XANES Spectroscopy Theory and applications

Transcription

1 XANES Spectroscopy Theory and applications Carlo Meneghini

2 XANES signal is dense of electronic, magnetic and structural information Near Edge Region (XANES) Extended Region (EXAFS) XANES signal is stronger than EAXFS - Less sensitive to sample quality, - Can be measured on less concentrated samples than EXAFS - Can be measured faster than EXAFS (Dispersive-XAS, Quick-XAS) - Magnetic effect (XMCD) are strong in the XANES region, weak and hard to model in the EXAFS region. - XANES measurements can be done at lower energies (light elements like C, O, N ) XANES features are prone to simple interpretation - Valence state of the photo-absorber - Phase mixture as a linear combination of XANES of reference compounds (LCA, PCA) Accurate, reliable and precise interpretation of XANES is a difficult task - Long computation time, - Hard theory - Correlation among the parameters - Failure of simplification in the edge region -The accuracy of theory is continuously progressing From M. Newville Tutorial at

3 -Simple measurements -Weak temperature dependence (Debye-Waller) -Limited energy region: fast scans, quick-xas mode: in-situ monitoring of reactions. -Micorfocus: extreme conditions measurements: high magnetic fields (H >> 10 T), high temperatures (T > 2500 K), high pressures (P>100 Gpa). Sulfur K-edge XANES used toidentify and quantify the form of sulfur in heavy petroleum, coals, soils, etc. 11 ev edge shift from S 2- to S 6+. Spectra of S in similar environments are similar: benzothiophene, dibenzothiophene. Can be used as fingerprint.

4 Physics Chemistry Biophysics Medicine Achaeometry Cultural Heritage etc... Sensitivity µ XRF: elemental µ XAS: elemental & chemical state Micro-XANES I. J.Pickering & G. N. George Proc. XAFS13 conference (2006)

5 High pressure XAS with DAC High pressure structure of molecular solids Energy dispersive XAS set-up: Small x-ray spot size (10x10 mm 2 ) High positional stability Intermolecular compression inter-molecular Br-Br EXAFS XRD? Transitions to σ* antibonding states of Br 2 New phase transition above 60 GPa Intramolecular behaviour: expansion, then compression intra-molecular Br-Br EXAFS

6 Single Energy - XAS Melting of In µ-particles and structure of undercooled In phase In K edge Undercooled phase A. Filipponi et al. J. Sinchr. Rad. 8, 81 (2001)

7 halide (x - ) 200 nm laser beam Transient states of hydrated atoms halogen (x o ) t ~ 10 ns production of Br 2 Br 2 Reaction with water

8 XANES analysis: semi-quantitative approach (LCA) Linear Combination Analysis XANES features are fingerprint of the valence state and local structure of the absorber XANES spectra of phase mixtures can be simply analyzed as a linear combination of reference compounds XANES. As adsorption in Natural Calcite samples Confined Co NPs As 2 O 5 model for As V As 2 O 3 model for As III P. Lattanzi et al. R. Torchio et al.

9 Simon R. Bare Linear combination analysis

10 XANES analysis: semi-quantitative approach (PCA) Principal Component Analysis PCA is widely used in pattern recognition problems. It is based on a simple linear algebra: each reference spectrum (component) represent a vector, the data are reproduced by vectorial sum. The algorithms automatically determine (statistics) the relevant components (principal) out a given ensamble and reject the others Automatic procedure to select principal components J.W. Sobczak J. of All. and Comp. 362 (2004) 162 Local structure of a Pd-doped polymer Warning How many standards are necessary? Which standards are required? Which standards are unphysical? Is the final fit reasonable? - Use all the a-priori knowledge on the sample (physics, chemistry, etc ) - Check carefully the results J. Agric. Food Chem. 2008, 56,

