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Theory of X-Ray Absorption Spectroscopy
J. J. Kas, University of Washington
EXAFS 2014, Brookhaven National LaboratoryNovember 13 – 15, 2014
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Outline
• Introduction to XAS
• Theory of EXAFS
• Theory of XANES
• Key Approximations
• Summary and Conclusion
2
Introduction: What is XAS?
3
Fine Structure – EXAFS and XANES
EFermi4
Qualitative Interpretation of EXAFS
Sayers, Stern, and Lytle 1970 EXAFS Fourier transform-> Shifted Radial distribution
RNN
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Theory of EXAFS
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Theory of EXAFS
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The real EXAFS Equation
• Phase Shifts
• Debye Waller Factors
• Inelastic losses
• Many-Body S02
• Curved Wave Scattering
• Multiple Scattering
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Phase Shifts
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Disorder: Debye-Waller Factors
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Inelastic losses
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Many-Body Effects S02
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Multiple Scattering and Curved Wave Scattering Amplitudes
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χ 𝑘 =
𝑖
𝑁𝑖𝑆02|𝑓𝑖𝑒𝑓𝑓𝑘 |sin[2𝑘 𝑅𝑖 + 𝛿𝑐(𝑘) + 𝛿𝑖(𝑘)]
𝑘𝑅𝑖2 𝑒
−𝑅𝑖λ(𝑘)𝑒−2𝜎𝑖
2𝑘2
χ𝑅
𝑅
Phase corrected EXAFS FT – RDF
Quantitative Theory of EXAFSJ. J. Rehr and R. C. AlbersRev. Mod. Phys. 72, 621 (2000)
X-Ray Absorption: Principles, Applications, Techniques of EXAFS, SEXAFS and XANESD. C. Koningsberger and R Prins(Editors)Chapter 1 – Theory of EXAFS, E. A. Stern
References
Theory of XANES
• Back to Fermi’s Golden Rule
• |i>, |f> are Many-Body states
– Effective Single Particle Theory, DFT, Quasiparticle, etc.
– BSE, TDDFT, CI, CAS SCF, …
– Multiplet theory
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DFT/Quasiparticle Methods
• Basis set methods
– Periodic: Quantum Espresso, WIEN2K, …
– Localized: Stobe, ORCA, …
• Real space grid
– Real Space Grid: FDMNES
• Green’s function methods
– RSMS: FEFF, SPRKKR, …
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SCF
DFT/Quasiparticle Methods
Spectrum: Golden Rule
r (0)
Veff(i)
r (i)
y (i+1) /G(i+1) Þ r(i+1)
r(i+1) = r(i) ?
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Green’s Functions and Absorption
SCF:18
Real-Space Multiple Scattering
• 𝐺 = 𝐸 − 𝐻 −1 = 𝐺0 + 𝐺0𝑉𝐺0+𝐺0𝑉𝐺0𝑉𝐺0 +⋯
• Muffin-Tin Potential: 𝑉 = 𝑖 𝑣𝑖
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Real-Space Multiple Scattering
• 𝐺 = 𝐸 − 𝐻 −1 = 𝐺0 + 𝐺0𝑉𝐺0+𝐺0𝑉𝐺0𝑉𝐺0 +⋯
• Muffin-Tin Potential: 𝑉 = 𝑖 𝑣𝑖
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Real-Space Multiple Scattering
• 𝐺 = 𝐸 − 𝐻 −1 = 𝐺0 + 𝐺0𝑉𝐺0+𝐺0𝑉𝐺0𝑉𝐺0 +⋯
• Muffin-Tin Potential: 𝑉 = 𝑖 𝑣𝑖
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Real-Space Multiple Scattering
• +
• Muffin-Tin Potential: • Scattering Matrix:
• Move to site and angular momentum basisSeparates structural and chemical dependence
Strutural dependence: Potential dependence
Structural dependence: Go
iL, jL '
Chemical dependence: t iL22
Path Expansion and Full Multiple Scattering
• Path Expansion – EXAFS Equation
• Full Multiple Scattering - XANES
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XANES and LDOS
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Beyond DFT – Quasiparticle Self-Energy Effects
• 𝐺 = 𝐸 − 𝐻 − Σ −1
• Σ: Self-Energy operator Quasi-particles
Quasi-Toddler
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Core-Hole Interaction• Photo-electron and hole interact
• Self-Consistent calculation without core electron: Final State Rule
• Linear response: 𝑊𝑐ℎ = 𝜖−1𝑣𝑐ℎ
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FEFF: Key Approximations
• Spherical muffin-tin potentials
• Local Density approximation
• Quasi-particle approximation
• Core-hole treatment
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When might approximations break down?
