Introduction to EXAFS III: XANES, Distortions, Debye-Waller, Glitches

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Introduction to EXAFS III: XANES, Distortions, Debye-Waller, Glitches F. Bridges Chalmers 2011 F. Bridges Physics Dept. UCSC, MC2 Chalmers Scott Medling Michael Kozina Brad Car Yu (Justin) Jiang Lisa Downward C. Booth G. Bunker Outline Review Brief introduction to XANES (NEXAFS) Why understand distortions ? Phonon vibrations and correlations Modeling thermal vibrations : Correlated Debye and Einstein models Polarons, Jahn-Teller distortions Glitches – monochromator glitches, sample Bragg peaks.

description

Introduction to EXAFS III: XANES, Distortions, Debye-Waller, Glitches. Scott Medling Michael Kozina Brad Car Yu (Justin) Jiang Lisa Downward C. Booth G. Bunker. F. Bridges Physics Dept. UCSC, MC2 Chalmers. Outline Review Brief introduction to XANES (NEXAFS) - PowerPoint PPT Presentation

Transcript of Introduction to EXAFS III: XANES, Distortions, Debye-Waller, Glitches

Page 1: Introduction to EXAFS  III:         XANES, Distortions, Debye-Waller, Glitches

Introduction to EXAFS III: XANES, Distortions, Debye-Waller, Glitches

F. Bridges Chalmers 2011

F. Bridges Physics Dept. UCSC,

MC2 Chalmers

Scott MedlingMichael KozinaBrad CarYu (Justin) JiangLisa Downward

C. Booth G. Bunker

Outline• Review• Brief introduction to XANES (NEXAFS)• Why understand distortions ?• Phonon vibrations and correlations• Modeling thermal vibrations : Correlated

Debye and Einstein models• Polarons, Jahn-Teller distortions• Glitches – monochromator glitches, sample

Bragg peaks.

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Chalmers 2011 F. Bridges

Review - EXAFS Equations

))()(22sin(]/),([

/),(),(Im)(

))(1)((0)(

,02,0,0

2

2))()(22(,0

22

kkkrrrkFAe

drrerrgrkFAkk

EEE

iciiiii

ik

i

kkkriiii

edge

i

ic

))()(22sin(]/),([)( 02

002 22

kkkrrrkFAekk ck

OMn

Fi(k,r), c , i --calculated using FEFF

σi – width of pair distribution function, g(ro,i,r)

Ai = Ni So2

(ħk)2/2m = E-Eo

))()(22sin(]/),([')( 02

0022)(2 22222

kkkrrrkFNSeAekk cokk

OMnoo

Experimental standard

Simplify to first neighbor peak only (we will fit in r-space – Fourier transform space)Use either:

FEFF to generate a theoretical standard (calculate Fi(k,r), c , i )

or an experimental standard

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XANES

F. Bridges Chalmers 2011

7600 7800 8000 8200 84000.0

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)

Region around edge; typical up to 30 eV above, but can be higher. Includes small pre-edge features at bottom of edge.

7700 7710 7720 7730 7740 7750 77600.0

0.3

0.6

0.9

1.2

1.5

1.8Bulk LSCO

7705 7710 7715 77200.00

0.05

0.10

0.15

0.20

Abs

orpt

ion

(a.u

.)

E (eV)

Abs

orpt

ion

(a.u

.)

E (eV)

0% 15% 20% 30%

“White” line at top of edge varies with environment – from film days, high intensity line would make film “white” on photo.

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How do you describe XANES?

F. Bridges Chalmers 2011

Same matrix elements – dominated by electric dipole transitions, Δℓ = ± 1, but small quadrupole transition sometimes in in pre-edge.K-edges , 1s → np ; often mixture (hybridization) of states on neighboring atoms.L3, L2 edges, 2p → nd or ms; 2p → nd dominate.

Dipole Quadrupole

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Final state of photoelectron difficult to determine

F. Bridges Chalmers 2011

Problem: is it localized (atomic-like), or extended (band-like) ; should it be treated as a scattering process rather than localized or band states ?

Complex theoretical problem

Edge position – usually depends on valence and local environment – bond lengths and symmetry.

Only Qualitative agreement between theory and experiment (numerical calculation are intensive). No theoretical model to do direct fit of XANES --

EXAFS can do quantitative fits

Structure in edge – depends on environment – often little structure for small clusters/molecules. More XANES structure as make cluster large (FEFF8 -- scattering approach) and include multiple scatterings.

