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1. Developments in the Theory of X-ray Absorption and Compton Scattering Egor Klevak University of Washington May 10, 2016 2. Research topics Charge transfer satellites…
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  • 1. Developments in the Theory of X-ray Absorption and Compton Scattering Egor Klevak University of Washington May 10, 2016
  • 2. Research topics Charge transfer satellites in x-ray spectra of transition metal oxides PRB 89, 085123 (2014) Finite temperature calculations of the Compton profile of Be, Li and Si. To be submitted to PRB Compton profile of Be in WDM regime, role of disorder 0 10 20 30 40 50 60 ω (eV) CoO(c) 5.3 eV Exp. This work FEFF Intensity(arb.units) 7.4 eV (b) NiO Exp. This work FEFF (a) 7.2 eV CuO Exp. This work FEFF 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Si liquid 1787K (a) J(p)(a.u.-1 ) Our theory Exp. Okada et al. Exp. Matsuda et al. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 Si solid 298K (b) J(p)(a.u.-1 ) p (a.u.) 0 2 4 6 8 10 -4 -3 -2 -1 0 1 2 3 4 850K 650K 300K 40K Be 1.815 g/cm3 1.832 g/cm 3 1.856 g/cm3 1.856 g/cm3 J(pq)(10 -2 a.u. -1 ) pq (a.u.) Our theory Huotari Theory Huotari Exp. FEFF 0K 0.2 0.4 0.6 0.8 1.0 1.2 1.4 p/pf 0 1 2 n(p) E. Klevak (Final Exam) Thermal disorder and CP May 2016 2 / 58
  • 3. Goals Compton profile at conventional temperatures: Material properties Electronic properties Chemical bonding Many body effects Study of Warm Dense Matter using Compton profile EOS from Compton profile Compton profile: CP Warm Dense Matter: WDM E. Klevak (Final Exam) Thermal disorder and CP May 2016 3 / 58
  • 4. WDM: ICF Inertial Confinement Fusion (outer layer oblates interior implodes) Rapidly heat outer layer of target (Be or C-H plastic coating) using 500 TW, 20 ns laser pulse. Heat and compress for fusion to occur E. Klevak (Final Exam) Thermal disorder and CP May 2016 4 / 58
  • 5. WDM: ICF Some of the challenges Laser energy Compressing Heating Ionization EOS difficult to measure or model in WDM regime Need method to extract EOS density temperature ionization E. Klevak (Final Exam) Thermal disorder and CP May 2016 5 / 58
  • 6. WDM: Diagnostic techniques System of interest: t ∼ 1 ns and T > 10000 K Non-contact thermometry Black-body radiation (Stefan-Boltzmann Law P = σT4) Optical Thomson scattering (Doppler broadening of elastic scatter) High energy probe Low energy (optical) probes can not penetrate or escape the sample. Use bulk sensitive high energy (x-ray) probe E. Klevak (Final Exam) Thermal disorder and CP May 2016 6 / 58
  • 7. WDM: X-ray based diagnostics Probe indirectly: Non-resonant inelastic X-ray Scattering (NRIXS) Light elements (H, Li, Be, C),ideal for low Z elements or low binding energies Only works for heavy elements (Z > 10): X-ray Absorption and Emission, binding energies > 1 keV X-ray absorption spectroscopy (XAS) Resonant inelastic X-ray scattering (RIXS) X-ray fluorescence (XRF) Scattered Energy (keV) Glenzerm et al PRL 90, 175002 (2003) Cho et al. PRL 106, 167601 (2011) WD Cu L-edge XAS Inferring temperature from the shape of edge(MD-DFT) Dorchies et al. PRB 92, 085117 (2015) XAS K-edge of WD Al, diagnostic of electronic temperature Ciricosta et al. PRL 109, 065002 (2012) determination of ionization state in core-shell binding energies Al Kα RIXS Hoarty, et al. PRL 110, 265003 (2013) Ionized Al Kβ XRF Observed delocalization of 3p states as density increases (continuum lowering) E. Klevak (Final Exam) Thermal disorder and CP May 2016 7 / 58
