Introduction to Powder X-Ray Diffraction

History Basic Principles

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.

X-Ray

Cathode Fast electrons

Anode

focus

Ejected electron (slowed down and changed direction) nucleus

Fast incident electron electrons

Atom of the anodematerial X-ray

Photoelectron Emission

M Kα-Quant

Electron Lα-Quant

Kβ-Quant

Bohr`s model

energy levels (schematic) of the electrons

Intensity ratios K K K L α1 : α2 : β = 10 : 5 : 2

K Kα1 Kα2 Kβ1 Kβ2

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

Emission Spectrum of a Molybdenum X-Ray Tube

Bremsstrahlung = continuous spectra

characteristic radiation = line spectra

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

Tube Tube

Crystal

Collimator Film

Tube

Powder

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

nλ = 2d sinθ

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°

a = b = c o α ===β γ 90

α a β γ

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

coherent scattering wavelength λPr λPr(Bragg´s-scattering) absorption Beer´s law I = I0*e-µd intensity Io fluorescence λ> λPr

photoelectrons

C. Gordon Darwin, grandson of C. Robert Darwin (picture) developed 1912 dynamic theory of scattering of X-rays at crystal lattice

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

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

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.

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.

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

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.

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.

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.

Tube Detector

q focusing- Sample 2q circle

measurement circle

Mono- chromator Divergence slit Antiscatter- slit

Detector- slit Tube

Sample

Bragg-Brentano Parallel Beam Geometry Geometry generated by Göbel Mirrors

Göbel Detector mirror

Soller Slit Tube

Sample

Detector Soller slit

Tube Sample

Measurement circle

Soller slit Detector

Göbel mirror

Tube Sample

Measurement circle

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

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

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.