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Folie.1© 2001 Bruker AXS All Rights Reserved
Introduction to Powder X-Ray Diffraction
HistoryBasic Principles
Basics-in-XRD.2© 2001 Bruker AXS All Rights Reserved
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 Postedited a stamp, dedicated to W.C. Röntgen.
Basics-in-XRD.3© 2001 Bruker AXS All Rights Reserved
The Principles of an X-ray Tube
Anode
focus
Fast electronsCathode
X-Ray
Basics-in-XRD.4© 2001 Bruker AXS All Rights Reserved
The Principle of Generation Bremsstrahlung
X-ray
Fast incident electron
nucleus
Atom of the anodematerial
electrons
Ejectedelectron
(slowed down and changed
direction)
Basics-in-XRD.5© 2001 Bruker AXS All Rights Reserved
The Principle of Generation the CharacteristicRadiation
Kα-Quant
Lα-Quant
Kβ-Quant
K
L
M
EmissionPhotoelectron
Electron
Basics-in-XRD.6© 2001 Bruker AXS All Rights Reserved
The Generating of X-rays
Bohr`s model
Basics-in-XRD.7© 2001 Bruker AXS All Rights Reserved
The Generating of X-rays
M
K
L
Kα1 Kα2 Kβ1 Kβ2
energy levels (schematic) of the electrons
Intensity ratiosKα1 : Kα2 : Kβ = 10 : 5 : 2
Basics-in-XRD.8© 2001 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, λ [Angström]
Kα1 : 0,70926
Kα2 : 0,71354
Kβ1 : 0,63225
Kß-Filter
Kα1 : 1,5405
Kα2 : 1,54434
Kβ1 : 1,39217
Kα1 : 1,78890
Kα2 : 1,79279
Kβ1 : 1,62073
Kα1 : 1,93597
Kα2 : 1,93991
Kβ1 : 1,75654
Zr0,08mm
Mn0,011mm
Fe0,012mm
Ni0,015mm
Basics-in-XRD.9© 2001 Bruker AXS All Rights Reserved
The Generating of X-rays
Emission Spectrum of aMolybdenum X-Ray Tube
Bremsstrahlung = continuous spectra
characteristic radiation = line spectra
Basics-in-XRD.10© 2001 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(cosα-cosα0)=hλb(cosβ-cosβ0)=kλc(cosγ-cosγ0)=lλ
Basics-in-XRD.11© 2001 Bruker AXS All Rights Reserved
Laue’s Experiment in 1912Single Crystal X-ray Diffraction
Tube
Collimator
Tube
Crystal
Film
Basics-in-XRD.12© 2001 Bruker AXS All Rights Reserved
Powder X-ray Diffraction
Tube
Powder
Film
Basics-in-XRD.13© 2001 Bruker AXS All Rights Reserved
Powder Diffraction Diffractogram
Basics-in-XRD.14© 2001 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.
θλ
sin2 ⋅⋅
=nd
Basics-in-XRD.15© 2001 Bruker AXS All Rights Reserved
Another View of Bragg´s Law
nλ = 2d sinθ
Basics-in-XRD.16© 2001 Bruker AXS All Rights Reserved
Crystal SystemsCrystal 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°
Basics-in-XRD.17© 2001 Bruker AXS All Rights Reserved
Reflection Planes in a Cubic Lattice
Basics-in-XRD.18© 2001 Bruker AXS All Rights Reserved
The Elementary Cell
a
b
c
αβ
γ
a = b = cβα γ= = = 90 o
Basics-in-XRD.19© 2001 Bruker AXS All Rights Reserved
Relationship between d-value and the Lattice Constants
λ = 2 d s i n θ Bragg´s lawThe wavelength is knownTheta is the half value of the peak positiond will be calculated
1/d2= (h2 + k2)/a2 + l2/c2 Equation for the determination of the d-value of a tetragonal elementary cell
h,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 position if the peak position is known it is possible to calculate the lattice parameter
Basics-in-XRD.20© 2001 Bruker AXS All Rights Reserved
Interaction between X-ray and Matterd
wavelength λPr
intensity Io
incoherent scatteringλCo (Compton-Scattering)
coherent scatteringλPr(Bragg´s-scattering)
absorptionBeer´s law I = I0*e-µd
fluorescenceλ> λPr
photoelectrons
Basics-in-XRD.21© 2001 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
Basics-in-XRD.22© 2001 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).
θλ
sin2 ⋅⋅
=nd
λθ 2
1sin d=
λ
σθ 12sin
⋅=
Folie.23© 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
Basics-in-XRD.24© 2001 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.
Basics-in-XRD.25© 2001 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 θ.The angle between incident and scattered beam is 2θ .The angle 2θ of maximum intensity is called the Bragg angle.
Basics-in-XRD.26© 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
or
θλ sin2 ⋅⋅=⋅=∆ dns
θλ
sin2 ⋅⋅
=nd
Basics-in-XRD.27© 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.28© 2001 Bruker AXS All Rights Reserved
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 2θ 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.
θλ
sin2 ⋅⋅
=nd
Basics-in-XRD.29© 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.
Basics-in-XRD.30© 2001 Bruker AXS All Rights Reserved
The Bragg-Brentano Geometry
Tube
measurement circle
focusing-circle
qq2
Detector
Sample
Basics-in-XRD.31© 2001 Bruker AXS All Rights Reserved
The Bragg-Brentano Geometry
Divergence slit
Detector-slitTube
Antiscatter-slit
Sample
Mono-chromator
Basics-in-XRD.32© 2001 Bruker AXS All Rights Reserved
Comparison Bragg-Brentano Geometry versus Parallel Beam Geometry
Bragg-BrentanoGeometry
Parallel Beam Geometry generated by Göbel Mirrors
Basics-in-XRD.33© 2001 Bruker AXS All Rights Reserved
Parallel-Beam Geometry with Göbel Mirror
Göbelmirror
Tube
Soller Slit
Detector
Sample
Basics-in-XRD.34© 2001 Bruker AXS All Rights Reserved
“Grazing Incidence X-ray Diffraction”
Tube
Measurement circle
Detector
Sample
Soller slit
Basics-in-XRD.35© 2001 Bruker AXS All Rights Reserved
Tube
Measurement circle
Detector
Sample
Soller slit
Göbel mirror
“Grazing Incidence Diffraction” with Göbel Mirror
Basics-in-XRD.36© 2001 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 (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 + background
The calculated intensity yci at the data point i is the result ofyci = structure model + sample model + diffractometer model
+ background model
Basics-in-XRD.37© 2001 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 sizescaling factor quantitative phase amountdiffuse background false ordermodulated background close order
Basics-in-XRD.38© 2001 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.