  • 8. Non resonant inelastic x-ray scattering (NRIXS) Dynamic structure factor: Allows probing: Low-Z absorption edges Electron momentum distribution “A quantum theory of the scattering of x-rays by light elements” Compton, Phys. Rev 21 483 (1923) “Experimental Confirmation for Sommerfeld-Fermi-Dirac Degenerate Gas Theory of Conduction Electrons” DuMond, Science 68 452 (1928) E. Klevak (Final Exam) Thermal disorder and CP May 2016 8 / 58
  • 9. Contributions to NRIXS spectrum Energy transfer (eV) S(q,ω)1/eV 1s binding energy 122 eV Be Z=4 θ = 171 q~ 10 Å 1s electrons core shell Valence electron (2s) Doppler broadened Compton peak Mattern, et al. PRB 85 115135 (2012) E. Klevak (Final Exam) Thermal disorder and CP May 2016 9 / 58
  • 10. Previous work: Mattern et al. All calculations are done in perfect crystal structure Excellent agreement with exp. + theory Solid state effects important at high T and ρ Disorder not included in calculations Mattern, et al. PRB 85 115135 (2012) Mattern, et al. arXiv: 1308.2990 (2013) E. Klevak (Final Exam) Thermal disorder and CP May 2016 10 / 58
  • 11. Beyond perfect crytal CP calculations End goal: WDM High temperature High density High distortion Need method to predict disorder: Molecular Dynamics 0 2 4 6 8 10 12 3 4 5 6 7 8 9 Be g(r) 40K 200K 300K 400K 650K 800K 850K 1000K 1400K 1800K 2000K 2500K 0 1 2 3 4 5 6 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Li g(r) r (a.u.) 95K 300K 400K 500K E. Klevak (Final Exam) Thermal disorder and CP May 2016 11 / 58
  • 12. Theory of Compton Scattering Compton Scattering S(q, w) = I,F F| j exp(iq · rj )|I 2 δ(EF − EI − ω) d2σ d Ω d ω2 = d σ d Ω Th S(q, ω) Impulse Approximation S(q, ω) = d3 p ρ(p) δ( q2 2 − p · q − ω)) J(pq) = d3 p ρ(p) δ(p · ˆq − pq) pq ≡ ω/q − q/2, J(pq)d pq = d3 pρ(p) = N Spherically avg case: ρ(p) = ρ(p) J(pq) = 2π ∞ |pq| dp p ρ(p) Eisenberger et al. PRA 2, 415 (1970) E. Klevak (Final Exam) Thermal disorder and CP May 2016 12 / 58
  • 13. Theory of Compton Scattering Compton: Fermi gas model Fermi gas electron density: ρ(p) = (2V /(2π)3)Θ(pF − p) n(p) = ((2π)3 /V )ρ(p) Fermi momentum: pF = (2π3N/V )1/3 J(pq) = 2π ∞ |pq| dp p ρ(p) Compton profile for Fermi Gas: J(pq) = 2 π2 (p2 F − p2 q)Θ(pF − p) E. Klevak (Final Exam) Thermal disorder and CP May 2016 13 / 58
  • 14. RSGF Green’s function: L expansion J(pq) = dx dy dz eıpq(z−z ) ρ(r, r ) ρ(r, r ) = − 2 π Im ∞ Ec dEG(r, r , E)fT (E) G(r, r , E) = −2k δn,n L HE Ln(r>)¯RE Ln (r<) + L,L RE Ln(rn) eıδLn gE Ln,L n eıδL n ¯RE L n (r n )   RE Ln(rn),HE Ln(rn): regular/irregular solution gE Ln,L n : FMS matrix (consist of free-propagator) Rehr, et al. Rev. Mod. Phys. 72, 621 (2000) Mattern, et al. PRB 85 115135 (2012) RSGF is perfectly suited for calculation of CP in disordered system Easy to couple with MD simulations: Output of MD position of ions → FEFF input We can study large systems not limited to the size of supercell of MD simulations Dirac HF solver Ψatomic n Uatomic muffin-tin G DOS ρ FEFF μ E. Klevak (Final Exam) Thermal disorder and CP May 2016 14 / 58
