Question

What's the “effective potential” for photons in $X$-ray diffraction?

Answer 1

The slickest way to introduce $X$-ray diffraction is to invoke scattering theory in quantum mechanics. One treats the incoming photon as just another particle in a scattering problem; by Fermi's golden rule and the Born approximation, the scattering rate from $\mathbf{k}$ to $\mathbf{k}'$ is $$\Gamma(\mathbf{k}', \mathbf{k}) \propto |\langle \mathbf{k}'| V | \mathbf{k} \rangle|^2$$ where $V$ is the potential experienced by the photon. Since $V$ has the periodicity of the lattice, it follows that the scattering rate vanishes unless $\mathbf{k} - \mathbf{k}'$ is a reciprocal lattice vector.

However, most sources do not say much about this "photon potential" or justify its form. For example, David Tong's lecture notes simply avoid the issue:

Firing a beam of particles — whether neutrons, electrons or photons in the X-ray spectrum — at the solid reveals a characteristic diffraction pattern. [...] Our starting point is the standard asymptotic expression describing a wave scattering off a central potential.

Tong declines to comment on the potential at all. Steve Simon's solids textbook says a bit more:

If we think of the incoming wave as being a particle, then we should think of the sample as being some potential $V(r)$ that the particle experiences. [...] X-rays scatter from the electrons in a system. As a result, the scattering potential is proportional to the electron density.

That is, the potential is the electron density, and nuclei don't contribute because they are heavier.

This sounds plausible, but I have no idea how to derive this from first principles. From the QFT side, I imagine such a calculation would start from the QED interaction $$\mathcal{L}_{\text{int}} = \bar{\psi} \gamma^\mu A_\mu \psi.$$ Treating the field $\psi$ as a classical, static background field we have $$\mathcal{L}_{\text{int}} = j^\mu A_\mu = \rho A_0$$ since the components $j^i$ are zero. But I'm not sure how this is supposed to be a "potential for the photon"; it seems to only affect one component of $A_\mu$. And it's unclear where the mass of the fermion is going to come in, to make the electrons count more than the nuclei.

On the other end, I suppose one could start from classical electromagnetism. In that case we're talking about Thomson scattering, and heavier particles indeed contribute less. The challenge is then exporting quantum mechanical ideas, such as partial waves and the Born approximation, to this classical context. Maybe this is manageable, but I've never seen this done anywhere either.