X-ray absorption spectroscopies : the monoelectronic approach

1. X-ray absorption spectroscopies : the monoelectronic approach Yves Joly Institut Néel, CNRS/UJF, Grenoble www.neel.cnrs.fr yves.joly@grenoble.cnrs.fr HERCULES 2008

2. Outline A- Introduction A-1) X-ray absorption and radiation – material interaction A-2) The different spectroscopies A-3) Characteristic times A-4) Mono and multi electronic approaches A-5) Available codes B- General equations B-1) Transition matrices B-2) Selection rules B-3) Application to XANES calculation B-4) Application to RXS calculation B-5) Tensor approach B-6) Concluding remarks C- Final states calculation C-1) About the potential C-2) The multiple scattering theory C-3) Band structure approach D- Applications D-1) Formal examples D-2) Some oxides D-3) Organic molecule : acrylonitrile D-4) Other compounds

3. Electrons ħwF, IF Fluorescence ħw, I0 ħw, I Transmission ħw’, I’ Elastic or inelastic scattering A-Introduction A-1 X-ray absorption and radiation-material interaction X-ray absorption spectra are a signature of the probed material

4. Resonant scattering Electrons Transmission f Fluorescence EF g Electronic transition between a core level and non occupied states Whatever is the detection mode, - one measures the transition probability between an initial state g and a final state f - Thus one measures the state density at all energy - The state density depends on the electronic and geometric surrounding of the absorbing atom

5. EF Absorption cross section g • X-ray absorption spectroscopies are • local spectroscopies • - Selective on the chemical specie • - Process involved are complex… Pertubated electronic structure f Core hole

6. Fermi M5 ℓ = 2 M4 M3 n = 3 ℓ = 1 M2 M1 spin-orbit spreads level between 2 ℓ = 0 L3 ℓ = 1 L2 n = 2 L1 ℓ = 0 K ℓ = 0 n = 1 For any chemical element there is a set of absorption edges Some edges (eV) Deeper is the edge Shorter is the time life  Broader is the edge Experiment Ch. Den Auwer et al.

7. A-2) The different spectroscopies Real absorption X-ray absorption : XAS and XANES ( < 50 eV) Linear dichroism (dependence on the polarization of the light) Circular dichroism Magnetism : magnetic circular dichroism Without magnetism : natural dichroism Photodiffraction (angular analysis of the photoelectron) Photoemission (valence state) Electron energy loss spectroscopy (EELS) The incident particle is an electron

8. Virtual absorption DAFS, DANES, RXS, MAD : diffraction mode Resonant process where the photon is absorbed, the electron goes up to an empty level, then immediately goes back to its initial core state, emitting another photon which can have another polarization Closely connected to the real x-ray absorption f EF g InAs

9. Atom shell 2 Atom shell 1 Final states are calculated by simple interference between the outgoing wave and the backscattered waves by the different shells EXAFS gives information on - The number of atoms per shell, - The distance of the shells EXAFS and XANES EXAFS XANES XANES gives information on - 3D arrangement - Local symmetry, - Electronic and magnetic structure

10. A-3) Characteristic times 1 – Time of the process « absorption of the photon » t1 = 1/Wfi, Wfi absorption probabilty t1 < 10-20 s multi-electronic process can be seen at low energy of the photoelectron 2 – Time life of the core hole t2 = ħ /DEi, DEi width of the level for 1s for Z = 20 up to 30, Ei ≈ 1 eV t2 ≈ 10-15 à 10-16 s 3 – Relaxation time of the electron Effect on all the electrons of the field created by the hole and the photoelectron. Many kinds of process, multielectronic. t3 ≈ 10-15 à 10-16 s 4 – Transit time of the photoelectron outward from the atom Depends on the photoelectron kinetic energy, for Ec = 1 à 100 eV t4 ≈ 10-15 à 10-17 s X-ray absorption takes a snap shot of the pertubated material 5 – Thermic vibration t5 ≈ 10-13 à 10-14 s

