Structural characterization Part 1 Experimental methods • X-ray diffraction • A number of other • Electron diffraction methods can give • Neutron diffraction important additional information such as: • Light diffraction • Electron spin • EXAFS-Extended X- resonance ray absorption fine structure • Nuclear magnetic resonance • XANES-X-ray absorption near edge • Mössbauer structure spectroscopy X-ray and electron diffraction • X-ray scattering: • Electron scattering: • Elastic scattering from • Elastic scattering from electrons screened Coulomb fields • Powders or thick (∼µm) of atoms films • Thin films in an electron • Thin films at glancing microscope incidence • Strong multiple scattering • Atomic form factor f(Q) for thicker films decreases with scattering • Inelastic background vector Q, but can be • Quantitative analysis computed. difficult Neutron scattering • Inelastic scattering from atomic nuclei • Short range nuclear interaction • Scattering length b, independent of Q. • b varies between elements and is dependent on isotope • Wavelength and energy of thermal neutrons comparable to atomic spacings and vibrational excitations • Both structure and dynamics can be studied • Bulk materials, powders • Isotope substitution • Scattering from magnetic structure • Few large international facilities Diffraction from crystals Bragg’s law Detector λ = 2dhkl sinθ θ θ d hkl Perfect crystal All atom positions are defined I by the lattice parameters Diffraction pattern consists of b Bragg-peaks which are delta functions. a 2θ Crystal with thermal vibrations of the atoms. Lattice parameters I define a mean position of the atoms and we obtain broadened Bragg- peaks. In polycrystals broadening occurs primarily from small grain sizes. 2θ Amorphous material Long range order lost, but some short range order still exists I Diffraction pattern consists of broad peaks 2θ Amorphous materials • No crystalline order – the reciprocal lattice does not exist! • An interpretation of diffraction experiments in terms of atomic planes or reciprocal lattice vectors is not possible • We must sum up the waves scattered from each atom in the whole sample • Gives experimental information on atomic distances and their distributions, as specified by the radial distribution function Ex: X-ray scattering detector • Elastic scattering lkil=lkfl=k m • Single scattering r • Momentum transfer: m 2θ Q=ki-kf, Q=2k sin(θ) O . • Phase difference Q rm • Atomic form factor f (Q) m Schematic geometry • Scattering amplitude A(Q) • Intensity I(Q)=A*(Q)A(Q) Scattering from an atom • Atomic form or scattering factor f(Q) fm (Q) = ne (r)exp(iQ •r) dr ∫ atom b Scattering factor • n (r) is electron e Q concentration around an atom • This is the same equation • Comparison of X-ray as in the theory of X-ray and neutron factors diffraction from crystals! • fe larger that fm, but similar shape Scattered intensity • Scattering amplitude A(Q) = ∑ fm exp(−iQ •rm ) • Intensity m I(Q) = ∑∑ fm fn exp(iQ • (rm − rn )) m n • Normalized to the intensity scattered by a single electron 2 Ie(Q)=lfe(Q)l • Isotropic material: Average over all orientations of rm-rn I(Q) = ∑∑ fm fn sin(Qrmn ) / Qrmn m n Monatomic solid - general • N atoms in sample, we put fm=fn=f • Sum terms with m=n separately 2 I(Q) = Nf 1+ ∑exp(iQ • (rm − rn )) m≠n • The sum over the atoms m around a given atom n can be converted to an integral over the pair distribution function g2(r). Note that r=rm-rn. = 2 + • I(Q) Nf (1 n0 ∫ g2 (r)exp(iQ r) dr) Monatomic isotropic solid • Isotropic materials I(Q) = Nf 2 (1+ n 4πr 2 g (r)((sinQr) / Qr) dr) 0 ∫ 2 n = 2 + 0 π + π 2 − I(Q) Nf 1 ∫ 4 r sin Qr dr n0 ∫ 4 r (g2 (r) 1)((sin Qr) / Qr) dr Q • The second term gives forward scattering, which cannot be separated from the incident beam • The first and third terms constitute the structure factor, S(Q) Pair distribution function • The pair distribution function can be written in terms of the structure factor by an inverse Fourier transformation = + π 2 − S(Q) 1 4 n0 ∫ r (g2 (r) 1)((sin Qr) / Qr) dr g (r) =1+ (8π 3n )−1 4πQ2 (S(Q) −1)((sin Qr) / Qr) dQ 2 0 ∫ • PDF can be inferred from experimental data • Truncation errors from integral – data for a restricted range of Q. Amorphous vs. crystalline • Amorphous metal alloy • Amorphous Si • Crystallized by heating • Partial crystallization Source: