Spettroscopia di assorbimento di raggi X

Transcription

1 VIII Scuola Nazionale di Luce di Sincrotrone Frascati, ottobre 2005 QuickTime and a TIFF (LZW) decompressor are needed to see this picture. Spettroscopia di assorbimento di raggi X Dipartimento di Fisica Università di Trento

2 VIII Scuola Nazionale di Luce di Sincrotrone Frascati, ottobre 2005 Spettroscopia di assorbimento di raggi X Dipartimento di Fisica Università di Trento

3 Introduction The absorption of X-rays EXAFS: theoretical background EXAFS: experimental details EXAFS: data analysis, examples Edge structures and XANES QuickTime and a TIFF (LZW) decompressor are needed to see this picture.

4 Structural techniques X-rays Scattering Spectroscopy Neutrons Absorption XAFS Energy Long-range order (crystals) Short-range order

5 Attenuation of X-rays Φ 0 x Φ source monochromator sample detectors Exponential attenuation Φ =Φ 0 exp[ µ ( ω) x] Attenuation coefficient µω ( )= 1 x ln Φ 0 Φ

6 Atomic cross sections µω () = N aρ A µ a() ω Cross section (Å 2 ) Coherent sc. Incoherent sc Ge Photoelectric absorption E (kev) Pair prod. nucl. field elec. field

7 Photoelectric absorption (a) E p (ev) 0 Hydrogen E p (ev) 0 Copper - 3, ,

8 Photoelectric absorption (b) Log. scale σ E () Z 4 () E 3 + Edges C ross section (Å 2 ) M L K 32 - Ge E (kev) Ge E (kev) Log. scale Linear scale

9 Photoelectric absorption (c) = K edge

10 Absorption edges K 1s µ/ρ (cm 2 /g) M Nd, 65-Tb, 70-Yb L 1 2s L 2 2p 1/2 L 3 2p 3/2 M 1 3s M 2 3p 1/2 M 3 3p 3/2 Z > 9 M 4 3d 3/ L 1-3 K hω (kev) Binding energy (kev) M 5 3d 5/ Z > Z K L 3 M 5

11 Fine Structure: Atoms hν < binding energy Ar E b = ev Core electron µ (a.u.) unoccupied levels Edge Fine Structure E - E b (ev) hν > binding energy Core electron continuum µ (a.u.) Kr Smooth µ Photon energy (kev)

12 Fine Structure: Molecules and Condensed systems hν binding energy Core electron unoccupied levels µ (a.u.) Ge Edge Fine Structure Photon energy (kev) hν > binding energy Core electron continuum Outgoing wave-function Back-scattering from neighbouring atoms INTERFERENCE Incoming wave-function Extended Fine Structure

13 XAFS: X-ray Absorption Fine Structure X-ray Absorption Near Edge Structure Near Edge X-ray Absorption Fine Structure Electronic transitions Photo-electron multiple scattering XANES NEXAFS Extended X-ray Absorption Fine Structure Mainly photo-electron single scattering EXAFS Edge Photon energy (kev)

14 EXAFS: the mechanism B A X-ray photon absorption. Photo-electron emission. Photo-electron propagation. Photo-electron back-scattering. k χ(k) 0.4 Interference at the core site Modulation of absorption coefficient Photo-electron wavenumber (Å -1 )

15 EXAFS: a structural probe R = 4 Å N = 2 σ 2 = 0.01 Å 2 R = 2 Å N = 4 σ 2 = Å k (Å -1 ) Frequency k (Å -1 ) Amplitude k (Å -1 ) Damping Inter-atomic distance Coordination number Disorder Selectivity of atomic species Insensitivity to long-range order

16 EXAFS applications Non-crystalline materials mono-atomic many-atomic Active sites embedded in a matrix Inorganic heterogeneos catalysts Metallo-proteins Impurities in semiconductors Luminescent atoms Local properties different from average properties Crystalline ternary random alloys Lattice dynamics studies Negative thermal expansion

17 Introduction The absorption of X-rays EXAFS: theoretical background EXAFS: experimental details EXAFS: data analysis, examples Edge structures and XANES QuickTime and a TIFF (LZW) decompressor are needed to see this picture.