11. EF g A-4) Mono and multi-electronic approaches Localized final states Non localized final states spatialy signal back quickly to 0 In energy EF Absorption cross section Absorption cross section g - Interaction with the hole mono-electronic approach - Several possible electronic states…

12. First theories : Dan Dill and J. L. Dehmer (1974), P. A. Lee and J. Pendry (1975) First calculations : C. R. Natoli (1980), L. F. Mattheiss et R. E. Dietz (1980) In principal : Multiplet : multi-electronic but mono-atomic Edge L2, L3 of 3d elements edge M4, M5 of rare earth Multi-atomic but mono-electronic Improvements in progress : LDA +U Bethe Salpeter Time-Dependent DFT

13. A-5) Available codes Using the mono-electronic approach

14. f o g B-General equations B-1) Transition matrices Signal depends on : - initial states g - final states f - a transition operator o The Fermi golden rule gives the transition matrices : g being localized, the integral is to perform only inside the absorbing atoms

15. The final states are calculated with different techniques : - the multiple scattering theory - the finite difference method - other methods In a cluster approach Using the 3D periodicity With or without self consistency We have to calculate them in a relatively wide range from the Fermi level up to 50 or 100 eV Inside the absorbing atom (non magnetic case) : Spherical harmonic Solution of the radial Schrödinger equation Amplitudes. Contains the main dependence on the energy. Contains the information on the density of state

16. LII edge : Photon wave vector Photon polarization The initial states are completely localized K edge : The transition operator Electromagnetic field magnetic operator : Negligible for X-rays electric operator : Electric octupole component Electric dipolecomponent Electric quadrupole component

17. Photon energy in Rydberg Fine structure constant The dipole, quadrupole and octupole components can be separated : quadrupole octupole dipole Bellow 4400 eV (Sc K-edge) quadrupole negligible… Above, it is small but interesting… Octupole can be detected only in very heavy elements

18. B-2 Selection rules Here we forget the spin… The expansion of and in real spherical harmonics gives : For example, polarization along z, wave vector along x : mo= 0 mo= 1 The transition matrix is then : Gaunt coefficient (tabulated constant related to the Clebch-Gordon coefficient Radial integral Slowly varying with E Strong dependence with ℓo Non zero, only for some ℓ and mgives the selection rules

19. Angular integral non zero only for : ℓ : same parity than ℓg + ℓo |ℓg + ℓo| ≤ ℓ ≤ ℓg + ℓo Dipole : Dℓ = ± 1 Quadrupole : Dℓ = 0, ± 2 with complex spherical harmonics : m = mo + mg with real spherical harmonics : when mo*mg = 0 : m = mo + mg when mo*mg > 0 : m = |mo + mg| and m = |mo - mg| when mo*mg < 0 : m = -|mo| - |mg| and m = -|mo + mg|

20. z x e k B K edge case : dipole component and polarization along z : one probes the pz states projected onto the absorbing atom quadrupole component, polarization along z, wave vector along x : one probes the dxz states projected onto the absorbing atom XANES is very sensitive to the 3D environment

21. B-3) Application to XANES calculation The photon absorption is real. The absorption cross section is given by : density of states photon energy Then the state density can be integrated in the amplitudes, thus for the dipole-dipole term : Angular integral x Radial integral Signal from one site : symmetry of the site Signal from the material ( sum over the atoms ) : symmetry of the material (space group)

22. B-4) Application to RXS calculation Very close formalism The emitted photon can have a different polarization with respect to the incident one Virtual absorption : during the process, difference on the energy conservation given by Scattering amplitude from one atom : outgoing photon incoming photon broadening DE The imaginary part is proportional to the absorption cross section