11 Toward a quantitative interpretation of XANES XANES development Lee & Pendry Phys. Rev. B 11, 2795 (1975) (initial theory) C.R. Natoli et al. Phys. Rev. A, 22, (1980) (first calculations) First-principles calculation of x-ray absorption-edge structure in molecular clusters C. R. Natoli, D. K. Misemer, S. Doniach, and F. W. Kutzler Tyson, Hodgson, Natoli, Benfatto Phys. Rev. B (1992) A. Filipponi et al. Phys. Rev. B 52, (1995) Ankudinov et al. Phys Rev. B 58, 7565 (1998) J. Rehr Rev. Mod. Phys. 72, 621 (2000) Available programs for XANES: C. R. Natoli and M. Benfatto CONTINUUM, MXAN Freeware J. Rehr FEFF, IFEFFIT leonardo.phys.washington.edu/feff/html/obtain.html Feff6: freeware freeware cars9.uchicago.edu/~ravel Y. Joly FDMNES (wikipedia) Further reading about XANES:

12 I I o µ = Linear absorption coefficient σ = absorption cross section n a = atomic density Final state Interaction Ground state E F E F magnetic quadrupole The final states modifies as a results of the cor-hole Electric dipole E.D./E.Q.~10 2 Electric quadrupole Definitively negligible below 4-5 kev Electric octupole Only for very heavy elements

13 Electric dipole approximation One electron approximation Many body effects (intrinsic losses): in the EXAFS region it is the S o 2 parameter, in the XANES may be a function of the energy Time scales: Photon absorption t a ~ s Core-hole life time (Z = 20-30) t h ~ s Relaxation time of the N-1 electrons t r ~ s Flight time of the photoelectron t f (E)~ s (E = ev) Thermal vibration t r ~ s XANES EXAFS Multielectronic effects become relevant at low energies Snapshot of the perturbed system

14 delocalized wave (continuum) localized wave (core electron) Two equivalent formalisms The scattering matrix τ (Natoli, Benfatto, Filipponi CONTINUUM, MXAN, GNXAS) The Green function matrix G (Rehr, Ankudinov, Ravel FEFF, IFEFFIT) Deep core hole (K, L edges) the initial state is strongly localized, the integral in just around the absorber r r V(r) = V (r) + V c exc r (r) r j r 1/ 3 c = V (r R j) r 3 r V j exc (r) = 6α ρ(r) V (r) Coulomb Atomic potentials Slater exchange factor (energy dependent) Exchange and correlation 8π Electron density

15 Calculating V: Muffin-tin scheme Spherical symmetry inside the atoms Constant between the atoms Atomic potential depends on charge density ρ = ρ ο + ρ N due mainly to the atom plus a weak contribution the from surrounding Exact calculation: time consuming (FDM) Overlapping potential: working approximation Mattheiss Phys. Rev. B, 133, A1399 (1964) Recipe: Place charge density from neutral atom on each atomic center Direct superimposition to a given cut-off of charges to obtain the total charge distribution around the atom. The muffin-tin charge is mainly that of the central ion, but on the edges neighbour charges remove the symmetry allowing some non-spherical effects to be taken into account. If the overlap is small the approximation is reasonable Spherical atomic atomic like potentials centred on each atomic site. They provide the scattering potentials described by the t matrices Constant value in the interstitial region in which the photoelectron propagate via the G o propagator Overlap Charge densities: tables (Clementi e Roetti, Herman & Skillman ) or LDA calculations Potentials: must describe the Coulomb potential + the exchange and correlation part, different theories are available: - X α, Hedin-Lundqvist,

16 Atomic potentials and XAS calculations Na K-edge in NaCl V r 3 exc (r) = 6α ρ(r) 8π r 1/3 Edin-Lundqvist Slater exchange factor Electron density Tables, or self consistence (FEFF) Dirac-Hara H.L.: Complex potentials

17 Muffin-Tin approximation I region Only the L=0 is considered: spherical symmetry I III II region I II I III region V II is a constant value, averaged over the interstitial volume. V III is a spherical average respect respect to the atomic cluster center It depends to the physical problem to be solved. In the extended continuum scheme V III = V II : the same energy scale apply to continuum and bound states (pre-edge) states.