18 X-ray attenuation Energy density u = ε 0 E = ε 0ω 2 A Φ 0 Φ Linear attenuation coefficient µω ( )= 1 u du dx = 1 x ln Φ 0 Φ = N a ρ A µ a ω ( ) N a = Avogadro number A = atomic weight r = mass density µ a = atomic cross section x Mass attenuation coefficient Elements µ ρ = N a A µ a Chemical compounds P x Q y... µ ρ tot = x µ ρ P A P M + y µ ρ Q A Q M + K A i = atomic weights, M = molecular weight

19 Transition probability W if µω ()= 2h ε 0 ωa n 2 0 f W if n = atomic density A 0 = vector potential amplit. Initial atomic state Interaction Final atomic state Ψ i Stationary ground state Ψ ( t) Ψ f Stationary excited state W if =?

20 Golden rule Weak interaction Time-dependent perturbation theory (1st-order) Initial atomic state Final atomic state H I Ψ i Stationary ground state Interaction with e.m. field Ψ f Stationary excited state 2 W if Ψ i H I Ψ f ρ ( E ) f matrix element density of final states

21 Interaction Hamiltonian Sum over electrons Electron spin r r A =0 Φ=0 r J = 0 Radiation gauge r E = A r / t r B = r A r r A r = A r r ˆ H I = j e m r p j A r r j,t ( ) e m r S j B r r j,t ( ) e2 ( ) 2m A 2 r j,t r A = A 0 ˆ η e i k r r + c.c. W if = πhe2 m 2 A o 2 Ψ i e ir k r j ˆ η r 2 j Ψ f j ρ( E f )

22 One-electron approximation µ ( tot ω )= µ ( el ω ) + µ ( inel ω ) 1 core electron excited N-1 passive electrons relaxed 1 core electron excited Other electrons excited EXAFS coherent signal µ el ω () Ψ i N 1 ψ i e ir k r ˆ η p r N 1 ψ f Ψ f 2 ρε ( f )

23 Electric dipole approximation e ir k r = 1 + ik r r K 1 H I e ir η p r k r ˆ ˆ η p r = ω 2 ˆ η r Dipole selection rules: l = ± 1 s = 0 j =±1, 0 m = 0 µ el () ω Ψ N 1 i ψ i ˆ η r N 1 ψ f Ψ 2 f

24 Sudden approximation No photoelectron influence on passive electrons Ψ N 1 ψ =Ψ N 1 ψ 1 active electron N-1 passive electrons µ el ( ω) ψ i ˆ η r ψ f 2 ρε ( ) f Ψ N 1 N 1 i Ψ 2 f S Structural information

25 De-excitation mechanisms Radiative: fluorescence X X X Fluorescence yield e - η = X X + A η η K X Non-radiative: Auger A e - e η L η M Z

26 Core-hole lifetime Lifetime of the excited state τ h ~ s τ h 1/Γ h Energy width of the excited state Γ h Γ h (ev) K edges 10 5 L 3 edges τ h Z Γ h Contribution to photo-electron life-time Energy resolution of XAFS spectra

27 Introduction The absorption of X-rays EXAFS: theoretical background EXAFS: experimental details EXAFS: data analysis, examples Edge structures and XANES QuickTime and a TIFF (LZW) decompressor are needed to see this picture.

28 Photon photo-electron µ x χ (k) hν (kev) k (Å -1 ) Photon energy Photo-electron wavenumber 2m h 2 ( hν E b ) = 2m h 2 ε = k = 2π λ -

29 Photo-electron parameters Wave-number 20 Wave-length 3 k (Å -1 ) λ (Å) ε f (ev) ε f (ev) Energy

30 Angular emission of photo-electron θ Photoelectron emission Photon polarisation asimmetry parameter N(θ) 1+ β ( 2 3cos2 θ 1) β = 2 θ ˆ η Emission from s orbitals N(θ) 3cos 2 θ = 3 ˆ η r ˆ 2

