What is the quantum mechanical explanation of X-ray diffraction? One topic in crystallography that I've found a surprising dearth of information on is what the fundamental interaction behind the interaction of the X-ray and atom. Pretty much every book just treats them as classical waves instead of explaining the quantum mechanics behind it. By this I don't mean that interference that leads to diffraction, but what actually happens when the X-ray interacts with the electron cloud. 

The only description I've found so far is in Glusker, Lewis and Rossi that talks about:


  "When X rays hit an atom, the rapidly oscillating electric field of the radiation sets the electrons of the atom into oscillation about their nuclei. This oscillation has the same frequency as that of the incident radiation. The result is that the electron acts as an oscillating dipole that serves as a source of secondary radiation with the frequency of the incident beam."


and goes on to note that this is coherent scattering and there is no wavelength shift.

Which doesn't make a ton of sense, isn't oscillating about the nuclei an old pre-quantum explanation of orbitals? Is this saying that this is similar to fluorescence/phosphorescence where the photo is absorbed, then another emitted? That is what most of the other grad students I've talked to have assumed, but if that was the case I'd expect a Stokes shift, the direction of emitted radiation to be random, and I don't see why you'd get a $180^\circ$ phase shift. 

The other explanation I've heard is that you are actually sloshing electrons around within their orbitals, with the same frequency as the X-ray, similar to when you are dealing with optical properties, but I'm not sure what the mechanism for the interaction would be in this case.
The description given in your question pretty much explains things. I would put it in slightly different words which may or may not help, which is that the x-ray photon induces a dipole moment in the atom (a field induced dipole) which then radiates and so the x-ray is scattered. This is pretty much what happens in scattering  by visible photons and we must assume that the photon is distant in energy from any transitions. (In Compton scattering part of the x-ray energy is imparted to the electron, so the frequency of the scattered x-ray is changed).

The field induced dipole moment is proportional to the polarisability which for a single bound electron is 
$$ \alpha = \frac{e^2}{m}\left ( \frac{1}{\omega_0^2-\omega^2} \right )$$

where $\omega_0$ is the resonance frequency of the bound electron and $\omega$ the x-ray frequency and so $\omega &gt;&gt; \omega_0$ which in the limit gives 
$$\alpha \rightarrow -\frac{e^2}{m\omega^2} $$ 
Thus the calculation reduces to understanding the QM calculation for the polarisability. See for example Atkins &amp; Friedmann 'Molecular Quantum Mechanics' Ch $12$.

(  By considering that the x-ray is a plane wave the intensity of scattered radiation relative to that incident is given by 
$$ \frac{I}{I_0}= \frac{e^4}{m^2c^4R^2}P(\theta)$$

where R is the distance from the scattering centre to the detector, m the electron mass and $\theta$ the scattering angle and $P(\theta)=(1+\cos^2(\theta))/2$ is called the polarisation factor.  This formula is sometimes named after Thomson. The $m^{-2}$ shows why significant scattering occurs only from electrons and not from protons. The book by Flygare 'Molecular Structure and Dynamics' section 8.5 gives a detailed description of x-ray scattering in general and details such as calculating atomic scattering factors. )