Spectroscopie d absorption X dans les systèmes désordonnés: méthodes numériques d analyse Guillaume Ferlat

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1 Spectroscopie d absorption X dans les systèmes désordonnés: méthodes numériques d analyse Guillaume Ferlat Institut de Minéralogie et de Physique des Milieux Condensés (IMPMC) Université P. & M. Curie (UPMC), France Modélisation des verres, Journées Thématiques CNRS, Marcoule, 1-13 Mai 11

2 XAS (in disorded systems) brings structural and electronic information a complementary (to diffraction) and useful structural probe a chemically-selective probe (wide applicability to almost all the elements) highly sensitive to dilute quantities «easy to do» experiments sometimes, XAS is the only structural probe which might be doable: diluted samples, chemically complex systems, high pressure and/or high temperature environments But: interpreting/analysing the data can be tricky or even misleading XAS is of limited interest if you know nothing (or almost) of your system The use of external information (e.g. MD, diffraction) is highly desirable

3 Basics of x-ray absorption spectroscopy (XAS) transmission experiment 1,6 K-edge x-ray I I 1 sample I 1 I e d μd (arb. units) 1,,8,4 Rb in aqueous solution 15, 15, 15,4 15,6 15,8 Incident photon energy (kev)

4 Basics of x-ray absorption spectroscopy (XAS) x-ray transmission experiment I I 1 sample I 1 I e d μd (arb. units) 1,6 1,,8 1 K-edge 3,4 Rb in aqueous solution 15, 15, 15,4 15,6 15,8 Incident photon energy (kev) E c = E - E b E F 1s E < E b E ~ E b E > E b

5 Basics of x-ray absorption spectroscopy (XAS) 1,6 μd (arb. units) 1,,8,4 15, 15, 15,4 15,6 15,8 Incident photon energy (kev)

6 Basics of x-ray absorption spectroscopy (XAS) x-rays atom B atom A

7 Basics of x-ray absorption spectroscopy (XAS) x-rays atom B atom A photo-electron wave

8 Basics of x-ray absorption spectroscopy (XAS) x-rays atom B atom A scattered electron wave photo-electron wave Fermi s Golden Rule (in one-electron approximation): E f E F ( E ) f Hint i ( E E f E i ) f i is an initial deep core state (e.g. 1s). f is an unoccupied state in the presence of a core hole. H int is the electron transition operator.

9 Basics of x-ray absorption spectroscopy (XAS) Electron mean-free path XANES EXAFS Photo-electron kinetic energy (ev)

10 Basics of x-ray absorption spectroscopy (XAS) Electron mean-free path XANES EXAFS Photo-electron kinetic energy (ev) - dominated by single-scattering (SS) paths (+ short multiple-scattering (MS) paths) single-scattering EXAFS: Multiple-scattering (3-leg)

11 Basics of x-ray absorption spectroscopy (XAS) Electron mean-free path XANES EXAFS XANES: - dominated by multiple-scattering terms - close to the edge : reflects the edos of empty levels at the absorbing sites convoluted by core-hole effects Photo-electron kinetic energy (ev) - dominated by single-scattering (SS) paths (+ short multiple-scattering (MS) paths) single-scattering EXAFS: Multiple-scattering (3-leg)

12 ( k) The standard EXAFS formula (restricted to single-scattering contributions) Ri N i ( k ) S fi ( k) e sin kri i ( k) c ( k) kr shell i i e k i k m( E E) fi, (k) S i : scattering amplitude and phase of atom type i : photo-electron mean free path : reduction factor (.75-1.) due to many-body effects N i, R i, i : structural parameters of shell i Dale E. Sayers, Edward A. Stern, and Farrel W. Lytle, New Technique for Investigating Noncrystalline Structures : Fourier Analysis of the Extended X-Ray Absorption Fine Structure, Phys. Rev. Lett. 7, (1971).

13 ( k) The standard EXAFS formula (restricted to single-scattering contributions) Ri N i ( k ) S fi ( k) e sin kri i ( k) c ( k) kr shell i i e k i k m( E E) Non-structural parameters fi, (k) S i : scattering amplitude and phase of atom type i : photo-electron mean free path : reduction factor (.75-1.) due to many-body effects N i, R i, i : structural parameters of shell i Dale E. Sayers, Edward A. Stern, and Farrel W. Lytle, New Technique for Investigating Noncrystalline Structures : Fourier Analysis of the Extended X-Ray Absorption Fine Structure, Phys. Rev. Lett. 7, (1971).

