Introduction-to-XRD.1 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved Introduction to Powder...
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Transcript of Introduction-to-XRD.1 © 1999 R. Haberkorn and BRUKER AXS All Rights Reserved Introduction to Powder...
Introduction-to-XRD.1© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
Introduction to Powder X-Ray Diffraction
History
Basic Principles
Introduction-to-XRD.2© 1999 R. Haberkorn and 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 Post edited a stamp, dedicated to W.C. Röntgen.
Introduction-to-XRD.3© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
The Principles of an X-ray Tube
Anode
focus
Fast electronsCathode
X-Ray
Introduction-to-XRD.4© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
The Principle of Generation Bremsstrahlung
X-ray
Fast incident electron
nucleus
Atom of the anodematerial
electrons
Ejected electron(slowed
down and changed
direction)
Introduction-to-XRD.5© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
The Principle of Generation the Characteristic Radiation
K-Quant
L-Quant
K-Quant
K
L
M
EmissionPhotoelectron
Electron
Introduction-to-XRD.6© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
The Generating of X-rays
Bohr`s model
Introduction-to-XRD.7© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
The Generating of X-rays
M
K
L
K K K K
energy levels (schematic) of the electrons
Intensity ratios KKK
Introduction-to-XRD.8© 1999 R. Haberkorn and 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 Angström
K1 : 0,70926
K2 : 0,71354
K1 : 0,63225
Filter
K1 : 1,5405
K2 : 1,54434
K1 : 1,39217
K1 : 1,78890
K2 : 1,79279
K1 : 1,62073
K1 : 1,93597
K2 : 1,93991
K1 : 1,75654
Zr0,08mm
Mn0,011mm
Fe0,012mm
Ni0,015mm
Introduction-to-XRD.9© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
The Generating of X-rays
Emission Spectrum of aMolybdenum X-Ray Tube
Bremsstrahlung = continuous spectra
characteristic radiation = line spectra
Introduction-to-XRD.10© 1999 R. Haberkorn and 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(cos-cos)=hb(cos-cos)=kc(cos-cos)=l
Introduction-to-XRD.11© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
Laue’s Experiment in 1912 Single Crystal X-ray Diffraction
Tube
Collimator
Tube
Crystal
Film
Introduction-to-XRD.12© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
Powder X-ray Diffraction
Tube
Powder
Film
Introduction-to-XRD.13© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
Powder Diffraction Pattern
Introduction-to-XRD.14© 1999 R. Haberkorn and 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.
sin2
n
d
Introduction-to-XRD.15© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
Another View of Bragg´s Law
n = 2d sin
Introduction-to-XRD.16© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
Crystal SystemsCrystal systems Axes system
cubic a = b = c , = = = 90°
Tetragonal a = b c , = = = 90°
Hexagonal a = b c , = = 90°, = 120°
Rhomboedric a = b = c , = = 90°
Orthorhombic a b c , = = = 90°
Monoclinic a b c , = = 90° , 90°
Triclinic a b c , °
Introduction-to-XRD.17© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
Reflection Planes in a Cubic Lattice
Introduction-to-XRD.18© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
The Elementary Cell
a
b
c
a = b = c = = = 9
0
o
Introduction-to-XRD.19© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
Relationship between d-value and the Lattice Constants
=2dsin Bragg´s law The wavelength is known
Theta is the half value of the peak position
d 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 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
Introduction-to-XRD.20© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
Interaction between X-ray and Matter
d
wavelength Pr
intensity Io
incoherent scattering
Co (Compton-Scattering)
coherent scattering
Pr(Bragg´s-scattering)
absorbtionBeer´s law I = I0*e-µd
fluorescense
> Pr
photoelectrons
Introduction-to-XRD.21© 1999 R. Haberkorn and 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
Introduction-to-XRD.22© 1999 R. Haberkorn and 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).
sin2
n
d
2
1sin d
12
sin
Introduction-to-XRD.24© 1999 R. Haberkorn and 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.
Introduction-to-XRD.25© 1999 R. Haberkorn and 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 .
The angle between incident and scattered beam is 2 .
The angle 2 of maximum intensity is called the Bragg angle.
