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Negative absolute temperatures

Suppose that by some artificial mean it is possible to put more electrons in the higher energy state than in the lower energy state of a two level system (this sort of situation occurs in a laser, for example). Now it is clear that this cannot be an equilibrium situation, but nevertheless, for the time that the system is in the strange state we could, if we wished, still express the ratio of the populations in the upper and lower states by some parameter we can think of as an effective temperature.

(i) Show that for such a population inversion to exist, the effective temperature must be negative.
(ii) Imagine I have electrons that populate two states in the normal manner at room temperature. I then somehow swap the populations (i.e. all the ones that were in the lower state go into the upper state, and vice versa). What is this new effective temperature?
(iii) What is the effective temperature if I put all the electrons in the upper energy state?


Solution Preview

The probability that some state r with energy E_r is occupied is:

P_r = Exp(- beta E_r)/Z (1)

Suppose that an electron in an atom can be in two states (r = 1, and r = 2 and E_2 > E_1). If there are N atoms, then the total number of atoms with an electron in state r is nr = N P_r. The ratio of n2 and n1 is thus:

n2/n1 = N P_2/(NP_1) = P_2/P_1 = (using (1) ) = [Exp(- beta E_2)/Z] / [ Exp(- beta E_1)/Z ] =

Exp[-(E_2 - E_1)/(k T) ] (2)

We can see from this equation that n2/n1 < 1 if E_2 > E_1. Here we assume ...

Solution Summary

The following posting helps with a problem that involves negative absolute temperatures.