Introduction to Powder X-Ray Diffraction History Basic Principles

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1 Introduction to Powder X-Ray Diffraction History Basic Principles Folie.1

2 History: Wilhelm Conrad Röntgen Wilhelm Conrad Röntgen discovered 1895 the X-rays he was honoured by the Noble prize for physics. In 1995 the German Post edited a stamp, dedicated to W.C. Röntgen. Basics-in-XRD.2

3 The Principles of an X-ray Tube X-Ray Cathode Fast electrons Anode focus Basics-in-XRD.3

4 The Principle of Generation Bremsstrahlung nucleus Ejected electron (slowed down and changed direction) Fast incident electron electrons Atom of the anodematerial X-ray Basics-in-XRD.4

5 The Principle of Generation the Characteristic Radiation Photoelectron Emission M Kα-Quant L K Electron Lα-Quant Kβ-Quant Basics-in-XRD.5

6 The Generating of X-rays Bohr`s model Basics-in-XRD.6

7 The Generating of X-rays energy levels (schematic) of the electrons M L Intensity ratios Kα1 : Kα2 : Kβ = 10 : 5 : 2 K Kα1 Kα2 Kβ1 Kβ2 Basics-in-XRD.7

8 The Generating of X-rays Anode (kv) Wavelength, λ [Angström ] Kß-Filter Mo 20,0 Kα1 : 0,70926 Kα2 : 0,71354 Kβ1 : 0,63225 Zr 0,08mm Cu 9,0 Kα1 : 1,5405 Kα2 : 1,54434 Ni 0,015mm Kβ1 : 1,39217 Co 7,7 Kα1 : 1,78890 Kα2 : 1,79279 Kβ1 : 1,62073 Fe 0,012mm Fe 7,1 Kα1 : 1,93597 Kα2 : 1,93991 Kβ1 : 1,75654 Mn 0,011mm Basics-in-XRD.8

9 The Generating of X-rays Emission Spectrum of a Molybdenum X-Ray Tube Bremsstrahlung = continuous spectra characteristic radiation = line spectra Basics-in-XRD.9

10 History: Max Theodor Felix von Laue Max von Laue put forward the conditions for scattering maxima, the Laue equations: a(cosα-cosα 0 )=hλ b(cosβ-cosβ 0 )=kλ c(cosγ-cosγ 0 )=lλ Basics-in-XRD.10

11 Laue s Experiment in 1912 Single Crystal X-ray Diffraction Tube Tube Crystal Collimator Film Basics-in-XRD.11

12 Powder X-ray Diffraction Film Tube Powder Basics-in-XRD.12

13 Powder Diffraction Diffractogram Basics-in-XRD.13

14 History: W. H. Bragg and W. Lawrence Bragg W.H. Bragg (father) and William Lawrence.Bragg (son) developed a simple relation for scattering angles, now call Bragg s law. d λ = n 2 sin θ Basics-in-XRD.14

15 Another View of Bragg s Law nλ = 2d sinθ Basics-in-XRD.15

16 Crystal Systems Crystal systems Axes system cubic a = b = c, α = β = γ = 90 Tetragonal a = b c, α = β = γ = 90 Hexagonal a = b c, α = β = 90, γ = 120 Rhomboedric a = b = c, α = β = γ 90 Orthorhombic a b c, α = β = γ = 90 Monoclinic a b c, α = γ = 90, β 90 Triclinic a b c, α γ β Basics-in-XRD.16

17 Reflection Planes in a Cubic Lattice Basics-in-XRD.17

18 The Elementary Cell a = b = c α = β = γ = 90 o c a β γ α b Basics-in-XRD.18

19 Relationship between d-value and the Lattice Constants λ =2dsinθ Bragg s law The wavelength is known Theta is the half value of the peak position d will be calculated 1/d 2 = (h 2 + k 2 )/a 2 + l 2 /c 2 Equation for the determination of the d-value of a tetragonal elementary cell h,k and l are the Miller indices of the peaks a and c are lattice parameter of the elementary cell if a and c are known it is possible to calculate the peak position if the peak position is known it is possible to calculate the lattice parameter Basics-in-XRD.19

20 Interaction between X-ray and Matter d incoherent scattering λco (Compton-Scattering) wavelength λpr intensity Io coherent scattering λpr(bragg s-scattering) absorption Beer s law I = I0*e-µd fluorescence λ> λpr photoelectrons Basics-in-XRD.20