J. SOLID STATE CHEM  141   294-297   (1998)

LiMn2O4

FEFF calculations for Mn K-edge

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Other considerations• Core-hole life time 10-15 sec. Some other electron drops into

core-hole and emits a fluorescence photon – ends the process. Entire absorption and multi-scattering of photo-electron takes place within life time. Locally the lattice is frozen on this time scale.

• Core-hole lifetime becomes shorter for higher Z - by uncertainty principle short life time gives an energy broadening of spectra. At Cu edge (9 keV) a few eV, but at high Z, (I or Ba) large broadening. Resonant Inelastic X-ray Scattering (RIXS) can partially avoid this broadening.

• White lines, particularly for L edges, can be very high and can distort fluorescence data (self-absorption); also important to have small particles because if particle size is too large, it distorts white line. Particle diameter < absorption length.

F. Bridges Chalmers 2011

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Empirical approaches

F. Bridges Chalmers 2011

• Edges shift with increasing valence (same number of neighbors) for K edges 3-4 eV per valence unit for L3 edges can be larger 7eV/unit

• Shift appears to be mainly a bond length change – higher valence, shorter the bond length. Estimates of charge transfer are quite small.

• Edge positions differ for different configurations -- tetrahedral vs octahedral.

• For mixed samples (geological) - may be a sum of known compounds. If have good XANES for the references, can fit data to a weighted sum of reference files. Often works well, - can provide relative abundances quickly. Requires that you know all compounds -- if missing one results aren’t reliable.

• Principal component analysis • Based on linear algebra; treat each XANES spectra as a vector. Can one find a

set of vectors (components) such that data is a linear weighted sum of the component? Turns out YES – and can find a minimum number - robust.

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Chalmers 2011 F. Bridges

Distortions -- Motivation• All systems have local distortions from lattice vibrations. In large-unit-cell systems

an atom may be weakly bonded to the rest of the crystal – can have large vibration amplitudes – called a “rattler”. This disorder can strongly scatter thermal phonons and lead to a glass-like, low thermal conductivity.

• Some systems have a Jahn-Teller (JT) distortion – e.g. the six O atoms around Mn+3 in LaMnO3 are not equivalent; there is a distortion with two long bonds and 4 shorter bonds (the four are slightly split). A similar JT splitting is expected for Cu2+. In contrast for CaMnO3, the 6 Mn+4-O bonds are equal within 0.01Ǻ. The competition between distorted and undistorted sites determines the magnetic and transport properties in substituted manganites (La1-xCaxMnO3) which are metallic and ferromagnetic at low T for some concentrations x, but non-metallic and insulating at high T.

• All these properties require knowledge about the broadening of the atom-pair distribution functions – usually described in terms of the width σ. σ2 = sum over all broadening mechanisms

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Chalmers 2011 F. Bridges

Thermal vibrationsSimple example – isolated atom pair

kTEnE

xxxvME

xvME

R

R

2/121

21

21

21

22222

22

Use reduced mass – MR = 1/2m

x is the variation in the bond length, about r0

κ is the spring constant

only one mode of vibration -- Einstein model

m m

at high T

Tlowat

ThighatTkE B

22

2

zero-point motion

κ = MRω2

ħω = kBΘ

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Chalmers 2011 F. Bridges

Split peaks and σWhen one has an unresolved split peak (i.e. small splittings) it contributes to the broadening (See EXAFS book by B. Teo); easiest to see at low T.

r ~ π/(2kmax) to resolve

N

j

oj

Nrr

1

22 )()(

Equal splitting of 6 bonds into two groups (3+3) split by Δr; and each split peak has same σj

δ(σ2) = (Δr/2)2 ; σone peak fit = σj2 + (Δr/2)2

Splitting into three peaks with equal splittings, Δr.

Then δ(σ2) = ((2/3)Δr)2

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F. Bridges

Long Wavelength Acoustic PhononPositive correlations

Short Wavelength Optical Phonon Negative correlations

Different Phonon Modeshttp://physics.ucsc.edu/~bridges/simulations/index.html

Polaron transportation

Motions of neighboring atoms are correlated in

dynamical examples

Chalmers 2011

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Debye-Waller factors, Diffraction, and Correlations

rdkr

kkkrrgekfrNSki

ici

kriii

2)(/222

0)()(22sin)(),()ˆˆ()(

2

2

2)(

21)(

irr

i erg

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

jijiij

jiji

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• Φ is the correlation factor – can be

positive or negative.

