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

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

This article needs additional citations for verification. (November 2006)

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".^[1]