Excitation temperature

The Excitation Temperature ($T_{\rm ex}$) is defined for a population of particles via the Boltzmann factor. It satisfies

$\frac{n_{\rm u}}{n_{\rm l}} = \frac{g_{\rm u}}{g_{\rm l}} \exp{(-\frac{\Delta E}{k T_{\rm ex}})},$

where n_u and n_l represent the number of particles in an upper (e.g. excited) and lower (e.g. ground) state, and g_u and g_l 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 thermodynamical 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, the apparent value of the excitation temperature is often called the "spin temperature".^[1]

References

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. doi:10.1086/308953.