1. Introduction to EXAFS III: XANES, Distortions, Debye-Waller, Glitches F. BridgesChalmers 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.

2. F. BridgesChalmers 2011 Review - EXAFS Equations Review - EXAFS Equations F i (k,r),  c,  i --calculated using FEFF σ i – width of pair distribution function, g(r o,i,r) A i = N i S o 2 (ħk) 2 /2m = E-E o  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 F i (k,r),  c,  i ) or an experimental standard

3. XANES F. BridgesChalmers 2011 Region around edge; typical up to 30 eV above, but can be higher. Includes small pre-edge features at bottom of edge. “White” line at top of edge varies with environment – from film days, high intensity line would make film “white” on photo.

4. How do you describe XANES? F. BridgesChalmers 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. L 3, L 2 edges, 2p → nd or ms; 2p → nd dominate. DipoleQuadrupole

5. Final state of photoelectron difficult to determine F. BridgesChalmers 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) LiMn 2 O 4 FEFF calculations for Mn K-edge

6. 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. BridgesChalmers 2011

7. Empirical approaches F. BridgesChalmers 2011 Edges shift with increasing valence (same number of neighbors) for K edges 3-4 eV per valence unit for L 3 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.

8. F. BridgesChalmers 2011 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 LaMnO 3 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 Cu 2+. In contrast for CaMnO 3, 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 (La 1-x Ca x MnO 3 ) 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. σ 2 = sum over all broadening mechanisms

9. F. BridgesChalmers 2011 Thermal vibrations Simple example – isolated atom pair Use reduced mass – M R = 1/2m x is the variation in the bond length, about r 0 κ is the spring constant only one mode of vibration -- Einstein model m m at high T zero-point motion κ = M R ω 2 ħω = k B Θ

10. F. BridgesChalmers 2011 Split peaks and σ When 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/(2k max ) to resolve Equal splitting of 6 bonds into two groups (3+3) split by Δr; and each split peak has same σ j δ(σ 2 ) = (Δr/2) 2 ; σ one peak fit = σ j 2 + (Δr/2) 2 Splitting into three peaks with equal splittings, Δr. Then δ(σ 2 ) = ((2/3)Δr) 2

11. F. Bridges Long Wavelength Acoustic Phonon Positive correlations Short Wavelength Optical Phonon Negative correlations Different Phonon Modes http://physics.ucsc.edu/~bridges/simulations/index.html Polaron transportation Motions of neighboring atoms are correlated in dynamical examples Chalmers 2011

12. Debye-Waller factors, Diffraction, and Correlations Debye-Waller factors, Diffraction, and Correlations Important: EXAFS measures MSD differences in position (in contrast to diffraction!!) Harmonic approximation: Gaussian U i 2 are the position mean-squared displacements (MSDs) from diffraction Φ is the correlation factor – can be positive or negative. PiPi PjPj

13. F. BridgesChalmers 2011 Comparison between Einstein and Correlated Debye models – T Dependence I (Simple systems) Einstein model (local modes, optical modes) M R – reduced mass κ – Spring constant Θ E – Einstein Temperature Θ cD – Correlated Debye Temperature; Θ cD = ℏ ω cD /k B c – effective speed of sound = ω cD /k D Correlated Debye Model (All modes; sometimes restricted to Acoustic modes) R ij is for atom pair ij At T~0, σ 2 E (0) = ħ 2 /(2M R k B Θ E ) = k B Θ E /(2κ)

14. F. BridgesChalmers 2011 Temperature Dependence of σ 2 Einstein vs Correlated Debye Einstein, Θ E = 750K Debye model, Θ D = 950K Einstein, Θ E = 950K Some general properties: For thermal vibrations σ 2 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 σ 2 thermal at low T – for Einstein Model, should correlate with appropriate Raman mode. If static disorder present (σ 2 static ), produces a rigid vertical shift [σ 2 = σ 2 static + σ 2 thermal ].

15. Other types of distortions F. BridgesChalmers 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: σ 2 total = σ 2 thermal + σ 2 static + σ 2 polarons + σ 2 JT + σ 2 off

16. 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. Mn 3+ (LaMnO 3 ) has 3d 4 configuration – one e g elect. The oxygen displacements about Mn 3+ for a Jahn– Teller distortion are indicated by arrows Assumes one quasi-localized e g electron is present on Mn; localized for times long compared to optical phonon period (~10 -13 sec.). Mn 4+, 3d 3 config., (CaMnO 3 ) - no e g electrons, no JT. (What happens for La 1-x Ca x MnO 3 ??) Cu + - 3d 10 ; Cu +2 - 3d 9, JT active Simplified energy level diagram; e g and t 2g split by crystal field. Large exchange energy (Hubbard U), so each level can only be singly occupied. For only one e g electron, a JT distortion of the surrounding O 6 octahedron can occur spontaneously; this splits the e g doublet by an energy E JT, and lowers total energy by ΔE = -E JT /2 +E strain If ΔE < 0, Jahn-Teller distortions form. F. Bridges

17. Polarons! UCSC April 2011 A charge “dressed” by a local lattice distortion: the distortion follows the charge. Dimeron; e g electron shared between two Mn sites; low JT distortion JT polarons: small lattice distortion in a distorted lattice

18. Glitches in EXAFS spectra F. BridgesChalmers 2011 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 causes Harmonics 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.

19. Monochromator Glitch library (SSRL) F. BridgesChalmers 2011 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 I o 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.

20. Examples of multiple diffractions in 2-D (exist in 3-D) F. BridgesChalmers 2011 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

21. Glitches -Variations in ln(I o /I 1 ): Harmonics or non-uniform sample F. Bridges Chalmers 2011 Assume no harmonics, and uniform t I 1 = I o e -µt µt = ln(I o /I 1 ) Non-uniform t - consider i elements: µt i = ln(I oi /I 1i ) for each element Harmonics present – I H Usually 2 nd (220’s) or 3 rd (111) are main harmonics. Intensity above 30 keV small I o = I o ' + I H ; I H / I o ' small. I 1 = I o ' e -µt + α I H ln(I o /I 1 ) ≈ µt + I H / I o ' (1- αe µt ) Bragg scattering: 2dsinθ = nλ n= 1 fundamental; n > 1 harmonics

22. Examples of glitches in unfocused beam: multiple diffractions F. Bridges Chalmers 2011 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

23. F. BridgesChalmers 2011 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. 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 ) = t o (1 + αy); F(y,E) = F o (E)(1 + βy) and μ(E)t o αy <<1; exp(-μ(E)t o αy) ~ (1- μ(E)t o αy) Non-uniformities couple when both Io and t vary spatially For small α and β, correction 10 -3 to 10 -4

24. Pinholes and tapers F. BridgesChalmers 2011 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 upWedge down Single 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).

25. 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. BridgesChalmers 2011

26. 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. BridgesChalmers 2011

27. More caveats II Don’t go too low in k-space in choosing the FT range. Remember k = 0.512 (E-E o ) ½ ; so for k = 3 Å -1, E-E o = 34.3 eV, and for k = 2 Å -1, E-E o = 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 k min = 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. k n weighting; depends on backscattering atom. Usually k 2 or k 3 make EXAFS spectra sharper – but be careful of noise at high k. F. BridgesChalmers 2011

28. 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