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Introduction-to-XRD.1 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved Introduction to Powder X-Ray Diffraction History Basic Principles

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Introduction-to-XRD.2 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved 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.

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Introduction-to-XRD.3 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved The Principles of an X-ray Tube Anode focus Fast electrons Cathode X-Ray

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Introduction-to-XRD.4 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved The Principle of Generation Bremsstrahlung X-ray Fast incident electron nucleus Atom of the anodematerial electrons Ejected electron (slowed down and changed direction)

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Introduction-to-XRD.5 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved The Principle of Generation the Characteristic Radiation K -Quant L -Quant K -Quant K L M Emission Photoelectron Electron

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Introduction-to-XRD.6 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved The Generating of X-rays Bohr`s model

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Introduction-to-XRD.7 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved The Generating of X-rays M K L K K K K energy levels (schematic) of the electrons Intensity ratios K K K

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Introduction-to-XRD.8 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved The Generating of X-rays Anode Mo Cu Co Fe (kV) 20,0 9,0 7,7 7,1 Wavelength Angström K 1 : 0,70926 K 2 :0,71354 K 1 :0,63225 Filter K 1 : 1,5405 K 2 :1,54434 K 1 :1,39217 K 1 : 1,78890 K 2 :1,79279 K 1 :1,62073 K 1 : 1,93597 K 2 :1,93991 K 1 :1,75654 Zr 0,08mm Mn 0,011mm Fe 0,012mm Ni 0,015mm

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Introduction-to-XRD.9 © 1999 R. Haberkorn and 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

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Introduction-to-XRD.10 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved History: Max Theodor Felix von Laue Max von Laue put forward the conditions for scattering maxima, the Laue equations: a(cos -cos )=h b(cos -cos )=k c(cos -cos )=l

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Introduction-to-XRD.11 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved Laue’s Experiment in 1912 Single Crystal X-ray Diffraction Tube Collimator Tube Crystal Film

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Introduction-to-XRD.12 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved Powder X-ray Diffraction Tube Powder Film

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Introduction-to-XRD.13 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved Powder Diffraction Pattern

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Introduction-to-XRD.14 © 1999 R. Haberkorn and 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.

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Introduction-to-XRD.15 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved Another View of Bragg´s Law n = 2d sin

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Introduction-to-XRD.16 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved Crystal Systems

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Introduction-to-XRD.17 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved Reflection Planes in a Cubic Lattice

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Introduction-to-XRD.18 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved The Elementary Cell a b c a = b = c === 90 o

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Introduction-to-XRD.19 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved Relationship between d-value and the Lattice Constants Bragg´s law The wavelength is known Theta is the half value of the peak position d will be calculated 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

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Introduction-to-XRD.20 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved Interaction between X-ray and Matter d wavelength Pr intensity Io incoherent scattering Co (Compton-Scattering) coherent scattering Pr (Bragg´s-scattering) absorbtion Beer´s law I = I 0 *e-µd fluorescense > Pr photoelectrons

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Introduction-to-XRD.21 © 1999 R. Haberkorn and 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

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Introduction-to-XRD.22 © 1999 R. Haberkorn and 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).

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Introduction-to-XRD.23 © 1999 R. Haberkorn and 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

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Introduction-to-XRD.24 © 1999 R. Haberkorn and 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.

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Introduction-to-XRD.25 © 1999 R. Haberkorn and 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 . The angle between incident and scattered beam is 2 . The angle 2 of maximum intensity is called the Bragg angle.

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Introduction-to-XRD.26 © 1999 R. Haberkorn and 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 or

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Introduction-to-XRD.27 © 1999 R. Haberkorn and 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.

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Introduction-to-XRD.28 © 1999 R. Haberkorn and 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 2 between primary beam and scattered radiation. This relation is quantified by Bragg’s law. A powder sample gives cones with high intensity of scattered beam.

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Introduction-to-XRD.29 © 1999 R. Haberkorn and 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.