  • 15. Theory of MD simulation Kohn-Sham and ab initio DFT MD Total energy of many body ground state: Min of Kohn-Sham energy (single particle) min Ψ0|He |Ψ0 = min EKS [ψi ] EKS [ψi ] = Ts [ψi ] + Vext + Vhar + Exc [n] + Eion(R) Born-Oppenheimer MD Lagrangian: Potential energy replace by KS energy LBO (RI , ˙RI ) = I 1 2 MI ˙R2 I − EKS [{ψi }; RI ] Equation of motion Mi ¨RI = − ∂EKS ∂RI = FI [RJ (t)] E. Klevak (Final Exam) Thermal disorder and CP May 2016 15 / 58
  • 16. Computational details Converge MD calculations (supercell size, ∆t): VASP Calculate CP for number of snapshots (FEFF) Average Compton calculations over snapshots 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.5 1 1.5 J(pq)(a.u. -1 ) pq (a.u.) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.5 1 1.5 BeJ(pq)(a.u. -1 ) pq (a.u.) avg. E. Klevak (Final Exam) Thermal disorder and CP May 2016 16 / 58
  • 17. MD results and analysis Supercell sizes: Be: 384 (4 × 4 × 6 rep. of conv. cell) Li: 250 (5 × 5 × 5 rep. of conv. cell) Si: 512 (4 × 4 × 6 rep. of conv. cell) 50 100 150 200 250 300 350 400 450 500 550 600 0 1 2 3 4 5 6 7 Be Equilibration Sampling T(K) t (ps) 300K Pair distribution function 0 1 2 3 4 5 6 7 8 4 5 6 7 8 9 (a) Be g(r) T = 300K, ρ=1.856 g/cm3 T = 650K, ρ=1.832 g/cm3 T = 850K, ρ=1.815 g/cm 3 0 1 2 3 4 5 5 6 7 8 9 10 11 12 13 (b) Li g(r) 95K, ρ = 0.55 g/cm3 300K, ρ = 0.534 g/cm 3 400K, ρ = 0.526 g/cm 3 500K, ρ = 0.518 g/cm3 0 1 2 3 4 5 6 7 8 4 5 6 7 8 9 10 (c) Si g(r) r (a.u.) 300K, ρ = 2.33 g/cm3 973K, ρ = 2.31 g/cm 3 1573K, ρ = 2.30 g/cm3 1787K, ρ = 2.56 g/cm3 3000K, ρ = 2.56 g/cm3 E. Klevak (Final Exam) Thermal disorder and CP May 2016 17 / 58
  • 18. Results: Li CP as a function of temperature 0.0 0.2 0.4 0.6 0.8 1.0 1.2 J(pq)(a.u.-1 ) 0K 95K 300K 400K 500K Exp. 1 Exp. 2 KKR GWA 1 2 3 4 5 0.8 1.0 1.2 1.4 Li q=[110] J(pq)(10-2 a.u.-1 ) pq (a.u.) -6 -4 -2 0 2 4 6 [110]-[100] ∆J(pq)(10-2 a.u.-1 ) -10 -8 -6 -4 -2 0 2 4 6 8 0.0 0.2 0.4 0.6 0.8 1.0 [110]-[111] ∆J(pq)(10 -2 a.u. -1 ) pq (a.u.) Our results are close to KKR theory Many body effects are important High momentum components are smeared Directional anisotropy: Qualitative agreement Exp1: Sakurai, et al. PRL 74, 2252 (1995) Exp2, KKR: Sch¨ulke, et al. PRB 54, 14381 (1996) GWA: Kubo, et al. J. Phys. Soc. Jpn. 66 2236 (1997) E. Klevak (Final Exam) Thermal disorder and CP May 2016 18 / 58
  • 19. Results: Li CP differences -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 q=[110] Li CP (T=95K)-CP (T=300K) ∆J(pq)(10 -2 a.u. -1 ) pq a.u. Exp Sternemann et al. Theory Quantitative agreement Lattice expansion: narrowing of the CP E. Klevak (Final Exam) Thermal disorder and CP May 2016 19 / 58
  • 20. Results: Si CP 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Si liquid 1787K (a) J(p)(a.u. -1 ) Our theory Exp. Okada et al. Exp. Matsuda et al. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 Si solid 298K (b) J(p)(a.u. -1 ) p (a.u.) Solid → liquid transition Semiconductor → metal transition Density increase by 10% Okada, et al. PRL 108, 067402 (2012) Matsuda, et al. PRB 88, 115125 (2013) E. Klevak (Final Exam) Thermal disorder and CP May 2016 20 / 58
  • 21. Results: Si CP differences -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.0 0.5 1.0 1.5 2.0 2.5 3.0 SiCP(298K)-CP(1787K) Jsolid(pq)-Jliquid(pq)(a.u. -1 ) pq a.u. Theory l=4 Theory l=5 Exp. Okada et al. Exp. Matsuda et al. CPMD Okada et al. CPMD: Car-Parinello MD, same MD dynamics CPMD: CP calculate using plain waves Lattice contraction does not broaden CP E. Klevak (Final Exam) Thermal disorder and CP May 2016 21 / 58