Pi

Pj

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Chalmers 2011 F. Bridges

Comparison between Einstein and Correlated Debye models – T Dependence I

(Simple systems)

Thigh at ;//

)2

coth(2

)2

coth(2

22

TkEnergy

Tk

TkM

B

EEB

E

EBRE

Einstein model(local modes, optical modes)

MR – reduced massκ – Spring constant

ΘE – Einstein Temperature

ΘcD – Correlated Debye Temperature; ΘcD =ℏ ωcD/kB

c – effective speed of sound = ωcD/kD

Correlated Debye Model (All modes; sometimes restricted

to Acoustic modes)

;)/(

))/sin((1

)(2

coth32 0

32

ij

ijij

ijBcDR

cD

RcRc

C

dCTkM

cD

Rij is for atom pair ij

At T~0, σ2E(0) = ħ2/(2MRkBΘE)

= kBΘE/(2κ)

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Chalmers 2011 F. Bridges

Temperature Dependence of σ2

Einstein vs Correlated Debye

Einstein, ΘE = 750K

Debye model, ΘD = 950K

Einstein, ΘE = 950K

Some general properties:• For thermal vibrations σ2

thermal vs T has a positive slope; linear with T at high T. Einstein model has

a sharper bend with T.

• Zero-point motion determines σ2thermal

at low T – for Einstein Model, should correlate with appropriate Raman mode .

• If static disorder present (σ2static), produces a rigid vertical shift [σ2 = σ2

static + σ2thermal ].

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Other types of distortions

F. Bridges Chalmers 2011

Contributions to 2 • Thermal phonons – Einstein or Debye

models.• Static distortions – distribution of pair

distances from strains, impurities, etc.• Polarons – a distortion associated with a

partially localized charge.• Jahn-Teller distortions e.g. Mn+3, Cu+2

• Off-center displacements (ferroelectric)• Dynamic distortions – time scale?

Polaron

Distortion produced by charged carrier and follows carrier; dynamic.Can move very fast if charge carrier moves rapidly.If too fast, lattice response is small.

For uncorrelated mechanisms:

σ2total = σ2

thermal + σ2static + σ2

polarons + σ2JT + σ2

off

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UCSC April 2011

The Jahn-Teller (JT) Distortion

A.J. Millis, Nature 392, 147 (1998)

• JT distortions lead to a distribution of Mn-O (or Co-O) bond lengths.

• Mn3+ (LaMnO3) has 3d4 configuration – one eg elect.• The oxygen displacements about Mn3+ for a Jahn–

Teller distortion are indicated by arrows • Assumes one quasi-localized eg electron is present on

Mn; localized for times long compared to optical phonon period (~10-13 sec.).

• Mn4+, 3d3 config., (CaMnO3) - no eg electrons, no JT.• (What happens for La1-xCaxMnO3??)• Cu+ - 3d10 ; Cu+2 - 3d9, JT active

• Simplified energy level diagram; eg and t2g split by crystal field.• Large exchange energy (Hubbard U), so each level can only be singly

occupied.• For only one eg electron, a JT distortion of the surrounding O6 octahedron

can occur spontaneously; this splits the eg doublet by an energy EJT, and lowers total energy by ΔE = -EJT/2 +Estrain

• If ΔE < 0, Jahn-Teller distortions form. F. Bridges

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Chalmers 2011

Glitches in EXAFS spectra

F. Bridges

• Spikes/peaks in EXAFS data that are not part of EXAFS oscillations

• Some obvious, others not; change shape of EXAFS k-space/r-space data and can introduce significant error.

• Several causesHarmonics in X-ray beam or non-uniform samples +multiple diffraction in monochromator crystals .Bragg diffraction from sample (single crystal , crystal substrate,

or oriented thin film).

Need to be able to minimize and/or remove them

Only important when diffraction is possible, and peaks relatively large.Usually not important for soft X-rays < 2 keV; (depends on crystal).Glitches also become very small at high energies E > 20-25 keV.

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Monochromator Glitch library (SSRL)

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Si(220) phi=0 (SSRL BL 7-3)Si(220) phi=90 (SSRL BL 7-3)

http://www-ssrl.slac.stanford.edu/userresources/index.html

Changes in Io intensity as scan energy

Quite extensive library for 111 and 220 crystals

Covers range from 2.5 to ~ 23 keV for some crystals

In practice we choose mono-crystals that have the lowest amplitude glitches in the energy range for our samples.

