Introduction to Powder X-Ray Diffraction History Basic Principles
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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.
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The Principles of an X-ray Tube X-Ray Cathode
Fast electrons
Anode focus
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The Principle of Generation Bremsstrahlung Ejected electron (slowed down and changed direction) nucleus
Fast incident electron electrons
Atom of the anodematerial X-ray
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The Principle of Generation the Characteristic Radiation Photoelectron
M
Emission Kα-Quant
L K Electron
Lα-Quant
Kβ-Quant
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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 Kα1 : Kα2 : Kβ = 10 : 5 : 2
L K
Kα1
Kα2
Kβ1
Kβ2
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The Generating of X-rays Anode
(kV)
Wavelength, λ [Angström] Kα1 : 0,70926
Mo Cu
20,0 9,0
Kα2 : 0,71354 Kβ1 :
Kß-Filter Zr 0,08mm
0,63225
Kα1 : 1,5405 Kα2 : 1,54434
Ni 0,015mm
Kβ1 : 1,39217
Co
Kα1 : 1,78890
7,7
Kα2 : 1,79279 Kβ1 :
Fe
Fe 0,012mm
1,62073
Kα1 : 1,93597
7,1
Kα2 : 1,93991 Kβ1 :
1,75654
Mn 0,011mm 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
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History: Max Theodor Felix von Laue
Max von Laue put forward the conditions for scattering maxima, the Laue equations:
a(cosα-cosα0)=hλ b(cosβ-cosβ0)=kλ 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
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Powder X-ray Diffraction Film Tube
Powder
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Powder Diffraction Diffractogram
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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 ⋅ sin θ 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°
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Reflection Planes in a Cubic Lattice
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The Elementary Cell a=b=c o α = β = γ = 90
c a
α
β γ
b
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Relationship between d-value and the Lattice Constants λ = 2 d s in θ
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
1/d = (h + k )/a + l /c
2
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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Interaction between X-ray and Matter d incoherent scattering
λCo (Compton-Scattering) wavelength λPr intensity Io
coherent scattering
λPr(Bragg´s-scattering) absorption Beer´s law I = I0*e-µd fluorescence
λ> λPr
photoelectrons
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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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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). d=
n⋅λ 2 ⋅ sin θ
1
sin θ = d 2
λ
sin θ =
σ 2⋅ 1
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
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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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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 ⋅ sin θ or
d =
n ⋅λ 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. This relation is quantified by Bragg’s law.
d =
n ⋅λ 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.
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The Bragg-Brentano Geometry
Detector
Tube
q focusingcircle
Sample
2q
measurement circle
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The Bragg-Brentano Geometry
Antiscatterslit
Divergence slit
Monochromator
Detectorslit
Tube Sample
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Comparison Bragg-Brentano Geometry versus Parallel Beam Geometry
Bragg-Brentano Geometry
Parallel Beam Geometry generated by Göbel Mirrors
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Parallel-Beam Geometry with Göbel Mirror Göbel mirror
Detector
Soller Slit Tube
Sample
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“Grazing Incidence X-ray Diffraction”
Soller slit
Detector
Tube Sample
Measurement circle
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“Grazing Incidence Diffraction” with Göbel Mirror
Soller slit
Detector
Göbel mirror
Tube Sample
Measurement circle
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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
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Which Information does a Powder Pattern offer?
peak position peak intensity peak broadening scaling factor diffuse background modulated background
dimension of the elementary cell content of the elementary cell strain/crystallite size quantitative phase amount false order close order
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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. Basics-in-XRD.38 © 2001 Bruker AXS All Rights Reserved