Excitation temperature

In statistical mechanics, the excitation temperature ($Tex$) 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}{k T_{\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;
 * $ΔE$ 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".