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I have read this:

https://en.wikipedia.org/wiki/Free-electron_laser

This states that electrons are free.

Now there is a debate on this site over whether a truly free electron in vacuum can or cannot emit a photon.

Free (unbound, so not bound to a nucleus) accelerated electron cannot emit/absorb a real photon?

Can an accelerated "free" electron absorb a photon?

Based on the answers, I believe that a free electron in a vacuum cannot emit a photon. There needs to be a third party in the equation, to whom the electron relatively accelerates.

Electrons can only emit a photon when they are :

  1. either bound to an atomic system

  2. or are accelerating in a metal conductor, but these electrons are not free, they are loosely bound to the atoms

Now based on the wiki article, this is not true. Free electron lasers emit photons, and the electrons are here said to be free in a vacuum. So they are not bound to any atomic system.

The electron beam must be maintained in a vacuum, which requires the use of numerous vacuum pumps along the beam path.

Question:

  1. Now how do these truly free electrons in a vacuum emit the photons of the laser?

  2. are these electrons in the free electron laser the same as accelerating electrons in a metal conductor? So are they loosely bound to the atoms? But there are no atoms in the vacuum so how are they emitting photons?

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  • $\begingroup$ Related: Do free-electron lasers actually lase? $\endgroup$
    – Emilio Pisanty
    Commented Apr 14, 2019 at 17:34
  • $\begingroup$ The incoming photon is free. The only condition to get EM radiation inside the FEL is the kinetic energy of the electrons. In the FEL it is not free because get deflected by the interaction of the magnetic dipole moment fo the electron and the magnetic field of the FEL. This time quanta by quanta the kinetic energy gets converted into EM radiation. $\endgroup$
    – HolgerFiedler
    Commented Dec 15, 2020 at 4:45

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You're mixing-and-matching different uses of the word "free" in ways which deeply undermine your question. Since your question does not contain a unique reference definition of that term, it is ultimately unanswerable.

Nevertheless, there's plenty to be said towards it.

  • The electrons are free in the sense that there is nothing else inside the vacuum chamber other than the electrons.

  • However, the electrons are not 'free' in the sense of the results you're finding contradictions with, which would require those 'free' electrons to move in perpetual uniform rectilinear motion in the absence of collisions. The electrons in an FEL are not in rectilinear motion - they are being driven by the magnets in the undulator, so they are accelerating, and in the process they emit synchrotron radiation.

    There is nothing specific to FELs as regards your query here: you get exactly the same apparent contradictions if you consider the emission of radiation in any synchrotron. You have magnets making the electrons accelerate, and in the process they emit radiation.

    The results you've quoted are based on the fact that you cannot add a null four-vector to a timelike four-momentum and get another timelike four-momentum with the same mass. (This is pretty technical, but it is extremely important. If you do not fully and deeply understand that reasoning, then everything else here is just words in vacuum.) Electrons in vacuum do not need to be "bound" in order to absorb or emit photons - they just need to have some other system, with a timelike four-momentum, with which to exchange momentum and energy. For electrons in a synchrotron or an FEL, that slot is filled by the magnets in the synchrotron or the undulator, respectively.

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  • $\begingroup$ thank you so much! so the electrons do not need to be bound to an atomic system. they just need to interact with the magnetic field via virtual photons right? that is where they gain the kinetic energy from to accelerate, thus making them able to emit photons. So when they emit the photons, they lose kinetic energy, but they gain kinetic energy from the magnetic field's virtual photons. the magnetic field has a timelike four momentum, and so the electrons can exchange momentum and energy with it when emitting a photon? $\endgroup$
    – Árpád Szendrei
    Commented Apr 14, 2019 at 17:47
  • $\begingroup$ "the magnetic field has a timelike four momentum" - no. The magnetic field mediates the transfer of momentum between the electrons and the magnet, which has a timelike four-momentum. The virtual-photon picture has its uses, but it's prone to leading down all sorts of wrong alleys (many of which you seem to be taking). $\endgroup$
    – Emilio Pisanty
    Commented Apr 14, 2019 at 19:13

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