Introduction-to-XRD.26© 1999 R. Haberkorn and 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
sin2 dns
sin2
n
d
Introduction-to-XRD.27© 1999 R. Haberkorn and 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.
Introduction-to-XRD.28© 1999 R. Haberkorn and 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 2 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.
sin2
n
d
Introduction-to-XRD.29© 1999 R. Haberkorn and 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.
Introduction-to-XRD.30© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
The Bragg-Brentano Geometry
Tube
measurement circle
focusing-circle
qq2
Detector
Sample
Introduction-to-XRD.31© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
The Bragg-Brentano Geometry
Divergence slit
Detector-
slitTube
Antiscatter-slit
Sample
Mono-chromat
or
Introduction-to-XRD.32© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
Comparison Bragg-Brentano Geometry versus Parallel Beam Geometry
Bragg-BrentanoGeometry
Parallel Beam Geometry generated by Göbel Mirrors
X-ray Source
Motorized Slit
Sample
Introduction-to-XRD.33© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
“Grazing Incidence Diffraction” with Göbel Mirror
X-ray Source
Göbel Mirror
Sample
Soller slit
Scintillationcounter
Measurement circle
Introduction-to-XRD.34© 1999 R. Haberkorn and 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 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
Introduction-to-XRD.35© 1999 R. Haberkorn and 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 size
scaling factor quantitative phase amount
diffuse background false order
modulated background close order
Introduction-to-XRD.36© 1999 R. Haberkorn and 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.
Introduction-to-XRD.37© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
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:
S w y obs y calci i ii
2min
Introduction-to-XRD.38© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
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
y calc SF M P F LP yb obsik
k k k k k i k i 2 2 2 2
Introduction-to-XRD.39© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
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)
Introduction-to-XRD.40© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
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
Introduction-to-XRD.41© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
Effects of Sample Displacement
Sample displacement
Peak shift
Sample
X-ray tube
Introduction-to-XRD.42© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
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
Introduction-to-XRD.43© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
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
Introduction-to-XRD.44© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
Peak Profile Shape of NIST 1976 (1)
Introduction-to-XRD.45© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
Peak Profile Shape of NIST 1976 (2)
Introduction-to-XRD.46© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
Peak Profile Shape of NIST 1976 (3)
Introduction-to-XRD.47© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
2 0 .0 0 4 0 .0 0 6 0 .0 0 8 0 .0 0 1 0 0 .0 0 1 2 0 .0 0 1 4 0 .0 0
0 .7 0
0 .8 0
0 .9 0
1 .0 0
1 .1 0
1 .2 0
1 .3 0
IA / I B
Instrument Response Function
D5005 Theta/2ThetaGöbel Mirror, 0.2 mm divergence slit, 2° vertical Soller slit and 0.15° collimator.
Introduction-to-XRD.48© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
2 0 .0 0 4 0 .0 0 6 0 .0 0 8 0 .0 0 1 0 0 .0 0 1 2 0 .0 0 1 4 0 .0 0
0 .5 0
0 .6 0
0 .7 0
0 .8 0
0 .9 0
1 .0 0
1 .1 0
1 .2 0
1 .3 0
1 .4 0
1 .5 0
A
D5005 Theta/2ThetaGöbel Mirror, 0,2 mm divergence slit, 2° vertical Soller slit and 0.15° collimator.
Peak Shape Asymmetry
Introduction-to-XRD.49© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved
Instrument Resolution Functions
2 0 .0 0 4 0 .0 0 6 0 .0 0 8 0 .0 0 1 0 0 .0 0 1 2 0 .0 0 1 4 0 .0 0
0 .0 0
0 .0 5
0 .1 0
0 .1 5
0 .2 0
0 .2 5
F W H M
D S :0 .3 m m0 .2 m m0 .1 m m0 .3 °0 .3 °
A S :0 .3 m m -- --0 .3 °0 .3 °
R S :0 .1 m m -- --0 .0 1 8 ° --
D 5 0 0 5 T h e ta /T h e ta D 5 0 0 5 T h e ta /2 T h e ta G ö b e lD 5 0 0 5 T h e ta /2 T h e ta G ö b e lD 5 0 0 , G e -P rim ., S ZD 5 0 0 , G e -P rim ., P S D