Suppose an electron hits a metal target after being accelerated in an X-ray tube. The production of the continuous X-ray is explained by stating that the electron decelerates continuously. In that process, it loses kinetic energy which is converted into X-ray photon. My question is, in the short wavelength limit, the energy of the emitted X-ray is equal to the initial kinetic energy of the electron. This means all the kinetic energy of the electron has been converted into X-ray energy. Now, if the electron has no kinetic energy, does it stand still? Will this electron not be attracted by the nucleus?
At the limit, the electron makes a transition from the continuum to the lowest unoccupied state in the metal. These are the states at the Fermi level. The process is called inverse photoemission. This can be used in spectroscopy to measure the unoccupied density of states (the DOS above the Fermi level).
The process is completely analogous to photoemission from occupied states to the continuum, where an electron absorbs a photon. Just the other way around. For exact measurements, one needs to take into account the work function of the metal.
For very high quantum numbers, there is Bohr's correspondence principle connecting quantum behaviour (Schrödinger equation) to classical electrodynamics.