18 The total w.f. is written as: I III I Ψ = In each atomic region (I) the w.f. is developed into spherical harmonics for each atom (J) Ψ = J J J I r) BLR l (E; r)ylm ) L ( r i Ψ I + Ψ II + Ψ III Radial solution with angular momentum l (rˆ Spherical harmonic Spherical Harmonic I II I II V is constant in II, then Ψ II is a combination of Bessel and Neumann functions III In the outer sphere region (region III) III III III III ΨIII = [A L f l (kr0 ) δ LL' + BLL' g l' (kr0 )] ΥL' (rˆ 0 ) L,L' regular part r o referred to the center of the whole molecule irregular part Spherical Harmonic One function is obtained for each angular momentum L ={l,m} of each atom and of the outer sphere. The coefficients B are calculated imposing the continuity of Ψ and its first derivate at the border of the different regions

19 Total Radial part of the final EM interaction state at r o (the w.f. is (dipole approximation) scattering localized) amplitude Selection rules l = l o +1, l o -1 Initial state From the Lippmann-Schwinger equation and using the Green s theorem: Total scattering amplitude of momentum L at the site i Scattering of waves coming from neighbour scattering Scattering of Exciting wave The scattering amplitude at the site i with angular momentum L is t i l times the amplitude of the exciting wave J plus the scattering of all the waves from neighbouring sites Tyson, Hodgson, Natoli, Benfatto Phys. Rev. B (1992)

20 Inside the i-th MT sphere Radial solution with angular momentum l Spherical harmonic Deep core hole (K, L edges) the integral in just around the absorber Total scattering amplitudeof momentum L at the site i Scattering of waves coming from neighbour scattering Scattering of Exciting wave Phase shift of the potential inside the MT sphere ~ G ij LL' Scattering matrix for the atom located at the site i for a wave of angular momentum L Free propagator between i and j of a spherical wave of angular momentum L at I and L around j = G ij LL' L'' J i0 LL'' t 0 l '' J oj L'' L' Correction due to the OS: region III Free propagator between i and j taking into account for the distortions due to the Outher sphere (OS) scattering Amplitude of the incoming wave at the site i Tyson, Hodgson, Natoli, Benfatto Phys. Rev. B (1992)

21 Optical theorem Diagonal matrix τ: scattering path operator, it contains all the structural and electronic information about the atomic cluster around the absorber.. G ji ( t i l ) 1 G ij.. 1

22 Atomic part Rarial dipole matrix element, the integral just inside the MT volume die to the localization of the w.f. The dipole allowed transition Structural part exact solution The matrix inversion allows obtaining all the structural and electronic information about the atomic cluster MS expansion EXAFS region

23 The MS series G ii LL' 0 For all the i neighbours For all the i and j neighbours For all the i,j,k neighbours

24 Expansion in terms of γ n signals The GnXAS approach A. Filipponi et al. Phys. Rev. B 52, (1995) gnxas.unicam.it/ r Continuous fraction expansion r 1, r 2,θ 12 r 1, r 2, r 3 θ 12, θ 23, ψ ψ θ 12 r r 3 1 r 2 θ 23

25 Advantages of MS Atomic clusters (relatively small) representative of the local structure around the absorber can be used to model the XAS of periodic as well aperiodic systems

26 The MS serie The absolute convergence of the MS series is determined by the spectral radius (maximum modulus of the eigenvalues) The scattering amplitude goes to 0 at high energy The G is singular on 0

