Excitation temperature

The Excitation Temperature ( $T_{ex}$ ) (often called the spin temperature for the 21 cm line) is defined for a population of particles via the Boltzmann factor. It satisfies

$\frac{n_u}{n_l} = \frac{g_u}{g_l} \exp{(-\frac{\Delta E}{k T_{ex}})},$

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).