How can a free electron and an ion in a plasma recombine without violating conservation of energy and momentum?

This question is the inverse of How does ionization of gas molecule affect the translational kinetic energy of the molecule?

The answer to that question is that a molecule cannot spontaneously ionize: it can only ionize by colliding with another particle. In this way momentum and energy can be conserved. Remember that since we are converting kinetic energy into ionization energy, so the total kinetic energy will change. Without colliding with another particle, we will end up violating conservation of momentum. Whilst it might be possible solve the simultaneous quadratic equation for kinetic and momentum in a single frame of reference, without involving a collision, it would not work for other frames of reference, for example one in which the one single particle is not initially moving.

However, what about the case when we wish an ion to recombine with a free electron? By symmetry, the same considerations hold, as for ionization. So... does that mean we need three particles to come together simultaneously? A free electron, an ion, and at least one other particle? But that seems vanishingly unlikely? So, how does that happen? Is there some misunderstandings in how I'm thinking about this?

Edit: hmm, unless there is a photon emitted perhaps?

ion + electron => neutral atom + photon


?

(Edit2: and that would also explain why plasmas glow. Like, in concrete terms, rather than hand-wavingly)

(Edit3: I've decided that emitting a photon is almost certainly the answer. I'll wait 24 hours or so, and then post that answer, if no-one has posted an alternative explanation by then).

Edit4. Some clarification points:

• all action should take place in otherwise empty space
• similarly, there are no other input particles or similar
• I'd like confirmation please that either or both:
• it's impossible for a free electron and an ion to combine without creating a photon, in the absence of any other input particles; and/or
• a free electron and an ion combining will always create a photon; in the absence of any other input particles

I'm putting "in the absence of any other input particles", since by reversability, and since we know that two neutral atoms can collide, and ionize one of those atoms, giving one neutral atom, one ion, and one free electron; so therefore by reversability, we can take one neutral atom, one free electron, and one ion, and recombine them, giving two neutral atoms.

Edit 5: actually, even if we emit a photon, don't we still violate conservation of momentum and/or conservation of energy?

Let's move in a frame of reference such that the total momentum of the electron and the ion are zero. They are moving towards each other, but total momentum zero.

When they collide, they form an atom and a photon.

The total momentum of the system should be 0. But the photon has momentum, so now the momentum of the atom should be non-zero. That is the atom should be moving.

However, the total energy before was the kinetic energy of the electron and the ion, plus the potential energy of the electron and ion being separated.

After the collision, the total energy will be the kinetic energy of the atom, plus the energy of the photon. The energy of the photon should exactly equal that of the potential energy before. So, the kinetic energy should be identical.

Does this mean then that the kinetic energy in our frame of reference needs to be non-zero, so that some of that kinetic energy can be used to move the atom in the opposite direction of the photon, and thus preserve conservation of momentum?

But presumably the kinetic energy before the collision needs to exactly equal that required to move the atom afterwards with equal and opposite momentum to the photon? This seems vanishingly unlikely?

Edit 6: so, I foudn out hte answer to the conundrum in edit 5 from ChatGPT: basically it is not required that the energy of the photon exactly equals that of the ionization energy. It can be greater or lower, so it can absorb any excess KE.

Yes, you have the suggested the answer correctly. When a free electron combines with a positive ion, a photon is emitted. Emission and Absorption are related by the Einstein A and B Coefficients. Just like an atom can absorb a phonon (mechanical collision) or a photon (electromagnetic adsorption) to reach an excited level, it can also be completely ionized by absorption of mechanical energy or by electrodynamics energy.

One example of this is the photoelectric effect where light alone ionizes the electron. Because the Einstein A and B coefficients relationships, the opposite reaction can also occur, that a free-electron combines with a metal to eject a photon.

Finally, Feynman diagrams can be used to represent an initial ejection of an electron by absorption of energy (electric or mechanical), and then its later recombination resulting in the ejection of a photon in the Quantum Electrodynamics Theory (QED) which Richard Feynman received a Nobel Prize For.

• Thanks. Could you confirm that it is impossible for an ion and a free electron to somehow recombine without photon emission? Commented Jun 6 at 10:26
• (I feel that 1. some kind of particle must be created, and 2. the only plausible particle is a photon; but would appreciate some confirmation of these points and conclusion, please) Commented Jun 6 at 10:27
• Added clarification questions/points to the question. Commented Jun 7 at 8:36
• Added update 5, discussing the frame of reference where total momentum of ion and electron is zero, before the collision. Thoughts? Commented Jun 17 at 8:31

This isn't quite what you asked, but in a semiconductor electrons and holes propagate through the solid. They can combine at a junction between differently doped regions. Sometimes this produces light, as in an LED. It is also possible to produce phonons, vibrations that travel through the lattice. This is like you example of electron + ion + 3rd particle.

So is the propagation of an electron or hole from atom to atom, as this does not create photons either. .

• Thanks. I'm meaning in otherwise empty space basically. Commented Jun 7 at 8:29
• Added clarification point to the question, to state this. Commented Jun 7 at 8:36
• Added update 5, discussing the frame of reference where total momentum of ion and electron is zero, before the collision. Thoughts? Commented Jun 17 at 8:31
• You are right, except for "vanishingly unlikely". Energy and momentum are conserved every time. It isn't coincidence. Commented Jun 17 at 13:31
• Oh, I checked iwth ChatGPT. I'll update the question with the answer. Edit: updated the question with update 6, which provides the answer to the conundrum in update 5, adn without needing 'vanishingly unlikely' things to happen. Commented Jun 17 at 17:09