14 ( k) The standard EXAFS formula (restricted to single-scattering contributions) Ri N i ( k ) S fi ( k) e sin kri i ( k) c ( k) kr shell i i e k i k m( E E) fi, (k) S i : scattering amplitude and phase of atom type i : photo-electron mean free path : reduction factor (.75-1.) due to many-body effects N i, R i, i : structural parameters of shell i strong correlations between E R and N limited k range [.5:] Å -1. Structural solutions hardly univoque standard formula valid only in the case of small and gaussian disorder Dale E. Sayers, Edward A. Stern, and Farrel W. Lytle, New Technique for Investigating Noncrystalline Structures : Fourier Analysis of the Extended X-Ray Absorption Fine Structure, Phys. Rev. Lett. 7, (1971). S

15 Theory of XAS Reduction of the photo-absorption many-body problem into that of an electron moving in an optical potential (that takes into account both the extrinsic and intrinsic losses). Aim : construction of the effective optical potential and solution of the associated Green s function equation for the calculation of the one-electron cross-section. Construction of the local charge density by overlapping neutral atomic charge densities; relaxation of the density around the core hole (optionally in a self-consistent way). Ansatz for the optical potential (e.g. Hedin-Lundqvist: self-energy of an electron embedded in a locally uniform interacting electron gas). Real part : scattering from atomic sites described through k-dependent partial wave phase shifts for different angular momenta l. Imaginary part : gives mean free path. Solve the Schrödinger equation for continuum states. Often (but not necessarily) using the muffin-tin scheme (reasonable approx. for EXAFS but might be too coarse at low energies). Polarisation averaged XAS cross-section for transitions to a dipole selected final state: ( E) Im 1 1 l 1 1 L, L T (1 GT ) l Im( t ) m, atomic cross-section scattering and propagator matrices A. Filipponi et al., Phys. Rev. B (1995)

16 Multiple-Scattering Theory of XAFS ( E) Im 1 1 l 1 1 L, L T (1 GT ) l Im( t ) m, if GT 1, then T( I GT) 1 T( I GT GTGTGTGTGT) ( E) 1 i i i j i, j ij 3 i j, k i k ijk 4 (SS) (3-leg) (4-leg) with ( k) A( k, R)sinkR ( k) n tot Limitations: valid only in the weak scattering regime, i.e. at high energies (EXAFS) path proliferation problem (need for ad hoc filtering criteria) A. Filipponi et al., Phys. Rev. B (1995)

17 Multiple-Scattering Theory of XAFS ( E) ( E) 1 ( E) can be factored in terms of an atomic background and an oscillatory part with 3 4 (SS) (3-leg) (4-leg) For each path : f eff ( k ) k ( k) S f ( k) e sin kr ( k) ( k e, eff 1 R ) c kr : effective scattering amplitude and phase of path R : effective path length (= R tot /) : mean-square fluctuation in path FEFF code : Ankudinov et al. Phys. Rev. B 58, 7565 (1998)

18 Theory of XAS: irreducible n-body contributions ( E) ( ) 1 i ij E 3 i i j i, i j, k i j k 1 (SS) (3-leg) (4-leg) (, i) j) ijk 4 () (3) (4) (, i, (, i, j, k) i ( i, j ) ( i, j, k ) ( n) ( n) / two-body three-body four-body (, i) (, i, j) : atomic cross-section : cross-section of the structure including atoms and i only : cross-section of the structure including atoms, i and j Irreducible n-body cross-sections: (, i) (, i) () (3) () () (, i, j) (, i, j) (, i) (, j) () 4 6 (3) ij 3 i j iji 4 4 jij 4 A. Filipponi et al., Phys. Rev. B (1995)

19 Theoretical approaches to XAS Band-structure and exact diagonalisation methods (e.g. XSpectra, FDMNES) Full Multiple Scattering Cluster (matrix inversion or recursion method) (e.g. Feff, Gnxas, ) Multiple Scattering expansion (e.g. Feff, Gnxas, Excurve, ) 5 ev ev Pros: - accurate electronic charge densities and potentials - accurate Fermi energy - exact treatment of curved-wave effects and MS to all orders Cons: - computationally inefficient; nb of basis-set functions and orbital angular momentum components rapidly increase with E - structural and electronic information not disentangled Energy Pros: - computational efficiency - geometrical interpretation of XAFS Cons: - untractable and inaccurate at low energies

