Introduction to Powder X-Ray Diffraction

Introduction to Powder X-Ray Diffraction History Basic Principles

Folie.1 © 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 Post edited a stamp, dedicated to W.C. Röntgen.

Basics-in-XRD.2 © 2001 Bruker AXS All Rights Reserved

The Principles of an X-ray Tube X-Ray Cathode

Fast electrons

Anode focus


The Principle of Generation Bremsstrahlung Ejected electron (slowed down and changed direction) nucleus

Fast incident electron electrons

Atom of the anodematerial X-ray


The Principle of Generation the Characteristic Radiation Photoelectron

M

Emission Kα-Quant

L K Electron

Lα-Quant

Kβ-Quant


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


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


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


Powder X-ray Diffraction Film Tube

Powder


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.

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°


Reflection Planes in a Cubic Lattice


The Elementary Cell a=b=c o α = β = γ = 90

c a

α

β γ

b


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


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


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). 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

Folie.23 © 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.


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.


The Bragg-Brentano Geometry

Detector

Tube

q focusingcircle

Sample

2q

measurement circle


The Bragg-Brentano Geometry

Antiscatterslit

Divergence slit

Monochromator

Detectorslit

Tube Sample


Comparison Bragg-Brentano Geometry versus Parallel Beam Geometry

Bragg-Brentano Geometry

Parallel Beam Geometry generated by Göbel Mirrors


Parallel-Beam Geometry with Göbel Mirror Göbel mirror

Detector

Soller Slit Tube

Sample


“Grazing Incidence X-ray Diffraction”

Soller slit

Detector

Tube Sample

Measurement circle


“Grazing Incidence Diffraction” with Göbel Mirror

Soller slit

Detector

Göbel mirror

Tube Sample

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 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


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

Introduction to Powder X-Ray Diffraction

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