Introduction to Powder X-Ray Diffraction

$: Introduction to Powder X-Ray Diffraction$
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.


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

Ejectedelectron

(slowed down and changed

direction)


The Principle of Generation the CharacteristicRadiation

Kα-Quant

Lα-Quant

Kβ-Quant

K

L

M

EmissionPhotoelectron

Electron


The Generating of X-rays

Bohr`s model



M

K

L

Kα1 Kα2 Kβ1 Kβ2

energy levels (schematic) of the electrons

Intensity ratiosKα1 : Kα2 : Kβ = 10 : 5 : 2



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



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(cosα-cosα0)=hλb(cosβ-cosβ0)=kλc(cosγ-cosγ0)=lλ


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

Tube

Collimator

Tube

Crystal

Film


Powder X-ray Diffraction

Tube

Powder

Film


Powder Diffraction Diffractogram


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


Another View of Bragg´s Law

nλ = 2d sinθ


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°


Reflection Planes in a Cubic Lattice


The Elementary Cell

a

b

c

αβ

γ

a = b = cβα γ= = = 90 o


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


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


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).

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


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 θ.The angle between incident and scattered beam is 2θ .The angle 2θ 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

θλ sin2 ⋅⋅=⋅=∆ dns

θλ

sin2 ⋅⋅

=nd


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

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


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-chromator


Comparison Bragg-Brentano Geometry versus Parallel Beam Geometry

Bragg-BrentanoGeometry

Parallel Beam Geometry generated by Göbel Mirrors


Parallel-Beam Geometry with Göbel Mirror

Göbelmirror

Tube

Soller Slit

Detector

Sample


“Grazing Incidence X-ray Diffraction”

Tube

Measurement circle

Detector

Sample

Soller slit


Tube

Measurement circle

Detector

Sample

Soller slit

Göbel mirror

“Grazing Incidence Diffraction” with Göbel Mirror


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


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


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 Powder X-Ray Diffraction

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