31 EXAFS normalisation µ 0 µ 0 () ω ψ i ˆ η r 0 ψ f 2 µ hν χ(k) = µ µ 0 µ 0 hν µω () ψ i ˆ η r 2 ψ f

32 The EXAFS function χ (k) χ(k) = µ µ 0 µ 0 µ 0 ω () ψ i µ ω () ψ i ˆ η r 0 ψ f 2 ˆ η r 2 ψ f k (Å -1 ) (weak interaction) ψ f = ψ f 0 +δψ f? Quantum states wavefunctions χ k ()= ( ) 2Re dr r ψ i ˆ η r 0 ( ψ * ) f ψ i* ˆ η r δψ f dr ψ i* ˆ η r 2 0 ψ f Core orbital = source & detector

33 EXAFS: Two-atomic system b Scattering theory in plane-wave approximation a R c exp( ikr) δψ f ψ 0 f i e iδ 2kR f( k, π) exp( ikr) R e iδ polarisation back-scattering amplitude central atom phase-shift χ()= k 3 ˆ η R ˆ 2 1 kr Im 2 [ f ( k,π ) exp( 2iδ 1 ) exp( 2ikR) ] spherical wave attenuation back & forth path

34 Basic interference effect χ()= k 3 ˆ η R r 1 Im f 2 ( k,π)exp( 2iδ 1 )exp 2ikR kr [ ( )] f( k,π)e 2iδ = f( k,π) e iφ R χ()= k 3 ˆ η R ˆ 2 1 kr 2 ( ) sin 2kR + φ() k f k,π [ ] R = 4 Å R = 2 Å EXAFS frequency Inter-atomic distance k (Å -1 )

35 Amplitudes and phase-shifts Central-atom Back-scattering (rad) 6 Phase-shift (a.u.) Amplitude (rad) 0 Phase-shift Ge 78 - Pt Ge 78 - Pt C 32 - Ge C k (Å -1 ) C k (Å -1 ) Pt k (Å -1 ) Z dependence [Calculated by FEFF 6.01]

36 Many-atomic system Coordination shells Scattering paths SS = Single scattering MS = Multiple scattering Single scattering approximation Isotropic samples: 3 ˆ η R ˆ 2 = 1 χ k ()= 1 k N s Im f s k, π shell ( )e 2iδ 1 1 R s 2 ( ) exp 2ikR s Coordination number

37 Intrinsic losses µ µ ( tot ω )= µ ( el ω ) + µ ( inel ω ) From experiment One-electron theory hν χ exp k ()= µ µ 0 () µ 0 < χ th k S 0 2 = Ψ i N 1 Ψ f N Attenuation factor S 0 2

38 Photo-electron mean-free-path Core-hole lifetime τ h λ h λ h λ (Å) 20 λ e Photo-electron lifetime τ e λ e k (Å -1 ) 1 λ = 1 λ h + 1 λ e Attenuation factor exp 2R λ k ()

39 k (Å -1 ) EXAFS and inelastic effects Intrinsic inelastic effects Photo-electron mean-free-path ()= S 2 0 χ k k N s Im f s k,π shell e ( 2R )e 2iδ s /λ (k) 1 R s 2 ( ) exp 2ikR s Atoms frozen in equilibrium positions! χ (k) Disorder effects?

40 Local disorder Thermal disorder Structural disorder Distorted coordination shells τ vib s τ exafs s Free-volume models r Sites disorder r + + Distance Distribution of distances

41 EXAFS and local disorder Real distribution ρ(r) Thermal + structural disorder distribution of distances Effective distribution P(r,λ) r ()= S 2 0 χ k k N s Im f s k,π shell 0 ( )e 2iδ 1 ρ s (r) e 2r s /λ(k) r s 2 e 2ikr s dr

42 The inversion problem ()= S 2 0 χ k k N s Im f s k,π shell 0 ( )e 2iδ 1 ρ s (r) e 2r s /λ(k) r s 2 e 2ikr s dr Characteristic function k EXAFS χ() k ρ() r?