20 For a mono-atomic system: Configurational average using the n-body contributions ( k) 4 with R 8 (n) max r g ( r) () ( k, r) dr (3) r1 r sin( ) g3( r1, r, ) ( k, r1, r, ) dr1 dr d : irreducible n-body contributions the above configurational average is of general validity (crystals, glasses, liquids) the ability to probe g n (n>) is a unique feature of XAS however, the inversion of to get the g n functions is a ill-defined problem The radial distribution function probed by x-ray absorption spectroscopy A. Filipponi, J. Phys.: Condens. Matter, 6, (1994)

21 ( k) Configurational average i) The standard EXAFS formula (don t use it!): Ri Ni ( k ) S fi ( k) e sinkri i ( k) c ( k) e kr shell i i N i, R i, i : structural parameters of shell i valid only in the case of small and gaussian disorder k ii) Cumulant expansion of the prob. distribution p(r) (refrain from using it ): e using weak-disorder approximations ik ( rr) (1) () (3) exp ( r R) ( r R) n ( r R) 3 (ik) n! n ( n) exp( W i) () W ( k) k 3 (1) (3) ( k) k 4 k 3 (4) 3 valid only in the case of weak anharmonic disorder may produce unphysical distributions i

22 Configurational average using weak-disorder approximations iii) Gamma-like distributions (gnxas) p(r) (4 / ) 4 (r R) (4/ )1 exp 4 (r R) (4/ ) : Euler function R,, : average, variance and skewness ( = C 3 / 3 ) of the distribution Then, <> obtained using Taylor expansions for amplitudes and phases around the average distance R. not to be used for broad distributions (wider than ~ 1 Å)

23 Diffraction versus XAS For a mono-atomic system: Diffraction: S(k) 4 k r( g (r) 1)sin(kr)dr probes the full r range obeys the compressibility sum rule multi-component system : N(N+1)/ partials contributing XAS: ( k) with 4 () R max r g ( r) A( k, r)sin(kr () ( k, r) dr ( k, r)) short-sighted (R max ~ ) EXAFS k twice Diffraction k multi-component system : N partials contributing The radial distribution function probed by x-ray absorption spectroscopy A. Filipponi, J. Phys.: Condens. Matter, 6, (1994)

24 Multi-electronic excitations Taken from positions in energy are well-known but not their shape and intensity (~ few %). taken into account empirically (usually by arc-tangent like functions). shake-up, shake-off resonances produce an amplitude reduction factor S with respect to the one-electron calculation.

25 MD/MC simulations: To sample the system phase space and get an energetically constrained model To assist the XAS analysis : - in a complex system : which species are contributing to the XAS signal and how - configurational average (intrinsic account of the disorder) - avoids the proliferation problem of fitting parameters: all structural parameters are fixed by the model (only two unknowns: E and S ) - to provide an initial model which may be refined (e.g. RMC/EPSR, gnxas) - to probe the data sensitivity To get information beyond the first shells of neighbours

26 Configurational average from the n-body distributions: in practice For each partial X-Y: partial g X-Y (r) partial R () X Y ( k) 4Y r g X Y ( r) X Y ( k, r) dr phagen grxas () X Y ( k, r) Pros: staighforward calculation (very fast) straightforward evidence of each atomic type contribution Cons: cumbersome for n-body contributions > Then, summation over the partials to get the total signal: X ( k ) ( k) Y X Y The Gnxas package: free but not open-source

27 Configurational average from a series of paths MS paths average over a clusters ensemble (k) (arb.un.) k (Å -1 ) snapshot #15 For each absorbing atom in each snapshot: cluster ( k) 1 kr R eff ( k ) f ( k) e sinkr ( k Note that = in cluster ) Direct Modelling of EXAFS Spectra from Molecular Dynamics Simulations, Bruce J. Palmer, David M. Pfund, John L. Fulton, J. Phys. Chem (1996)

28 Configurational average from a series of paths MS paths average over a clusters ensemble + (k) (arb.un.) k (Å -1 ) snapshot #15 snapshot # For each absorbing atom in each snapshot: cluster ( k) 1 kr R eff ( k ) f ( k) e sinkr ( k Note that = in cluster ).1 (k) (arb.un.) N cluster = N abs * N snapshot k (Å -1 ) +... (k) (arb.un.) average (1 snap.) cluster ( k ) ( k) N clusters k (Å -1 ) Direct Modelling of EXAFS Spectra from Molecular Dynamics Simulations, Bruce J. Palmer, David M. Pfund, John L. Fulton, J. Phys. Chem (1996)

