title: Excitation temperature
text: # 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;
  * 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".
  
  