Fundamentals of X-ray diffraction

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1 Fundamentals of X-ray diffraction Elena Willinger Lecture series: Modern Methods in Heterogeneous Catalysis Research

2 Outline History of X-ray Sources of X-ray radiation Physics of X-ray scattering Fundamentals of crystallography X-ray diffraction methods X-ray diffraction in material science: examples

3 History of X-rays Anna Bertha Röntgen Wilhelm Röntgen In 1895 Röntgen discovers X-rays 1901: The First Nobel Prize in Physics Wilhelm Conrad Röntgen Prize motivation: "in recognition of the extraordinary services he has rendered by the discovery of the remarkable rays subsequently named after him"

4 History of X-ray Max von Laue 1914: The Nobel prize for physics Prize motivation: "for his discovery of the diffraction of X-rays by crystals Director at the FHI ( ) Diffraction pattern X-ray Crystal diffracted beams

5 History of X-ray W. H. Bragg and his son W. L. Bragg and the diffraction of x-rays by crystals. 1915: The Nobel prize for physics 2dsinq=nl Prize motivation: "for their services in the analysis of crystal structure by means of X-rays"

6 The progress in crystalline structure determination Bragg: In sodium chloride there appear To be no molecules represented by NaCl. The equality in number of sodium and chloride atoms is arrived at by a chess-board pattern of these atoms; it is a result of geometry and no of a pairing off of the atoms NaCl fcc lattice Fm-3m a= 5.64 Å Henry Armstrong This statement is absurd to the n th degree; not chemical cricket. Chemistry is neither chess nor geometry, whatever X-ray physics may be. [ ] William H. Bragg It is time that chemists took charge of chemistry once more and protected neophytes against the worship of false gods

7 The progress in crystalline structure determination 1912 NaCl (2 atoms) Iron Pyrite FeS2 (3 atoms) Calcite CaCO3 (5 atoms) 1945 Penicillin, Dorothy Hodtkins (41 atoms) 1953 DNA, James Watson and Francis Crick (77 atoms, diameter~20 Å) 1954 Vitamin B12, Dorothy Hodtkins (181 atoms) 1962 Hemoglobin structure (diameter 50 Å ) 2011 Yeast Ribosome structure (diameter ~ Å) atoms

8 Evolution of X-ray techniques DNA NaCl The Yeast Ribosome

9 Outline History of X-ray Sources of X-ray radiation Physics of X-ray scattering Fundamentals of crystallography X-ray diffraction methods X-ray diffraction in material science: examples

10 X-ray X-rays are electromagnetic waves

11 Conventional X-ray laboratory source X-ray sources Electrostatic potential 20 to 60 kv Anode current 10 to 50 ma Input power ~ 0.5 to 3 kw Only 0.1% transforms into X-ray beam The wavelengths, most commonly used in crystallography: Å e- X-ray

12 X-ray spectra A typical x-ray spectrum from a Cu target ev= h = hc λ c = 12.4 V, kv Å

13 Characteristic X-ray lines Anode Kα Kβ Cu Å Å Mo Å Å

14 Electron beam Synchrotron X-ray Sources The Berlin Electron Storage Synchrotron radiation (BESSY) X-ray beam Storage ring Injection point Advantages: High brightness small samples High collimation high resolution of 2q Continuous spread of wavelengths Bending magnet Disadvantage: the synchrotron sources are not available on a daily basis

15 Outline History of X-ray Sources of X-ray radiation Physics of X-ray scattering Fundamentals of crystallography X-ray diffraction methods X-ray diffraction in material science: examples

16 Elementary scattering Scattering intensity distribution k 0 k 1 E z z e 2q x R k 0 0 I 2θ k 1 E y y e acceleration direction if E z Thomson scattering formula e I = I 4 0 ( 1+cos2 2 ) r 2 m 2 c 4 2

17 X-ray scattering physics Plane wave approximation the source and detector are far away from the object the object is small k 0 2θ k 1 Elastic - no energy loss coherent - no random phase shift scattering

18 X-ray scattering physics k 0 R 2θ S S = 2 k sinq scattering vector k 1 Δφ = S R phase shift Scattering intensity I(S)= ΣA j e -2pi (S R j ) ΣA k e 2pi (S R k ) = ΣΣA j A k e -2pi (S (R j - R k )) = A 2 ΣΣe -2pi (S (R j - R k )) {R j - R k } Object property (unique set)

19 X-ray scattering physics Atomic scattering amplitude E(s) E(s) is the sum of all the electrons scattering amplitudes A at (s) = z 1 j=0 A e (s)e 2πi(sr) Atomic scattering factor f(s) f(s) = A at (s)/ A e (s) f(s) = z 1 j=0 e 2πi(sr) s=0, f(s)=z r j =0, f(s)=z and doesn t depend on s Since electrons are not concentrated in one point f(s) depends on s = 2sinq/l

20 Atomic scattering factor O, z=8, atomic radius =0.66 Cl, z=17, atomic radius=1 Cl-, z=18, radius= 1.81 K, z = 19, atomic radius=2.02 K+ =18, radius= 1.33 X-ray atomic factors of O, Cl, Cl - and K + ; smaller charge distributions have a wider atomic factor.