Page 19: Introduction to EXAFS  III:         XANES, Distortions, Debye-Waller, Glitches

Examples of multiple diffractionsin 2-D (exist in 3-D)

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Desired monochromator Bragg planes

Other sets of planes (Blue and Green) that meet Bragg condition over a tiny rotation angle

Ewald sphere in K-space

Incident beam

2d sin θ = nλ; k = 2π/λ

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Glitches -Variations in ln(Io/I1):Harmonics or non-uniform sample

F. Bridges Chalmers 2011

Assume no harmonics, and uniform t

I1 = Io e-µt

µt = ln(Io/I1)

Non-uniform t - consider i elements:µti = ln(Ioi/I1i) for each element

Harmonics present – IHUsually 2nd (220’s) or 3rd (111) are main harmonics. Intensity above 30 keV small

Io = Io' + IH ; IH/ Io

' small.I1 = Io

'e-µt + α IH

ln(Io/I1) ≈ µt + IH/ Io'(1- αeµt)

Bragg scattering: 2dsinθ = nλn= 1 fundamental; n > 1 harmonics

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Examples of glitches in unfocused beam:multiple diffractions

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Expanded view of large glitch – subtracted scan at 6765 eV from rest of scans

Typical slits; slope of intensity varies as glitch passes. These data collected with small slits 0.2mm

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Chalmers 2011 F. Bridges

Coupling between Beam Intensity non-uniformity and sample non-uniformity

The signal obtained from a detector is an integral over the cross-sectional area of the beam.

2/

2/

),()(2/

2/1

2/

2/

2/

2/

),,(

),,(

b

b

yxtEa

a

b

b

a

ao

dydxeEyxFI

dydxEyxFI

F(x,y,E) is the X-ray flux (I/area)

slit width a, slit height b

μ(E) absorption coefficient

t(x,y) sample thickness.

Simple case: One dimension and assume F(y,E) and t(y) vary linearly with y; t(y ) = to(1 + αy); F(y,E) = Fo(E)(1 + βy)

and μ(E)to αy <<1; exp(-μ(E)toαy) ~ (1- μ(E)toαy)

)12/)(1(

)12/)(1(

))(1)(1(

2)(

2)(

2/

2/

)(1

btEeI

btEebF

dyytEyeFI

otE

o

otE

o

b

bo

tEo

o

o

o

Non-uniformities couple when both Io and t vary spatially

For small α and β, correction 10-3 to 10-4

Page 23: Introduction to EXAFS  III:         XANES, Distortions, Debye-Waller, Glitches

Pinholes and tapers

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Wedge plexiglas sample;t(y) = 1.5 +0.2y( 1 mm slit, t(y) varies from 1.4 to 1.6 mm)Dotted lines in (b) show model

Wedge up Wedge downSingle pinhole in foil.

Note sign change.Sum of +y and –y is almost zero.

Uniform distribution of tiny pinholes has low glitch amplitude.

GG. Li etal. Nucl. Instr. & Meth. A 340, 420 (1994)

Nucl. Instr. & Meth. A 320, 548 (1992).

Page 24: Introduction to EXAFS  III:         XANES, Distortions, Debye-Waller, Glitches

Minimizing Glitches• Make samples as uniform as possible; small variation in thickness and few

pinholes.• Eliminate harmonics (harmonic rejection mirror or “detuning” mono)• Use narrow vertical slits -- reduces glitch amplitude and can improve energy

resolution.• Use different monochromator crystals

• If you have significant glitches, you have a non-uniform sample or significant harmonic contamination

• Small, narrow glitches (1-3 points) can be removed; but be careful F. Bridges Chalmers 2011

Page 25: Introduction to EXAFS  III:         XANES, Distortions, Debye-Waller, Glitches

Sample glitches: single crystals, thin films• For anisotropic single crystals or thin films may want to do

polarized EXAFS with E polarization along different crystal axes or directions; usually detect using fluorescence.

• Over a typical EXAFS scan, usually several Bragg diffractions from sample – reduced absorption in sample. These move if sample is rotated slightly. Series of data sets at several slightly different angles, usually allows removal but time consuming.

• For thin films, if incident beam is Bragg scattered from substrate and passes through sample again – get increased fluorescence. Again will move if sample is rotated slightly.

Don’t use single crystals of cubic materials, use powders!

F. Bridges Chalmers 2011

Page 26: Introduction to EXAFS  III:         XANES, Distortions, Debye-Waller, Glitches

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 eV, 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.

F. Bridges Chalmers 2011

Page 27: Introduction to EXAFS  III:         XANES, Distortions, Debye-Waller, Glitches

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