EXAFS data analysis. Extraction of EXAFS signal

Transcription

1 EXAFS data analysis Extraction of EXAFS signal 1

2 Experimental EXAFS spectrum Φ x Φ µ (ω) source monochromator sample detectors Ge Photon energy hω (kev) 1-1 Ge, 1 K µ tot x Photon energy (kev) 2

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

4 Data analysis - The EXAFS signal Atomic absorption coefficient EXAFS signal 2 µ x µx.6.3 χ (k) Photon energy hν (kev) E s.6 (Å -1 ).3 k χ(k) Edge energy χ( k)= µ µ µ -.3 k = 2m h 2 ( hν E s ) Photoelectron wavenumber Photoelectron wavenumber k (Å -1 ) 4

5 Two distances seen by EXAFS 5

6 The phase s effect 6

7 The back-scattering function 7

8 Their effect on EXAFS signal 8

9 The Debye-Waller damping effect 9

10 The m.f.p. damping effect 1

11 Weighting the EXAFS signal 11

12 EXAFS signals: examples Amorphous Germanium Crystalline Germanium T = 77 K T = 77 K T = 3 K.5 χ(k) (Å -1 ) k (Å -1 ) k (Å -1 ) k (Å -1 ) 1 coord. shell Several coord. shells Temperature effect 12

13 Quantitative analysis χ(k) = i A i (k)sinφ i (k).5 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 13

14 EXAFS data analysis Fourier transform 14

15 Data analysis - Fourier Transform k r 4 k 3 χ(k) W(k) weight window F(r) = k max k min χ( k) k n W( k) e 2ikr dk k (Å -1 ) 1st Ge, 1 K 2nd Modulus 3rd Peak's position and shape influenced by: - total phaseshifts - disorder - Fourier transform window Imaginary part r (Å) 15

16 Fourier Transform and distribution σ =.5 Å EXAFS simulation (Ge phases and amplit.) σ =.1 Å k χ(k) k (Å -1 ) r(a) F.T.: k= K 3, square w. 16

17 26 - Iron: bcc structure i N i R i (Å) Modulus of F.T. (a.u.) Fe (T = 3K) r (Å) a = 2.86 Å 8 8 Peak shift 6 6 Superposition of shells 17

18 29 - Copper: fcc structure i N i R i (Å) Modulus of F.T. (a.u.) Cu (T=4K ) a = 3.61 Å Peak shift Focussing effect r (Å) 18

19 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 (1 K) r (Å) 6 a = 5.66 Å 19

20 32-Ge: crystalline and amorphous.4 1 K c-ge 1 K a-ge k χ(k). -.4 EXAFS signals.4 3 K 3 K k χ(k) k (Å -1 ) k (Å -1 ) Fourier transforms F(r) (arb.u.) K 3 K c - Ge 1 K 3 K a - Ge r (Å) r (Å)

21 EXAFS data analysis Fourier back-transform 21

22 Data analysis - Fourier Back-transform r k Ge, 1 K 1st 2nd 3rd - Peak superposition - Multiple scattering - F.T. artifacts r (Å) ( ) = ( 2 π) F( r) W' ( r) χ' k r max r min e 2ikr dr k χ(k) k (Å -1 ) k (Å -1 ) k (Å -1 ) 22

23 The cumulants Cumulant expansion of EXAFS ln P(r,λ ) exp( 2ikr) dr = n= ( 2ik) n n! C n S χ( k) = 2 2 e kc sin 2 C / λ f ( k) Nexp 4 3 [ kc k C φ( k) ] Amplitude: even C n [ 2k C + k C +...] exp C ( ) exp ( 2C 1 / λ) C 1 2 Normalisation Degrees of disorder 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 23

24 The cumulants Degrees of disorder χ( k) = N f ( k,π ) S 2 e 2 C 1 / λ k C 1 2 exp 2k 2 C 2 ( ) exp 2 3 k 4 C sin 2 kc k 3 C φ (k) Harmonic approx. Standard formula C 3 = C 4 =... = 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 } Cumulant series fastly convergent High-order anharmonic terms Cumulant series: slowly convergent... not convergent 24

25 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 S 2 e kc 2C / λ ( k) = f k, π) Nexp 2k C + k C + K sin 2kC k C + K+ φ( k) ( Coordination number N Even cumulants C 2 C 4 Odd cumulants C 1 C 3 P (r,λ) 25

26 Data analysis - Independent parameters k r k r N ind = 2 k r π + 1 Maximum number of independent parameters Correlation of parameters 26

27 EXAFS data analysis Phase and amplitude analysis 27

28 Phase and amplitude χ(k) k χ( k) = A( k)sinφ( k) = 1 2i A ( k )e iφ(k) + 1 2i A ( k )e +iφ(k) 2C / λ ( ) = f ( k, π) Nexp 2k C + k C + K A k 2 S e kc Φ( k)= 2kC 1 ( 4 / 3)k 3 C φ( k) 2kC 1 +ϑ( k) 28

29 Direct Fourier transform F = ( r) k k max min = k k max min ( ) iφ( k ) iφ( k ) [ e e ] ( ) 2ik ( C + r ) + i ( k ) 2ik ( C r ) A k 2i A k 2i dk [ 1 θ 1 + iθ ( k ) e e ]dk e 2ikr < > Real F(r) for r C 1 r C 1 Imaginary r (symmetric) r (antisymmetric) 29

30 Inverse Fourier transform Real r Imaginary Full r-range Only r > r Real Real k k Imaginary Imaginary k k 3

31 Real and imaginary part Real original signal 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) k Complex filtered signal 31

32 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 A( k)= S 2 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)?? 32

33 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 = C 1 C 3 > k 2 k 2 33

34 Ratio method - amplitudes If suitable model compound available s = sample m = model ln A A s m = ln N N s m + s m 2 s m 2 4 s m ( C C ) 2k ( C C ) + k ( C C ) intercept Linear slope ln As A m C 4 = C 4 > s N ln + m N s m ( C C ) k 2 k 2 34

35 Ratio method - results Ratio of coordination numbers N s N m C s C m = 2 C s m 1 C 1 λ 2 lnc 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? 35

36 Ratio method - OK when Only Single Scattering Only one distance Suitable reference model available χ( k) = A( k)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 36

37 Copper 1st shell - ratio method Phases Amplitudes 37

38 EXAFS data analysis Fitting with a theoretical model 38

39 The FeO example 39

40 The first shell 4

41 The first shell 41

42 The second shell 42

43 A free program for EXAFS data analysis 43

44 Loading data 44

45 Reading raw data 45

46 B.G subtraction and FT 46

47 Enlarged view of the edge region 47

48 Legend 48

49 Details about the FT 49

50 Isolation of the first shell 5

51 First shell BFT 51

52 Amplitude and phase calculation 52

53 BFT details with calculated amplitude and phase 53

54 Athena: another (free) program for EXAFS 54

55 BG removal 55

56 FT of the two samples 56

57 Back FT of the first shell 57

58 Amplitude-Phase analysis 58

59 Amplitude analysis: ln(n s / N m ) 59

60 Phase analysis : Φ s Φ m 6

61 The end Thank you for your attention 61