Structural characterization. Part 1

Transcription

1 Structural characterization Part 1

2 Experimental methods X-ray diffraction Electron diffraction Neutron diffraction Light diffraction EXAFS-Extended X- ray absorption fine structure XANES-X-ray absorption near edge structure A number of other methods can give important additional information such as: Electron spin resonance Nuclear magnetic resonance Mössbauer spectroscopy

3 X-ray and electron diffraction X-ray scattering: Elastic scattering from electrons Powders or thick ( µm) films Thin films at glancing incidence Atomic form factor f(q) decreases with scattering vector Q, but can be computed. Electron scattering: Elastic scattering from screened Coulomb fields of atoms Thin films in an electron microscope Strong multiple scattering for thicker films Inelastic background Quantitative analysis difficult

4 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

5 Diffraction from crystals Detector Bragg s law λ = d sinθ hkl θ θ d hkl

6 b a Perfect crystal All atom positions are defined I by the lattice parameters Diffraction pattern consists of Bragg-peaks which are delta functions. θ Crystal with thermal vibrations of the atoms. Lattice parameters define a mean position of the atoms and we obtain broadened Braggpeaks. In polycrystals broadening occurs primarily from small grain sizes. Amorphous material Long range order lost, but some short range order still exists Diffraction pattern consists of broad peaks I I θ θ

7 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

8 Ex: X-ray scattering Elastic scattering lk i l=lk f l=k Single scattering Momentum transfer: Q=k i -k f, Q=k sin(θ) Phase difference Q. r m Atomic form factor f m (Q) Scattering amplitude A(Q) Intensity I(Q)=A*(Q)A(Q) r m O detector m θ Schematic geometry

9 Scattering from an atom Atomic form or scattering factor f ( Q ) = ne ( r) exp( iq r dr m ) atom n e (r) is electron concentration around an atom This is the same equation as in the theory of X-ray diffraction from crystals! Scattering factor f(q) Q Comparison of X-ray and neutron factors f e larger that f m, but similar shape b

10 Scattered intensity Scattering amplitude Intensity A Q ) = f exp( i Q r ) ( m m m I( Q) = f f exp( iq ( r m rn )) m n m n Normalized to the intensity scattered by a single electron I e (Q)=lf e (Q)l Isotropic material: Average over all orientations of r m -r n I Q) = f f sin( Qr ) / Qr ( m n m n mn mn

11 Monatomic solid - general N atoms in sample, we put f m =f n =f Sum terms with m=n separately I ( Q) = Nf 1 + exp( iq ( r m r n )) m n The sum over the atoms m around a given atom n can be converted to an integral over the pair distribution function g (r). Note that r=r m -r n. ( 1+ n ) 0 g ( r)exp( iq r d I Q) = Nf ) r (

12 Monatomic isotropic solid Isotropic materials I I ( + ) 1 n0 4πr g ( r)((sin Qr) / Qr) ( Q) = Nf dr n0 Q) = Nf 1+ 4πr sin Qr dr + n0 4πr Q ( ( g ( r) 1 )((sin Qr) / Qr) dr 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)

13 Pair distribution function The pair distribution function can be written in terms of the structure factor by an inverse Fourier transformation S( Q) g ( r) = 1+ 4πn 0 3 = 1+ (8π n r 0 ) ( g ( r) ) 1 4πQ 1 ((sin Qr) / Qr) dr ( S( Q) 1)((sin Qr) / Qr) dq PDF can be inferred from experimental data Truncation errors from integral data for a restricted range of Q.

14 Amorphous vs. crystalline Amorphous metal alloy Crystallized by heating Amorphous Si Partial crystallization Source: Zallen, The Physics of amorphous solids

15 Amorphous metals: RSP model Actually an alloy Ni 76 P 4. Radii of Ni and P similar, hence comparison with RSP model reasonable Reduced pair distribution function is plotted G(r)=4πrn 0 (g (r)-1) Very good agreement even for example splitting of second peak! Source: Zallen, The Physics of amorphous solids

16 Amorphous Ge: CRN model Source: Zallen, The Physics of amorphous solids Radial distribution function 4πr n 0 g (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

17 CRN vs. microcrystalline model a-ge Exp_microcryst exp-dashed Exp-CRN F( Q) ( I / ) 1) = Q Nf Source: C. Kittel: Introduction to Solid State Physics

18 Compounds Materials that consist of more than one kind of atom In general we must use the partial pair distribution functions g,ij (r) to describe the structure needs EXAFS measurements! One can define an effective total g (r) from the partial functions For binary compounds one can analyze diffraction experiments in a way analogous to the case of a monatomic material.

19 Polyatomic solid -1 Sum over p different kinds (i) of atoms of number N i and with atomic fractions x i Scattering intensity for isotropic material (m,n enumerate the atoms) I I Q) = f f sin( Qr ) / Qr ( m n m Perform first the summation over the N terms with m=n. The term Nf becomes now a sum over N i f i. The next sum is when m and n are not equal. The sum over the N atoms at the origin becomes a sum over N i f i and the integral over each partial pair distribution function g,ij (r) is weighted by f j for each atom type j. n mn mn ( 1+ n 4 r g ( r)((sin Qr) / Qr) ) π ( 0 Q) = Nf dr

20 Polyatomic solid - Hence the relation for a monatomic solid I is generalized to (i-atom at origin) ( 1+ n 4 r g ( r)((sin Qr) / Qr) ) π Q) = Nf dr ( 0 I p p p ( Q) / N xi fi + xi fi f jn0 4πr g, ij i= 1 i= 1 j= 1 = ( r)((sin Qr) / Qr) dr 4πr n 0 g,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.

21 Total pair distribution function Define a total pair distribution function by g r) xi fi f j g, ij ( r) / ij ( = f We obtain an expression similar to the one for a monatomic solid Inversion difficult when f depends on Q Radial distribution function (RDF) n 0 ρ(r) (in units of e - /Å) ( g ( r) ) Def: I( Q) / N = f + f n0 4πr 1 ((sin Qr) / Qr) dr f f = = p i= 1 p x i= 1 ρ ( r) = 4πr f g i x f i i f i ( r)

22 Ex: Binary compound - 1 Two kinds of atoms A,B Formula unit A x B y 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 n ρ( r) = 4πr 0 + xf A f B g, AB n 0 ( xf ( r) + yf A A g f, AA B g ( r) +, BA ( r)) yf B g, BB ( r) + Note that g BA /g AB =x/y Integration over a peak in the RDF gives the number of electrons giving rise to it

23 Binary compound - First coordination shell: Contributions from AB and BA terms Number of B atoms surrounding an A atom: n AB Number of A atoms surrounding a B atom: n BA They are related by the stoichiometry n BA /n AB =x/y Area under the first peak of the generalized RDF A = f f ( xn + yn ) = xf A B AB BA A f B n AB Second peak: AA, BB or maybe both. May be difficult to resolve

24 Vitreous SiO Scattered X-ray intensity Radial distribution function Source: C. Kittel: Introduction to Solid State Physics

25 Structural modeling of amorphous WO 3 RDF from X-ray diffraction for WO 3 films evaporated at different substrate temperature. Compared to models based on connected WO 6 octahedra Good agreement with nanocrystalline model and more so at higher substrate temperatures. Source: Nanba and Yasui, J. Solid state Chem. 83 (1989) 304

26 Sputtered WO 3-x thin films More amorphous than evaporated films C. Triana, lic. thesis G(r)=4πr n 0 (g(r)-1) (reduced radial distribution function)

27 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