43 Use of structural models χ(k) Experimental EXAFS Prametrised structural model Change parameters k ρ (r) Distribution of distances FIT Compare r kχ(k) 0 ρ(r) e 2r / λ r 2 e 2ikr dr

44 Cumulant expansion of EXAFS ln P(r,λ 0 ) exp( 2ikr) dr = 0 n=0 ( 2ik) n n! C n Amplitude: even C n χ(k) = S 2 0 e 2C 1 /λ f (k) N exp 2 [ 2k 2 C C k4 C ] 1 sin[ 2kC k3 C φ(k) ] ( ) exp( C 0 ) exp 2C 1 / λ C 1 2 Normalisation Phase: odd C n C 1 = mean value C 2 = variance Position width C 2 1/2 C 3 asymmetry C 4 flatness shape C 1

45 Degrees of disorder Wavenumber k Å -1 Exchanged momentum 2k χ( k)= N f( k,π ) S 2 0 e 2 C 1 / λ 2 exp 2k 2 C 2 kc 1 ( ) exp 2 3 k 4 C sin 2 kc k 3 C φ (k) Harmonic approx. Standard formula C 3 = C 4 =... = 0 - Disorder exp ( 2k 2 C 2 ) exp( 2k 2 σ 2 ) (EXAFS Debye-Waller factor) Low-order anharmonic terms C 3, C 4, {C 5, C 6 } High-order anharmonic terms Cumulant series fastly convergent Cumulant series: slowly convergent... not convergent

46 EXAFS for one shell Approx.: Single Scattering Plane waves Theory (interaction potentials + scattering theory) Experiment (reference samples) Inelastic terms Back-scattering amplitude Total phase-shift χ() k = S 2 0 e 2C 1 / λ 2 C 1 f (k,π) N exp 2k 2 C k4 C 4 + K sin 2kC k3 C 3 + K + φ(k) Coordination number N Even cumulants C 2 C 4 Odd cumulants C 1 C 3 P (r,λ)

47 Introduction The absorption of X-rays EXAFS: theoretical background EXAFS: experimental details EXAFS: data analysis, examples Edge structures and XANES QuickTime and a TIFF (LZW) decompressor are needed to see this picture.

48 XAFS: experimental layout monochromator sample holder storage ring mirror Source Optical apparatus detectors Experim. apparatus storage ring beamlines monochromators mirrors sample holder detectors Alternative layouts dispersive EXAFS refl-exafs... Sample conditioning: cryostat oven reactor manipulators Detection: transmission fluorescence electron yield...

49 XAFS: experimental Monochromators & mirrors

50 X-ray crystal monochromators Bragg law 2 d hkl sinθ = n λ θ d Incidence angle wavelength 2d Si (111) Si (220) 3.84 Si (311) 3.28 Si (331) 2.5 Si (511) 2.08 Ge (111) Ge (220) Forbidden reflections - Harmonics - Spurious reflections

51 Crystal reflectivity θ B Rocking curve (from dynamical theory of diffraction) Darwin width θ (arc sec) Higher order reflections have narrower rocking curves.

52 Energy resolution Beam divergence Darwin width (Intrinsic resolution) + = Total angle Θ E E = λ λ = Θcotgθ B E (ev) 8 6 Si (111) Si (220) Core-level width 4 Darwin width E (kev) E (kev)

53 Two-crystal monochromators Channel-cut Independent crystals Mechanical simplicity Stability Harmonics Spurious reflections Detuning: harmonic reduction Possibility of focussing Mechanical complexity Instability

54 Crystals detuning Silicon crystal (E B = 10 kev) (333) (111) E-E B (ev) 3.5 arcsec

55 X-ray mirrors Complex refractive index n = 1 δ iβ δ for x - rays absorption Total external reflection : θ < θ c Surface bending θ θ c = 2δ λ ρ grazing incidence θ 10-3 rad harmonics rejection Beam collimation and focalisation

56 XAFS: experimental Detection schemes

57 XAFS: direct transmission measurements Sample: Powders or thin films Thickness 10 µm No holes or inhomogeneities Ι 0 Ι 1 Detectors: ionisation chambers V 100 V/cm I A

58 Direct transmission measurements OK NO Bulk information (not fom surface) from: Thick samples Thin samples Diluted samples Non-diluted samples Homogeneous samples Surface information High accuracy attainable

59 Indirect detection methods Ι 0 Detection of decay products X-ray fluorescence FLY = FLuorescence Yield AEY = Auger Electron Yield Electrons PEY = Partial Eletron Yield TEY = Total Electron Yield Optical luminescence XEOL-PLY = X-ray Ecxited Optical Luminescence Photo Luminescence Yield

