A Brief Guide to X-Ray Diffraction

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. 

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

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

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 d_hkl.

Bragg's Law;  = 2d_hklsin (x)

Crystalline samples - defined peaks

Amorphous - broad humps in background

Purity determined by fingerprint analysis.

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. 

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 + l² / c²

Orthorhombic;

1/d² = h² / a² + k² / b² + l² / c²

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.

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.