X-rays scattered from the regular planes of atoms in a crystal interfere, giving sharp peaks at angles that reveal the spacings between those planes – the same effect that throws a rainbow off the fine tracks of a CD. For a layered material the first thing it tells you is how far apart the layers sit.
Going deeper
X-rays reflected from successive planes add up only where the extra path is a whole number of wavelengths. Flakes of a layered crystal lie flat, so a standard scan shows mainly the 00l series, which measures the interlayer spacing – and moves to lower angles when something is inserted between the layers.
What the peaks say
X-rays scattered from parallel planes of atoms interfere constructively only when the extra path between planes, 2d sin θ, equals a whole number of wavelengths: Bragg’s law, 2d sin θ = nλ. Laboratory instruments usually use copper Kα radiation with λ = 1.5406 Å.
Each part of a pattern carries information. Peak positions give lattice spacings and identify phases by comparison with reference patterns. Peak widths grow as crystallites shrink, and the Scherrer relation turns width into a size – a lower bound, because and the instrument broaden peaks too. Intensities depend on which atoms sit where and on preferred orientation. A full structure needs diffraction, which measures thousands of reflections from one crystal.
Layered crystals and intercalation
Platelets of a layered material settle flat on a sample holder or a , so in a standard scan only planes parallel to the layers are in the right orientation to diffract. The pattern is dominated by the evenly spaced 00l series, which measures the directly: for MoS2, the (002) peak at 14.4° corresponds to a spacing of 6.15 Å.
Inserting ions, molecules or water between the layers expands that spacing and shifts the series to lower angles, which is the standard check for and for swelling of and clays. The 00l series alone cannot distinguish such as 2H and 3R, whose layer spacings are nearly identical; that needs reflections from planes inclined to the layers. Strong preferred orientation also distorts intensities used to quantify phases.
Thin films and single layers
A scatters very little, and in a conventional scan the substrate dominates. Grazing-incidence diffraction sends the beam in at a shallow angle so that it travels along the film rather than into the substrate, and in-plane scans then give the in-plane and how domains are rotated relative to the substrate. For the thinnest films, synchrotron sources are often needed.
X-ray reflectivity, measured at even smaller angles, gives film thickness, density and roughness. Diffraction averages over areas of millimetres, which complements : TEM sees individual defects and in a tiny region, XRD tells whether that region is typical of the whole sample.
For specialists
Elastic scattering from lattice planes, giving phase identification, lattice parameters and – from peak width through the Scherrer relation – a lower bound on crystallite size. In layered materials the 00l series measures the interlayer spacing directly, which is how intercalation and polytype changes are confirmed. Thin films need grazing incidence to keep the beam inside the film rather than the substrate.