29 Configurational average from a set of clusters: in practice MD/MC trajectory (xyz file) resize clusters and prepare feff input cards iterative calling Feff Compute the average 1 cluster ( k ) ( k) N clusters Feff input card : Pros: easy to evidence the MS contributions allows to study the influence of fluctuations (space/time) allows a site-by-site study Cons: need a lot of clusters for convergence (~ 1 1) (Each non-scf calculation ~ 1 mn; each SCF calculation ~ mn) the identification of the contribution from a given atomic type of neighbours is not straightforward The Feff code: open-source but not free

30 Refining the initial (MD) model Liquid InSe (AIMD) g(r) 3 11 K SeSe InIn InSe r (Å) G. Ferlat et al. Phys. Rev. B, 69, 155 (4)

31 Refining the initial (MD) model g(r) K EXAFS SeSe InIn InSe (k) (arb. units) In K-edge InSe (k) InIn (k) MD (k) / exp (k) k (Å) -1 Se K-edge SeIn (k) SeSe (k) MD (k) / exp (k) k (Å) r (Å) G. Ferlat et al. Phys. Rev. B, 69, 155 (4)

32 Refining the initial (MD) model g(r) K r (Å) EXAFS SeSe InIn InSe (k) (arb. units) The initial model (AIMD) needs to be refined G. Ferlat et al. Phys. Rev. B, 69, 155 (4) ND In K-edge InSe (k) S(q) (arb. units) k (Å) -1 InIn (k) MD (k) / exp (k) Se K-edge SeIn (k) SeSe (k) MD (k) / exp (k) k (Å) -1 S(q) AIMD S(q) Expt. ND (Barnes 1996) q (Å) -1

33 Refining the initial (MD) model: g(r) refinement (using gnxas) g(r) 4 3 g MD (r) (k) (arb. units) MD (k) exp (k) k (Å) r(å) The initial model g(r) s (which do not well reproduce the XAS data)...

34 Refining the initial (MD) model: g(r) refinement (using gnxas) 4 3 g MD (r) g(r) 1 N 1, R 1, σ 1, β 1 N, R, σ, β fixed tail r(å)... are decomposed into short-range peaks plus a long-range tail. The parameters describing the short-range peaks are then freed in a fitting procedure...

35 Refining the initial (MD) model: g(r) refinement (using gnxas) g(r) g MD (r) g refined-model (r) r(å)... until agreement is achieved between the experimental exp (k) and the XAS signal computed from the refined model refined-md (k). (k) (arb. units) refined-model (k) exp (k) k (Å) -1

36 Refining the initial (MD) model: g(r) refinement (using gnxas) (k) (arb. units),15,1,5, -,5 AIMD / EXAFS In K-edge InSe (k) InIn (k),6,4, refined (k) / exp (k) refined (k) / exp (k) residual, -, Se K-edge SeIn (k) SeSe (k) residual In Se Multiple-edge analysis : ( k) ( k) ( k) InSe InIn ( k) ( k) ( k) SeIn SeSe better constrain of the In-Se contribution k (Å) k (Å) -1 The refinement can be driven by a multiple-edge fitting of the XAS data

37 Refining the initial (MD) model: g(r) refinement (using gnxas) AIMD / EXAFS AIMD / ND (k) (arb. units),15,1,5, -,5 In K-edge InSe (k) InIn (k) refined (k) / exp (k) residual,6,4,, -, Se K-edge SeIn (k) SeSe (k) refined (k) / exp (k) residual S(q) (arb. units) S(q) AIMD S(q) Exp. ND (Barnes 96) q (Å) k (Å) k (Å) -1 The refinement (which was driven by XAS only) is validated a posteriori by the ND data. S(q) 1..9 S InSe (q) MD S tot (q) Exp. ND S InSe (q) refined G. Ferlat et al. Phys. Rev. B, 69, 155 (4)

38 Refining the initial (MD) model: g(r) refinement (using gnxas) 6 5 g InSe (r) 4 AIMD More on the methodology, see: g(r) 3 refined-model - A. Filipponi, J. Phys.: Condens. Matter, 6, (1994) - P. D Angelo et al., J. Chem. Phys. 1, 985 (1994) - A. Trapananti & A. DiCicco, Phys. Rev. B, 7, 1411 (4) r(å) Drawback of this method: empirical refinement of the g(r) shape: the microscopic 3-D information is lost -> RMC-GnXAS: A. Di Cicco et al. Phys. Rev. Letters (3)