23. Summation over the atoms with the Bragg factor + Thomson (non resonant) term : The resonant scattering amplitude depends on the polarization and wave vector orientations, thus the scattering is not scalar (spherical) but tensorial (anisotropic) Example : NaV2O5 The XANES spectra present a linear dichroism with a pre-edge Some reflections present spectra around the edge with a strong energy and angular dependency

24. B-5) Tensor approach Dipole Quadrupole Signal amplitude : Dipole-Dipole Dipole-Quadrupole Quadrupole Quadrupole rank 2 tensor rank 3 tensor rank 4 tensor Each component probes a specific projection of the density of state For the K-edge, the dipole-quadrupole tensor probes the hybridized p-d states ( non centro-symmetric material)

25. B-6) Concluding remarks X-ray absorption spectroscopies are selective probes of the local electronic structure around the absorbing atoms Often used for the associated sensitivity on the geometrical structure and the symmetry around the absorbing atoms Study of the electronic properties are increasing (charge exchange, charge ordering, orbital hybridization… The photoelectron probes an excited environment (hole in a core level). The measured signal can be sensitive on this perturbation Selection on - the chemical specie by the value of the energy edge - the angular momentum by the selection rules. This selection is re-enforced by a convenient choice on the polarization and orientation of the sample. - the atomic site when working in the diffraction mode All these selections make the strength of these techniques

26. C- Final state calculation C-1) About the potential As in most electronic structure calculations the choice of the potential is important Depends just on the electron density One body calculation = local density approximation (LDA) Potential = Coulomb potential + exchange-correlation potential Example of calculation Xanes spectra of the copper K-edge in copper fcc Energy dependent Hedin and Lundqvist Different theories Xa Hedin and Lundqvist Perdew….. Xa Depends also on the electron kinetic energy

27. Empty sphere And about the shape of the potential The muffin-tin approximation the MT of the LMTO program (quite) always used in the multiple scattering theory Before approximation After approximation Spherical symmetry inside the atoms Overlap Constant between atoms With the muffin-tin, there are always 2 parameters : overlap and interstitial constant

28. Vacuum density of state Atomic scattering amplitude Amplitude Solution of the radial Schrödinger equation Bessel Photoelectron wave vector Outgoing Hankel C-2) The multiple scattering theory Two ways to explain it : the Green function approach the scattering wave approach Just one atom : We build a complete basis in the surrounding vacuum (Bessel and Hankel functions) We look how the atom scatters all the Bessel functions (phase shift theory)

29. Several atoms ( cluster ) Each atom receives not only the central Bessel function but also all the back scattered waves from all the other atoms The problem is not anymore spherical We have to fill a big matrix with the scattering atomic amplitudes of each atom and the propagation function from one atom to another Matrix containing the atomic scattering amplitudes Matrix containing the geometrical terms corresponding to the scattering from any site “a” of the harmonic L=(ℓ,m) towards any site “b” with the harmonic L’

30. Then one gets the scattering amplitude of the central atom in the presence of its neighboring atoms. When considering only one scattering process : EXAFS When considering a limited expansion of scattering processes : path expansion XANES without the first eV When considering all the scattering processes : XANES including the edge It can be demonstrated that the absorption cross section is then given by : Multiple scattering amplitude Green’s function

31. C-3) Band structure approach Any method which computes the final states can serve to calculate the matrix elements, and therefore the XANES or RXS spectra Band structure approach is useful because it is self-consistent Some commercial codes also allow to compute the absorption spectra LAPW : Wien code Full potential KKR : H. Ebert Pseudo potential : D. Cabaret

32. 2.06 Å Fe D- Applications D-1) Formal examples Analysis of the influence of some parameters Typical shape for an octahedron ! FeO6 octahedron Calculated (and experimental !) spectra are isotropic For a cubic symmetry XANES signal is isotropic (dipole approximation) No difference between calculations with a full and a spherical average potential

33. Fe 1.86 Å 2.26 Å FeO6 with a 10% contraction along z Signal is not anymore isotropic FeO6 with a 0.2 Å vertical displacement of Fe Influence of the 3D structure on the shape More important difference between calculation with different potential shape