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Introduction-to-XRD.30 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved The Bragg-Brentano Geometry Tube measurement circle focusing- circle q q 2 Detector Sample

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Introduction-to-XRD.31 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved The Bragg-Brentano Geometry Divergence slit Detector- slit Tube Antiscatter- slit Sample Mono- chromator

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Introduction-to-XRD.32 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved Comparison Bragg-Brentano Geometry versus Parallel Beam Geometry Bragg-Brentano Geometry Parallel Beam Geometry generated by Göbel Mirrors X-ray Source Motorized Slit Sample

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Introduction-to-XRD.33 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved “Grazing Incidence Diffraction” with Göbel Mirror X-ray Source Göbel Mirror Sample Soller slit Scintillation counter Measurement circle

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Introduction-to-XRD.34 © 1999 R. Haberkorn and 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 (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

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Introduction-to-XRD.35 © 1999 R. Haberkorn and 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

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Introduction-to-XRD.36 © 1999 R. Haberkorn and 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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Introduction-to-XRD.37 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved Principles of the Rietveld method Hugo M. Rietveld, 1967/1969 The Rietveld method allows the optimization of a certain amount of model parameters (structure & instrument), to get a best fit between a measured and a calculated powder diagram. The parameter will be varied with a non linear least- squares algorythm, that the difference will be minimized between the measured and the calculated Pattern:

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Introduction-to-XRD.38 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved Basis formula of the Rietveldmethod SF: Scaling factor M k : Multiplicity of the reflections k P k : Value of a preffered orientation function for the reflections k F k 2 : Structure factor of the reflections k LP: Value of the Lorentz-Polarisations function for the reflections k F k : Peak profile function for the reflections k on the position i yb i : Value of the background at the position i k: Index over all reflexes with intensity on the position i

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Introduction-to-XRD.39 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved Comparison of Profile Shape and Intensity Accuracy between Parallel Beam Göbel Mirror and Bragg-Brentano Parafocusing Diffractometers A. Seyfarth, A. Kern & G. Menges AXS GmbH, Östliche Rheinbrückenstr. 50, D Karlsruhe Fifth European Powder Diffraction Conference, EPDIC-5, Abstracts, p. 227 (1997) XVII Conference on Applied Crystallography, CAC 17, Abstracts, p. 45 (1997)

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Introduction-to-XRD.40 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved Göbel Mirrors for parallel Beam Graded and bent multilayers optics Capture a large solid angle of X-rays emitted by the source Produce an intense and parallel beam virtually free of Cu Kß radiation

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Introduction-to-XRD.41 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved Effects of Sample Displacement Sample displacement Peak shift Sample X-ray tube

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Introduction-to-XRD.42 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved Sample Displacement Effects on Quartz Peak Positions with Parafocusing Geometry No Sample Displacement 0.2mm Downward Displacement 0.4mm Downward Displacement 1.0 mm Downward Displacement 1.2mm Downward Displacement 0.5mm Upward Displacement

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Introduction-to-XRD.43 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved Sample Displacement Effects on Peak Positions with Göbel Mirror No Sample Displacement 0.2mm Downward Displacement 0.4mm Downward Displacement 1.0 mm Downward Displacement 1.2mm Downward Displacement 0.5mm Upward Displacement

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Introduction-to-XRD.44 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved Peak Profile Shape of NIST 1976 (1)

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Introduction-to-XRD.45 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved Peak Profile Shape of NIST 1976 (2)

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Introduction-to-XRD.46 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved Peak Profile Shape of NIST 1976 (3)

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Introduction-to-XRD.47 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved Instrument Response Function D5005 Theta/2Theta Göbel Mirror, 0.2 mm divergence slit, 2° vertical Soller slit and 0.15° collimator.

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Introduction-to-XRD.48 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved D5005 Theta/2Theta Göbel Mirror, 0,2 mm divergence slit, 2° vertical Soller slit and 0.15° collimator. Peak Shape Asymmetry

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Introduction-to-XRD.49 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved Instrument Resolution Functions

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