Introduction to the Theory of X-ray Absorption Spectra
J. J. RehrDepartment of Physics,
University of WashingtonSeattle, WA, USA
Supported by the DOE and NIH-SSRL
Introduction to XAFS: Experiment, Theory, Data Analysis
NSLS, Brookhaven National Laboratory Oct 30-Nov 1, 2008
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Acknowledgments
• J. Kas (UW)• M. Prange (UW)• A. Sorini (UW)• Y. Takimoto (UW)• F. Vila (UW)• K. Jorissen (UW, U Antwerp)• A.L. Ankudinov (APD)• R.C. Albers (Los Alamos)• W. Bardyszewski (ITP, Warsaw)• Z. Levine (NIST)• E. Shirley (NIST)• A. Soininen (U. Helsinki) • G. Hug (ONERA/CNRS)• M. Jaouen (U. Poitiers)• S.R. Bare (UOP)• H. Krappe (HMI)
Collaborators Supported by
DOE BES
NIH-SSRL
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Introduction to the Theory of X-ray Absorption Spectra
• GOAL: ab initio Theory of XAS (and EELS)• No adjustable parameters• Accuracy ~ experiment
• GOAL: Quantitative interpretation• Inverse problem: What’s in a spectrum?• Atomic structure, chemistry, …
• TALK:1. EXAFS and XANES Theory, History2. Theoretical parameters: MFP, S0
2, DW Factors, …3. Interpretation of XANES Bayesian Analysis
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``If I can’t calculate it,
I don’t understand it.”
R.P. Feynman
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Full spectrum X-ray Absorption Spectra
Photon energy (eV)
fcc Al
UV X-ray
arXiv:cond-mat/0601242
http://leonardo.phys.washington.edu/feff/opcons
theory vs expt
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1.EXAFS and XANES
X-ray energy ω (eV)
XANES EXAFS Extended X-ray Absorption Fine Structure
EF
Cu K- edge 10 keV x-rays
E
EK
ω
ω photoelectron
core level
M. Newville
X-ray Absorption Near Edge Structure
Cu
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J. J. Rehr & R.C. AlbersRev. Mod. Phys. 72, 621 (2000)
Quantitative
Theory of XAS
FEFF codes
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Curved Wave Scattering Theory feff
Inelastic Losses Σ(E)
EXAFS Debye Waller Factors σ2
1. Quantitative EXAFS Theory
Must extend ground state methods!
Three Key Developments
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cf.Ground-state vs FEFF vs Expt
ground state: no loss, no DW WRONG Amplitudes!
Cu
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Σ(E) replaces Vxc !
Beyond DFT!
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Paradigm shift:
Use Green’s functions not wave functions!
Efficient!
Ψ
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not k-space!
“Real-space KKR”
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REVISED XAFS Formula
χ δ φ λ σΓ ( ) ( )
( ).sin( ( ) ( )) / ( )E S E N
f kkR
kR k k e eeffc
R k k= + + − −02
22 22 2
2 2
Multiple Scattering Expansion: sum over paths Γ
Same form as Stern-Sayers-Lytle BUT
ALL PARAMETERS RENORMALIZED !
N = coordination number R = ½ path lengthExtrinsic losses: λk = XAFS mean free path Intrinsic losses: S0
2 = Many body amplitude factor
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No shift!
Path Expansion 15 paths
Rnn= 2.769 fcc Pt
Theoretical phases accurate distances to < 0.01 Å
Χ(R)
R (Å)
Phase Corrected EXAFS Fourier Transform
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1. XANES Theory Key Developments
● Beyond Ground State Density Functional Theory
● Inelastic losses, self-energy Σ
● Core-hole effects +Σ
Quasi-particle Theory
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FEFF8 ab initio, Relativistic, Self-consistent, Full Multiple Scattering XANES Code
BNCore-hole, SCF potentials
Essential!
89 atom cluster
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XAS and Pt Catalysts: FEFF8 vs. experiment Pt L3-edge Pt L2-edge (S. Bare, UOP)
• Good agreement: Relativistic FEFF8 code reproduces all spectral features, including absence of white line at L2-edge.
• Self-consistency essential: small changes in position of Fermi level strongly affect white line intensity.
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FAST! Parallel Computation FEFFMPI
MPI: “Natural parallelization”
Each CPU does few energies
Lanczos: Iterative matrix inverse
Smooth crossover between
XANES and EXAFS!1/NCPU 18 of 40
2. Theoretical parameters in XAS
A. Self-energy and Mean Free Path
B. Multi-electron excitations
C. Debye Waller factors
D. Core-hole effects
E. Local Field Effects
GOAL: ab initio calculations
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A. ab initio Mean Free Paths
Plasmon-pole model many-pole model
Plasmon-pole model has too much loss!
