Excitation temperature

In statistical mechanics, the excitation temperature ( $T ex$ ) is defined for a population of particles via the Boltzmann factor. It satisfies

{\frac {n_{\rm {u}}}{n_{\rm {l}}}}={\frac {g_{\rm {u}}}{g_{\rm {l}}}}\exp {\left(-{\frac {\Delta E}{kT_{\rm {ex}}}}\right)},

where

$n u$ is the number of particles in an upper (e.g. excited) state;
$g u$ is the statistical weight of those upper-state particles;
$n l$ is the number of particles in a lower (e.g. ground) state;
$g l$ is the statistical weight of those lower-state particles;
$exp$ is the exponential function;
$k$ is the Boltzmann constant;
$Δ E$ is the difference in energy between the upper and lower states.

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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[1] 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.