X-ray Diffraction and Bragg’s Law: Solving Simple Crystal Structures

Why X-rays, specifically

Diffraction — constructive and destructive interference of waves scattered by a regular array of objects — only produces a clear, measurable pattern when the wavelength of the radiation is comparable in size to the spacing of the array doing the scattering. Interplanar spacings in typical crystals fall in the roughly 1–10 Å range, and X-rays (commonly Cu-Kα radiation, λ ≈ 1.54 Å) sit in exactly that range, which is why X-rays — rather than visible light, whose wavelength is hundreds of times too large — are the practical tool for probing crystal structure directly.

Bragg's law, derived from a simple picture

Picture a crystal as stacks of parallel atomic planes separated by a spacing d. X-rays reflect off successive planes, and the reflected beams interfere constructively only when the extra distance travelled by the beam reflecting off the lower plane is a whole number of wavelengths. That extra path length works out to 2d sinθ, where θ is the angle between the incident beam and the crystal plane (not the angle from the normal, a common point of confusion). Setting this equal to nλ for constructive interference gives:

where n is an integer (the "order" of reflection), λ is the X-ray wavelength, d is the interplanar spacing, and θ is the glancing angle. Constructive interference — and therefore a detectable diffracted beam — only occurs at specific angles satisfying this relationship for a given d and λ; every other angle gives destructive interference and no signal, which is exactly why a diffraction pattern consists of sharp peaks at specific angles rather than a smooth continuous response.

Worked numerical

Suppose Cu-Kα radiation (λ = 1.54 Å) is diffracted from a crystal, and a first-order (n = 1) reflection is observed at θ = 15°. Find the interplanar spacing d.

The method generalises directly: given any two of {n, λ, d, θ}, Bragg's law solves for the third (or fourth, with n usually given or taken as 1 unless stated). The most common exam variant gives λ and a measured θ and asks for d, exactly as above; a second common variant gives d (from a known lattice parameter and Miller indices) and asks at what angle a particular reflection should appear.

Where this connects to the rest of solid-state chemistry

Once d is known for several different sets of lattice planes (indexed by Miller indices hkl), those spacings can be combined with the crystal system's geometric relationships to determine the unit cell dimensions themselves — which is how X-ray diffraction, in the hands of a crystallographer, moves from "a single spacing number" to a full three-dimensional structure determination. For exam purposes, though, the Bragg's-law numerical in isolation, as worked above, is what is most frequently tested.

FAQs

Why is θ measured from the crystal plane and not from the normal in Bragg's law?

Bragg's law is conventionally stated using the glancing angle (from the plane itself), consistent with how the path-length difference 2d sinθ is geometrically derived. Confusing this with the angle from the normal is the most common source of numerical errors on this topic.

What does the integer n in Bragg's law represent?

It is the order of reflection, corresponding to the extra path length being a whole-number multiple of the wavelength. Unless a question specifies otherwise, first-order (n = 1) reflections are usually assumed.

Is Bragg's law enough to determine a full crystal structure?

Not by itself — it gives interplanar spacing for one set of planes from one measured angle. Full structure determination combines spacings from multiple reflections (indexed by Miller indices) with the crystal system's geometry.