# A Brief Guide to X-Ray Diffraction

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• Created by: E.H13
• Created on: 24-05-20 16:14

## Generation of X-Rays

Typically generated by directing an electron beam at a metal target, leading to the ejection of a core electron. An outer electron drops down to fill the hole, releasing energy (the x-ray). Characteristic of each metal.

X-Rays are scattered by electrons in an atom. Degree of scattering is proportional to the number of electrons, so the heavier the atom - the more scattering.

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## Diffraction Pattern

From the diffraction pattern, we ca;

• Fingerprint analysis; compare it to a database to identify compound
• Determine unit cell size and shape of compound
• Determine full structure of compound
• Determine average particle size in a powder
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## Unit Cells

Simplest unit which has the overall symmetry of a crystal and can build the entire lattice by repeating just that unit.

7 possible shapes/crystal systems of unit cell and 4 different types of lattice give us the 14 Bravais lattice types.

Crystal systems;

• Cubic
• Tetragonal
• Orthorhombic
• Monoclinic
• Hexagonal
• Rhombohedral
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## Lattice Planes; Miller Indices

Regard crystals as parallel planes of atoms, and assume X-Rays are reflected from these planes.

These planes are defined by their Miller indices, (hkl), with spearation dhkl.

Bragg's Law;  = 2dhklsin (x)

Crystalline samples - defined peaks

Amorphous - broad humps in background

Purity determined by fingerprint analysis.

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## Indexing - Determining Unit Cell Parameters 1

This method is used when no information about the unit cell is known.

First, determine (hkl) values for each reflection, then detemine unit cell size.

Cubic systems

In a table;

• Given 2X values; divide by 2 to get values for X
• Find values of sin2(x)
• sin2(x)/HCF where HCF = Highest common factor
• Set this equal to (h2 + k2 + l2)
• Use this to work out (hkl)

Then use d = lambda / 2*sin(x) to find values for d. Use a = d*sqrt(h2 + k2 + l2) to find values of a. Average these values to find the overall value for unit cell parameter, a.

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## Equations - Determining Unit Cell Parameters 2

This method is used when you know the Miller Indicies (hkl values) and peak positions.

Use the following equations;

Cubic; where a = b = c

1/d2 = (h2 + k2 + l2)/a2

Then use d = a / sqrt(h2 + k2 + l2). For more complex cells, you have to take into account angles.

Tetragonal; where a = b

1/d2 = (h2 + k2) / a2 + l/ c2

Orthorhombic;

1/d2 = h2 / a2 + k2 / b2 + l/ c2

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## Determining Unit Cell Symmetry

Systematic absences; where certain collections of reflections will be missing/have zero intensity.

Primitive - all reflections observed

Body Centred Cubic (BCC) - first reflection is (110) reflection

Face Centred Cubic (FCC) - first reflection is (111) reflection.

If cell is not cubic, the individual cell lengths will affect which peak is first observed.

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## Peak Multiplicity

Reflections are made up of multiple peaks overlapping in the same position

Cubic; (100), (010), (001), (-100), (0-10), (00-1) - multiplicity of 6.

Tetragonal; (100), (010), (-100), (0-10) - multiplicity of 4. (001), (00-1) - multiplicity of 2.

Orthohorombic; (100), (-100) - multiplicity of 2. (010), (0-10) - multiplicity of 2. (001), (00-1) - multiplicity of 2.

Large difference in cell length -> large shift in peak position.

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