60 XAFS: fluorescence detection (FLY) I f I 0 Ω z s θ i θ f z n I f ()dz z n = I 0 (ω) exp µ s ω sinθ i ()z n η f µ a () ω dz sinθ i exp µ s( ω f )z n sinθ f Ω 2π Absorption Fluorescence Absorption

61 Fluorescence: total intensity I f = I 0 ω ()η f Ω 4π µ s ω ( ) µ a ω ( ) ()+ µ s ω f Sample of thickness z s θ i = θ f = 45 { 1 exp() A } A = 2z s µ s ω [ ()+ µ s ( ω f )] Thin samples Thick samples 1 exp A () 1 1 A = A I f µ a () ω I f = I 0 ω ()η f 1 exp( A) 1 Ω 4π µ s ω () µ a ω ( ) ()+ µ s ω f OK for diluted samples (< 1%)

62 Fluorescence signal I f I 0 Background signals Elastic scattering Compton scattering Other fluorescences Energy selective detectors Filters + Soller slits

63 XAFS: electron detection (a) Photo-electrons: Energy varies with hν background hν Ι 0 Auger Electron Yield Auger electrons: Fixed energy atomic selectivity Intensity µx XAFS signal Electron mean free path: adsorbates thin layers

64 Indirect processes and escape depth Primary Auger mean free path ~ 5-10 Å Secondary electrons escape depth ~ Å Photons penetration depth ~ 500 Å

65 XAFS: electron detection (b) electron detector AEY = Auger Electron Yield - narrow energy window - only direct Auger electrons - spurious structures from photoelectrons AEY - PEY - TEY PEY = Partial Eletron Yield - large energy window - direct + secondary Auger = XAFS signal - direct + secondary photoel. = background TEY TEY = Total Electron Yield - all electrons collected - direct + secondary Auger = XAFS signal - direct + secondary photoel. = background

66 XAFS: experimental Alternative layouts

67 Dispersive XAFS (a)

68 Dispersive XAFS (b) Curved crystal monochromator 2 d sinθ = λ θ 1 θ 2 θ 3 E 1 E 2 E 3 S.R. incoming white beam Position-sensitive detector

69 Dispersive XAFS (c) No mechanical movements (no dead times) Simultaneous acquisition of all data points Acquisition time determined by acceptable statistics OK for time-resolved measurements Crytical in terms of temporal and spatial beam stability and sample presentation Only trasmission mode X-ray beam not perfectly focussed through the sample No reference measurements during acquisition NO accurate quantitative results

70 Introduction The absorption of X-rays EXAFS: theoretical background EXAFS: experimental details EXAFS: data analysis, examples Edge structures and XANES QuickTime and a TIFF (LZW) decompressor are needed to see this picture.

71 EXAFS data analysis Extraction of EXAFS signal

72 Data analysis - Absorption coefficient Experimental signal 1 0 Ge, 10 K Extrapolation of pre-edge behaviour -1 µ tot x Photon energy (kev) 0-1 µx µ tot x Specific absorption coefficient Photon energy (kev) µ n x 2 1 µx Photon energy (kev)

73 Data analysis - The EXAFS signal Atomic absorption coefficient EXAFS signal 2 µ 0 x µx χ (k) Photon energy hν (kev) E s 0.6 (Å -1 ) 0.3 k χ(k) Edge energy χ()= k µ µ 0 µ k = 2m h 2 ( hν E s ) Photoelectron wavenumber Photoelectron wavenumber k (Å -1 )

74 EXAFS signals: examples Amorphous Germanium Crystalline Germanium T = 77 K T = 77 K T = 300 K 0.5 χ(k) (Å -1 ) k (Å -1 ) k (Å -1 ) k (Å -1 ) 1 coord. shell Several coord. shells Temperature effect

75 Quantitative analysis χ(k) = i A i (k)sinφ i (k) kχ(k) Sum over: S.S. paths (coord. shells) M.S. paths k (Å -1 ) Input for each path: backscattering amplitude phaseshifts inelastic terms Different analysis procedures

76 EXAFS data analysis Fourier transform

77 Data analysis - Fourier Transform k r 40 k 3 χ(k) W(k) weight window 0 F(r) = k max k min χ()k k n Wk () e 2ikr dk k (Å -1 ) 1st Ge, 10 K 2nd Modulus 3rd Peak's position and shape influenced by: - total phaseshifts - disorder - Fourier transform window Imaginary part r (Å)

78 Fourier Transform and distribution σ = 0.05 Å EXAFS simulation (Ge phases and amplit.) σ = 0.1 Å k χ(k) k (Å -1 ) r(a) F.T.: k= K 3, square w.