21 Atomic scattering factor LaSrCrFeMoO3_cub 881BR ED.R AW X-ray Z neutron

22 X-ray scattering physics X-ray scattering by crystalline (periodic) ensemble of atoms c b a R mnp = ma + nb + pc Periodic arrangement of atoms is given by lattice with the basis vectors a,b,c Scattering amplitude A(S)= A e Σf j e i2p(s R j ) = -A e fσe i2p(s R mnp ) In general A(S) ~ 0 for infinite crystal, i.e. m, n and p But if (S R mnp ) = (S a)m + (S b)n + (S c)p = q integer then e i2p(s R mnp ) = 1 and A(S) = N A e f(s), N number of atoms, There is the only solution: S = {G hkl }= {ha * + kb* + lc*} with a * = [bxc]/v, b * = [cxa]/v, c * = [axb]/v, V = (a[bxc])

23 Ewald construction Space of wave vectors - Ewald construction G hkl = ha * + kb* + lc* c * k o 2q k sphere G hkl = 2 k sinq G a * G -10-1

24 Outline History of X-ray Sources of X-ray radiation Physics of X-ray scattering Fundamentals of crystallography X-ray diffraction methods X-ray diffraction in material science: examples

25 Crystal periodic structures Quartz Pyrite Rocksalt Trigonal cubic cubic

26 Bravais lattices Bravais_table.jpg

27 Symmetry elements m o

28 Interaction of symmetry elements m 2+1 m m 2+m 2+m+4 m m

29 32 Point groups We want to know where in the 3D periodic structure the symmetry element is located!

30 Translational symmetry elements Reflection + translation = glide planes

31 reflection Glide planes c b a translation vector, b 1/2b shift, 1/2b shift, 1/2b

32 Translational symmetry elements Rotation + translation = screw axis 3 1 Order Screw axes X y The degree of rotation is 360/x The translation is y/x units along screw axis , , 4 2, , 6 2, 6 3, 6 4, 6 5

33 Screw axis in Quartz 3 1 c 2 a 1 3 c o o c c S.G.: P trigonal Z: 3

34 Space group The combination of all available symmetry operations (32 point groups), together with translation symmetry, within the all available lattices (14 Bravais lattices) lead to 230 Space Groups that describe the only ways in which identical objects can be arranged in an infinite lattice.

35 Intensity, counts Intenational tables for Crystallography LaMnO 3 We know specific density, r, of the sample Pm-3m (221) cubic a=3.905 Å V=59.56 Å q, degree Z = ρv m Z = 1 Z - number of molecules in the unit cell m molecular mass

36 Intenational tables for Crystallography La Mn O La 1a m-3m Mn 1b m-3m ½ ½ ½ O 3c 4/mm.m 0 ½ ½ ½ 0 ½ ½ ½ 0

37 Outline History of X-ray Sources of X-ray radiation Physics of X-ray scattering Fundamentals of crystallography X-ray diffraction methods X-ray diffraction in material science: examples

38 The diffraction scheme Laue diffraction 2d hkl sinq hkl = l q hkl is fixed polychromatic radiation The crystal position is fixed G hkl = 2 k sinq X-ray Crystal polychromatic diffracted beams 2q 2q 2q hkl

39 The diffraction scheme Laue diffraction Historical X-ray photograph Modern XRD pattern of Hemoglobin obtained with the CCD detector Silicon crystal, 3-fold [111] * axis S.Gr.: Fd-3m face-centered cubic (fcc)

40 The diffraction scheme Bragg diffraction (oscillating crystal) 2d hkl sinq hkl = l l is fixed monochromatic radiation The crystal is rotating (oscillating) G hkl = 2 k sinq

41 The diffraction scheme Bragg diffraction (the crystal rotating)

42 The diffraction scheme Debye-Scherrer diffraction 2d hkl sinq hkl = l Quartz l is fixed monochromatic radiation sinq hkl = l 2d hkl

43 The diffraction scheme Debye-Scherrer diffraction CCD detector LaSrCrFeMoO3_cub 881BR ED.R AW X-ray

44 Outline History of X-ray Sources of X-ray radiation Physics of X-ray scattering Fundamentals of crystallography X-ray diffraction methods X-ray diffraction in material science: examples

45 Intensity, counts Structure factor La Mn O La1 Mn1 O3 S.G: Pm-3m cubic a= Å c q, degree F 100 = f La +f O -f Mn -2f O =f La -f Mn -f O = 22 d 100 d 200 a F 200 = f La +f Mn +3f O =75 F 110 = f La +f Mn -f O =60

46 Systematic absences AuCu- face-centred cubic (F), fcc disordered solid solution AuCu3- primitive cubic (P) ordered solid solution Fm-3m a= Å Pm-3m a= Å

47 Perovskites emoo3_cub D.R AW z y x z z x y x y Pm-3m (221) cubic a=3.905 Å V=59.56 Å 3 Z =

48 Perovskites Pm-3m, cubic 4368 LSCrFMoO3_ortho 701BR ED.R AW Pnma, orthorhombic I/ rel. b a p t 2a p 2a p Pnma, а = 2а p, b = 2а p, с = 2а p Pnma (62) a=5.539 Å, b=5.489 Å, c=7.793 Å V= Å 3 Z= 4

49 Conclusions

50 X-ray scattering physics Finite crystal N 3 I ~ sin2 ( N 1 φ 1 ) sin 2 φ 1 sin2 ( N 2 φ 2 ) sin 2 φ 2 sin2 ( N 3 φ 3 ) sin 2 φ 3 N 1 N 2 1 = p (S a), 2 = p S b, 3 = p (S c) N 1 = 10 There are high peaks in I(S) when (S a*) - integer N I N 12 (S-n a*)

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