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.1© 1999 R. Haberkorn and BRUKER AXS All Rights Reserved

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

Anode

focus

Fast electronsCathode

X-Ray


The Principle of Generation Bremsstrahlung

X-ray

Fast incident electron

nucleus

Atom of the anodematerial

electrons

Ejected electron(slowed

down and changed

direction)


The Principle of Generation the Characteristic Radiation

K-Quant

L-Quant

K-Quant

K

L

M

EmissionPhotoelectron

Electron


The Generating of X-rays

Bohr`s model



M

K

L

K K K K

energy levels (schematic) of the electrons

Intensity ratios KKK



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



Emission Spectrum of aMolybdenum 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(cos-cos)=hb(cos-cos)=kc(cos-cos)=l


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

Tube

Collimator

Tube

Crystal

Film


Powder X-ray Diffraction

Tube

Powder

Film


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.

sin2

n

d


Another View of Bragg´s Law

n = 2d sin


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 , °


Reflection Planes in a Cubic Lattice


The Elementary Cell

a

b

c

a = b = c = = = 9

0

o


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


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


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).

sin2

n

d

2

1sin d

12

sin


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 .

The angle between incident and scattered beam is 2 .

The angle 2 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

sin2 dns

sin2

n

d


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 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


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

measurement circle

focusing-circle

qq2

Detector

Sample


The Bragg-Brentano Geometry

Divergence slit

Detector-

slitTube

Antiscatter-slit

Sample

Mono-chromat

or


Comparison Bragg-Brentano Geometry versus Parallel Beam Geometry

Bragg-BrentanoGeometry

Parallel Beam Geometry generated by Göbel Mirrors

X-ray Source

Motorized Slit

Sample


“Grazing Incidence Diffraction” with Göbel Mirror

X-ray Source

Göbel Mirror

Sample

Soller slit

Scintillationcounter

Measurement circle


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


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


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:

S w y obs y calci i ii

2min


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


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

Sample displacement

Peak shift

Sample

X-ray tube


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


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


Peak Profile Shape of NIST 1976 (1)


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.


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


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

introduction-to-xrd.1 © 1999 r. haberkorn and bruker axs all rights reserved introduction to powder...

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