The spacing of rotational energy levels is very much less than that between vibrational energy levels. At long wavelengths, electric dipole rotational transitions may be promoted by the absorption or emission of a photon provided, as mentioned earlier, that the molecule has an electric dipole moment defined as
For homonuclear diatomic molecules such as H2, and symmetric linear polyatomic molecules such as CO2, the electric dipole moment M is zero and thus pure rotational transitions are forbidden. However, heteronuclear diatomic molecules and all polyatomic molecules other than symmetric linear ones do have an electric dipole moment and thus may absorb photons subject to the selection rule AJ = l and may emit a photon provided A J = —l (Hanel et al. 2003). Hence, a molecule in the J th rotational energy level may only absorb a photon with an energy needed to promote it to the (J + l)th energy level. This required energy, known as the transition energy, is given by the difference between the (J + l)th and J th rotational energy levels: that is,
AE = Ej+1 — Ej = 27 [(J + l)(J + 2)—J (J + l)]= y (J + l). (6.22)
Thus, since the transition energy is proportional to the total angular momentum J of the lower state, the frequencies of the rotational absorption lines are to a first approximation all equally spaced. In reality, effects such as centrifugal distortion lead to rotational energy levels becoming more closely spaced at higher J.
The strength of a rotational absorption line, or the line strength, depends both on the transition probability derived earlier from quantum mechanics and also on the number of molecules which are in the J th rotational energy level at any particular time. For rotation bands it is found that the dominant factor affecting line strengths is the population of the lower rotational state. Assuming thermodynamic equilibrium, the population of states varies with energy according to the Boltzmann distribution
where kB is the Boltzmann constant; EJ is the energy of the J th energy level; and qj is the degeneracy of the J th level (i.e., the number if individual states with the same energy EJ). Hence, for linear rotors, substituting for the energy EJ from Equation
(6.16) and setting the degeneracy gJ equal to 2 J + 1, the number of molecules actually in the K th rotational energy state is given by:
which has a distribution as shown in Figure 6.1. The measured line strengths of the rotation band of the heteronuclear linear rotor CO are shown in Figure 6.2 and it can be seen that the variation in strength closely resembles the population curve of Figure 6.1 and that the lines are equally spaced as expected.
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