Introduction to Powder X-Ray Diffraction History Basic Principles

Introduction to

Powder X-Ray Diffraction

History

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 electrons Cathode 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

Kα-Quant Lα-Quant K L M Emission Photoelectron Electron

The Generating of X-rays

The Generating of X-rays

M

K

L

Kα1 Kα2 Kβ1 Kβ2

energy levels (schematic) of the electrons

Intensity ratios

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 Zr 0,08mm Mn 0,011mm Fe 0,012mm Ni 0,015mm

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

Laue’s Experiment in 1912

Single Crystal X-ray Diffraction

Tube

Collimator Tube

Crystal

Powder X-ray Diffraction

Tube

Powder

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.

θ

λ

sin

2

⋅

⋅

=

n

d

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°

The Elementary Cell

a

b

c

α β γ

a = b = c

β α

_{= = = 90}

γ o

Relationship between d-value and the Lattice

Constants

λ = 2 d s i n θ

Bragg´s law

The wavelength is known

Theta is the half value of the peak position d will be calculated

1/d

2

= (h

2

+ k

2

)/a

2

+ l

2

/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

Interaction between X-ray and Matter

d

wavelength

λ

Pr

intensity Io

incoherent scattering

λ

Co (Compton-Scattering) coherent scattering

λ

Pr(Bragg´s-scattering) absorption

Beer´s law I = I0*e-µd fluorescence

λ

>

λ

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).

θ λ sin 2⋅ ⋅ = n d θ 2 1 sin = d θ σ₁ 2 sin ⋅ =

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).

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 .

Bragg’s Law

A powder sample results in cones with high intensity of scattered beam.

Above conditions result in the Bragg equation

or

θ

λ

=

2

⋅

⋅

sin

⋅

=

∆

s

n

d

θ

λ

sin

2

⋅

⋅

=

n

d

Film Chamber after

Straumannis

The powder is fitted to a glass fibre or into a glass capillary.

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.

A powder sample gives cones with high intensity of scattered beam.

θ

λ

sin 2 ⋅ ⋅ = 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

The Bragg-Brentano Geometry

Tube measurement circle focusing-circle

q

q

2

Detector Sample

The Bragg-Brentano Geometry

Divergence slit Detector-slit Tube Antiscatter-slit Sample Mono-chromator

Comparison Bragg-Brentano Geometry

versus Parallel Beam Geometry

Bragg-Brentano

Parallel-Beam Geometry with Göbel Mirror

Göbel mirror 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

What is a Powder Diffraction Pattern?

a powder diffractogram is the result of a convolution of

a) the diffraction capability of the sample (F

_hkl

) and

b) a complex system function.

The observed intensity y_oi at the data point i is the result of

y_oi = ∑ of intensity of "neighbouring" Bragg peaks + background

The calculated intensity y_ci at the data point i is the result of y_ci = structure model + sample model + diffractometer 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

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.