Neutron and X-ray diffraction

Transcription

1 Neutron and X-ray diffraction Laurent Cormier Institut de Minéralogie, Physique des Matériaux et Cosmochimie Université Pierre et Marie Curie CNRS Paris, France Cargese School

2 First structural studies by X-ray diffraction Warren, B.E., The diffraction of X-rays in glass. Phys. Rev. B, 45, Warren, B.E., X-ray determination of the structure of glass. J. Am. Ceram. Soc., 17, Warren, B.E., Biscoe, J., Fourier analysis of X-ray patterns of sodasilica glass. J. Am. Ceram. Soc., 21,

3 Structural model for covalent glasses Rules for glass formation 1. No O atoms bonded to more than 2 cations 2. Coordination cation is small (3,4) 3. O polyhedra are corner-shared (no edge- or face-sharing) 4. To obtain 3D network, at least 3 corners must be shared Zachariasen model (1932) Continuous random network Zachariasen, J. Am. Ceram. Soc., 54(1932)3841 Huang et al., Nano Lett. 12 (2012)1081

4 Note the difference in intensity Bragg peaks and amorphous signal

5 Diffraction Diffraction Diffuse scattering Small angle scattering Scattering Diffraction Diffusion diffuse Diffusion aux petits angles Diffusion diffraction crystals disorder in crystals > 1nm < 1nm glass - liquid

6 Diffraction Or simply use the term PDF (Partial distribution function)

7 A diffraction experiment k i Detector Q k d 2q Sample (surtout en RX) Source Modulus scattering vector k i = 2π/l Scattering vector Elastic diffusion k i = 2π/l = k d = k l: wavelenth of the incident particule 2q : diffraction angle Q = 2ksinq Q = 4 p sinq / l Q sometimes noted k (mainly for X-ray)

8 Fonctions in Q-space S(Q) Structure factor F(Q)/<b> 2 =S(Q)-1 Interference function S(Q ) 1 S(Q) 0 F(Q ) S(Q) F(Q) Q Q (Å -1 ) Q (Å -1 ) D.A. Keen, A comparison of various commonly used correlation functions for describing total scattering, J. Appl. Cryst (2001).

9 Low-Q features Q 1.d 2-3 => arrangements of network-forming motifs on an intermediate range (first sharp diffraction peak) Q 2.d => size of the local network-forming motifs Q 3.d => nearest-neighbour separation Zeidler & SalmonPhys. Rev. B 93 (1016) Note : some low-q features are not present for some classes of glasses (e.g. Q 1 and Q 2 are absent in metallic glasses) or are not observable for some diffraction methods (e.g. Q 2 is present in neutron diffraction data for SiO 2 but absent in the X-ray diffraction data).

10 Low-Q features Q 1.d 2-3 => arrangements of network-forming motifs on an intermediate range (first sharp diffraction peak) Q 2.d => size of the local network-forming motifs Q 3.d => nearest-neighbour separation A dense random packing structure (typical to metallic glasses) a tetravalent structure (e.g. a-si) a 4:2 structure (e.g., a- GeSe2) Filled circles in (b) and (c) are fourcoordinated atoms, filled squares are twocoordinated atoms, and open circles are voids Dixmier J. Phys. I 2 (1992) 1011

11 Low-Q features Correlations between, Q3 and the reduced volume Ehrenfest relation The position of this peak is associated with the principal diffraction peak (Q3). As a consequence, the position Q3 is inversely proportional to the mean atomic spacing and the third power of Q3 scales inversely with the volume

12 Low-Q features Correlations between, Q 3 and the reduced volume Q 3.V a = 9.3 ± 0.2 power law relationship between the reverse of the principal peak position, 2π/Q 3, and the glass volume Va = ρ 0 / (N A M) Ma et al., Nat. Mater. 8 (2009) 30 convenient to understand the relative volume (density) change with pressure

13 Low-Q features First sharp diffraction peak Q 1 Ø Important and sometimes anomalous changes with composition, temperature or pressure Ø Associated with the presence of medium range order Q 1 ~2p/d Ø Not associated with precise features in r

14 Low-Q features Various interpretation Ø Quasi-crystalline organization, quasi-bragg peak or quasi-periodic arrangement Ø Correlations between clusters and voids

15 Low-Q features group-iv (GexSe1 x, SixSe1 x) and group-v (AsxSe1 x, PxSe1 x alloys) => strong variation in position and intensity chainlike group-vi alloy TexSe1 x => no dramatic evolution Bychkov et al., Phys. Rev. B 72 (2005) NFC : network-forming cations random distribution of the NFC-related structural units network formation by NFC-related structural units homopolar NFC-NFC bonds Ge and Si no longer NFC but depolymerize the network

16 Faber-Ziman formalism Partial structure factor Partial pair distribution function (PPDF) Atomic fraction of component a Neutron scattering length or X-ray form factor f(q) System of n chemical species n (n+1)/2 independent PPDF g ab (r) r0 (atome Å -3 ) is the average atomic density which can be obtained from the macroscopic density d(g cm -3 ) : A = Atomic mass of the sample