79 26 - Iron: bcc structure i N i R i (Å) Modulus of F.T. (a.u.) Fe (T = 300K) r (Å) Peak shift 6 6 Superposition of shells

80 29 - Copper: fcc structure i N i R i (Å) Modulus of F.T. (a.u.) Cu (T=4K ) Peak shift Focussing effect r (Å)

81 32 - Germanium: diamond structure i N i R i (Å) 1 4 a( 3)/ a/ a( 11)/ a a( 19)/ a( 6)/ Modulus of F.T. (a.u.) Ge (10 K) r (Å) 6 a= 5.66 Å

82 32-Ge: crystalline and amorphous K c-ge 10 K a-ge k χ(k) EXAFS signals K 300 K k χ(k) k (Å -1 ) k (Å -1 ) Fourier transforms F(r) (arb.u.) K 300 K c - Ge 10 K 300 K a - Ge r (Å) r (Å)

83 EXAFS data analysis Fourier back-transform

84 Data analysis - Fourier Back-transform r k Ge, 10 K 1st 2nd 3rd - Peak superposition - Multiple scattering - F.T. artifacts r (Å) ()=( 2 π) Fr χ' k r max r min () ()W' r e 2ikr dr k χ(k) k (Å -1 ) k (Å -1 ) k (Å -1 )

85 EXAFS for one shell Approx.: Single Scattering Plane waves Theory (interaction potentials + scattering theory) Experiment (reference samples) Inelastic terms Back-scattering amplitude Total phase-shift χ() k = S 2 0 e 2C 1 /λ f (k,π) N exp 2k 2 C C 1 4 k4 C 4 +K sin 2kC k3c +K+φ(k) 3 Coordination number N Even cumulants C 2 C 4 Odd cumulants C 1 C 3 P (r,λ)

86 Data analysis - Independent parameters k r k r N ind = 2 k r π + 1 Maximum number of independent parameters Correlation of parameters

87 EXAFS data analysis Phase and amplitude analysis

88 Real and imaginary part Real original signal 0 χ(k) k Complex Fourier transform F.T. artifacts A ˆ χ (k) = ˆ (k) 2i exp[ iφ ˆ (k)] Real Imaginary = 1 2 [ ] A ˆ (k) sinφ ˆ (k)+ i cosφ ˆ (k) 0 k Complex filtered signal

89 Calculation of phase and amplitude Amplitude ˆ A (k) = 2 [ Re ˆ χ (k)] 2 + Im ˆ χ (k) [ ] 2 Total phase Φ ˆ Re ˆ χ (k) (k) = tan 1 Im ˆ χ (k) Amplitude Total phase (rad) k k Ak ()= S 2 0 e 2C 1 /λ f (k,π) N exp 2k 2 C C 1 4 k4 C 4 +K? Φ()= k 2kC k 3 C φ() k?

90 Ratio method - phases If suitable model compound available Φ s Φ m = 2k C s m ( 1 C 1 ) 4 3 k 3 C s m ( 3 C 3 ) s = sample m = model Φ s Φ m 2k = C s m ( 1 C 1 ) 4 3 k 2 C s m ( 3 C 3 ) Φ s Φ m 2k C 1 C 3 = 0 C 1 C 3 > 0 k 2 k 2

91 Ratio method - amplitudes If suitable model compound available s = sample m = model ln As A = ln N s m N + C s m ( m 0 C 0 ) 2k 2 C s m ( 2 C 2 )+ 2 3 k 4 C s m 4 C 4 ( ) intercept Linear slope ln As A m C 4 = 0 C 4 > 0 C 1 k 2 k 2

92 Ratio method - results Ratio of coordination numbers N s N m C s 0 C m 0 = 2 C s m 1 C 1 λ 2lnC s m [ 1 lnc 1 ] Relative values of cumulants δc i = C i s C i m δc 1 δc 2 δc 3... Thermal expansion Width Asymmetry Absolute values? Physical meaning?

