Scattering and diffraction described using the momentum representation

• A simple demonstration of the fundamental similarity between X-ray and neutron scattering.
• A demonstration of the basic difference between scattering/diffraction of X-rays relative to that of visible light photons.
• An explanation of diffraction based on quantized momentum transfer rather than interferenceof (matter) waves.

We present a unified analysis of the scattering and diffraction of neutrons and photons using momentum representation in a full quantum description. The scattering event is consistently seen as a transfer of momentum between the target and the probing particles. For an elastic scattering process the observed scattering pattern primarily provides information on the momentum distribution for the particles in the target that cause the scattering. Structural information then follows from the Fourier transform relation between momentum and positional state functions. This description is common to the scattering of neutrons, X-ray photons and photons of light. In the quantum description of the interaction between light and the electrons of the target the scattering of X-rays is dominated by the first order contribution from the vector potential squared. The interaction with the electron is local and there is a close analogy, evident from the explicit quantitative expressions, with the neutron scattering case where the nucleus–neutron interaction is fully local from a molecular perspective. For light scattering, on the other hand, the dominant contribution to the scattering comes from a second order term linear in the vector potential. Thus the scattering of light involves correlations between electrons at different positions giving a conceptual explanation of the qualitative difference between the scattering of high and low energy photons. However, at energies close to resonance conditions the scattering of high energy photons is also affected by the second order term which results in a so called anomalous X-ray scattering/diffraction. It is also shown that using the momentum representation the phenomenon of diffraction is a direct consequence of the fact that for a system with periodic symmetry like a crystal the momentum distribution is quantized, which follows from Bloch's theorem. The momentum transfer to a probing particle is then also quantized resulting in a discrete diffraction pattern.

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