Cu
Inel
astic
mea
n fre
e pa
th
JJR et al. arXiv:cond-mat/060124120 of 40
B. Intrinsic losses: Multi-electron Excitations
Multi-electron excitations → satellites in A(k,ω)
Explains intrinsic losses
S02=0.9
Energy Dependent Spectral Function A(k,ω)
Beyond quasiparticles!
Quasi-boson Model
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Vn → -Im ε-1(ωn,qn)22 of 40
S02 Many Body Amplitude
• XAS = Convolution with spectral function
• Spectral function
= S02 μqp (ω)
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( ) ⎩⎨⎧
∂∂∂
=ssian03ABINIT/Gau from
Matrix Dynamical1
''
2
21'
'',βα
βαljjljj
ljjl uuE
mmD
( ) ωωβωρμ
σ di 2
coth0
22 hh∫∞
=
( ) ( ){ }recursion Lanczos step6
22
−=
−= ii QDQ ωδωρ
C. Ab initio XAS Debye Waller Factors
e-2σ2 k2
Replaces correlated Debye Model !
many pole model of vibrational response
Phys. Rev. B76, 014301 (2007)
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Ab Initio Debye-Waller Factors in Rubredoxin
Convert Full System into Smaller Model
Path Theory Exp.Fe-S1 2.9 2.8±0.5
Fe-S2 2.9 2.8±0.5
Fe-S3 3.3 2.8±0.5
Fe-S4 3.5 2.8±0.5
σ2 (in 10-3 Å2)
Fe
S
SS
S
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D. BSE - Screened Core Hole
Corrections to final state rule
Static RPA screened core hole: W = ε-1 Vch
Bethe-Salpeter Equation (2-particle Green’s function)
ε2 = Im χ χ = (1 – K χ0) -1 χ0 K = V + W
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Screened core-hole potential
0 1 2 3R (bohr)
-6
-4
-2
0
Pote
ntia
l (H
artre
e)
RPA – Stott Zaremba
Fully screened FEFF8
Unscreened
W metal (Y. Takimoto)
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Expt
BSE*
feff(q)
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E. TDDFT: Local Field Effects
• Transition operator not external x-ray field φext
Must include polarization φ = φext + φinduced
= ε-1 φext
ε = 1 – KL χ0 Dielectric matrixKL = TDDFT Kernel = V + fxc
→ Golden rule with screened matrix elementsModified L3/L2 ratio
Important for soft edges e.g., L-shell29 of 40
TDDFT Calculation of XAS of W*
*Data: Z. Levine et al. J. Research. NIST 108, 1 (2003)
TDDFT
FEFF8 FSR
Data*No adjustable parameters
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FEFF8 / TDDFTA. L. Ankudinov et al. Phys. Rev. B. 67, 115120 (2003)
Local fields
can explain
anomalous
L3/L2 ratio.
L3 L2
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3. Interpretation of XAS• EXAFS measures structure
→ distances, coordination numbers, disorder and vibrations, …
• XANES electronic structureρ(E) = - Im G ~ μ(E)
→ LDOS, charge counts, valence …
• Need NEW TOOLS to analyze XANES
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Cu pDOS vs XANES and XES
XANES vs Projected Density of States (LDOS)
XANES
XESpDOS
Fermi energy EF Final state energy E33 of 40
Pre-edge Peaks and Valence/Coordination
Pre-edge peak height vs coordination XAS vs cluster size
Pre-edge peak
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Bayesian Analysis of EXAFS+XANES
Approach: Minimize
χ2=Σi |μi theory(X) - μi
expt|2
+ xAx (a priori)→ [Q + A] x = b
Q information matrixA a priori matrixb normalized signalx parameters R,N,… μ0
J. Synchrotron Rad. 12,70 (2004)
Natural separation into
Relevant (Q dominates)or
Irrelevant (A dominates)
parameters
Combined fit: XAFS+XANES
and refs therein
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Implementation in FEFFITJ. Kas
• Add-on to FEFFIT (M. Newville et. al.)
• Uses for XAS calculation• Several modes of operation:
– EXAFS only:R-space, k-space (default)– XANES+EXAFS, Multi-Edge; E-Space
• Structural input– Internal coordinates, xyz, DW factors, …
• Background correction using splines
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ab initio XANES Spectrum of Rubredoxin
Ab initio
SPSE FEFF8
without fitting!
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FT of EXAFS Spectrum of Rubredoxin (Fitted)
R=2.26 (.01)
E0=1.3 (0.7)
S02=0.91
σ2=2.8(.5)
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CONCLUSIONS
• EXAFS – Quantitative! Reliable – Parameter free theory
• XANES – Semi-quantitative – Parameter free theory
• NEW: Many body correctionsAb initio Self-energies, Debye-Waller factors, etc.
• NEW: Bayesian EXAFS/XANES analysis codes
• GOALS – Achieved or within sight!-- closer to BLACK BOX analysis
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That’s all folks
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