39 Refining the model: RMC/EPSR First RMC-XAS: S. J. Gurman, R. McGreevy, J. Phys. : Condens. Matter,, 9463 (199) RMC should not be driven by XAS only. It should include long-range constraints such as diffraction data : A. Di Cicco et al., Phys. Rev. Letters, 91, (3): RMC-GnXAS(ND+EXAFS) M. V. Coulet et al., Phys. Rev. B, 7, 1749 (5) : RMC(ND + EXAFS) D. T. Bowron, Pure Appl. Chem., 8, 111 (8): EPSR(ND/XRD + EXAFS)

40 MD-XAS studies of liquids and glasses: selected examples I) Zr environment in borosilicate glasses (weak disorder, multi-component system, many-body contributions) II) Ions in aqueous solutions (strong disorder)

41 Environment of Zr in borosilicate glasses Empirical MD simulations: SiO -B O 3 -Na O-Al O 3 -ZrO (4% - 3% - % - 5% - 5%) Born-Mayer-Huggins (BMH) potentials + 3-body terms (on network formers) Time step: 1 fs ; Equilibration at 3 K: 7 ps N = 4 atoms; Box length ~ 3 Å g(r) ZrO ZrB ZrSi ZrAl ZrNa n(r) r (Å) r (Å)

42 Environment of Zr in borosilicate glasses MD-EXAFS calculations: g(r) (Grxas) χ(k) (arb. units),16,1,8,4, r (Å) k (Å -1 ) ZrO ZrSi ZrB ZrNa ZrAl MD simulations allow to evidence the dominant contributions to the XAS signal

43 Environment of Zr in borosilicate glasses MD-EXAFS versus Expt.:,1 χ(k) (arb. units),5, -,5 Zr K-edge Expt. / MD -, k (Å -1 ) Qualitative agreement. The MD structural model needs to be refined

44 Environment of Zr in borosilicate glasses Refining the MD model,1 14 g ZrO (r) Refined model MD χ(k) (arb. units),5, -,5 -,1 Expt. Refined model (-bodies only) Residual r(å) Sharp 1 rst peak in the Zr-O distribution N ZrO ~ 6 ZrO 6 octahedra -, k (Å -1 ) Significant residual: evidence for three-body contributions

45 Environment of Zr in borosilicate glasses Refining the MD model g 3 (r):,1 O O Zr O Zr O χ(k) (arb. units),5, -,5 -,1 Expt. Refined model (- and 3-bodies) Residual O Zr O -, Significant three-body contributions from intra-octahedra paths

46 Environment of Zr in borosilicate glasses Refining the MD model χ(k) (arb. units),4,35,3,5,,15,1,5, -,5 -,1 -, k (Å -1 ) ZrO (g ) Zr-O-O (g 3 ) ZrSi ZrB ZrNa Expt. / Refined Residual The 3-body contributions are larger than the nd shells -bodies signals Determining the next-nearest neighbours distributions from XAS is still challenging G. Ferlat et al., Phys. Rev. B, 73, 147 (6)

47 Environment of Zr in borosilicate glasses Site by site inspection of the individual spectra:,1,1 χ(k) (arb. units),5, -,5 -,1 -,15 Expt. Site # χ(k) (arb. units),5, -,5 -,1 -,15 Expt. Site # k (Å -1 ) k (Å -1 ) bad site good site Moderately disorded environment: an individual site by site analysis is possible.

48 Environment of Zr in borosilicate glasses Site by site inspection of the radial distribution functions: refined model (gnxas) 14 1 refined model (gnxas) g ZrO (r) MD model (average) Site # 36 g ZrO (r) MD model (average) Site # 8 1,9,,1,,3,4,5 r(å) bad site 1,9,,1,,3,4,5 r(å) good site The site by site analysis corroborates the g(r) refinement.