• Spherical potentials: non-symmetric systems, water, graphene, benzene, …
• Treatment of the core hole
• Self-energy approximations
• Many-body effects– Charge transfer excitations: transition metal
oxides, cuprates, …
– Multiplet effects: L-edges in transition metals, L/M edges in f-state systems
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Muffin-Tin Approximation
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
13470 13480 13490
m(E
)
E (eV)
FEFF-FPExperiment
Muffin-Tin
Br2
-10
-8
-6
-4
-2
0
-3 -2 -1 0 1 2 3
V(x
)
x
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Many-Body Effects: Charge Transfer
NiO CoO
Calandra et al, PHYSICAL REVIEW B 86, 165102 (2012) 30
Conclusions
• RSGF method– Pros
• No restriction on symmetry
• Large systems
• Wide energy range
• Relativistic/All electron
• Can include self-energy effects
• Easy to use
– Cons• Spherical muffin tin approximation
• Difficult to incorporate many-body effects
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Thank You!
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• Rehr group members– John Rehr– Fernando Vila– Kevin Jorrissen– Egor Clevac– Andrew Lee– Shauna Story
Supported by DOE DE-FG03-97ER45623 and the DOE CMCSN throughGrant No. DE-FG02-08ER46540
Workshop
• http://leonardo.phys.washington.edu/feff/BNL_XANES_WORKSHOP_2014.zip
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Theory of RIXS
• Core excitation: similar to XANES
• Probes unoccupied and occupied states
• Many-Body effects more evident
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1J.J. Kas et. al, PRB 83, 23114 (2011)
2Fujikawa et al., J. Electron Spectrosc. Relat. Phenom. 134, p 195 (2004)39
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XANES is sensitive to …
• Unoccupied local DOS
• Symmetry
• Charge transfer (oxidation state)
• Occupation (L-edges)
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Eu2O3 L3 XANES
0 1 2 3 4 5 6960 6970
6980 6990
7000 7010
7020 7030
m(E)
E (eV)
Thy.HERFD XAS
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Eu2O3 L3 XANES
0 1 2 3 4 5 6960 6970
6980 6990
7000 7010
7020 7030
m(E)
E (eV)
f-DOSd-DOS
Thy.HERFD XAS
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Symmetry and Pre/Near-Edge Features
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Symmetry and Pre/Near-Edge Features
Ti K edge XANESBaTiO3
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Edge Position and Charge Transfer
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Edge Position and Charge Transfer
EFermiECore
EEdge
EFermiECore
EEdge
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Edge Position and Charge TransferF. D. Vila
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Simulation of XANES with FEFF
• Structure – coordinates, species, vibrational character
• Important options for XANES calculations
– Self-consistent potentials
– Core hole treatment
– Self-energy
• Convergence: FMS/SCF radii, Maximum angular momentum
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FMS Convergence
• Guess: Inelastic mean free path
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SCF Convergence
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Lmax Convergence
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Potentials in FEFF
• Symmetry enforced using “unique potentials”
• Add unique potentials to relax restraints on symmetry
• Potential of absorbing atom is always unique
– Must average spectrum over absorbers
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Augmenting FEFF with DFT
• Find possible structures
• Check full potential effects
• Check accuracy of Fermi level
• Fine chemical shifts
• Find dynamical properties
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