21 History (4): C. Gordon Darwin C. Gordon Darwin, grandson of C. Robert Darwin (picture) developed 1912 dynamic theory of scattering of X-rays at crystal lattice Basics-in-XRD.21

22 History (5): P. P. Ewald P. P. Ewald 1916 published a simple and more elegant theory of X-ray diffraction by introducing the reciprocal lattice concept. Compare Bragg s law (left), modified Bragg s law (middle) and Ewald s law (right). λ d = n 2 sinθ sinθ = 1 d 2 λ σ sinθ = 2 1 λ Basics-in-XRD.22

23 Introduction Part II Contents: unit cell, simplified Bragg s model, Straumannis chamber, diffractometer, pattern Usage: Basic, Cryst (before Cryst I), Rietveld I Folie.23

24 Crystal Lattice and Unit Cell Let us think of a very small crystal (top) of rocksalt (NaCl), which consists of 10x10x10 unit cells. Every unit cell (bottom) has identical size and is formed in the same manner by atoms. It contains Na + -cations (o) and Cl - - anions (O). Each edge is of the length a. Basics-in-XRD.24

25 Bragg s Description The incident beam will be scattered at all scattering centres, which lay on lattice planes. The beam scattered at different lattice planes must be scattered coherent, to give an maximum in intensity. The angle between incident beam and the lattice planes is called θ. The angle between incident and scattered beam is 2θ. The angle 2θ of maximum intensity is called the Bragg angle. Basics-in-XRD.25

26 Bragg s Law A powder sample results in cones with high intensity of scattered beam. Above conditions result in the Bragg equation s or = n λ = 2 d sinθ d λ = n 2 sin θ Basics-in-XRD.26

27 Film Chamber after Straumannis The powder is fitted to a glass fibre or into a glass capillary. X-Ray film, mounted like a ring around the sample, is used as detector. Collimators shield the film from radiation scattered by air. Basics-in-XRD.27

28 Film Negative and Straumannis Chamber Remember The beam scattered at different lattice planes must be scattered coherent, to give an maximum of intensity. Maximum intensity for a specific (hkl)-plane with the spacing d between neighbouring planes at the Bragg angle 2θ between primary beam and scattered radiation. n λ This relation is quantified by Bragg s law. d = 2 sin θ A powder sample gives cones with high intensity of scattered beam. Basics-in-XRD.28

29 D8 ADVANCE Bragg-Brentano Diffractometer A scintillation counter may be used as detector instead of film to yield exact intensity data. Using automated goniometers step by step scattered intensity may be measured and stored digitally. The digitised intensity may be very detailed discussed by programs. More powerful methods may be used to determine lots of information about the specimen. Basics-in-XRD.29

30 The Bragg-Brentano Geometry Tube Detector focusingcircle Sample q 2q measurement circle Basics-in-XRD.30

31 The Bragg-Brentano Geometry Divergence slit Detectorslit Antiscatterslit Monochromator Tube Sample Basics-in-XRD.31

32 Comparison Bragg-Brentano Geometry versus Parallel Beam Geometry Bragg-Brentano Geometry Parallel Beam Geometry generated by Göbel Mirrors Basics-in-XRD.32

33 Parallel-Beam Geometry with Göbel Mirror Göbel mirror Detector Soller Slit Tube Sample Basics-in-XRD.33

34 Grazing Incidence X-ray Diffraction Soller slit Detector Tube Sample Measurement circle Basics-in-XRD.34

35 Grazing Incidence Diffraction with Göbel Mirror Soller slit Detector Göbel mirror Tube Sample Measurement circle Basics-in-XRD.35

36 What is a Powder Diffraction Pattern? a powder diffractogram is the result of a convolution of a) the diffraction capability of the sample (F hkl ) and b) a complex system function. The observed intensity y oi at the data point i is the result of y oi = of intensity of "neighbouring" Bragg peaks + background The calculated intensity y ci at the data point i is the result of y ci = structure model + sample model + diffractometer model + background model Basics-in-XRD.36

37 Which Information does a Powder Pattern offer? peak position dimension of the elementary cell peak intensity content of the elementary cell peak broadening strain/crystallite size scaling factor quantitative phase amount diffuse background false order modulated background close order Basics-in-XRD.37

38 Powder Pattern and Structure The d-spacings of lattice planes depend on the size of the elementary cell and determine the position of the peaks. The intensity of each peak is caused by the crystallographic structure, the position of the atoms within the elementary cell and their thermal vibration. The line width and shape of the peaks may be derived from conditions of measuring and properties - like particle size - of the sample material. Basics-in-XRD.38

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