49 Ions in aqueous solutions: hydration A Simulations: (NaCl) Expt (EXAFS): (1) (RbBr) Ambient n ~ 6 n ~ 7 Supercritical ~ 6 ~ 1) S.L. Wallen et al., J. Phys. Chem., 11, 963 (1997)

50 Ions in aqueous solutions: complexation A C Simulations: (NaCl) Ambient ~ Supercritical 3% () Expt (EXAFS): (1) (RbBr) ~? (m=.-1.5 kg/mol) 1) S.L. Wallen et al., J. Phys. Chem., 11, 963 (1997) ) E. Oelkers, H. Hegelson, Science, 61, 888 (1993)

51 Ions in aqueous solutions: hydration MD simulations of diluted 1:1 aqueous electrolytes 3 Br-O Ambient / Supercritical (RbBr).4 -(H O) 99.6 (m=.) g(r) 1 Ambient: ~ 1. g.cm -3, T = 3 C Supercritical: ~.4 g.cm -3, T = 45 C potentials: LJ + Coulomb (ions); SPC/E (water) Box size: 45 H O + RbBr pairs Time step: fs Simulation duration: ~ 1 ns,5 3, 3,5 4, 4,5 5, r (Å) Expansion of the first hydration shell in supercritical conditions

52 Ions in aqueous solutions: hydration MD simulations of diluted 1:1 aqueous electrolytes 1 3 Br-O Ambient / Supercritical n BrO ~ 7 8 (RbBr).4 -(H O) 99.6 (m=.) g(r) 1 n BrO ~ n(r) Ambient: ~ 1. g.cm -3, T = 3 C Supercritical: ~.4 g.cm -3, T = 45 C potentials: LJ + Coulomb (ions); SPC/E (water) Box size: 45 H O + RbBr pairs Time step: fs Simulation duration: ~ 1 ns,5 3, 3,5 4, 4,5 5, r (Å) Expansion of the first hydration shell in supercritical conditions Locally, slight dehydration (at constant cutoff)

53 Ions in aqueous solutions: hydration MD simulations of diluted 1:1 aqueous electrolytes 3 Br-O Ambient / Supercritical n n BrO ~ 7 BrO ~ (RbBr).4 -(H O) 99.6 (m=.) g(r) n(r) Ambient: ~ 1. g.cm -3, T = 3 C Supercritical: ~.4 g.cm -3, T = 45 C potentials: LJ + Coulomb (ions); SPC/E (water) Box size: 45 H O + RbBr pairs Time step: fs Simulation duration: ~ 1 ns,5 3, 3,5 4, 4,5 5, r (Å) Expansion of the first hydration shell in supercritical conditions Locally, slight dehydration (at constant cutoff) However, total number of water molecules unchanged (~ 7)

54 Ions in aqueous solutions: MD-XAS Supercritical RbBr-water χ(k) (arb. units),3,,1, -,1 n = 5 (6 %) n = 4 ( %) n = 3 (34 %) n = (9 %) n = 1 (1 %) Cluster analysis: - 8 individual MD-XAS computed spectra (sampled every. ps). - clusters classified according to the nb of neighbours at a given cutoff (3.55 Å). -, Br K-edge -, k (Å -1 ) Drastically different local environments (none of them being representative of the average) Significantly interfering signals (~ 5 % of the clusters will give a ~ zero contribution) G. Ferlat et al., Phys. Rev. B, 63, 134 (1); ibid,high Pressure Research ()

55 Ions in aqueous solutions: MD-XAS Supercritical RbBr-water χ(k) (arb. units),3,,1, -,1 n = 5 (6 %) n = 4 ( %) n = 3 (4 %) Cluster analysis: - 8 individual MD-XAS computed spectra (sampled every. ps). - clusters classified according to the nb of neighbours at a given cutoff (3.55 Å). -, Br K-edge -, k (Å -1 ) Significantly interfering signals (~ 5 % of the clusters) The inversion of the average EXAFS signal is ill-defined (can give different solutions) G. Ferlat et al., Phys. Rev. B, 63, 134 (1); ibid,high Pressure Research ()

56 Ions in aqueous solutions: hydration MD-XAS calculations at the Br K-edge (RbBr).4 -(H O) 99.6 Br ( k) ( k) ( k) ( k) BrO BrH BrRb R () Br ( k) 4O r g BrO ( r) BrO ( k, r) dr with () BrO A O (k,r)sin(kr BrO (k,r))

57 Supercritical aqueous solutions: hydration MD-XAS calculations (RbBr).4 -(H O) 99.6 g(r) 3 1 Br-O Supercritical χ(k) (arb. units),8,4, -,4 EXAFS (Br K-edge) Supercritical Expt. / MD -, r (Å) k (Å -1 ) The MD model reproduces quite well the experimental data. However, let s check the sensitivity of the XAS data to the g(r)...

