Difrakce X-ray scattering

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Miloš Pospíšil
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1 Difrakce X-ray scattering

2 Interaction of X-rays with samples Scattering Important because no lenses for imaging Need understanding of how atomic structure affects the scattering of materials Absorption

3 Electromagnetic waves. E i = E Oi exp[2πiν(t-x/c)] = E Oi [cos2πν(t-x/c)+isin2πν(t-x/c)] E i is field, t time, x position - from origin O. i = -1 E i represented as complex vector. Field accelerates a charged particle with frequency ν. Max. acceleration as particle passes node max E i. Thus (electron) particle displacement π/2 from E i. The accelerating orbital electron initiates a second electromagnetic wave with a 2nd phase change of π/2. E i E Oi

4 Elastic Scattering Coherent, Thompson. No loss of energy (λ unchanged). Phase changes exactly π. Dominates diffraction. Mechanism to be discussed a little later.

5 Coherent Scattering Incident X-ray photon Scattered X-ray photon Low energy photons only scattering, no ionization

6 Thompson scatter. Consider the electric field at an observation position r: E i = E Oi exp[2π iν(t - r/c) iπ] Thompson scattered wave (E d ): E d = (1/r) E Oi (e 2 /mc 2 ) sin φ mass, m in denominator nuclei unimportant φis the angle between electron acceleration and direction of scatter

7 Another implication of the Thompson Equation Integration over all directions For crystal of A typical size Perfect condition Shows 2% of x-rays diffracted Weak data

8 Polarization Assuming polarized x-rays Electrons oscillate E Oi and E Oi s oscillation Along direction of electron oscillation, no E d Orthogonal to e- oscillation: E d (1/r) E Oi Non-polarized x-rays = sum of two orthogonallypolarized waves. Partially polarized: weighted sum Most experimental sources Degree of polarization depends on apparatus, not structure Correct as if data collected with sin φ = 1.0.

9 Compton Scattering Inelastic, Incoherent. In collision w/ e (rarely nucleus)... Fraction of energy transferred Change in wavelength Phase unpredictable. weak background non- Bragg scatter. Usually a nuisance, requiring correction E Energy of incident photon E - - scattered photon mc 2 mass equivalent energy of 1electron

10 Example Photoelectric effect Photon with energy 40keV enters Photoelectron from K-shell with energy ( )=6.8keV exits Electron from M- to K-shell Characteristic radiation at ( )= 31.6KeV in a random direction. The Atom now has positive charge Iodine Energy levels K -33.2keV L -4.3keV M -0.6keV K L M

11 X-rays Fluorescence. Excited atoms return to minimal state. Release series of energy quanta. Sometimes of x-ray energy. Characteristic lines.

12 Diffraction

13 Diffraction No lenses available for X-ray

14 Constructive interference Two coherent wave sources produce a diffraction image on a distant screen. Coherent sources are produced by passing waves from a single source S through two slits, S 1 and S 2, of size of the order of the wave s wavelength. According to Huygens principle that every point on a wave front behaves like a point source, the slits produce two coherent sources.

15 Interference I Bright r 2 r 1 S 1 S d θ n Bright S 2 Bright λ=wavelength d=distance between slits S=slits r=optical paths of wave fronts n=diffraction number r r = d sinθn = nλ 1 2

16 Interference II. λ d n = 2 n = 1 n = 0 n = -1 n = -2 grating screen In a diffraction grating for visible light, constructive interference between light rays passing through slits of the grating leads to light intensity ONLY at certain locations on the screen:

17 Interference III. s o φ ψ S detector λ a crystal lattice k = s/ λ, k =vlnový vektor s=ednotkový vektor Laue equations hλ = a(cos ψ 1 cosφ 1 ) = (s.a s 0.a) kλ = a(cos ψ 2 cosφ 2 ) = (s.b s 0.b) lλ = a(cos ψ 2 cosφ 2 ) = (s.c s 0.c)

18 Original von Laue Formulation of X-Ray Diffraction s=ñ-ednotkový vektor

19 Bragg s Law 2d hkl sinθ = nλ Correlates X-ray wave length, λ, interplanar spacing, d, and reflection angle, θ.

20 Θ A BD = d sin Θ DC = d sin Θ B C d hkl BD + DC = n λ D Bragg rovnice: 2d hkl sin Θ = n λ

21 Reálný Prostor Reciproký prostor h,k,l (uzlové roviny) h,k,l (mřížkové body) Vektor reciproké mřížky : H hkl = ha * + kb * + lc * a*,b*, c* - bázové vektory reciproké mřížky h, k, l Miller Indexy roviny Reciproký vektor e kolmý na osnovu hkl rovin Délka reciprokého vektoru e 1/d. d mezirovinná vzdálenost Reciproká mřížka (opakování)

22 Laue-ho podmínky pro difrakci Laue rovnice hλ = (s.a s 0.a) kλ = (s.b s 0.b) lλ = (s.c s 0.c) H hkl = ha*+kb*+lc* skalární součin a.h hkl = h Spoením obou rovnic dostáváme ednoduchou rovnici pro difrakci: k = s/ λ, Laue rovnice h = (k.a k 0.a) k = (k.b k 0.b) l= (k.c k 0.c) a.h hkl = h b.h hkl = k c.h hkl = l k - k o =H hkl k směr rozptýleného záření k 0 směr dopadaícího záření H vektor reciproké mřížky h,k,l indexy roviny odpovídaící vektoru H

