Introduction to EXAFS I

Introduction to EXAFS I

Outline• Brief history• Why EXAFS? Some general differences

between local and average structure• What information can one extract?• Experimental set-up and requirements• EXAFS equation• Data reduction • Data analysis

F. Bridges Chalmers 2011

F. Bridges Physics Dept. UCSC,

MC2 Chalmers

Scott MedlingMichael KozinaBrad CarCarley CorradoJin Zhang Yu (Justin) JiangLisa DownwardJohn J. Neumeier T. TysonCollin BroholmSatoru Nakatsuji

Brief History of EXAFS(Extended X-ray Absorption Fine Structure)


• Every atom has well defined absorption steps.

• For a solid, above each step there is structure in the absorption as a function of energy; known since ~ early ’20s-’30s Kossel , Kronig.

• Many explanations proposed – some were multi-electron one was single electron (Kronig).

• Stern and Sayers (Stern’s student) developed a useful model to use the oscillations to understand structure ~ 1970-71 – used Lytle’s data.

• Argument as to who first suggested using Fourier Transforms will never be resolved.

1 10 1000.01

0.1

1

10

M

LIII, LII, LI

K

Xenon

(cm

-1 )

E (keV)

What information does EXAFS provide?

• Can measure bond lengths and further pair distances. Usually for bond lengths, absolute distances to 0.01Å; relative change even smaller.

• Types of neighbors – but need change in Z (4 or more)• Coordination numbers – generally better than 20%; sometimes to

10%. Limited by correlations between parameters. • Disorder/Distortions – static, thermal phonons, lattice distortions

(Jahn-Teller), polarons etc F. Bridges Chalmers 2011

• Local structure about a selected atom type; determined by X-ray energy used. (Moseley’s Law for fluorescence)

• E.g. for Cu use X-ray energies near 9000 eV; Rb, 15,200 eV; Ag, 25,500eV

Bunker

Types of materials/Samples• Solids – usually complex; often distorted• Nanoparticles; small – and doped• Thin films (nano-sized grains)• Amorphous materials• Liquids (and gases); solutions• Need samples to be homogeneous and uniform

in thickness. Uniform thickness – no pinholes, tapers etc., no concentration gradients. (Address in later lecture) Choose appropriate thickness for edge energy. Step height <1.


Experimental set-up for X-ray spectroscopy (solids)


Top View

Side View

Source

Source

Typical set-up for transmission

µt = ln Io/I1

µReft = ln I1/I2

Set-up for fluorescence detection.For low concentrationsµ ~ If/Io

Need very linear detectors and amps. Because use ratios, gains not important, but need dynamic range.

Typical set-up (CH Booth)

“white” x-rays from

synchrotron

double-crystal monochromator

collimating slits

ionization detectors

I0 I1 I2

beam-stop

LHe cryostatsample

reference sample


Windows in detectors – thin Kapton – often aluminized; 50 –75 µm .

• Choose gas in detectors for energy range (He, N2, Ar, Kr, etc.) Can use mixtures to optimize. Not too much absorption in ionization detectors.

• Choose edge/energy; check no other edges overlap

• Prepare uniform sample

Leave gap between Io and sample – and I1 and sample.Keep reference sample away from I1

Reduce higher harmonics(2dsinθ = nλ)

X-ray Absorption SpectroscopyExtended X-ray Absorption Fine structure (EXAFS)

• 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.• Core-hole lifetime ~ 10-15 sec sets response time – a very fast probe – but data

collection slow.

1 10 1000.01

0.1

1

10

M

LIII, LII, LI

K

Xenon

(cm

-1 )

E (keV) From McMaster Tables 1s

filled 3d

continuum

EF

core hole

unoccupiedstates

K-edge

From G. Bunker


EXAFS /XANES overview

7600 7800 8000 8200 8400

-2.4

-2.1

-1.8

-1.5

-1.2 raw data pre-edge fit

a)

Abs

orpt

ion

(a.u

.)

