Excitation temperature

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

$\frac{n_{\mathrm{u}}}{n_{\mathrm{l}}}=\frac{g_{\mathrm{u}}}{g_{\mathrm{l}}}\exp (-\frac{\mathrm{\Delta }E}{k{T}_{\mathrm{e}\mathrm{x}}}),$

where

• Template:Math is the number of particles in an upper (e.g. excited) state;
• Template:Math is the statistical weight of those upper-state particles;
• Template:Math is the number of particles in a lower (e.g. ground) state;
• Template:Math is the statistical weight of those lower-state particles;
• Template:Math is the exponential function;
• Template:Mvar is the Boltzmann constant;
• Template:Math 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".

References

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