reciprocal lattice

Picture the rows of trees in an orchard, all spaced evenly apart. Your eye naturally notices the spacings: how far apart the rows are, and which way they run. The reciprocal lattice is a way of writing down exactly those spacings and directions as a tidy grid of its own. It is a second lattice, living not in ordinary space but in 'wave-space', that carries all the information about how regularly the real crystal repeats.

Here is the precise idea. A crystal repeats its pattern after fixed steps in real space. For each direction of repetition, the reciprocal lattice has a matching point, placed at a distance that is large when the real spacing is small, and small when the real spacing is large — that is what 'reciprocal' means. Mathematically you build it from the crystal's repeat vectors so that each reciprocal point corresponds to a set of evenly spaced planes of atoms. The closer those planes sit, the farther out the reciprocal point lands.

This matters because waves — X-rays, neutrons, electrons — do not 'see' single atoms one by one; they respond to the whole repeating pattern at once, and the reciprocal lattice is the natural map of that response. Every bright spot a crystal scatters lands exactly on a reciprocal lattice point. A common confusion: the reciprocal lattice is not a physical object you could touch, and its points are not atoms — they are bookkeeping markers for the crystal's periodicity, measured in units of one-over-length.