E (eV)7600 7800 8000 8200 8400

0.0

0.3

0.6

0.9

1.2

1.5

1.8 normalized data 7-knots spline

b)

Abs

orpt

ion

(a.u

.)

E (eV)

0 2 4 6 8-0.8-0.6-0.4-0.20.00.20.40.60.8

LaCoO3 r-space data

FFT

( k

(k) )

r (Å)

0 2 4 6 8 10 12 14-0.8-0.6-0.4-0.20.00.20.40.60.8

LaCoO3 k-space data

k(k

)

k (Å-1)

)()()()(

0

0

EEEE

FT

4K

Dotted background constrained by Victoreen Equ. and the step height.

0512.0 EEk

XANESµt = ln Io/I1

Raw absorption data, pre-edge fit


7600 7800 8000 8200 8400

-2.4

-2.1

-1.8

-1.5

-1.2 raw data pre-edge fit

a)

Abs

orpt

ion

(a.u

.)

E (eV)

Co K-edge

• Pre-edge region –absorption from all other material in beam; includes other atom in sample.

• Varies as AE-3+BE-4 (Victoreen) -- tabulated in McMaster tables.

• Using measured step height and Victoreen equation below edge and well above edge, can extract background above edge.

• Keep step height below 1 (between 0.3 and 0.7)

• Need to also know absorption from rest of sample!!

Pre-edge subtracted data (Co K-edge LaCoO3)


7600 7800 8000 8200 84000.0

0.3

0.6

0.9

1.2

1.5


b)

Abs

orpt

ion

(a.u

.)

E (eV)

• Eo – we define as the half step energy.• Slope at high energy agrees with

Victoreen formula.• Errors in slope add (or subtract) to

σ2.• If slope varies from trace to trace

(e.g. in a temperature dependant study) get fluctuations in σ2.

• EXAFS function χ(E) – the oscillations on top of a background function µo – red line.

)()()()(

0

0

EEEE

0512.0 EEk

E-Eo =ħ2k2/2m

Co k-space data


0 2 4 6 8 10 12 14-0.8-0.6-0.4-0.20.00.20.40.60.8

LaCoO3 k-space data

k(k

)

k (Å-1)

• Usually plot as knχ(k); here kχ(k). • Would like kmax as high as

possible – but limited by noise and time to collect data.

• Sum of sine waves of form sin(2kr + Φ(k))

• Take Fourier Transform to get an r-space plot.

i i

bcikkri kr

kkkreekfrNSk i2

2)(/2220

)]()(22sin[),()ˆˆ()(22

r-space (Co)


0 2 4 6 8-0.8-0.6-0.4-0.20.00.20.40.60.8

LaCoO3 r-space data

FFT

( k

(k) )

r (Å)

Peaks in r-space correspond to different shells of neighbors

Usually can fit out to 4-6 Å depending on the structure.

EXAFS peaks shifted from real distances.

Co-O Co-LaCo-Co

More r-space (FT Spectra)Example: cubic ZnS:Cu,Mn


• Fast oscillation – real part , R, of transform

• Imaginary part , I, not shown.• Envelope function ±√(R2

+ I2)• Fit to R and I • Peaks in EXAFS shifted in position by

well known amount - Δr (from phase shifts δc and δb in term

sin[2kr + 2δc(k) + δb(k)]

2δc(k) + δb(k) ≈ -2kΔr + f(k)RR

Zn-S; Mn-SZn-Zn,; Mn-Zn

0 5 10 15 20-5

0

5

10

Phas

e sh

ift (r

ad)

k(Å-1)

2c

a

Co-O, Co K edge

Bunker

Chalmers 2011

X-ray Absorption Spectroscopy

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

- Lord Rutherford

1s

filled 3d

continuumEF

core hole

F. Bridges

Time scale – 10-15 sec.