Zallen, The Physics of amorphous solids Amorphous metals: RSP model • Actually an alloy Ni76P24. • Radii of Ni and P similar, hence comparison with RSP model reasonable • Reduced pair distribution function is plotted • G(r)=4πrn0(g2(r)-1) • Very good agreement – even for example splitting Source: Zallen, The Physics of amorphous solids of second peak! Amorphous Ge: CRN model Source: Zallen, The Physics of amorphous solids 2 • Radial distribution function 4πr n0g2(r) • Very good agreement with CRN model • Microcrystalline models do not give so sharp first peaks without also giving to sharp peaks at larger r CRN vs. microcrystalline model a-Ge Exp_microcryst exp-dashed Exp-CRN F(Q) = Q((I / Nf 2 ) −1) Source: C. Kittel: Introduction to Solid State Physics Compounds • Materials that consist of more than one kind of atom • In general we must use the partial pair distribution functions g2,ij(r) to describe the structure – needs EXAFS measurements! • One can define an effective total g2(r) from the partial functions • For binary compounds one can analyze diffraction experiments in a way analogous to the case of a monatomic material. Polyatomic solid -1 • Sum over p different kinds (i) of atoms of number Ni and with atomic fractions xi • Scattering intensity for isotropic material (m,n enumerate the atoms) I(Q) = ∑∑ fm fn sin(Qrmn ) / Qrmn m n = 2 + π 2 I(Q) Nf (1 n0 ∫ 4 r g2 (r)((sin Qr) / Qr) dr) • Perform first the summation over the N terms with m=n. The term 2 2 Nf becomes now a sum over Nifi . • The next sum is when m and n are not equal. The sum over the N atoms at the origin becomes a sum over Nifi and the integral over each partial pair distribution function g2,ij(r) is weighted by fj for each atom type j. Polyatomic solid - 2 • Hence the relation for a monatomic solid 2 2 I(Q) = Nf 1+ n 4πr g (r)((sin Qr) / Qr) dr ( 0 ∫ 2 ) • is generalized to (i-atom at origin) p p p 2 2 I(Q)/ N = xi fi + xi fi f j n0 4πr g2,ij (r)((sin Qr) / Qr) dr ∑ ∑∑ ∫ i=1 i=1 j=1 2 • 4πr n0g2,ij(r) is the average number of j-atoms at a distance r from an i-atom. • Treat near forward scattering as for the monatomic case. Total pair distribution function • Define a total pair distribution function by • Def: p 2 = g (r) = x f f g (r) / f f ∑ xi fi 2 ∑ i i j 2,ij i=1 ij p 2 2 • We obtain an expression similar f = ∑ xi fi to the one for a monatomic solid i=1 = 2 + 2 π 2 − I(Q)/ N f f n0 ∫ 4 r (g2 (r) 1)((sin Qr) / Qr) dr • Inversion difficult when f depends on Q • Radial distribution function (RDF) 2 2 - ρ(r) = 4πr f g (r) • n0 ρ(r) (in units of e /Å) 2 Ex: Binary compound - 1 • Two kinds of atoms A,B • Formula unit AxBy – scattered intensity normalized by number of formula units M instead of number of atoms N • Sum PDF and RDF over a formula unit of the material 2 2 2 n0 ρ(r) = 4πr n0 (xf A g2,AA (r) + yfB g2,BB (r) + + xf A f B g2,AB (r) + yf A f B g2,BA (r)) • Note that gBA/gAB=x/y • Integration over a peak in the RDF gives the number of electrons giving rise to it Binary compound - 2 • First coordination shell: Contributions from AB and BA terms • Number of B atoms surrounding an A atom: nAB • Number of A atoms surrounding a B atom: nBA • They are related by the stoichiometry nBA/nAB=x/y • Area under the first peak of the generalized RDF A = f f (xn + yn ) = 2xf f n A B AB BA A B AB • Second peak: AA, BB or maybe both. May be difficult to resolve Vitreous SiO2 • Scattered X-ray intensity • Radial distribution function Source: C. Kittel: Introduction to Solid State Physics Structural modeling of amorphous WO3 • RDF from X-ray diffraction for WO3 films evaporated at different substrate temperature. • Compared to models based on connected WO6 octahedra • Good agreement with nanocrystalline model and more so at higher substrate temperatures. Source: Nanba and Yasui, J. Solid state Chem. 83 (1989) 304 Sputtered WO3-x thin films • More ”amorphous” than evaporated films • C. Triana, lic. thesis G(r)=4πr n0 (g(r)-1) (reduced radial distribution function) Summary • We have concentrated on X-ray scattering • Neutron scattering and EXAFS: Later lecture • Broadened diffraction peaks give short range order • Featureless spectrum at larger length scales for a disordered material • Spectrum can be inverted to obtain pair distribution function • Ex: Monatomic solids and binary compounds with amorphous structure .