17 Fourier transform M(Q) : modification function Þ limit effect of truncation of the data

18 üpair distribution function g(r) Fonctions dans l espace réel üpair correlation function G(r) or differential correlation function D(r) G(r) = D(r) = 4p r r 0 [g(r) 1] T(r) = RDF(r)/r = 4p r r 0 g(r) = 4p r r 0 + D(r) RDF(r) = 4p r 2 r 0 g(r) ütotal distribution function T(r) üradial distribution function RDF(r)

19 Information from the correlation function ü Peak position Distances and angles between atoms ü Peak area Coordination number 1 ère couche de coordinence 2 ème couche de coordinence ü Full width half maximum Disorder Unidimensional representation of a 3D structure

20 Lead silicate glasses Kohara (2010) PbO 4 tetrahedra are in majority, al- though PbO 3 and PbO 5 Takahashi (2005) PbO unit is dominant Kohara et al., Phys. Rev. B 82 (2010) Takahashi et al., J. Am. Ceram. Soc., 88 (2005) 1591

21 Fourier transform

22 Use of high energy X-rays E=100keV High resolution diffraction Distance Si-O=1.62 Å Distance Al-O=1.75 Å

23 High resolution diffraction Vitreous P 2 O 5 P 2s 2 2p 3 Possibility to distinguish O linked with P O T : terminal oxygen = end of chain O B : bridging oxygen Hoppe et al., Sol. St. Comm., 115(2000)559

24 Better to use neutron or X-rays? Use both! They are complementary SiO 2

25 Comparison between neutron and X-ray diffraction X-ray Interaction with electronic cloud f(q) form factor üstrong variation of scattered intensity with q Neutrons Interaction with the nucleus b neutron scattering length ü b not a monotonous function of Z q = 0, I = Z

26 Comparison between neutron and X-ray diffraction X-ray f(q) form factor ü information on high Z elements ü weak contrast for elements with close Z Neutrons b neutron scattering length ü b independent of Q = constant Þ light elements are visible (H, Li, N, O, etc) Þ possibility to distinguish elements with close Z üb vary among isotopes of the same elment Þ isotopic substitution (Ex: H/D) q = 0, I = Z

27 Comparaison entre diffraction des neutrons et rayons X Facteur de structure S(Q) : somme des facteurs de structure partiels, S ab (Q) : S(Q) = α,β W αβ (Q)S αβ (Q) RX Facteurs pondérants Neutrons W αβ (Q) = c α c β f α ( Q, E) f β Q, E ( ) W αβ (Q) = c α c β b α b ( β 2 δ ) αβ ( ) 2 δ αβ Fonction de corrélation RX Neutrons G(r) = TF W αβ G(r) = W αβ g αβ (r) α,β α ( ) g αβ (r) α,β α Fonctions de distribution de paires partielles : g ab (r)

28 Neutron GeO 2 X-ray GeO 2 glass From Benmore

29 Neutron diffraction for silica glass: radial distribution function Polyatomic solid Q(S(Q)-1) Glass SiO 2 Correlation function Overlapping of PPDF G ab (r) R. L. Mozzi, Warren, B.E., J. Appl. Cryst. 2, 164 (1969)

30 Extract information from diffraction data: numerical modelling Classical Molecular Dynamics Ab inito Molecular Dynamics RMC and EPSR (Monte Carlo methods)

31 Pair distribution function g(r) Measures the probability that two atoms are separated by a distance r 3 Interatomic potential Atomic repulsion g(r) 1 V(r) r r/å Minimum of the pair potential ü ü Structural test of the MD models Improvement of potentials

32 Comparison experiment / simulation Always good agreement in publication! Wright, J. Non-Cryst. Solids, 159 (1993)264

33 Lead silicate glasses RMC modeling => quantitative fit of the experimental data 34mol% PbO : large fraction of PbOx polyhedra do not participate in network formation non-network PbOx unit (isolated short chains or complex units) = yellow points Kohara et al., Phys. Rev. B 82 (2010)

34 Lead silicate glasses X-ray Neutron 34 mol% PbO 50 mol% PbO 65 mol% PbO inhomogeneous lead distribution Peak at Q P =0.4 Å -1 => distance 2p/Q p ~15 Å 34mol% PbO : large fraction of PbOx polyhedra do not participate in network formation non-network PbOx unit (isolated short chains or complex units) = yellow points Kohara et al., Phys. Rev. B 82 (2010)

35 Lead silicate glasses : a binary network-former glass Cyan= voids 34 mol% PbO 50 mol% PbO PbO-SiO 2 glass as a binary networkformer glass with large amounts of free volume (voids) 65 mol% PbO Kohara et al., Phys. Rev. B 82 (2010)

36 A new order for metallic glasses Random sphere packing with canonical polyhedra Also medium range order (with 5-fold symmetry) X-ray diffraction + RMC modeling Sheng et al., Nature, 439(2006)419

37 Extract information from diffraction data: experimental contrast methods System of n chemical species Number of independent S ab (Q): n(n+1)/2 We need N= n(n+1)/2 different experiments! Difference method Neutron: isotopic substitution X-ray: anmalous diffraction