58 Supercritical aqueous solutions: hydration MD-XAS calculations: 3 g(r) 1 g 1 (r) g (r) g 3 (r) g 4 (r) g 5 (r) g 6 (r) g BrO (r) g 7 (r) r (Å) n-coordinated g(r) To test the XAS sensitivity, the MD g(r) is decomposed into subgroups...

59 Supercritical aqueous solutions: hydration MD-XAS calculations:,8 g(r) 3 1 g 1 (r) g (r) g 3 (r) g 4 (r) g 5 (r) g 6 (r) g BrO (r) g 7 (r) χ(k) (arb. units),4, -,4 < 4 (k)> <3 (k)> < (k)> < 1 (k)> r (Å) -, k (Å -1 ) n-coordinated g(r) n-coordinated χ(k) To test the XAS sensitivity, the MD g(r) is decomposed into subgroups and the corresponding XAS signals are computed

60 Supercritical aqueous solutions: hydration How fast does the total XAS signal converged wrt to the partial 1-n (k)? 3,8 g(r) 1 g BrO (r) g 1- (r) χ(k) (arb. units),4, -,4 -,8 1- (k) BrO (k) r (Å) k (Å -1 )

61 Supercritical aqueous solutions: hydration 3,8 g(r) 1 g BrO (r) g 1-3 (r) χ(k) (arb. units),4, -,4 -,8 1-3 (k) BrO (k) r (Å) k (Å -1 )

62 Supercritical aqueous solutions: hydration 3,8 g(r) 1 g BrO (r) g 1-4 (r) χ(k) (arb. units),4, -,4 -,8 1-4 (k) BrO (k) r (Å) k (Å -1 )

63 Supercritical aqueous solutions: hydration 3,8 Converged! g(r) 1 g BrO (r) g 1-5 (r) χ(k) (arb. units),4, -,4 -,8 1-5 (k) BrO (k) r (Å) k (Å -1 )

64 Supercritical aqueous solutions: hydration 3,8 Converged! g(r) 1 g BrO (r) g 1-5 (r) χ(k) (arb. units),4, -,4 -,8 1-5 (k) BrO (k) r (Å) k (Å -1 ) The convergence is quite fast The high-r part of the g(r) is not contributing to the XAS signal

65 Supercritical aqueous solutions: hydration MD-XAS calculations: g(r) 3 1 g 1-4 (r) r (Å) g BrO (r) χ(k) (arb. units),8,4, -,4 -, k (Å -1 ) Expt. χ BrO (r) χ 1-4 (r) Considering only a fraction of the full g(r) is sufficient to reproduce the expt. data: this shows that the inversion of the EXAFS χ(k) is a ill-defined problem and that conventional approaches (cumulant or peak-fitting) should not be trusted! G. Ferlat et al., J. Phys. : Condens. Matter, 17, S145 (5)

66 Ions in aqueous solutions: complexation A «differential» XAS analysis XAS at Br K-edge in KBr-H O RbBr-H O CsBr-H O (k) BrO (k) BrX (k) Br - K +? G. Ferlat et al., J. Mol. Liquids, 11, 17 ()

67 Ions in aqueous solutions: complexation A «differential» XAS analysis XAS at Br K-edge in KBr-H O RbBr-H O CsBr-H O Br - Rb + (k) BrO (k) BrX (k)? G. Ferlat et al., J. Mol. Liquids, 11, 17 ()

68 Ions in aqueous solutions: complexation A «differential» XAS analysis XAS at Br K-edge in KBr-H O RbBr-H O CsBr-H O Br - Cs + (k) BrO (k) BrX (k)? G. Ferlat et al., J. Mol. Liquids, 11, 17 ()

69 .3 Differential XAS : Expt. (k) (unités arb.) (k) (unités arb.) KBr RbBr CsBr k (Å -1 ) KBr RbBr CsBr k (Å -1 ) Ambient Supercritical XAS at Br K-edge in KBr-H O RbBr-H O CsBr-H O No significative differentiation of the signals neither in ambient nor in SC. no differentiation <-> no ion-pairs? (what are the sensitivity limits?)