Simple model for absorption• Use Fermi’s golden rule µ ~ <f |(ε·r)2|i>• Final state f is modified by backscattering +

interference of outgoing and backscattered waves, i.e. f = fo + Δf ; dipole selection rules

• Can write µ = µo(1+χ); or χ = (µ-µo)/µo

µ, µo, and χ are all energy dependent


i i

bcikkri kr

kkkreekfrNSk i2

2)(/2220

)]()(22sin[),()ˆˆ()(22

How is final state wave function modulated?

• Assume photoelectron reaches the continuum within dipole approximation:

2)ˆˆ( r

From CH Booth LBL




krer

ikr

2)ˆˆ(




central atom phase shift c(k)

krer

ikr

2)ˆˆ(

krer

kiikr c )(2)ˆˆ(




electronic mean-free path (k)


krer

ikr

2)ˆˆ(

krer

kiikr c )(2)ˆˆ(

)(/)(

2)ˆˆ( kRkiikR

ekR

erc






complex backscattering probability f(,k)

krer

ikr

2)ˆˆ(

krer

kiikr c )(2)ˆˆ(

)(/)(

2)ˆˆ( kRkiikR

ekR

erc

)(/)(

2 ),()ˆˆ( kRkiikR

ekkfkR

erc






complex backscattering probability f(,k)

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

krer

ikr

2)ˆˆ(

krer

kiikr c )(2)ˆˆ(

)(/)(

2)ˆˆ( kRkiikR

ekR

erc

)(/)(

2 ),()ˆˆ( kRkiikR

ekkfkR

erc

kReekfk

kRer

kirRikkR

kiikR bc )()()(/

)(2 ),()ˆˆ(




krer

ikr

2)ˆˆ(

krer

kiikr c )(2)ˆˆ(

)(/)(

2)ˆˆ( kRkiikR

ekR

erc

)(/)(

2 ),()ˆˆ( kRkiikR

ekkfkR

erc

kReekfk

kRer

kirRikkR

kiikR bc )()()(/

)(2 ),()ˆˆ(

)(/22

)()(222 ),()ˆˆ(Im kR

kikikRi

ekfkR

erac



complex backscattering probability kf(,k)

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

final interference modulation per point atom!


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

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)

2

2

2)(

21)(

iRr

erg

i

bckkri kr

kkkreekfrNSk 22)(/222

0)()(22sin),()ˆˆ()(

22

where and S02 is an inelastic loss factor


Chalmers 2011

Contributions to 2 • Static distortions – distribution of pair distances

from strains, impurities, etc.• Thermal phonons – Einstein or Debye models.• Polarons – a distortion associated with a partially

localized charge.• An (unresolved) split peak – effective is ~ r/2

where r is the peak splitting.

• Atoms A and B displaced, can be static or dynamic.

• is the second moment of the pair -distribution function.

• Primarily sensitive to radial displacements

For uncorrelated mechanisms:

σ2total = σ2

static + σ2thermal + σ2

polarons +

F. Bridges

2

2

2)(

21)(

orr

erg

Definition of σ for Gaussian PDF

Fitting data I


0 2 4 6 8-0.8-0.6-0.4-0.20.00.20.40.60.8

LaCoO3 r-space data

FFT

( k

(k) )

r (Å)

• Parameters for each shell: r, σ, N• Quantities |f(π,k)|e-2r/λ(k) and phases δc, δb calculated using program FEFF (Rehr); or

obtained from experimental function for that atom-pair.• Two global parameters So

2 and ΔEo required when using FEFF; determined for highest amplitude data, usually at low T. ΔEo is the shift between the experimentally defined edge position and edge position where k = 0 in theory. These parameters are usually negligible when using experimental functions

Many codes are available for performing these fits:EXCURVE98 (Diamond England)EXAFSPAK (G. George)IFEFFIT (M. Neville)

-SIXPACK (Sam Webb)-ATHENA (Bruce Ravel)

GNXAS (Italy)RSXAP (Booth, Bridges)

i i

bcikkri kr

kkkreekfrNSk i2

2)(/2220

)]()(22sin[),()ˆˆ()(22


Fitting data IIQuandary: r-space or k-space fitting

LaCoO3 example

• Since FT is a linear operator, if you do fits correctly should get same answers.

