edges appear more complicated
• 2p1/2 or 2p3/2 core hole and a mixed s and d final state
)]}(2sin[2
10
)cos31)]((2sin[2
12
)cos31)]((2sin[1021{|),(|
1021
12
1)(
0
2
22
2
202222
kkr
kkr
kkrkr
kfk
Heald and Stern 1977
polarization vs. spherically averaged
• L2 and L3 mostly d final states (yeah!)
edges L and Lfor )cos31(2
1-
edges L andK for cos3)(cos
)](2sin[),(
)(cos)(
5010
12)cos31(5
10
12
IIIII2
12
2
2
2
2
P
kkrkr
kfPk
Stern 1974Heald 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
dxy z
Sample
point of absorption and fluorescence
incident photon trajectoryflu
ores
cing p
hoton
traje
ctory
EKE
1s
filled 3d
continuum
EF
core hole
unoccupiedstates
0I
It F
2p
8800 9000 9200 9400 9600 98000.0
0.2
0.4
0.6
0.8
1.0
1.2
Cu foil, Cu K edgeN
orm
aliz
ed a
bsor
ptio
n
E(eV)
transmission fluorescence
t=1.4
10
exp I
IF
• 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 fixedd
xy z
Sample
point of absorption and fluorescence
incident photon trajectoryflu
ores
cing p
hoton
traje
ctory
EKE
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:
}4]))1([
]))1(([{2
1
expexp
exp
2
a
a
sin
sin
d
a ed
where
sin1d
e
• In the thick limit (d), this treatment gives:
aa
exp
exp
1
FT
sin
sin
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 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 this 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, S0
2 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 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” 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
• Assuming Ndata are independent data points, and a fit range over k (and r!):
result EXAFS rule" sStern'" 22
resultFourier 2
2
ind
maxmaxdataind
maxmaxdatamax
1212
kr
krkr
rkk
kr
N
krNN
rkNr
kkrr
• Fit degrees of freedom =Nind-Nfit
• Generally should never have Nfit>=Nind (<1)
• But what does this mean? It means that
For every fit parameter exceeding Nind, there is another linear combination of the same Nfit 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 vs. Random error
Systematic vs. Random error
0 1 2 3 4
-6
0
6
data fit
FT o
f k3
(k)
r (Å)
Cp''2Ti(bipy)
Statistics:
freedom) of (degrees
)(
12
22
N
i i
ii dd
constant
2syst
2true
2meas
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 pi
— Fewer assumptions• random errors treated correctly as long as no nearby minima in 2(pi) exist
• systematic errors lumped into an unaccounted shift in <pi>
• 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=(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 of a 2-type test:
—don’t need to know the errors!
F-test how-to
function beta incompletean is ][ and level confidence theis where
]2
,2
[1)(
parametersfit points, datat independen ,with
)(1
)/(
/)(
residual a of definitionproper )(
2,,
0
1
220
20
21
1
22
y,zI
bmnIFFP
mnR
R
b
mn
mnR
bRRF
ddR
x
mnb
N
iii
Example and limitations
• consider 4 different samples with various amount of species TcSx: 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, R0=0.078 to 0.096
—model 1 has only S neighbors and m=10, R0=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 R0 and R1 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 Feff
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