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I wonder if this answer to What would happen if we supercool and then superheat an atom very abruptly? goes far enough.

I almost wrote the comment:

I don't think we can even talk about the temperature of a single atom, at least based on its kinetic energy. Now if there were several electrons in one atom simultaneously in higher orbits at the same time, then it might be possible.

but then decided to cast this in the form of a question.

One way to think of the temperature of a group or ensemble of atoms can be the average of a particular energy. There can be more than one temperature; e.g. kinetic energy, the distribution of excited states and nuclear spins (What is the ortho/para issue with LH2 as a fuel?) If they are different we can say that they are out of equilibrium, but we can still talk about each as a different temperature in cases where coupling between different partitions is slow compared to the timescale of interest.

In a strongly interacting quantum system like an atom, there can be some processes that can excited several electrons to higher energy states at once.

Question: Has a concept of temperature ever been (usefully) defined in the context of a single atom?

This would likely be an extreme situation and may not happen in conventional reactions, maybe an intense and ultra-short laser pulse or nuclear decay or reaction, or possibly in the case of some explosive reaction.

update: comments mention a related question in Physics SE Can a single molecule have a temperature? with several answers, though they don't directly answer this question because I'm trying to find out if this "has ever been defined in the context of a single atom", and not asking if you think it should/shouldn't or can/can't be. It is possible that at some point in some, (and possibly extreme) experimental contexts this may have been workable.

The answer there that I do find most helpful is @JohnRennie's. Still though, I'm asking primarily about excitations within a single atom rather than the rotational/vibrational states of a large molecule.

uhoh
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    This is already discussed in detail https://physics.stackexchange.com/questions/65690/can-a-single-molecule-have-a-temperature – AChem Dec 16 '19 at 05:59
  • @M.Farooq I've link back there and addressed those answers. Thanks! – uhoh Dec 16 '19 at 06:12
  • It wouldn't make sense. So it is hardly that someone used that idea. – Alchimista Dec 16 '19 at 07:46
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    To extend your comment on @JohnRennie's answer, in spectroscopy the term vibrationally ‘hot’ molecule is used sometimes. And, of course, there are hot atoms too. – andselisk Dec 16 '19 at 08:53
  • @Alchimista perhaps that's what they said about Onsager in 1949 as well? ;-) – uhoh Dec 16 '19 at 09:02
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    No. That is about system of N particles. A bound N particle system might be a single atom but then a concept of T would refer to its electrons. This "reduce" to the electronic energy of the atom and its quantistic treatment. Hot molecules and atoms pointed out by @andselisk still refer to members of a Boltzmann distribution. – Alchimista Dec 16 '19 at 10:48
  • @Alchimista It was meant as humor but perhaps it was too oblique, that the concept of "negative temperature" might have also elicited a similar "doesn't make sense" first reaction, but in hindsight can have meaning in a certain context. – uhoh Dec 16 '19 at 12:56
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    The important aspect of temperature is equilibrium. In system not in equilibrium, different kind of energies may be distributed according to different temperatures, e.g. by laser excitation. But if the system is in equilibrium, these formal "temperatures" are equal, or rather they fluctuate around the same value. For a single atom, one can speak about an equivalent temperature only, as the same atomic state may happen in huge temperature range. – Poutnik Dec 17 '19 at 06:59
  • @Poutnik thank you for your comment, it's quite helpful! If each electron in a high-Z atom had its own set of excited states unrelated to the other electrons, then an excitation temperature could be defined in a similar way to how it would be done for an ensemble of protons each with one electrons. But I'm thinking that once several electrons are excited, it may not be so easy because it may be impossible to say "electron #1 is 0.8 eV above its ground state, electron #2 is 1.2 eV above its ground state..." etc. – uhoh Dec 17 '19 at 07:14
  • Thus I've asked if this has ever been done because I wonder if it's possible to try at all, no matter how useful it may or may not be. – uhoh Dec 17 '19 at 07:14
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    P.S.:Additionally,, such 1 atom ( or molecule ) equivalent temperature would have different values for different kinds of atom k molecule ) energies. – Poutnik Dec 17 '19 at 07:33
  • Aside of equilibrium and the mean energy value aspects, another important aspect is the energy distribution. 1 entity has just 1 point value like the Delta function. – Poutnik Dec 17 '19 at 08:11
  • @Poutnik so in this case is would each entity be an electron, or an occupied orbit within the atom? – uhoh Dec 17 '19 at 10:15
  • The kinetic energy of the atom and the electron excitation energy, that would be compared to the mean energy of the particular statistical distribution with the equivalent temperature. – Poutnik Dec 17 '19 at 10:21
  • Let us continue this discussion in chat. – uhoh Dec 17 '19 at 10:27

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