The Reciprocal Lattice

a reciprocal lattice row

A reciprocal lattice row is simply a straight line of evenly spaced reciprocal lattice points — the reciprocal-space equivalent of a row of fence posts. Because the reciprocal lattice is a lattice, its points line up in perfectly regular rows in every direction, and these rows are the natural units you scan along when reading a diffraction pattern. In an electron diffraction spot pattern the rows are the obvious lines of dots marching across the screen.

Each row has a direction and a spacing, and both mean something in the real crystal. A row of reciprocal points is parallel to a direction in reciprocal space, and that direction is the plane normal shared by the family of reflections along the row — for example, the row (h00), meaning (100), (200), (300), and so on, marches outward along a* with a constant spacing of a* = 1/a between consecutive points. The spacing between points along a row is the reciprocal of a real-space repeat distance, so a tightly spaced row corresponds to a long real-space period and a widely spaced row to a short one. A whole plane of the reciprocal lattice, such as the layer you see in a single electron-diffraction pattern, is just a stack of parallel rows.

Rows matter in practice because diffraction effects often act along a whole row at once. Systematic absences from a screw axis or glide plane switch off ALTERNATE points along particular rows (for instance a 2-fold screw axis along b extinguishes the (0k0) reflections with k odd), so a gap-toothed row in the pattern is an immediate clue to the symmetry. Rows are also how you index a pattern: pick two non-parallel rows, measure their spacings and the angle between them, and you have effectively measured two reciprocal basis vectors and can name every spot. Thinking in rows turns a daunting field of dots into a small number of regular tracks to read along.

In an electron diffraction pattern from a cubic crystal down [001], the horizontal line of spots is the (h00) row, spaced 1/a = 0.25 angstrom^-1 apart for a = 4 angstrom; the vertical line is the (0k0) row. If a screw axis is present, every other spot along one such row is missing — a gap-toothed row that fingerprints the symmetry.

A row is one straight track of reciprocal points; its spacing reads a real-space period and its gaps reveal screw/glide symmetry.

A missing point in a row is usually a systematic absence (the structure factor is zero), not a flaw in the crystal or the film. Reading which points along a row are extinguished is exactly how you diagnose screw axes and glide planes.

Also called
systematic rowrow of reflections倒晶格排