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Suppose we have a plasma with corresponding plasma frequency $\omega_{\text{pl}}$. Next, assume that there is some scattering inside the plasma, due to which photons can be created. Is it possible to create a photon with the energy $E < \omega_{\text{pl}}$? Or such process is impossible?

Name YYY
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The short answer is yes. I wrote some more general responses at https://physics.stackexchange.com/a/138460/59023 and https://physics.stackexchange.com/a/264526/59023.

The longer answer is yes as well, but then one is concerned with whether the photons to which you refer exist as a freely propagating electromagnetic wave (i.e., the oscillating fields effectively do not interact with the charged particles) or as coupled plasma mode (i.e., the oscillating fields affect the charged particles).

A plasma is just an ionized gas and can be weakly ionized or strongly ionized. An example of the former is the Earth's ionosphere (at certain altitudes and times of day).

In the latter case (e.g., the solar wind), the charged particles (i.e., negatively charged electrons and positively charged ions) are allowed to oscillate. If we only examine things larger than the Debye length, then for most purposes the plasma is effectively charge neutral. However, this does not mean that electric and/or magnetic fields due to oscillating charges cannot exist. It just means that in a quasi-steady state, the gas is neutral over distances larger than the Debye length

There are a whole zoo of plasma waves that exist in both weakly and strongly ionized plasma and even more instabilities – the mechanism through which a plasma dissipates free energy by radiating an electromagnetic or electrostatic wave.

If you are asking about quark-gluon plasmas, then I am guessing somewhat similar rules apply but that is beyond my expertise.

honeste_vivere
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  • I don't think this right. The dispersion relation is $\omega_T^2=\omega_{pl}^2+c^2k^2$ ($c$ is weakly $k$ dependent). This means precisely what one would expect: There are no propagating modes with $\omega<\omega_{pl}$. – Thomas May 29 '17 at 01:30
  • @Thomas - That dispersion relation only applies to free modes, i.e., electromagnetic waves that are not coupled to the plasma which is why $c$ is weakly dependent upon $k$. There are hundreds if not thousands of papers on propagating modes below the electron plasma frequency and nearly all of my papers have focused on modes satisfying $\omega < \omega_{pl}$, all of which have finite phase and group velocities. – honeste_vivere May 29 '17 at 17:25
  • I'm not sure what you mean by "free". Obviously, if the charges are free there would no be a collective plasma oscillation. I agree that there are MHD modes below $\omega_{Pl}$, but it is not clear to me that these will couple to photons (in particular if the Plasma is not magnetized). – Thomas May 30 '17 at 03:17
  • Consider, in particular, the following question. Take a container filled with a neutral, non-magnetized plasma. The container has a transparent window. We shine light on the window with $\omega<\omega_{Pl}$. Do we get complete reflection, or will the photons penetrate? – Thomas May 30 '17 at 03:20
  • @Thomas - I am confused. Aren't the electromagnetic (EM) fields of even MHD modes made up of photons? In regards to your question/example, the lack of transparency does not mean that EM oscillations do not exist (which require photons), it's just that they are damped and/or evanescent. By free I mean an EM oscillation with $\omega > \omega_{pl}$, namely one that "sees" a plasma as transparent and its index of refraction has little dependence upon the plasma. – honeste_vivere May 30 '17 at 11:18
  • Evanescent = non-propagating. That's exactly the point. – Thomas May 31 '17 at 01:35
  • @Thomas - Not all modes are evanescent. – honeste_vivere May 31 '17 at 13:30
  • Below $\omega_{pl}$ there is ordinary sound (which does not couple to photons), and trans/long MHD waves which may couple to photons (not sure exactly how), but their velocity goes to zero in a non-magnetized plasma. – Thomas Jun 01 '17 at 03:15
  • @Thomas - I assure you, there are a whole zoo of instabilities that radiate another zoo of waves/modes, most below the local plasma frequency. For instance, whistler mode waves have a cutoff at the local electron cyclotron frequency, which is often below the plasma frequency except for regions of high magnetic field (e.g., near Earth or Jupiter). Ion acoustic waves are an electrostatic mode that exist well below the electron plasma frequency and are routinely observed in the solar wind. etc. – honeste_vivere Jun 02 '17 at 20:25
  • As I said, I'm happy to accept that there are MHD modes (Alfven, whistlers, etc), but their velocity goes to zero in a non-magnetized plasma. This means that in a non-magnetic plasma photons below $\omega_{\it pl}$ don't propagate. – Thomas Jun 04 '17 at 04:31
  • @Thomas - I can name a few instabilities that still propagate in the absence of a magnetic field (by the way, whistlers are not an MHD mode): two stream instabilities (multiple types exist), Buneman instability, etc. Ion acoustic waves are electrostatic and exist in non-magnetized plasmas as an electrostatic wave that propagates at the ion sound speed. – honeste_vivere Jun 04 '17 at 16:50
  • The original question was what happens to a photon with $\omega<\omega_{\it pl}$. You seem to argue that it excites a plasma instability which re-radiates the photon (as what? even lower energy?). I'm not sure this is plausible. – Thomas Jun 05 '17 at 16:45
  • @Thomas - Oh no, sorry for the confusion. No, I was trying to say that electromagnetic oscillations/waves/modes exist and propagate below the local plasma frequency. – honeste_vivere Jun 05 '17 at 18:31