The Scattering Vector

The wave vector of a wave ‘k’ is a vector quantity that carries information about the momentum and direction of a wave. The magnitude of the wave vector is related to the wavelength ‘λ’ by:

k=2π/λ.

In quantum mechanics, we can use the de Broglie wavelength to relate the magnitude of a wave’s momentum ‘p’ and its wavelength by:

p=h/λ.

where 'h' is Planck's constant h = 6.63 x 10⁻³⁴ m²kg/sh equal to 6.63 times 10 to -34 meters squared times kilograms divided by seconds. From here we can construct a vector equation that relates wave vector and momentum:

p = h/2π ⋅ k.

During a scattering process, the wave vector of an incident wave changes. Even in elastic scattering, the direction of the momentum changes so there is always a change in wavevector. This change in wavevector is known as the scattering vector ‘q’ and is given by the difference between the initial wave vector ‘k_i’ and the final scattered wave vector ‘k_f’:

q=k_i-k_f.

In the same way that Bragg’s Law relates scattering observations to the scattering material’s structure, in depth analysis into the scattering vector can infer structural information such as the distribution of charges and the atomic configuration in 3D space.