X-Ray Diffraction and Bragg’s Law: Reading a Powder Pattern

Why X-rays and not visible light

Diffraction only gives useful structural information when the wavelength is comparable to the spacing being probed. Interatomic distances in crystals are of the order of 100 to 300 pm, and X-rays have wavelengths in exactly that range. Visible light, at hundreds of nanometres, is thousands of times too long to resolve atomic planes.

Copper K-alpha radiation at 154 pm is the standard laboratory source and appears in most numerical questions, so it is worth remembering the value.

Bragg’s law

Treat a crystal as a stack of parallel planes of atoms separated by a distance d. X-rays reflecting from successive planes travel different distances, and they interfere constructively only when that path difference is a whole number of wavelengths.

Two points about the geometry are regularly examined. First, θ is measured from the plane, not from the normal, which is the opposite of the convention used in optics. Second, a diffractometer reports the angle between the incident and diffracted beams, which is 2θ, so the value read off the pattern must be halved before substituting.

Plane spacing in a cubic crystal

For a cubic lattice with cell edge a, the spacing of the planes with Miller indices (hkl) is

Combining this with Bragg’s law for first-order diffraction gives the relation that does the real work:

Since λ and a are fixed for a given pattern, sin²θ is directly proportional to (h² + k² + l²). Dividing every sin²θ by the smallest one gives a set of small numbers whose pattern identifies the lattice.

Systematic absences

Not every set of planes produces a peak. In centred lattices, the atoms at the centring positions scatter exactly out of phase with those at the corners for certain reflections, cancelling them completely. These missing reflections are systematic absences and they are the fingerprint of the lattice type.

The face-centred pattern is the easiest to spot because the first two peaks are close together — (111) and (200) give values of 3 and 4, a ratio of only 1.33. Body-centred gives a clean 1 : 2 : 3 sequence after division.

A worked route from pattern to structure

1. Halve each 2θ to get θ for every peak.
2. Compute sin²θ for each.
3. Divide all values by the smallest. If the results are close to whole numbers, you have simple cubic or body-centred; if they start 1, 1.33, they are face-centred and should be multiplied by 3 to clear the fraction.
4. Assign (hkl) to each peak from the table.
5. Find a from any single peak using a = λ√(h²+k²+l²) / (2 sinθ), and average across peaks for a better value.

Once a is known the density follows, which is often the second half of the question:

Here Z is the number of formula units per unit cell — 1 for simple cubic, 2 for body-centred, 4 for face-centred — and M is the molar mass. Getting Z wrong is the other frequent source of a wrong final answer, and it follows directly from the lattice type you just determined.

What powder diffraction can and cannot tell you

A powder contains crystallites in every orientation, so all sets of planes contribute at once and the result is a one-dimensional trace of intensity against angle. That gives the unit cell, the lattice type and the phase identity quickly, and it is excellent for identifying an unknown solid against a database.

It does not directly give the positions of individual atoms within the cell. Peak positions come from the lattice geometry, while peak intensities come from what is inside the cell through the structure factor, and extracting atomic coordinates from intensities requires single-crystal work or full profile refinement. Distinguishing what positions tell you from what intensities tell you is a standard conceptual question.

Frequently asked questions

Why is theta measured from the plane rather than the normal?

Bragg derived the law by considering the path difference between rays reflecting off successive parallel planes, and that geometry produces 2d sinθ when the angle is taken from the plane surface. It differs from the optics convention, so the definition must be stated carefully in a derivation.

What causes systematic absences?

Atoms at centring positions scatter exactly out of phase with those at the corners for particular sets of planes, so those reflections cancel entirely. The pattern of which reflections vanish is characteristic of the lattice type and is what allows body-centred and face-centred structures to be told apart.

Why is there no peak with h² + k² + l² equal to 7?

Because no three integers squared can sum to 7. The same applies to 15, 23 and 28. These gaps are arithmetic, not physical, and should not be read as evidence of a centred lattice.

How do I get Z for the density formula?

Z follows from the lattice type determined by the absences: 1 for simple cubic, 2 for body-centred and 4 for face-centred. Since the diffraction pattern gives you the lattice type, Z is not an independent piece of information you need to be told.

Can powder diffraction locate individual atoms in the cell?

Not directly. Peak positions give the unit cell dimensions and lattice type, while atomic positions are encoded in the peak intensities through the structure factor. Recovering them requires single-crystal diffraction or full-profile refinement of the powder data.