Undergraduate · 20 min

X-ray diffraction: measuring the spacing between planes

A powder pattern is a fingerprint of interplanar spacings. Move the Bragg peaks by changing the wavelength or the d-spacing and read 2θ off the diffractogram.

Goal

Apply Bragg's law $n\lambda = 2d\sin\theta$ to predict peak positions and see how a larger d or a shorter λ moves the pattern.

Apparatus and reagents

Virtual bench: X-ray tube (Cu, Mo or Co anode), powdered sample holder, goniometer and detector readout.

Procedure

Start with Cu Kα and d = 3.0 Å. Read the 2θ of the first-order peak and check it against $2\theta = 2\arcsin(\lambda/2d)$ .
Keep Cu Kα but shrink d to 1.8 Å. In which direction do all the peaks move?
Switch to Mo Kα (λ = 0.71 Å) at d = 3.0 Å. How do the peak positions compare with Cu Kα at the same d?
For the first preset, note how many orders appear before sin θ exceeds 1. Why does the pattern stop?

What to observe

Smaller d pushes every peak to larger 2θ: d = 1.8 Å puts the first order near 51° instead of 30°.
Shorter λ compresses the whole pattern to lower angles: with Mo Kα the first order for d = 3.0 Å sits near 13.6°.
Peaks fade with order (intensity ~1/n² here is schematic): in real data the envelope is set by atomic scattering factors.

Explanation

Constructive interference between waves bounced off parallel crystal planes occurs when the path difference is a whole number of wavelengths: $n\lambda = 2d\sin\theta$ . Everything is measured against 2θ because the detector moves twice as fast as the sample tilt. Since $\sin\theta \le 1$ , only orders with $n\lambda \le 2d$ exist — that is why the stick pattern simply stops at high n, and why Mo Kα (shorter λ) can probe smaller spacings than Cu Kα.

History of the experiment

Max von Laue proved X-ray diffraction by crystals in 1912; William Henry and Lawrence Bragg turned it into a measuring tool with the 1913 law bearing their name. Dorothy Hodgkin later pushed the method to chart vitamin B12 and insulin, and powder diffraction became the routine way to fingerprint crystalline materials.

Chemists behind it

Rosalind Franklin Dorothy Crowfoot Hodgkin