34. Planar FeO4 Fe Signal highly anisotropic Big difference upon the potential shape Comparison of the octahedral Fe and the tetrahedral Fe in Fe3O4 Occupiedstates One considers just one oxygen shell Very different shapes A typical pre-edge

35. Comparison with the density of states Effect of the radial integral Suppress the occupied stated Copper FCC Broadening - Core hole life time - Plasmon - Phonon - Experimental uncertainty Beforeand after broadening

36. Calculation versus cluster radius experiment

37. z (a) e k Influence of the core-hole Shift of the 3d x y e // z Full line : Calculation Dotted experiment A A A 2 e 3 perp. z 1 0 10 20 30 40 50 D-2) Some oxides (b) e k Rutile TiO2 Experiment by Poumellec et al. (c) k e Important linear dichroism Fit of the charge exchange between oxygen and titanium quadrupole dipole

38. dz2 pz Quantitative analysis of the pre-edge By the dipole component which see the p states, we also see the projection of the d states of the neighboring Fe With a precise analysis of the XANES features, we get a detailed description of the electronic structure

39. Normal e e grazing The molecules are deposited on a surface The experiment is perfomed along 2 directions Scheme of Acrylonitrile y H x N C D-3) Organic molecule : acrylonitrile For the light element - Long hole life time - Good energy resolution - Study of the first non occupied molecular orbitals The XANES permits to determine how are arranged the molecule Normal incidence, x-ray probe px and py orbitals, projections of the antibonding molecular orbitals py* and s Grazing incidence, x-ray probe pz orbitals, projections of the antibonding molecular orbitals pz*

40. Nitrogen K-edge p* y p* : normal z : grazing p* Calculation z s* s* Experiment ( Tourillon, Parent, Laffont, SACEMOR, LURE) -5 0 5 10 15 20 25 30 eV) Energy ( Carbon K-edge Sum of 3 different atoms

41. S S S S S S SH S S = = S S = S O O O S = S S = = O S S O S S Mixing identification Sufur edge in hydrocarbide Problem : is it possible to identify distillation residu using XANES ? with Ch. Pichon (IFP)

42. Residu References

43. Dibenzodithiophène Dibenzothiophène (SC6H4)2C2 S(C6H4)2 S : +0.30 S : +0.34 • Calcul Gaussian  Mulliken analysis • XANES calculation with FDM

44. Influence of supplementary aromatic cycles DBTP Little influence of second neighbor cycles… Identification of mixing by type of compound

45. 1.6 1.6 1.4 1.4 1.2 1.2 1.0 1.0 0.8 0.8 0.6 0.6 Br Experiment 0.4 0.4 Simulation Simulation 0.2 0.2 Experiment 0.0 0.0 Zn 9640 9650 9660 9670 9680 9690 9700 9710 9720 9730 Zn 9670 9680 9690 9650 9660 9700 9710 Energy (eV) Energy (eV) O Fit of bond angle bond distance D-4) Other compounds ZnBr2 in solution around critical conditions J. L. Hazemann et al study Experiment : 30°C, 250 bar Simulation : octahedral structure Experiment : 300°C, 250 bar Simulation : tetrahedral structure Resolution of the structure by XANES

46. Experiment Calculation (without broadening) Rydberg series H2O gaz Rydberg series Unoccupied bound states O H Unoccupied bond states

47. 1.77 Å 2.41 Å Works also for heavy atoms : Uranyle MV edge : transition to the f states UO2(OH2)5

48. General conclusion XANES is a useful complementary tool to solve geometric structure Calculations are necessary to determine quantitatively the corresponding parameters XANES alsogives precious information about the symmetry of a system and helps in the comprehension of the electronic structure Quantitative resonant x-ray scattering (RXS) is an emerging field enable to extract subtle electronic parameters A special thanks to C. R. Natoli