93 Ratio method - OK when Only Single Scattering Only one distance Suitable reference model available χ( k) = Ak ( )sinφ( k) First coordination shell, one distance Same sample-model chemical environment T or p-dep. Studies Amorphous.vs. crystalline samples 1st shell, different sample-model chemical environment Separated outer shells, weak M.S. Depending on sought accuracy 1st shell in bcc structure (2 distances) Superposed outer shells M.S. contributions

94 c-ge: 1st shell cumulants (effective distribution) Data analysis: Reference = 11 K spectrum C 1 (Å) C C 2 C i = C(T) - C(11 K) C 2 (Å 2 ) T-dependence: quality of exper. data convergence of series dynamical information C 3 (Å 3 ) C T (K)

95 EXAFS data analysis Interpretation of results

96 EXAFS and local dynamics R R r Absolute relative motion 3-dim. 1-dim. ρ(r) P(r,λ) =ρ(r) e -2r/λ /r 2 r P (r,λ) ρ (r) 2C C 1 C 1 λ + C * 1 C 1 = r Mean C 2 C 2 * = ( r r ) 2 Variance C 3 C 3 * = ( r r ) 3 Asymmetry -

97 Relative thermal motion u 0 R 0 u j r R 0 u u u 2 = r u j r ( u ) 2 0 = u u Perpendicular Parallel r R 0 Instantaneous distance r R 0 + u + u 2 2 R 0

98 Mean Square Relative Displacements u u r R 0 Mean values (harmonic approximation) u = 0 C 1 * = r R 0 + u 2 2R 0 MSRD C 2* = u 2 MSRD

99 MSRD and correlation C 2 = MSRD u 2 = R ˆ r = R ˆ R r r u j r ( u 0 ) [ ] 2 ( u ) 2 j + ( ˆ u ) ( ˆ u )( ˆ j u ) 0 R r R r R r u u MSD Mean Square Displacements DCF Displacement Correlation Function Uncorrelated motion (from Bragg diffraction) DCF > 0 DCF < 0

100 MSRD and lattice dynamics eigenvectors inter-cell phase-shift Harmonic approximation u 2 = 1 Nµ q,λ Q( r q,λ,t 2 r w j r ( q,λ)e ir m j / µ q R r r w 0 r ( q,λ) m 0 / µ R ˆ 2 normal coordinate projection Q( q r,λ,t 2 = E ( r q,λ) r ω 2 q,λ ( ) = h 2ω( q r,λ) r hω( q,λ) coth 2k B T = h ω q r 1 (, λ) exp hω ( q r,λ)/ kt [ ] T dependence

101 MSRD - Debye correlated model ω u 2 = 3h 2ω D3 µ ω D 0 ω coth hω 2 k B T ( ) 1 sin ωq R D ωq D R dω MSD DCF k 10-2 Å 2 Copper 2 MSD 2nd shell 4th shell 3rd shell 1st shell Best-fitting Debye temperatures θ D = 315 K θ M = 313 K θ 4 = 321 K θ 3 = 322 K θ 2 = 283 K θ 1 = 328 K T (K) 1 3

102 MSRD - Einstein correlated model m 1 m 2 µ u 2 = h coth hω E 2µω E 2kT ω k (Å 2 ) 0.02 Germanium 2 MSD T (K) 3rd 2nd 1st Debye θ D = 354 K θ M = 290 K θ 3 = 290 K θ 2 = 299 K θ 1 = 460 K Non-Bravais crystals ν =ω /2π (THz) ν 3 = 3.95 ν 2 = 4.21 ν 1 = 7.55 Einstein k = µω 2 (ev/å 2) k 3 = 3.95 k 2 = 4.21 k 1 = 7.55

103 Thermal expansion MSRD C 1 * R c + u 2 2 R 0 3 Cu - 1st shell 3 Ge - 1st shell -2 Å) EXAFS XRD 1 EXAFS 0 0 XRD T(K) T (K) G. Dalba et al., PRB, 70, (2004) G. Dalba et al., PRL 82, 4240 (1999)

104 Perpendicular MSRD Einstein frequencies Cu Ge THz THz u 2 u 2 MSRD (10-2 Å 2 ) Cu 1st shell Ge 1st shell T (K) T (K)

105 Amorphous and crystalline Germanium Parallel MSRD 8 a-ge 3 Nearest-neighbour distance a-ge MSRD (10-3 Å 2 ) c-ge C 1 * (10-2 Å) 2 1 c-ge EXAFS XRD T (K) T (K) Perpendicular MSRD MSRD (10-3 Å 2 ) T (K)

106 Introduction The absorption of X-rays EXAFS: theoretical background EXAFS: experimental details EXAFS: data analysis, examples Edge structures and XANES QuickTime and a TIFF (LZW) decompressor are needed to see this picture.