  • 22. Results: Si electron momentum density 0.0 0.5 1.0 Si liquid 1787K (a) ρ(p) Exp. Matsuda et al. Exp. Okada et al. Our theory 0.0 0.5 1.0 Si solid 298K (b) ρ(p) p (a.u.) ρ(p) ∼ d 1 p J(p) dp E. Klevak (Final Exam) Thermal disorder and CP May 2016 22 / 58
  • 23. Results for Be 0 2 4 6 8 10 -4 -3 -2 -1 0 1 2 3 4 850K 650K 300K 40K Be 1.815 g/cm 3 1.832 g/cm 3 1.856 g/cm 3 1.856 g/cm3 J(pq)(10-2 a.u.-1 ) pq (a.u.) Our theory Huotari Theory Huotari Exp. FEFF 0K Position of high momentum components agreement Huotari theory: fitted potentials Smearing of the peak as a function of temperature E. Klevak (Final Exam) Thermal disorder and CP May 2016 23 / 58
  • 24. Results: Be high momentum components differences -2 -1 0 1 2 q=[110] 300K - 850K (a) Be ∆J(pq)/J(0)(%) Theory Exp. Huotari et al. Theory Huotari et al. -2 -1 0 1 2 -3 -2 -1 0 1 2 3 q=[110](b) Be 300K - 650K ∆J(pq)/J(0)(%) pq (a.u.) Theory Exp. Huotari et al. Theory Huotari et al. Sharpens CP as lattice expands E. Klevak (Final Exam) Thermal disorder and CP May 2016 24 / 58
  • 25. High energy density space ”Frontiers in High Energy Density Physics:The X-Games of Contemporary Science (2003)” E. Klevak (Final Exam) Thermal disorder and CP May 2016 25 / 58
  • 26. High energy density space ”Frontiers in High Energy Density Physics:The X-Games of Contemporary Science (2003)” E. Klevak (Final Exam) Thermal disorder and CP May 2016 26 / 58
  • 27. WD Be: MD results, pair distribution function 1.0 1.5 2.0 2.5 3.0 3.5 4.0 2 3 4 5 6 7 8 9 V = 0.953 V0 V = V0 g(r) r (a.u.) 2 eV 1 eV First shell peak shifts More disorder at high temperature E. Klevak (Final Exam) Thermal disorder and CP May 2016 27 / 58
  • 28. WDM: Perfect crystal momentum density 0.2 0.4 0.6 0.8 1.0 1.2 1.4 p/pf 0 1 2 n(p) Momentum density as a function of temperature Plasma community models based on noniteracting Fermi gas T∗ temperature estimated from fitting Fermi function to n(p) E. Klevak (Final Exam) Thermal disorder and CP May 2016 28 / 58
  • 29. WDM: Perfect crystal momentum density 0.0 0.5 1.0 1.5 2.0 p/pf 0.0 0.5 1.0 1.5 2.0 n(p) V=V0 T=2 eV T∗ = 4.02 ± 0.01 eV Fermi gas disorder+electronic 0.0 0.5 1.0 1.5 2.0 p/pf 0.0 0.5 1.0 1.5 2.0 n(p) V=V0 T=1 eV T∗ = 3.06 ± 0.01 eV T∗ temperature in Fermi gas model Plasma community models based on noniteracting Fermi gas Does not fit well E. Klevak (Final Exam) Thermal disorder and CP May 2016 29 / 58
  • 30. Dependence on density and temperature 0 1 2 3 4 5 6 7 T (eV) 2 3 4 5 6 7 8 9 FermiparameterT∗ (eV) electronic V=1.23 V0 electronic V=V0 electronic V=0.953 V0 V0 +disorder 1.23 V0 +disorder Temperature with thermal disorder electronic temperature are quite differnt Gateway to improve theory Further investigation of parameter space needed E. Klevak (Final Exam) Thermal disorder and CP May 2016 30 / 58
  • 31. Conclusions Goals Ultimate: Study WDM using CP Realiable theory for calculation of CP as a function of temperature and thermal disorder Importance of disorder and anharmonic thermal effects Summary Study of thermal disorder and electronic temperature effects on CP Calculation of CP for Li, Si and Be at moderate temperature and melted structure. Good qualitative/quantitative agreement for low temperatures of CP Method good for determining temperature/density effects on CP Anharmonic and thermal disorder both important in the CP studies Preliminary: Calculation of CP at WDM regime E. Klevak (Final Exam) Thermal disorder and CP May 2016 31 / 58
