Theory of X-Ray Absorption · Quantitative Theory of EXAFS J. J. Rehr and R. C. Albers Rev. Mod....

Theory of X-Ray Absorption Spectroscopy

J. J. Kas, University of Washington

EXAFS 2014, Brookhaven National LaboratoryNovember 13 – 15, 2014

1

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

5

Theory of EXAFS

6

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, …

16

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

20

Real-Space Multiple Scattering

• 𝐺 = 𝐸 − 𝐻 −1 = 𝐺0 + 𝐺0𝑉𝐺0+𝐺0𝑉𝐺0𝑉𝐺0 +⋯

• Muffin-Tin Potential: 𝑉 = 𝑖 𝑣𝑖

21

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

24

Beyond DFT – Quasiparticle Self-Energy Effects

• 𝐺 = 𝐸 − 𝐻 − Σ −1

• Σ: Self-Energy operator Quasi-particles

Quasi-Toddler

25

Core-Hole Interaction• Photo-electron and hole interact

• Self-Consistent calculation without core electron: Final State Rule

• Linear response: 𝑊𝑐ℎ = 𝜖−1𝑣𝑐ℎ

26

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

29

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

31

Thank You!

32

• 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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36

37

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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

42

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

43

Symmetry and Pre/Near-Edge Features

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Symmetry and Pre/Near-Edge Features

Ti K edge XANESBaTiO3

45

Edge Position and Charge Transfer

46

Edge Position and Charge Transfer

EFermiECore

EEdge

EFermiECore

EEdge

47

Edge Position and Charge TransferF. D. Vila

48

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

49

FMS Convergence

• Guess: Inelastic mean free path

50

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

53

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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