EXAFS data analysis Extraction of EXAFS signal 1
Experimental EXAFS spectrum Φ x Φ µ (ω) source monochromator sample detectors 1 6 1 5 32 - Ge 1 4 1 3 1 2 1 1 1 1 1 Photon energy hω (kev) 1-1 Ge, 1 K µ tot x -2 11 11.5 12 Photon energy (kev) 2
Data analysis - Absorption coefficient Experimental signal 1 Ge, 1 K Extrapolation of pre-edge behaviour -1 µ tot x 1-2 11 11.5 12 Photon energy (kev) -1 µx µ tot x Specific absorption coefficient -2 11 11.5 12 Photon energy (kev) µ n x 2 1 µx 3 11 11.5 12 Photon energy (kev)
Data analysis - The EXAFS signal Atomic absorption coefficient EXAFS signal 2 µ x µx.6.3 χ (k) 1 -.3 -.6 11 11.5 12 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 -.6 5 1 15 2 Photoelectron wavenumber k (Å -1 ) 4
Two distances seen by EXAFS 5
The phase s effect 6
The back-scattering function 7
Their effect on EXAFS signal 8
The Debye-Waller damping effect 9
The m.f.p. damping effect 1
Weighting the EXAFS signal 11
EXAFS signals: examples Amorphous Germanium Crystalline Germanium T = 77 K T = 77 K T = 3 K.5 χ(k) (Å -1 ) -.5 5 1 15 k (Å -1 ) 5 1 15 k (Å -1 ) 5 1 15 k (Å -1 ) 1 coord. shell Several coord. shells Temperature effect 12
Quantitative analysis χ(k) = i A i (k)sinφ i (k).5 kχ(k) Sum over: S.S. paths (coord. shells) M.S. paths -.5 5 1 15 k (Å -1 ) Input for each path: backscattering amplitude phaseshifts inelastic terms Different analysis procedures 13
EXAFS data analysis Fourier transform 14
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 -4 5 1 15 2 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 2 4 6 r (Å) 15
Fourier Transform and distribution σ =.5 Å EXAFS simulation (Ge phases and amplit.) σ =.1 Å 1.5 2 2.5 3 3.5 k χ(k).2 -.2 2 6 1 14 k (Å -1 ) 1.5 2 2.5 3 3.5 r(a) F.T.: k=2.5-16 K 3, square w. 16
26 - Iron: bcc structure i N i R i (Å) 1 8 2. 48 2 6 2. 86 3 12 4. 5 4 24 4. 75 5 8 4. 96 6 6 5. 73 2 3 1 Modulus of F.T. (a.u.).6.4.2 Fe (T = 3K) 1 2 3 4 5 6 r (Å) 12 24 a = 2.86 Å 8 8 Peak shift 6 6 Superposition of shells 17
29 - Copper: fcc structure i N i R i (Å) 1 12 2. 55 2 6 3. 61 3 24 4. 42 4 12 5. 1 5 24 5. 7 6 8 6. 25 2 1 4 3 Modulus of F.T. (a.u.) 12 8 4 Cu (T=4K ) 12 6 24 12 a = 3.61 Å Peak shift Focussing effect 1 4 1 2 3 4 5 6 r (Å) 18
32 - Germanium: diamond structure i N i R i (Å) 1 4 a( 3)/4 2.45 2 12 a/ 2 4. 3 12 a( 11)/4 4.69 4 6 a 5.66 5 12 a( 19)/4 6.16 6 24 a( 6)/2 6.93 4 1 3 6 5 2 Modulus of F.T. (a.u.) 6 4 2 Ge (1 K) 4 12 12 12 24 1 2 3 4 5 6 7 r (Å) 6 a = 5.66 Å 19
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). -.4 5 1 15 2 k (Å -1 ) 5 1 15 2 k (Å -1 ) Fourier transforms F(r) (arb.u.) 6 4 2 1 K 3 K c - Ge 1 K 3 K a - Ge 2 4 6 r (Å) 2 2 4 6 r (Å)
EXAFS data analysis Fourier back-transform 21
Data analysis - Fourier Back-transform r k Ge, 1 K 1st 2nd 3rd - Peak superposition - Multiple scattering - F.T. artifacts 1 2 3 4 5 6 r (Å) ( ) = ( 2 π) F( r) W' ( r) χ' k r max r min e 2ikr dr k χ(k).2 -.2 2 6 1 14 18 k (Å -1 ) 2 6 1 14 18 2 6 1 14 18 k (Å -1 ) k (Å -1 ) 22
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 2 1 1 f ( k) Nexp 4 3 [ kc k C +... + φ( k) ] 1 3 3 Amplitude: even C n 2 2 4 [ 2k C + k C +...] 2 3 4 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
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 4 +... sin 2 kc 1 4 3 k 3 C 3 +... + 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
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 1 2 4 3 ( k) = f k, π) Nexp 2k C + k C + K sin 2kC k C + K+ φ( k) 2 1 2 4 ( 2 4 1 3 4 3 Coordination number N Even cumulants C 2 C 4 Odd cumulants C 1 C 3 P (r,λ) 25
Data analysis - Independent parameters k r k r N ind = 2 k r π + 1 Maximum number of independent parameters Correlation of parameters 26
EXAFS data analysis Phase and amplitude analysis 27
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 1 2 4 ( ) = f ( k, π) Nexp 2k C + k C + K A k 2 S e kc 2 1 2 2 4 4 Φ( k)= 2kC 1 ( 4 / 3)k 3 C 3 +...+φ( k) 2kC 1 +ϑ( k) 28
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
Inverse Fourier transform Real r Imaginary Full r-range Only r > r Real Real k k Imaginary Imaginary k k 3
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
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 2 2 + 2 C 1 4 k4 C 4 +K Φ( k)= 2kC 1 4 3 k 3 C 3 +...+φ( k)?? 32
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
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 ) 2 2 3 4 4 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
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
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
Copper 1st shell - ratio method Phases Amplitudes 37
EXAFS data analysis Fitting with a theoretical model 38
The FeO example 39
The first shell 4
The first shell 41
The second shell 42
A free program for EXAFS data analysis 43
Loading data 44
Reading raw data 45
B.G subtraction and FT 46
Enlarged view of the edge region 47
Legend 48
Details about the FT 49
Isolation of the first shell 5
First shell BFT 51
Amplitude and phase calculation 52
BFT details with calculated amplitude and phase 53
Athena: another (free) program for EXAFS 54
BG removal 55
FT of the two samples 56
Back FT of the first shell 57
Amplitude-Phase analysis 58
Amplitude analysis: ln(n s / N m ) 59
Phase analysis : Φ s Φ m 6
The end Thank you for your attention 61