  • 32. Acknowledgments John Rehr Jerry Seidler Rehr group: Joshua Kas Fernando Vila Andrew Lee (alumni) Shauna Story (alumni) Brian Mattern (alumni) Committee members: Anton Andreev Christine Luscombe Silas Beane David Cobden Funding: Physics Department: teaching assistance North Seattle Community College Supported from: DOE Computational resources: National Energy Research Scientific Computing Center E. Klevak (Final Exam) Thermal disorder and CP May 2016 32 / 58
  • 33. WD Be studies, MD simulation limits 0 2 4 6 8 10 12 2 3 4 5 6 7 8 u(r)(eV) r (a.u.) Gaussian Be dimer VASP Be dimer E. Klevak (Final Exam) Thermal disorder and CP May 2016 33 / 58
  • 34. Be high momentum components thermal expansion vs. disorder 0.5 1.0 1.5 2.0 2.5 850K 650K (a) J(pq)(10-2 a.u.-1 ) ambient expanded 0.5 1.0 1.5 2.0 2.5 2.0 3.0 4.0 5.0 (b) J(pq)(10-2 a.u.-1 ) pq (a.u.) 850K 650K 300K 40K E. Klevak (Final Exam) Thermal disorder and CP May 2016 34 / 58
  • 35. Li electron momentum density 0 0.5 1 0 0.5 1 ρ(p) p (a.u.) Our theory Kubo et al. el. gas Kubo et al. GW <100> Kubo et al. GW <111> E. Klevak (Final Exam) Thermal disorder and CP May 2016 35 / 58
  • 36. X-ray based diagnostics: Compton Profile Okada, et al. “Visualizing the Mixed Bonding Properties of Liquid Boron with High-Resolution X-Ray Compton Scattering” PRL 144, 177401 (2015) Okada, et al. “Persistence of Covalent Bonding in Liquid Silicon Probed by Inelastic X-Ray Scattering” PRL 108, 067402 (2012) Huotari, et al. “High-momentum components and temperature dependence of the CP of beryllium” PRB 66, 085104 (2002) E. Klevak (Final Exam) Thermal disorder and CP May 2016 36 / 58
  • 37. Theory of MD simulation Kohn-Sham and ab initio MD min Ψ0|He |Ψ0 = min EKS [ψi ] EKS [ψi ] = Ts [ψi ]+Vext +Vhar +Eex [n]+Eion(R) Born-Oppenheimer MD LBO (RI , ˙RI ) = I 1 2 MI ˙R2 I − EKS [{ψi }; RI ] Mi ¨RI = − ∂EKS ∂RI = FI [RJ (t)] Car-Parrinello L = i 1 2 µ Ω d3 r| ˙ψi |2 + i 1 2 MI ˙R2 I + ν 1 2 µν ˙αν 2 − EKS [{ψi }, {RI }, {αν }] + Constraints Real dynamics Mi ¨RI = − ∂EKS ∂RI = FI [RJ (t)] FI = − ∂EKS ∂R I Fictitious dynamics µ ¨ψi (r, t) = −Hψi (r, t) + k Λik ψk (r, t) µν ¨αnu = − ∂EKS ∂αν E. Klevak (Final Exam) Thermal disorder and CP May 2016 37 / 58
  • 38. Past work Introduction Motivation CT satellites: signature of strong local correlation effects Typically modeled by CT multiplet approach with crystal-field parameters Absorption coefficient Charge transfer model Lee et al. d shell to ligand, charge transfer satellites Theory XAS beyond sudden approximation Calculation of the parameters using FEFF code Results Calculation XAS of TMO’s Calculation polarized XAS for CuO E. Klevak (Final Exam) Thermal disorder and CP May 2016 38 / 58
  • 39. X-ray absorption and photoelectron spectroscopy 0 920 930 940 Energy (eV) 950 960 1 2 3 4 Normalizedabsorption 5 L3 L2 6 XAS XPS: probe elemental composition measurement electronic and chemical state XAS local geometric and electronic structure Siegbahn notation L2 – 2p1/2 L3 – 2p3/2 History: Manne Siegbahn, Nobel Prize in Physics 1924 ”for his discoveries and research in the field of X-ray spectroscopy” Kai Siegbahn son of Manne Siegbahn 1981 physics Nobel Prize ”for Electron Spectroscopy for Chemical Analysis” E. Klevak (Final Exam) Thermal disorder and CP May 2016 39 / 58
