Introduction to Powder X-Ray Diffraction History Basic Principles

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.

The Principles of an X-ray Tube Cathode Fast electrons Anode focus

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

The Principle of Generation the Characteristic Radiation Emission Photoelectron M K-Quant L K Electron L-Quant K-Quant

The Generating of X-rays Bohr`s model

The Generating of X-rays energy levels (schematic) of the electrons M Intensity ratios KKK L K K K K K

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 Zr 0,08mm K1 : 1,5405 K2 : 1,54434 K1 : 1,39217 Ni 0,015mm K1 : 1,78890 K2 : 1,79279 K1 : 1,62073 Fe 0,012mm K1 : 1,93597 K2 : 1,93991 K1 : 1,75654 Mn 0,011mm

The Generating of X-rays Emission Spectrum of a Molybdenum X-Ray Tube Bremsstrahlung = continuous spectra characteristic radiation = line spectra

History: Max Theodor Felix von Laue Max von Laue put forward the conditions for scattering maxima, the Laue equations: a(cosa-cosa0)=hl b(cosb-cosb0)=kl c(cosg-cosg0)=ll

Laue’s Experiment in 1912 Single Crystal X-ray Diffraction Tube Tube Crystal Collimator Film

Powder X-ray Diffraction Film Tube Powder

Powder Diffraction Pattern

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.

Another View of Bragg´s Law n = 2d sin

Crystal Systems

Reflection Planes in a Cubic Lattice

The Elementary Cell a = b = c o = = = 90 c a b

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

Interaction between X-ray and Matter incoherent scattering Co (Compton-Scattering) coherent scattering Pr(Bragg´s-scattering) wavelength Pr absorbtion Beer´s law I = I0*e-µd intensity Io fluorescense > Pr photoelectrons

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

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).

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

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.

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 q. The angle between incident and scattered beam is 2q . The angle 2q of maximum intensity is called the Bragg angle.

Bragg’s Law A powder sample results in cones with high intensity of scattered beam. Above conditions result in the Bragg equation or

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.

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 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.

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.

The Bragg-Brentano Geometry Tube Detector q q 2 focusing-circle Sample measurement circle

The Bragg-Brentano Geometry Mono- chromator Antiscatter- slit Divergence slit Detector- slit Tube Sample

Comparison Bragg-Brentano Geometry versus Parallel Beam Geometry Motorized Slit X-ray Source Sample Bragg-Brentano Geometry Parallel Beam Geometry generated by Göbel Mirrors

“Grazing Incidence Diffraction” with Göbel Mirror Measurement circle Scintillation counter Göbel Mirror Soller slit Sample X-ray Source

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 5

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 6

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.

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: 7

Basis formula of the Rietveldmethod SF : Scaling factor Mk : Multiplicity of the reflections k Pk : Value of a preffered orientation function for the reflections k Fk2 : Structure factor of the reflections k LP : Value of the Lorentz-Polarisations function for the reflections k Fk : Peak profile function for the reflections k on the position i ybi : Value of the background at the position i k : Index over all reflexes with intensity on the position i

A. Seyfarth, A. Kern & G. Menges 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-76187 Karlsruhe Fifth European Powder Diffraction Conference, EPDIC-5, Abstracts, p. 227 (1997) XVII Conference on Applied Crystallography, CAC 17, Abstracts, p. 45 (1997)

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

Effects of Sample Displacement X-ray tube Peak shift Sample displacement Sample

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

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

Peak Profile Shape of NIST 1976 (1)

Peak Profile Shape of NIST 1976 (2)

Peak Profile Shape of NIST 1976 (3)

Instrument Response Function D5005 Theta/2Theta Göbel Mirror, 0.2 mm divergence slit, 2° vertical Soller slit and 0.15° collimator.

Peak Shape Asymmetry D5005 Theta/2Theta Göbel Mirror, 0,2 mm divergence slit, 2° vertical Soller slit and 0.15° collimator.

Instrument Resolution Functions