27 XAS spectrum three regions Global information, Few scattering terms are relevant EXAFS region: only single (and very special MS) terms are necessary FMS IMS SS ρ 1 ρ < 1 ρ << 1 Many or infinite number of paths contribute to the shape of the spectrum - usually near the edge region (20-40 ev from the edge) for low Z scattering atoms The scattering path operator must be calculated exactly χ, χ 3, 4 2 χ... χ 2 Energy, k Typically from 30 ev above the absorption edge, Information on bond lengths and angles (n-body distribution functions, topological information) The photoelectron is sensitive to the relative position of two, three or more atoms at the time via the MS paths

28 Mn K-edge 50 mm MnCl 2 50 mm KMnO Å 1.63Å [Mn(OH 2 ) 6 ] 2+ MnO 4 - The energy scale are in the ratio 0.47 to account for the different distance between Mn and O in MnO 6 and MnO 4 M. Benfatto et al. PRB 34 (1986) The amplitude has been corrected for the different number of neighbours The two spectra are definitively different below 150 ev MS contributions

29 exp 2.17Å FMS IMS SS All the MS contributions are at the same energy Shape resonance M. Benfatto et al. PRB 34 (1986) FMS up to ev

30 exp 1.63Å Tetrahedral coordination Relevant MS contributions up to 150 ev

31 Advantages of MS Atomic clusters representative of periodic as well aperiodic systems How large must be the atomic cluster? Computation time

32 Full Multiple Scattering: (I-TG) -1 FMS is calculated taking into account an atomic cluster made by the absorber surrounded by scattering neighbours. FMS is CPU-time consuming Raising k increases the matrix size, computation effort becomes unreasonable above 3-4 Å -1 Each atom requires angular momentum states Centrifugal barrier cut off Memory requirements: ~ M^2 CPU time ~ M^3 (M=matrix size) Simple parallelization (few energy points per CPU): computation time scales linearly with the number of CPUs

33 Si K-edge crystal silicon S K-edge in ZnS 8 shells around the absorber Continuum (Natoli - Benfatto) Continuum (Natoli - Benfatto) Cu-fcc FEFF Y. Joly FDMNES B. Ravel J. of All. and Comp. 401 (2005)

34 We need to account for other physical processes inelastic excitations suffered by the photoelectron electronic excitations due to the creation on core-hole finite core hole width Self consistence (details of ρ(e)). They drain away amplitude from the elastic channel and must be included in any realistic calculation finite lifetime of the photoelectron in the final state many-body treatment of the photoabsorption process

35 Self Energy effects Cu fcc Multipole Self Energy Self Energy Dyson equation exp Core Hole Cu L 3 edge Excited state potential J. Sync. Rad. (1999) 6, 315 GS The form of Σ(E) depends on the approximation used (LDA is suitable) Inelastic losses: dynamically screened potential (Hedin equation) for a very simple description: J. Rehr Phys. Scripta, T115, 19 (2005) Photoelectron MFP (extrinsic losses) Final state broadening + energy shift J. Rehr Rev. Mod. Phys. 72, 621 (2000)

36 The Green function Matrix scheme is better suited for theory improvement Ankudinov et al. Phys Rev. B 58, 7565 (1998) Tyson, Hodgson, Natoli, Benfatto Phys. Rev. B (1992) (Rehr, Ravel FEFF)Better to treat the inelastic effects The Green function The optical theorem The Green s-functions matrix satisfies a set of coupled equations that contains the complete description of all the possible outcomes of a photoemission process

37 Kinetic operator In the free space: Free electron in the interstitial potential Atomic potentials Free propagator between r and r Spherical waves Dyson equation Scattering matrix: sum al the scattering events at the site i

38 We must use a sum that start and end on the central atom Total scattering potential can be separated in contributions from central (c) and all the other i-th neighbours ; Scattering propagator matrix to be calculated on the central atom i = i = c Ankudinov et al. Phys Rev. B 58, 7565 (1998)