  • 40. Strong correlations in TMO XPS 3S data Strong Satellite Strong Satellite E. Klevak (Final Exam) Thermal disorder and CP May 2016 40 / 58
  • 41. Charge transfer satellites in TMO spectra 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 21 HeH 4 5 6 7 8 9 103 Be B C N O F NeLi 12 13 14 15 16 17 1811 Mg Al Si P S Cl ArNa 20 21 22 23 24 30 31 32 33 34 35 3619 Ca Sc Ti V Cr 25 26 27 28 29 Mn Fe Co Ni Cu Zn Ga Ge As Se Br KrK 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 5437 AgSr Y Zr Nb Mo Tc Ru Rh Pd Cd In Sn Sb Te I XeRb 56 72 73 74 75 76 77 78 79 80 81 82 83 84 85 8655 Ba Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At RnCs 88 104 105 106 107 108 109 110 111 112 113 114 115 116 117 11887 Sg Rg Fl UupRa Rf Db Bh Hs Mt Ds Cn Uut Lv Uus UuoFr Group Period -10 -5 0 5 10 15 Intensity(arb.units) ω (eV) XPS 1s Edge Exp. Fit Satellite Main 0 10 20 30 40 50 60 Intensity(arb.units) ω (eV) XAS Exp. vs. single particle Main Ni K-edge=8333 eV Satellite 7.4 eV Exp. FEFF Transition metal oxides (TMO) Motivation: CT satellites: signature of strong local correlation effects Technique used in study of strongly correlated materials(hight-Tc) Charge transfer satellites are clearly seen in experimental XPS of TMO Single particle calculation (FEFF) of absorption spectra missing peak at about 6 eV from the edge E. Klevak (Final Exam) Thermal disorder and CP May 2016 41 / 58
  • 42. X-ray absorption vs photoemission } Incident x-ray Photoelectrons Scattered x-ray Transimitted x-ray Single particle absorption coefficient µ(ω) = 2π f f |p · ˆε eık·r |c 2 δ(Ef − Ec − ω) Dipole approximation eık·r ≈ 1 X-ray absorption and photoemission XAS near edge: |I = |core |si |F = |k |sf scattered states+many body states. µ(ω) XAS = k 2π i,f sf | k| ˆDε|c |si 2 δ(Ef − Ei − ω) µ(ω, k) XPS µ(ω) – x-ray absorption (XAS), µ(ω, ˆε) – polarized x-ray absorption, E. Klevak (Final Exam) Thermal disorder and CP May 2016 42 / 58
  • 43. Charge transfer model outline Lee et al. charge transfer model Calculation of Vsc (r) for several TMO Calculation of Matrix elements Solving for final state Adiabatic to sudden transition Application of CT to XAS of several TMO E. Klevak (Final Exam) Thermal disorder and CP May 2016 43 / 58
  • 44. LGH Charge transfer model Transition from the adiabatic to the sudden limit in core-level photoemission: A model study of a localized system J. D. Lee and O. Gunnarsson Max-Planck Institut fu¨r Festko¨rperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany L. Hedin Max-Planck Institut fu¨r Festko¨rperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany and Department of Theoretical Physics, University of Lund, So¨lvegatan 14 A, S-223 62 Lund, Sweden ͑ Received 23 April 1999͒ We consider core electron photoemission in a localized system, where there is a charge transfer excitation. Examples are transition metal and rare earth compounds, chemisorption systems, and high-Tc compounds. The system is modeled by three electron levels, one core level, and two outer levels. In the initital state the core level and one outer lev
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