The excitation temperature ( \( {\displaystyle T_{\rm {ex}}} \)) is defined for a population of particles via the Boltzmann factor. It satisfies
\( {\displaystyle {\frac {n_{\rm {u}}}{n_{\rm {l}}}}={\frac {g_{\rm {u}}}{g_{\rm {l}}}}\exp {(-{\frac {\Delta E}{kT_{\rm {ex}}}})},} \)
where nu and nl represent the number of particles in an upper (e.g. excited) and lower (e.g. ground) state, and gu and gl their statistical weights respectively.
Thus the excitation temperature is the temperature at which we would expect to find a system with this ratio of level populations. However it has no actual physical meaning except when in local thermodynamic equilibrium. The excitation temperature can even be negative for a system with inverted levels (such as a maser).
In observations of the 21 cm line of hydrogen, the apparent value of the excitation temperature is often called the "spin temperature".[1]
References
Dickey, J. M.; Mebold, U.; Stanimirovic, S.; Staveley‐Smith, L. (2000). "Cold Atomic Gas in the Small Magellanic Cloud". The Astrophysical Journal. 536 (2): 756. Bibcode:2000ApJ...536..756D. doi:10.1086/308953.
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