39 Density of electronic states J. Rehr Coord. Chem 249, 131 (2005) The Green funtion Matrix is directly related to the electron state density with a given angular momentum To be calculated in the space just around the absorber Filled states Theory E F Empty states Y. Joly (Hercules courses 2009) From J. KAS, Theory and calculation of X-Ray Absorption

40 Self Consistence Calculate τ or G matrices Initial Charge densities: tables (Clementi e Roetti, Herman & Skillman ) or LDA calculations Calculate χ l No Converged? Yes

41 Many Particles effects Antysimmetrization operator Ground state Core Electron w.f. w.f. of N-1 electrons in the ground state Final state σ( ω) G α, α' m, σ 0 0 d 3 r r (r, r';e) rd 3 r' φ Using the Green Functions matrix c L 0 f r φα(r) φ E E f α' = f f r r r (r) ε Im S α, α' r (r') iη E E f f = * α = E S E G α' f G + ω α, α' E G r r (r, r'; ω I α c r r ) ε r' φ c L 0 r (r') N 1 N 1 S Ψ α ΨG =

42 Many Particles effects EXAFS FEFF: Ab initio approach XANES MXAN: Phenomenological model Campbell et al. Phys. Rev. B

43 XANES fitting: MXAN method Starting from a possible geometrical arrangement around the absorber MXAN reaches the best-fit conditions minimizing the Res 2 function in the space of structural, electronic and normalization parameters. Best fit procedures exploiting the capabilities of MINUIT package of CENR libraries Generation of the MT potential. Exchange and correlation part of interatomic potentials: X a (energy independent) Hedin-Lundqvist (energy dependent) α exp (E) atomic coordinates MXAN MINUT select structure and polarization fixed potentials MINUIT command file recalculate the potentials VGEN Tyson et al 1992 M. Benfatto et al. J. Synchr. Rad. 8, 1087 (2001) P. D angelo et al. J. Am. Chem. Soc (2002). CONTINUUM code calculates the absorbing cross section in the framework of FMS approach within the muffin tin approximation. CONTINUUM Natoli et al 1980 χ 2 -test Data refinement

44 XANES fitting: MXAN method Refining the potential parameters Refining the atomic Structure Exp File + Command file + Coordinate file MINUIT Change potential parameters Exp File + Command file + Coordinate file MINUIT Change structure Adjust non structural parameters Adjust non structural parameters Calculate potential (VGEN) Calculate potential (VGEN) Calculate XANES Calculate XANES No Convergence? No Convergence? Yes Yes END END

45 Best Fit α exp (E) slope Structure VGEN slope correction (ae+b) normalization (N) E α ( E ) N CONTINUUM α ( E ) E n e r g y α th (E) E E n e r g y Broadinng: Γ=Γ c +Γ(E), Multiple Excitations MINUIT: least square deta refinement

46 Coordination geometry of Cu 2+ in solution - Fivefold or sixfold coordination? - J.T. distortion? Car Parrinello molecular dynamics calculation A. Pasquarello et al. Science (2001) XAFS probe: fs = s

47 GNXAS analysis (XAFS + MS terms) 0.1 M Cu 2+ water solution H atoms are included in the analysis Fivefold coordination sixfold coordination Two geometries with the same accuracy 4 equa. O at 1.96 Angs 1 axial O at 2.36 Angs N R σ 2 Cu-O eq (4) (5) Cu-O ax (2) 0.010(3) 4 equa. O at 1.96 Angs 2 axial O at 2.36 Angs N R σ 2 Cu-O eq (4) (5) Cu-O ax (2) 0.020(3)

48 MXAN analysis (XANES + Structural refinement) Fivefold coordination sixfold coordination Two different solutions 4 equa. O at 1.97(1) Angs 1 axial O at 2.39(6) Angs 4 equa. O at 1.99(1) Angs 2 axial O at 2.56(4) Angs

49 Combining the two possible solutions An average fivefold coordination geometry N R σ 2 Cu-O eq (4) (5) Cu-O ax (2) 0.010(3) A possible dynamic picture

