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Introduction to Powder X-Ray Diffraction

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Introduction to Powder X-Ray Diffraction Introduction to Powder X-Ray Diffraction History Basic Principles Folie.1 © 2001 Bruker AXS All Rights Reserved History: Wilhelm Conrad Röntgen Wilhelm Conrad Röntgen discovered 1895 the X-rays. 1901 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 © 2001 Bruker AXS All Rights Reserved The Principles of an X-ray Tube X-Ray Cathode Fast electrons Anode focus Basics-in-XRD.3 © 2001 Bruker AXS All Rights Reserved The Principle of Generation Bremsstrahlung Ejected electron (slowed down and changed direction) nucleus Fast incident electron electrons Atom of the anodematerial X-ray Basics-in-XRD.4 © 2001 Bruker AXS All Rights Reserved The Principle of Generation the Characteristic Radiation Photoelectron Emission M Kα-Quant L K Electron Lα-Quant Kβ-Quant Basics-in-XRD.5 © 2001 Bruker AXS All Rights Reserved The Generating of X-rays Bohr`s model Basics-in-XRD.6 © 2001 Bruker AXS All Rights Reserved The Generating of X-rays energy levels (schematic) of the electrons M Intensity ratios K K K L α1 : α2 : β = 10 : 5 : 2 K Kα1 Kα2 Kβ1 Kβ2 Basics-in-XRD.7 © 2001 Bruker AXS All Rights Reserved The Generating of X-rays Anode (kV) Wavelength, λ [Angström] Kß-Filter Kα1 : 0,70926 Zr Kα2 : 0,71354 Mo 20,0 0,08mm Kβ1 : 0,63225 Kα1 : 1,5405 Cu 9,0 Ni Kα2 : 1,54434 0,015mm Kβ1 : 1,39217 Kα1 : 1,78890 Co 7,7 Fe Kα2 : 1,79279 0,012mm Kβ1 : 1,62073 Kα1 : 1,93597 Fe 7,1 Kα2 : 1,93991 Mn 0,011mm Kβ1 : 1,75654 Basics-in-XRD.8 © 2001 Bruker AXS All Rights Reserved The Generating of X-rays Emission Spectrum of a Molybdenum X-Ray Tube Bremsstrahlung = continuous spectra characteristic radiation = line spectra Basics-in-XRD.9 © 2001 Bruker AXS All Rights Reserved History: Max Theodor Felix von Laue Max von Laue put forward a(cosα-cosα0)=hλ the conditions for scattering b(cosβ-cosβ0)=kλ maxima, the Laue equations: c(cosγ-cosγ0)=lλ Basics-in-XRD.10 © 2001 Bruker AXS All Rights Reserved Laue’s Experiment in 1912 Single Crystal X-ray Diffraction Tube Tube Crystal Collimator Film Basics-in-XRD.11 © 2001 Bruker AXS All Rights Reserved Powder X-ray Diffraction Film Tube Powder Basics-in-XRD.12 © 2001 Bruker AXS All Rights Reserved Powder Diffraction Diffractogram Basics-in-XRD.13 © 2001 Bruker AXS All Rights Reserved 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. n ⋅λ d = 2⋅sinθ Basics-in-XRD.14 © 2001 Bruker AXS All Rights Reserved Another View of Bragg´s Law nλ = 2d sinθ Basics-in-XRD.15 © 2001 Bruker AXS All Rights Reserved 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 © 2001 Bruker AXS All Rights Reserved Reflection Planes in a Cubic Lattice Basics-in-XRD.17 © 2001 Bruker AXS All Rights Reserved The Elementary Cell a = b = c o α ===β γ 90 c α a β γ b Basics-in-XRD.18 © 2001 Bruker AXS All Rights Reserved 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 2 2 2 2 2 2 Equation for the determination of 1/d = (h + k )/a + l /c 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 © 2001 Bruker AXS All Rights Reserved Interaction between X-ray and Matter d incoherent scattering λCo (Compton-Scattering) coherent scattering wavelength λPr λPr(Bragg´s-scattering) absorption Beer´s law I = I0*e-µd intensity Io fluorescence λ> λPr photoelectrons Basics-in-XRD.20 © 2001 Bruker AXS All Rights Reserved 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 © 2001 Bruker AXS All Rights Reserved 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). n ⋅λ 1 σ d = sinθ = d sinθ = 2⋅sinθ 2 2⋅ 1 λ λ Basics-in-XRD.22 © 2001 Bruker AXS All Rights Reserved Introduction Part II Contents: unit cell, simplified Bragg’s model, Straumannis chamber, diffractometer, pattern Usage: Basic, Cryst (before Cryst I), Rietveld I Folie.23 © 2001 Bruker AXS All Rights Reserved 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 © 2001 Bruker AXS All Rights Reserved 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 © 2001 Bruker AXS All Rights Reserved Bragg’s Law A powder sample results in cones with high intensity of scattered beam. Above conditions result in the Bragg equation ∆s = n ⋅λ = 2⋅d ⋅sinθ or n ⋅ λ d = 2 ⋅ sin θ Basics-in-XRD.26 © 2001 Bruker AXS All Rights Reserved 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 © 2001 Bruker AXS All Rights Reserved 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 d = This relation is quantified by Bragg’s law. 2 ⋅ sin θ A powder sample gives cones with high intensity of scattered beam. Basics-in-XRD.28 © 2001 Bruker AXS All Rights Reserved 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 © 2001 Bruker AXS All Rights Reserved The Bragg-Brentano Geometry Tube Detector q focusing- Sample 2q circle measurement circle Basics-in-XRD.30 © 2001 Bruker AXS All Rights Reserved The Bragg-Brentano Geometry Mono- chromator Divergence slit Antiscatter- slit Detector- slit Tube Sample Basics-in-XRD.31 © 2001 Bruker AXS All Rights Reserved Comparison Bragg-Brentano Geometry versus Parallel Beam Geometry Bragg-Brentano Parallel Beam Geometry Geometry generated by Göbel Mirrors Basics-in-XRD.32 © 2001 Bruker AXS All Rights Reserved Parallel-Beam Geometry with Göbel Mirror Göbel Detector mirror Soller Slit Tube Sample Basics-in-XRD.33 © 2001 Bruker AXS All Rights Reserved “Grazing Incidence X-ray Diffraction” Detector Soller slit Tube Sample Measurement circle Basics-in-XRD.34 © 2001 Bruker AXS All Rights Reserved “Grazing Incidence Diffraction” with Göbel Mirror Soller slit Detector Göbel mirror Tube Sample Measurement circle Basics-in-XRD.35 © 2001 Bruker AXS All Rights Reserved What is a Powder Diffraction Pattern? a powder diffractogram is the result of a convolution of a) the diffraction capability of the sample (Fhkl) and b) a complex system function. The observed intensity yoi at the data point i is the result of yoi = ∑ of intensity of "neighbouring" Bragg peaks + background The calculated intensity yci at the data point i is the result of yci = structure model + sample model + diffractometer model + background model Basics-in-XRD.36 © 2001 Bruker AXS All Rights Reserved 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 © 2001 Bruker AXS All Rights Reserved 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.
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