19 Relationship between d-value and the Lattice Constants Bragg´s lawThe wavelength is knownTheta is the half value of the peak positiond will be calculatedEquation for the determination of the d-value of a tetragonal elementary cellh,k and l are the Miller indices of the peaksa and c are lattice parameter of the elementary cellif a and c are known it is possible to calculate the peak positionif the peak position is known it is possible to calculate the lattice parameter
20 Interaction between X-ray and Matter incoherent scatteringCo (Compton-Scattering)coherent scatteringPr(Bragg´s-scattering)wavelength PrabsorbtionBeer´s law I = I0*e-µdintensity Iofluorescense> Prphotoelectrons
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
22 History (5): P. P. EwaldP. 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).
23 Introduction Part IIContents: unit cell, simplified Bragg’s model, Straumannis chamber, diffractometer, patternUsage: Basic, Cryst (before Cryst I), Rietveld I
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.
25 Bragg’s DescriptionThe 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 q.The angle between incident and scattered beam is 2q .The angle 2q of maximum intensity is called the Bragg angle.
26 Bragg’s LawA powder sample results in cones with high intensity of scattered beam.Above conditions result in the Bragg equationor
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.
28 Film Negative and Straumannis Chamber RememberThe 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 2q 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.
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.
30 The Bragg-Brentano Geometry TubeDetectorqq2focusing-circleSamplemeasurement circle
31 The Bragg-Brentano Geometry Mono-chromatorAntiscatter-slitDivergence slitDetector-slitTubeSample
32 Comparison Bragg-Brentano Geometry versus Parallel Beam Geometry Motorized SlitX-ray SourceSampleBragg-BrentanoGeometryParallel Beam Geometrygenerated by Göbel Mirrors
34 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 ofyoi = of intensity of "neighbouring" Bragg peaks + backgroundThe calculated intensity yci at the data point i is the result ofyci = structure model + sample model + diffractometer model + background model5
35 Which Information does a Powder Pattern offer? peak position dimension of the elementary cellpeak intensity content of the elementary cellpeak broadening strain/crystallite sizescaling factor quantitative phase amountdiffuse background false ordermodulated background close order6
36 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.
37 Principles of the Rietveld method Hugo M. Rietveld, 1967/1969The 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:7
38 Basis formula of the Rietveldmethod SF : Scaling factorMk : Multiplicity of the reflections kPk : Value of a preffered orientation function for the reflections kFk2 : Structure factor of the reflections kLP : Value of the Lorentz-Polarisations function for the reflections kFk : Peak profile function for the reflections k on the position iybi : Value of the background at the position ik : Index over all reflexes with intensity on the position i
39 A. Seyfarth, A. Kern & G. Menges Comparison of Profile Shape and Intensity Accuracy between Parallel Beam Göbel Mirror and Bragg-Brentano Parafocusing DiffractometersA. Seyfarth, A. Kern & G. MengesAXS GmbH, Östliche Rheinbrückenstr. 50, D KarlsruheFifth European Powder Diffraction Conference, EPDIC-5, Abstracts, p. 227 (1997)XVII Conference on Applied Crystallography, CAC 17, Abstracts, p. 45 (1997)
40 Göbel Mirrors for parallel Beam Graded and bent multilayers opticsCapture a large solid angle of X-rays emitted by the sourceProduce an intense and parallel beam virtually free of Cu Kß radiation
41 Effects of Sample Displacement X-ray tubePeak shiftSample displacementSample
42 Sample Displacement Effects on Quartz Peak Positions with Parafocusing Geometry No Sample Displacement0.2mm Downward Displacement0.4mm Downward Displacement1.0 mm Downward Displacement1.2mm Downward Displacement0.5mm Upward Displacement
43 Sample Displacement Effects on Peak Positions with Göbel Mirror No Sample Displacement0.2mm Downward Displacement0.4mm Downward Displacement1.0 mm Downward Displacement1.2mm Downward Displacement0.5mm Upward Displacement