• Need to define both an FT range and a range in r-space e.g. 4-14Å-1 and 1.2-2.0 Å.

• Straight forward in r-space for the Co-O peak. Here one fits a model to the real and imaginary parts of FT over a restricted range in r-space.

• k-space?? If fit the model for the Co-O peak to the full k-space data, poor fit.

• The other components (other peaks) act like noise or an oscillating background

• Need to go to r-space, and then back-transform the region of interest (Co-O peak) into k-space.

0 2 4 6 8 10 12 14-0.8-0.6-0.4-0.20.00.20.40.60.8

LaCoO3 k-space data

k(k

)

k (Å-1)

0 2 4 6 8-0.8-0.6-0.4-0.20.00.20.40.60.8

LaCoO3 r-space data

FFT

( k

(k) )

r (Å)

i i

bcikkri kr

kkkreekfrNSk i2

2)(/2220

)]()(22sin[),()ˆˆ()(22

F. Bridges

0 2 4 6 8-0.6

-0.4

-0.2

0.0

0.2

0.4

0.620% bulk LSCO(SA)

FFT

( k

(k) )

r (Å)

Example of a fit – first peak;Co-O

Chalmers 2011

0.0 0.5 1.0 1.5 2.0-0.6

-0.4

-0.2

0.0

0.2

0.4

0.630% Bulk LSCOT = 4 K

FFT

( k

(k) )

r (Å)

Co

O

La/Sr

• Fit r-range, 1.1 – 1.6 Å; k-range 3.3 – 13 Å-

1.• Space group R-3c is used to calculate

theoretical Co-O function using FEFF code.• From the fit, we extract the width, σ, of the

PDF of the Co-O bond; σ = 0.059 Å • Its bond length (1.92 Å) agrees well with

diffraction results, ~ 0.01 Å.• Below 1 Å, often poor agreement – all

errors in background subtractions etc. , end up in this low r range.

• Region between 1.8 and 2.2 Å, interference between 1st and 2nd neighbors

4 K

300 KNearly cubic (trigonal),Co-O-Co 163-167°

Some caveats• Many places to develop systematic

errors - pre-edge subtraction and using multiple splines to obtain µo are two main areas for such errors. Errors in pre-edge subtraction can lead to error in σ.

• Using multiple splines is a type of filter. Want to extract any slow variation in background (part of µo) but don’t want to also filter out part of EXAFS. Termination of spline fit near edge is very important.


7600 7800 8000 8200 84000.0

0.3

0.6

0.9

1.2

1.5


b)

Abs

orpt

ion

(a.u

.)

E (eV)

A spline is a cubic polynomial fit over a restricted range. For multiple splines, match value and slope where two splines join.

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)

More caveats II• Don’t go too low in k-space in choosing the FT range. Remember k =

0.512 (E-Eo)½; so for k = 3 Å-1 , E-Eo = 34.3 3V, and for k = 2 Å-1, E-Eo = 15.3 eV. XANES structure usually extends up to 20-30 eV above edge and sometimes higher, so dangerous to go below k = 3 Å-1. If not sure, do fits for various FT ranges- parameters should not change significantly. If large change in σ, say from kmin = 2.5 and 3 Å-1 then a problem.

• Strong correlations between N and σ. Don’t think of σ as a “throw-away” parameter, even when you are more interested in N and r. σ must be larger than zero-point motion value.

• kn weighting; depends on backscattering atom. Usually k2 or k3 make EXAFS spectra sharper – but be careful of noise at high k.


A view of Monterey bay from above UCSC


Documents

Introduction to EXAFS I