23 Difrakce. Geometrická interpretace k - k o = H hkl =H hkl

24 Ewald Construction Ewald Construction for XRD Ewald Construction

25 ELECTRON DIFFRACTION PATTERNS A SPOT PATTERN REPRESENTED A PARTICULAR PLANE OF THE RECIPROCAL LATTICE PASSING THROUGH OF THE POINT 000 H hkl = ha* + kb* = lc* a*, b*, c* are axial vectors, h,k,l are point indices

26 INDEXING OF DIFFRACTION PATTERN

27 Intenzita

28 Starting Assumptions For x-rays, electrons, and neutrons incident on a crystal, diffraction occurs due to interference between waves scattered elastically from the atoms in the crystal. If we treat the incident waves as plane waves and the electrons as ideal point scatterers, the scattered waves are spherical waves. We will assume they are also isotropic.

29 Physical Model for X-ray Scattering Consider two parallel plane waves scattered elastically from two nearby electrons A and B in a solid material: A P k v v v k ρ O ψ incident = φe elastic scattering: v v i( k r ωt) v k = B v k ψ scattered = φ f e ω R v i( k r v t) f = scattering power of electron

30 Phase Difference Between the Waves For the spherical waves scattered from electrons A and B : ψ A = φ f e ω R A v i( k r v t) ψ v i( k r v t+ Δ) R A = position of detector relative to A R A = position of detector relative to B r = position of B relative to A Δ= phase difference between Ψ A and Ψ A v r v r r v v r v Δ = k + k = ( k k ) = Δk B = φ f e ω R B scattering vector So the wave scattered from the -th electron is: R position of atom relative to A ψ = φ f R e v r v i( k r v ωt+ Δk )

31 Sum of Scattered Waves Thus the total scattered wave at the detector is: ψ = all el φ f. R e v v i( k r ωt) e r i( v Δk ) φ f R e v v i( k r ωt) all el. e r i( v Δk ) For a small sample, the distances R are all essentially the same ( R). Thus we see that constructive and destructive interference between the scattered waves that reach the detector is due to the atomic sum. The detector location is determined by the scattered wave vector k and thus Δk. We can see: General wave motion Sum represents Amplitude

32 Atomic scattering factors Atomový faktor Rozptyl na všech elektronech ednoho atomu Z sin(θ)/λ

33 Summing Over Lattice points Now assume a crystal whose lattice has base vectors a, b, c, with a total number of atoms along each axis M, N, and P, respectively: Thus the amplitude of the total wave at the detector is proportional to: ψ v M 1 N 1 P 1 i( r Δk ) v v v v i[( ma+ nb + pc ) Δk ] e = e all atoms m= 0 n= 0 p= 0 Replacing n 1 n 2 n 3 with the familiar hkl, we see by inspection that these three conditions are equivalently expressed as: v Δ r * * k = H hkl = ha + kb + The sum of the scattered x-rays from the crystal was found to be composed of two components: sum over the lattice and sum over the one unit cell A f all atoms e r v i( Δk ) = lc * lattice basis f e r r i2π ( H hkl )

34 Strukturní Faktor F hkl Strukturní faktor e suma rozptylových schopností (vln) všech atomů v základní buňce (součet komplexních čísel, vektorový součet) F hkl = f e r i2π ( r H hkl ) ( hx + ky lz ) r H r = + hkl f atomový (rozptylový) faktor Kde reciproký vektor a polohy všech atomů vbuňce sou: r H hkl = ha v * + v kb * + lc v * r = x a v + y v b + z c v

35 Structure factor The structure factor F(hkl) is (vector) sum of the scattering f of each atom in the unit cell b Im F(hkl) Re O a v F( hkl) = f exp π + { 2 i( hx + ky lz )}

36 Intenzita I hkl Intenzita e druhá mocnina (čtverec) absolutní hodnoty strukturního faktoru. Intenzita e reálné (ne komplexní číslo). Informace o fáze strukturního faktoru zaniká. I hkl = F hkl F * hkl Intenzita závisí také na experimentálních podmínkách: Polarizace rtg. záření, rychlostí pronikání vektoru difrakční sférou (Lp faktor). Absorpční faktor. Extinkční faktor. Integrální intenzita.

37 Difrakční snímek

38 Laue Metoda Laue Pattern

39 Prášková metoda

40 Záznam

41 Práškový záznam I Θ Braggova rovnice: 2d sin Θ = n λ

42 Scattering by elements of electron density F(r*) = Σ N =1 A exp 2πi r* r Let r be center of infinitely small element of electron density, ρ. Consider total scattering: F(r*) = V ρ(r)exp 2πir* r dr Right-hand side FT(ρ). Structure determination: measure amplitude determine phase throughout (continuous) function, F(r*) compute inverse FT electron density: ρ(r) = T -1 [F(r*)] = V* V* F(r*)exp -2πi r* rdr*

43 Atomic Factor eqn.(2) f at atomic scattering factor FT isolated atom (later): Depends on # electrons, thermal vibration. Tabulated theoretical or experimental values. Can be approximated by spherically symmetric Gaussian. But usually more sophisticated description Scattering FT(molecule)

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