X-ray Diffraction

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Electromagnetic Spectrum

Wavelength (m)10-15 10-12 10-9 10-6 10-3 1.0 103 106

Gamma Rays

X-rays

UV IR

Micro TV FM AM

Long Radio Waves

Frequency (Hz)1024 1021 1018 1015 1012 109 106 103

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What are X-rays?

X-Rays (Roentgen rays) were dicovered by Roentgen in 1895. They are a kind of electromagnetic radiation of very short wavelength and very high energy produced when high-speed electrons strike a solid target. The wavelength range lyes between 0.0001 and 10 nm i.e., the range between ‘gamma rays’ and ‘ultraviolet rays’

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History of X-ray and XRDWilhelm Conrad Röntgen discovered

X-Rays in 1895.1901 Nobel prize in Physics

Wilhelm Conrad Röntgen (1845-1923)

A modern radiograph of a hand Bertha Röntgen’s Hand 8 Nov, 1895 4

X-Rays penetrat matter but not all of them come out. Some get lost. The missing X-rays having very high energy get absorbed in the body and the X-ray's energy is released. This energy is transferred to an electron, which does a lot of damage in a small area.

X-Rays affect the DNA of cells.

For the levels of radiation one gets at the dentist and the doctor, the body is able to repair most all of the damage done.

Risks

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Origin of X-Rays• A vacuum tube consists of a cathode, an anode, an a

heating filament.• The heated cathode emits electrons (e-).• A field of ~50kV accelerates the electrons onto the anode.• The elctrons bump on atoms and slow down, emitting

radiation of a continuous distribution of wavelengths (``Bremsstrahlung'').

• Some electrons cause sharp atomic transitions, resulting in X-rays with definite wavelengths (0.1~50A).

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The X-ray generator

X-rays

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The X-ray generator

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Wavelengths for X-Radiation are Sometimes UpdatedCopperAnodes

Bearden(1967)

Holzer et al.(1997)

CobaltAnodes

Bearden(1967)

Holzer et al.(1997)

Cu Ka1 1.54056Å 1.540598 Å Co Ka1 1.788965Å 1.789010 Å

Cu Ka2 1.54439Å 1.544426 Å Co Ka2 1.792850Å 1.792900 Å

Cu Kb 1.39220Å 1.392250 Å Co Kb 1.62079Å 1.620830 Å

MolybdenumAnodes

ChromiumAnodes

Mo Ka1 0.709300Å 0.709319 Å Cr Ka1 2.28970Å 2.289760 Å

Mo Ka2 0.713590Å 0.713609 Å Cr Ka2 2.293606Å 2.293663 Å

Mo Kb 0.632288Å 0.632305 Å Cr Kb 2.08487Å 2.084920 Å

Often quoted values from Cullity (1956) and Bearden, Rev. Mod. Phys. 39 (1967) are incorrect.

Values from Bearden (1967) are reprinted in international Tables for X-Ray Crystallography and most XRD textbooks.

Most recent values are from Hölzer et al. Phys. Rev. A 56 (1997) Has your XRD analysis software been updated?

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What are the atomic processes that can produce X-rays?

(1) White radiation

electrons from an external source are deflected around the nucleus of a target atom and X-rays of multiple wavelengths are emitted, depending on the kinetic energy lost by the electron on deflection

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What are the atomic processes that can produce X-rays?

(2) Characteristic emission

An incident e- bumps onto an inner level e-. Both electrons fly out and energy is emitted as an electromagnetic wave (white). The “hole” is filled with a higher level electron and this is accompanied by X-ray emission (characteristic). These peaks are labeled K, K, L, L, etc, depending on the specific energy levels

involved. 11

X-Ray interactions with an object

X-Ray elastic scattering is the basis for crystallographic analysis WXRD and SAXS

X-Ray inelastic scattering is the basis for analytical methods such as XPS (X-ray Photoelectron Spectroscopy), XRF, AES (Auger Electron Spectroscopy)and EDAX (Energy Dispersive Analysis by x-ray)

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The first kind of scatter process to be recognised was discovered by Max von Laue who was awarded the Nobel prize for physics in 1914 "for his discovery of the diffraction of X-rays by crystals". His collaborators Walter Friedrich and Paul Knipping took the picture on the bottom left in 1912. It shows how a beam of X-rays is scattered into a characteristic pattern by a crystal. In this case it is copper sulphate.

The X-ray diffraction pattern of a pure substance is like a fingerprint of the substance. The powder diffraction method is thus ideally suited for characterization and identification of polycrystalline phases.

History of X-ray and XRD

Max von Laue (1897-1960)

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Bragg’s Law The father and son team of Sir

William Henry and William Lawrence Bragg were awarded the Nobel prize for physics "for their services in the analysis of crystal structure by means of Xrays“ in 1915.

Bragg's law was an extremely important discovery and formed the basis for the whole of what is now known as crystallography. This technique is one of the most widely used structural analysis techniques and plays a major role in fields as diverse as structural biology and materials science.

William LawrenceBragg (1890-1971)

Sir William HenryBragg (1862-1942)

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Bragg’s Law• Correlates X-ray wave length,

l, interplanar spacing, d, and reflection angle, q.

2222

2

222

4sin

sin2

lkha

N

a

lkh

ad

nd

hkl

nhkl

• Scattering atoms (circled in red) behave like slits in Young’s experiment.

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Crystal planes and Miller’s indices

• Each plane in a crystal is defined by Miller’s indices

l)k,(h, n normal plane crystal

z/al

y/ak

x/ah

x

y

z

y=1x=1

z=1

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Crystal Planes, Miller indices and diffraction

Since a crystal has an ordered 3-D periodic arrangement of atoms (ions or molecules) the atomic planes in any crystal can be related to the unit cell.

One can label each set of planes uniquely by considering their (fractional) intersection with the unit cell axes a,b,c and converting these to INTEGERS h, k, and l.

e.g. the planes that intersect the b-axis at ½ and are parallel to a and c. ( a/ , b/2, c/) are defined by the MILLER INDiCES (0 2 0)

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A reminder: finding Miller Indices of a plane

Extend the plane to make it cut the crystal axis system at points (a1, b1, c1)

Note the reciprocals of the intercepts, i.e.: Multiply or divide by the highest common factor to

obtain the smallest integer numbers. Replace negative integers with bar over the

number, i.e. we replace -h by Note:

If the plane is parallel to an axis, we say it cuts at and

If the plane passes through the origin, we translate the unit cell in a suitable direction.

We use round brackets ( ) to describe a single plane and curly brackets { } to describe a family of planes.

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Examples:

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The Interplanar Spacing, d(hkl) Can be found for each set of planes provided we know the unit cell

parameters.

e.g. for a CUBIC unit cell:

d(100) = a

d(010) = b (= a)

d(030) = ( )a

d(110) = ( )a

3

1

2

1

For unit cells with = = = 90º

2

2

2

2

2

2

2

1

c

l

b

k

a

h

d

2

222

2

1

a

lkh

d

Which for a CUBIC unit cell simplifies to

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Bragg Example

d = λ / (2 Sin θB) λ = 1.54 Ǻ

= 1.54 Ǻ / ( 2 * Sin ( 38.3 / 2 ) ) = 2.35 Ǻ

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Different Types of Unit cells in Materials

Bravis Lattices

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Bravais Lattices

Let’s go to another class

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