38 Isotopic substitution (neutron diffraction) M = substituted element First difference method Sample 1 Sample 2 Cormier et al., Chem. Geol. 174(2001)349

39 Isotopic substitution (neutron diffraction) M = substituted element First difference method Sample 1 Sample 2 with

40 For a binary system with species a,b : Matrix F(Q) Fixed composition: constant c a, c b Isotopes with a good contrast b ai : scattering length for isotope i of species a Matrix inversion allows the determination of partial structure factors S ab (Q) : Possibility to do isomorphic substitution: Two different chemical species with similar radius (same place within the structure?) but different b

41 Isotopic substitution (neutron diffraction) 3 different samples 3 different partial functions isolated Ge-Ge O-O Ge-O g ab (r) GeO 2 glass From Salmon et al. R (Å)

42 Homopolar bonds in chalcogenide glasses Example of glassy GeSe 2 Measure total structure factors for three samples with different isotopic enrichments e.g. N Ge N Se 2, 70 Ge N Se 2, 73 Ge 76 Se 2 Ge-Ge Observe: Ge-Ge and Se-Se homopolar bonds Egde- and corner-sharing Ge(Se 1/2 ) 4 tetrahedra Se-Se Ge-Se Petri et al., Phys. Rev. Lett.84(2000)2413

43 Isotopic substitution (neutron diffraction) Ti : TiK 2 Si 2 O 7 glass

44 Isotopic substitution (neutron diffraction) Ti-a pairs: first difference Ti : TiK 2 Si 2 O 7 glass coordination [5] Ti 4+ redox Cormier et al., Phys. Rev. B 58(1998)11322

45 Isotopic substitution (neutron diffraction) M = substituted element Second difference method Cormier et al., Chem. Geol. 174, 349 (2001)

46 0.02 G(r) (barns Å -2 ) 0 Ni-Ni Ni-Ni Ni-Ni r(å) Second difference functions for Ca 2 NiSi 3 O 9 = Ni-Ni distances

47 Distributionof non-network formers Second difference functions - cation-cation distribution Ca : CaSiO 3, CaNiSi 3 O 9 Ti : TiK 2 Si 2 O 7 Ni : CaNiSi 3 O 9 Li : Li 2 Si 2 O 5 SANDALS at ISIS (UK) Cormier et al., Chem. Geol., 174 (2001) 349 D4 at ILL (Grenoble)

48 Distributionof non-network formers Second difference functions - cation-cation distribution R1 short distance Non-homogeneous distribution Second and third neighbours important medium range order Ratio of distances Ordered cationic domains Cormier et al., Chem. Geol., 174 (2001) 349

49 Random modified network (Greaves) Separation regions enriched in network formers and region enriched in network modifiers Zones enriched with nonnetwork forming cations Silicate network (or network formers) Greaves, J. Non-Cryst. Solids 71(1985)203

50 Anomalous scattering (X-ray diffraction) Atomic form factor: f and f strong variation near the absorption edge of a given element f and f linked by the Kramer-Kronig relation :

51 Anomalous scattering (X-ray diffraction) 2 experiments at 2 different energies (different l): at the absorption edge (l1 or l2) and far from the edge (l3) Energie des rayons X (ev) 1 sample is needed Domain in Q defined by Q = 4 p sinq / l In practice, only interesting for element above ~Fe otherwise Q- range too small

52 Anomalous scattering (X-ray diffraction) Ge x Se 1-x Ge and Se K-edge Stiffness transition Short range order Hosokawa et al., Phys. Rev. B, 84(2011) Intermediate range order More Se6 rings at low x =>RMC models

53 Summary # Short and medium range order in the same experiments Þ structural models can be generated Þ or comparison with MD simulations # HP/HT structural transformations # contrast techniques to investigate specific elements

54 Bibliography Squires, G. L. Introduction to the theory of thermal neutron scattering, Cambridge University Press: Cambridge (1978). A.C. Wright, The structure of amorphous solids by x-ray and neutron diffraction, Advances in Structure Research by Diffraction Methods 5, 1 (1974). Chieux, P. In Neutron diffraction, Dachs, H., Ed., Springer-Verlag: Berlin, (1978). L. Cormier, La structure des verres étudiée par diffraction des neutrons, J. Phys. IV, 111, (2003). H. E. Fischer, A. C. Barnes, P. S. Salmon, Neutron and x-ray diffraction studies of liquids and glasses, Reports on Progress in Physics 69, (2006).

55 X-ray diffraction source Laboratory diffractometer Neutron diffraction Spallation source (pulsed source) Ex: Empyrean Panalytical (IMPMC, Paris) Source: Ag l=0.56 Å Q-range: Å-1 Ex: ISIS (UK), ESS (Sweden, 2020?) Synchrotron Ex: ESRF (Grenoble) Diffraction, anomalous scattering, temperature, pressure etc Q range up to 50 Å-1 Nuclear reactor Ex: LLB (France), ILL (Grenoble, european source) 7C2 diffractomter at LLB