70 χ(k) (arb. units) χ(k) (arb. units),3,,1, -,1 -, -,3,1,5, -,5 Differential XAS : MD-XAS Ambient k (Å -1 ) Supercritical KBr RbBr CsBr KBr RbBr CsBr XAS at Br K-edge in KBr-H O RbBr-H O CsBr-H O No differentiation in Ambient (in agreement with expt. observations) However, strong differentiation in SC : overestimation of the ion-pairs in the MD model -, k (Å -1 )

71 supercritical complexation: MD-XAS (KBr) g(r) 3 Br-O r (Å) χ(k) (arb. units),5, -,5 -,1 Expt. / MD (KBr).4 -(H O) k (Å -1 ) Br-O g(r) Br-K χ(k) (arb. units),5, -,5 Br-O + Br-K Expt. / MD (KBr).4 -(H O) r (Å) -, k (Å -1 )

72 supercritical complexation: MD-XAS (KBr) Refining the ion-ion model residual (*1-6 ) initial amplitude amplitude divided by 1.5 amplitude divided by. amplitude divided par 3. g(r) MD Br-K g(r) Refined Br-K g(r) (upper limit),3,35,4,45,5 r shift of the g(r) 1 st maximum (Å),5 3, 3,5 4, 4,5 5, r (Å) Ion ion correlations are overestimated in the MD model The MD-XAS combination allows to estimate an upper limit G. Ferlat et al., J. Mol. Liquids, 11, 17 ()

73 S(q) (arb. units) Beyond MD-XAS: MD-NMR, MD-Raman... Boroxol-poor (BP) and boroxol-rich (BR) models tested against expt. probes: 1,6 1,,8,4 Static structure factor NMR 11 B and 17 O (DAS, MAS) XANES at B and O K-edges Raman Infra-red Heat capacity Expt. MD (f=%) Raman intensity Expt. MD (f=%) rings 11 B DAS NMR Expt. MD (f=%) non rings S(q) (arb. units) 1,6 1,,8, q (Å -1 ) Expt. MD (f=75%) Raman intensity MD (f=75%) Expt. MD (f=75%) rings non rings Energy (cm -1 ) Isotropic shift (ppm)

74 Conclusions XAS has provided a very valuable, sometimes unique information in many disorded systems: triplet correlations, original coordination environments The MD-MC/XAS symbiosis played a key role, especially for determining very subtle contributions (MS, ), to assess the limits of the technique sensitivity or to evidence inaccuracies in the MD/MC potentials. There are still difficulties to get reliable information beyond the first coordination shells but progress is being made Critical reading of a XAS paper How was the experimental signal extracted (Fourier filtering, splines)? Were possible multi-electronic excitations considered? What is the available k range? Are the spectra in k space shown? Data analysis: how was the configurational average taken into account? Was some external information used to constrain the analysis?

75 References: (Lectures, Tutorials, ) A History of the X-ray Absorption Fine Structure by R. Stumm von Bordwehr, Ann. Phys. Fr. vol. 14, (1989) (real author's name is Ch. Brouder). "Theoretical approaches to X-ray absorption fine structure", J.J. Rehr and R.C. Albers, Review of Modern Physics, vol. 7, () A review of the current status of XAFS spectroscopy, E. Daryl Crozier, Nuclear Instruments and Methods in Physics, vol. 133, (1997) EXAFS for liquids A. Filipponi J. Phys.: Condens. Matter 13 R3-R6 (1) Combining EXAFS with numerical simulations for disordered systems, G. Ferlat et al., J. Phys.: Condens. Matter 17 S145-S157 (5)

76 The muffin-tin (MT) approximation The potential is spherically averaged inside MT spheres around the atoms and averaged to a constant in the interstitial region : Justification : photo-electrons with high kinetic energies are essentially scattered by the core-regions and are almost free elsewhere (weakly sensitive to the potential details). However, it is now largely superseded by full potential techniques (from low to moderate energies).

77 Ions in aqueous solutions A C A C Simulations: (NaCl) Expt (EXAFS): (1) (RbBr) Hydration Amb. SC n ~ 6 ~ 6 n ~ 7 ~ Complexation Amb. SC ~ 3% () ~? (m=.-1.5 kg/mol) 1) S.L. Wallen et al., J. Phys. Chem., 11, 963 (1997) ) E. Oelkers, H. Hegelson, Science, 61, 888 (1993)

78 Ion-pairing (with decreasing density) Low density -> low dielectric constant -> favours ion-pairing Copper(I) and Copper(II) coordination structure under hydrothermal conditions at 35 C: an x-ray absorption fine structure and molecular dynamics study J. L. Fulton et al., J. Phys. Chem. A, 14, ()

79 Ion-pairing (with decreasing density) Ga K-edge Br K-edge XAS study of solvation and ion-pairing in aqueous gallium bromide solutions at supercritical conditions C. Da Silva et al., J. Mol. Liquids, 147, 83 (9)