107 X-ray Absorption Near Edge Structure XANES: Electronic structure: Geometric structure: unoccupied electronic states formal oxidation states spin state local symmetry inter-atomic distances short-range structure XANES EXAFS Phenomenological interpretations Photon energy (kev) Theories molecular orbitals crystal bands multiple scattering

108 XANES - electronic structure: atoms µ (a.u.) Ar E b continuum Rydberg lev. empty full levels E - E b (ev) Core level width Resolution Direct probe of Rydberg levels Selectivity of angular momentum (dipole selection rules)

109 XANES: general properties 2 W if Ψ i H I Ψ f ρ ( E ) f Initial state Density of final states chemistry insensitive direct probe of final DOS localized site-projected DOS angular momentum symmetry electric dipole selection rules, angular momentum projected DOS energy width Γ (several ev ) Γ-convoluted DOS

110 XANES - electronic structure: crystals SbSI Sb L 1 (s p) L 3 (p s,d) Sb I µ (a.u.) I Total DOS E-E th Total DOS E-E th [Burattini, Dalba,, Nuovo Cim. (1986)] Core level width Resolution Direct probe of conduction band D.O.S. Selectivity of angular momentum Selectivity of site

111 Chemical shift µ 1 Cu 2 O CuO Y-Ba-Cu-O 0.5 Cu KCuO E (ev) (J.B. Boyce et al. Phys. Rev. B 1987) Oxidation Numbers (formal valences) I Cu 2 O II CuO III KCuO 2 Higher transitions energy are expected for higher valence states.

112 Core level chemical shift A Same atom in different chemical environments B Core level i - shift of binding energy E i = E i A () E i ( B) = k ( c q A q B )+ ( V A B i V i )+ ( E A B R E ) R valence charges potential from other atoms relaxation around core-hole Info on chemical state

113 XANES: electronic structure of N 2 Absorption Intensity ( arb. Units) N 1s 1π g * K-shell - gas-phase N 2 x10 N 1s Rydberg series Double excitations Shape resonance Photon energy (ev) Double excitations Double excitations associated to the 1s 1p g * transition. N 1s Rydberg series 6 1s Rydberg series N 1s 1π g * vibrational levels 1s 1π g * Photon Energy (ev) C.T. Chen and F. Sette, Phys. Rev. A 40 (1989)

114 XANES - electronic structure: molecules CH 4 CH 4 CF 4 CH 3 F CH 4 CH 2 F 2 Continuum resonances CHF E (ev) CF E (ev) CF 4 Transitions to: Rydberg states (narrow) Valence orbitals (broad) Resonances (broad)

115 XANES and multiple scattering χ()= k µ µ 0 [ ] Single Scattering µ ( k) = µ ( k) + χ( k) µ Multiple Scattering µ ( ) = µ ( k) [ + χ ( k) + χ ( k) + ( k) +K] k χ4 χ 2 χ n () k Contribution from all n-order paths χ 3 χ 4 g 2 g 3 g 4 g n = n-body correlation function

116 Multiple scattering series µ ()= k µ 0 ( k) 1 + χ 2 k [ ] ( )+ χ 3 k ()+Kχ n ( k)+k Full Multiple Scattering Intermediate Multiple Scattering Single Scattering Photo-electron wave-number k Set of coordinates Photo-electron path χ n ()= k Ak, ( { r })sin kr p +φ k, r ( { }) [ ] Neglecting thermal disorder! Functions of potential

117 MS calculation of XANES Structural model (atomic cluster) Change model parameters µ Exper. spectrum Sorting of relevant paths E Calculating amplitudes and phases FIT Compare χ p () k = p A ( p k, {} r p )sin kr p +φ p k, p r [ ( {})] p

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