How to read an XRD pattern — from Bragg's law to phase ID
An X-ray diffractogram looks like a forest of spikes. Each one is a crystal plane satisfying a simple geometric condition, and once you see that, the pattern reads as a fingerprint.
X-ray diffraction (XRD) identifies crystalline phases by the angles at which a sample scatters X-rays. Constructive interference occurs only when Bragg's law is satisfied: nλ = 2d·sinθ, where d is the spacing between atomic planes and θ is the incidence angle. The result is a pattern of intensity versus 2θ with sharp peaks at angles set by the lattice; the peak positions index the (hkl) planes and act as a fingerprint for phase identification, while peak width relates to crystallite size and strain.
Why peaks appear where they do
A crystal is stacks of parallel atomic planes separated by a distance d. X-rays reflecting off successive planes travel different path lengths; when that extra path equals a whole number of wavelengths, the reflections add up and you get a peak. That condition is Bragg's law, nλ = 2d·sinθ. Small d-spacings (tightly packed planes) satisfy it at large angles; large d-spacings at small angles. So the peak positions on the 2θ axis are a direct readout of the lattice geometry.
How to read the pattern
1 · Peak positions → which phase
The set of 2θ angles (and the d-spacings they imply) is unique to a crystal structure. Matching them against reference patterns identifies the phase — this is the core of phase identification. A new peak appearing, or an old one splitting, means a new phase or a distortion.
2 · Peak intensities → orientation and occupancy
Relative peak heights encode which planes scatter most strongly — affected by atomic positions and by preferred orientation (texture). Intensities that don't match the reference often mean the crystallites are aligned, not that the phase is wrong.
3 · Peak width → crystallite size and strain
Sharp peaks mean large, well-ordered crystallites; broad peaks mean small crystallites or microstrain (Scherrer). Broadening that grows with angle points to strain; uniform broadening points to size.
4 · Full-pattern fitting → quantitative phases
Fitting the entire pattern (Rietveld refinement) against calculated structures yields quantitative phase fractions and refined lattice parameters — the difference curve between observed and calculated should flatten as the fit converges.
Common ways an XRD pattern misleads you
- Sample displacement shifts every peak in 2θ and will throw off d-spacings — a flat, well-aligned sample matters before you trust positions.
- Preferred orientation distorts intensities; don't reject a phase match because the heights look off.
- Amorphous content is invisible to peaks — it shows as a broad hump, not lines, and is easy to miss when eyeballing.
- Peak overlap in multi-phase samples hides minor phases under a neighbor; full-pattern fitting separates them.
Where this gets slow by hand
Indexing one clean pattern is routine. A formulation or aging series is dozens of patterns, each needing background subtraction, peak finding, phase matching, and size/strain analysis — and the meaningful story is usually a subtle shift or a minor new phase appearing across the series, which is exactly what manual, one-at-a-time analysis tends to miss.
From raw diffractograms to identified, quantified phases
Niobia background-subtracts, finds peaks, and matches them to candidate phases across an entire series of patterns — then refines the fit for quantitative phase fractions, lattice parameters, and crystallite size, and tracks how those evolve from sample to sample. The subtle peak shift or the minor phase that appears at cycle 200 surfaces as a trend, not a needle you have to find by hand. It links composition to morphology and performance alongside Raman and electrochemical data.
Frequently asked
What is Bragg's law?
Bragg's law, nλ = 2d·sinθ, gives the condition for constructive interference of X-rays scattered by a crystal: the path difference between rays reflecting off adjacent atomic planes (spacing d) at angle θ must equal a whole number of wavelengths. It's why XRD peaks appear only at specific angles.
What does an XRD pattern tell you?
The positions of the peaks (in 2θ) identify the crystalline phases present, because each crystal structure scatters at a unique set of angles. Peak intensities relate to atomic arrangement and orientation, and peak width relates to crystallite size and strain.
How do you identify a phase from XRD?
You measure the angles of the diffraction peaks, convert them to d-spacings via Bragg's law, and match that set of d-spacings and relative intensities against reference patterns. A matching set of peak positions identifies the phase.