50 Thermal and structural disorder (DW) Problem: including the structural disorder in the XANES calculations. FEFF: Ab initio DW calculations (lattice dynamic) Possible approach: Molecular dynamics + configuration average XANES computation. MD allow generating thousands of geometrical configurations accordingly to the system dynamics (themperature, potentials etc ) Each snapshot is used to generated a single XANES spectrum Final XANES model is obtained averaging many (~10 4 ) geometrical configuration Merkling et al. JACS 124 (2002) Campbell et al. Jour.Synch.Rad. 6 (1999)

51 Total MD trajectories Sequential snapshot extraction around the absorber MXAN run MXAN run Parallel calculations Incremental Average averaged XANES spectrum yes R f < 10-4? no R f 1/ 2 N N 1 2 ( N ) = [ σ ( Ei ) σ ( Ei )] i

52 Ni 2+ in water Ni Kedge average Calculations for some particular snapshots

53 Comparison between the averaged theoretical spectrum and a single theoretical spectrum at the symmetrical first shell configuration Arrows indicate the damping - very weak effect - it can be included in the phenomenological treatment of the inelastic losses in MXAN

54 Transient structure during SCO Structural modifications in the light induced LS -> HS transition Spin CrossOver potential applications: - magnetic storage - bistable magnetic devices - fast data processing at the nanoscale U.V. ~1ps HS Pump probe XAS Measurements: at the µxas beamline at the SLS synchrotron facility Laser pulse: 100fs X-ray pulse: 100ps ~6000 cm -1 LS

55 effect of convolution with 100ps x-ray pulse width R FeN ~0.2Å LS HS Data analysis: MXAN: ab initio structural refinement in the XANES region. Refinement of differential spectra

56 HS-LS difference spectrum MXAN fit of transient XANES features

57 Beyond the MT scheme Finite Difference Method Near Edge Structures: FDMNES code Y. Joly Pys. Rev. B 63, (2001). FDM: general method to solve differential equations by discretizing them on a grid of points in the volume Spherical waves, r~ Å Multipolar expansion of the atomic potential. Spherical symmetry: only 0-th order = MT The curves are quite similar. The equivalence of the results shows together with the validity of the FDM method that nonmuffin-tin calculations are really not necessary in highly symmetrical dense metal! FDM requires a considerable CPU time!!!

58 Multipolar expansion of the atomic potential. Allow take into account for electronic configurations: covalent bonding MXAN fitting using FDMNES Not- MT potentials K 3 Fe(CN) 6 Hayakawa et al. Physica Scripta. Vol. T115, , 2005

59 The CO molecule case MT-MS FDM FDM is able to reproduce the continuum states as well the localized stated (Rydberg series) The MT approximation corresponds to a monopolar representation of the potential with constant value in the interstitial region. Overlapping MT spheres are used to take into scattering from interstitial area. The use of overlapping spheres is mathematically wrong but (Ov~ 10-15%) the benefit remains greater than the error. WARNING: relatively good artificial agreement can even be reached in some cases playing with the interstitial potential and the muffin-tin radius. These false improvements can hide structural or electronic information

60 Theory: MT may be inaccurate dealing with strongly asymmetric structures MT MT FDM FDM Symmetric FeO 6 Planar FeO 4

61 Experimental EXP Non-MT corrections are restricted in very near edge region FDM MT-MS 5 Pyrrole ligands + CO molecule Fe K edge on MbCO (Carbonmonoxy-myoglobin) protein

62 MT MXAN approach Photodissociation of CO from Mb can be induced by visible light. It can be described by a three step model MbCO Mb*CO MbCO -> Mb*CO -> Mb Low temperature measurements: frozen in Mb*CO state MbCO Mb*CO S. Della Longa et al. Phys. Rev. Lett. 2001

63 Thanks for your attention