scattering vector

Throw a ball at a wall and it comes back heading a different way. The change in its direction — how much its motion got redirected, and toward where — is the heart of what happened in the bounce. The scattering vector is that idea for a wave: it is the difference between the direction the wave was going before it scattered and the direction it goes afterward.

Precisely, the scattering vector points from the incoming wave's direction to the outgoing wave's direction, and its length grows with the scattering angle — small for a gentle deflection, large for a sharp turn-around. Crucially it lives in reciprocal space, measured in units of one-over-length, which is exactly the language of the reciprocal lattice. Diffraction's central rule, the Laue condition, then reads simply: a bright spot appears whenever the scattering vector equals a reciprocal lattice vector.

This matters because the scattering vector is the single quantity that ties together every diffraction idea: it is what the Ewald sphere is built around, what sets the angle in Bragg's law, and what the structure factor and form factor are functions of. Physicists usually call it Q. An honest note: because a moving wave carries momentum proportional to one over its wavelength, the scattering vector is also the momentum the wave hands to the crystal, which is why it is often called the momentum transfer.