the scattering vector

When a wave scatters off a crystal, it comes in heading one way and leaves heading another. The scattering vector captures that change of direction in a single arrow. Draw the incoming wave's direction as a vector k and the outgoing wave's as k' (both of length 1/lambda, since scattering here is elastic and does not change the wavelength). The scattering vector is their difference, usually written Q or s: Q = k' - k. It points along the bisector of the incoming and outgoing beams and measures how hard the wave was deflected.

Its length is fixed by the scattering angle alone. A little triangle geometry gives |Q| = 2 sin theta / lambda in the crystallographers' convention (or 4 pi sin theta / lambda if you carry the 2 pi factor, as physicists do). Small deflection means a short scattering vector; scatter the beam through a wide angle and Q grows. Notice this is the same quantity that appears in the length of a reciprocal lattice vector, |g_hkl| = 1/d — and that is no coincidence.

The scattering vector is the bridge between the experiment and the reciprocal lattice. The Laue condition says diffraction happens precisely when the scattering vector equals a reciprocal lattice vector, Q = g_hkl. So measuring the direction and angle of a diffracted beam is really measuring a reciprocal lattice vector; the detector reads the crystal's reciprocal lattice one Q at a time. This is why data are so often plotted against Q rather than angle — Q is the natural, wavelength-independent coordinate of diffraction.