In gas that is being heated by nearby young stars, the upper vibrational levels of CO become significantly populated. Equation (5.8) still gives the level energies to fair accuracy, but understanding the observed complex spectra requires that we consider the next higher correction. Specifically, the picture of CO oscillating in a parabolic potential well must break down at large amplitude, where the molecule is eventually torn apart. Thus, Evib does not increase precisely as v + 1/2, but contains a relatively small negative term proportional to (v + 1/2)2. As a consequence, the frequencies of photons emitted by the transitions with v =1 ^ 0, 2 ^ 1, etc., decrease slowly with the starting v-value. Such fundamental vibrational transitions have the largest A-values. Other overtone transitions with Av = -2, -3, etc. also occur. Figure 5.8 shows the system of first overtones observed toward the BN object in Orion. Here, the vibrational levels are collisionally excited in a gas with kinetic temperature near 4000 K.
The emission spikes in Figure 5.8 are actually bands consisting of many closely spaced lines. Each band corresponds to a pair of vibrational quantum numbers, say v' and v", while the lines within a band are individual rovibrotional transitions. The spacing between these lines gradually decreases toward higher frequency. Eventually, the lines merge in a band head. Let us see how this behavior, evident in the higher-resolution spectrum of Figure 5.9, can be understood in physical terms.
For downward dipole transitions within the same vibrational state (J ^ J - 1), equation (5.6) predicts that AErot is proportional to J, so that the lines are equally spaced. However, a CO molecule in a higher v-state has a slightly larger average separation between atoms. Its moment of inertia, Iv, is therefore greater and its rotational constant, Bv, is lower. If v' and v'' again denote the upper and lower vibrational states, respectively, then the J ^ J — 1 transition now yields an energy of
AE (v',J ^ v'',J — 1) = AEv>v» + (Bv> + Bv„) hJ + (Bv — Bv„) hJ2
« AEv>v>> + 2Bv> hJ + (Bv> — Bv") hJ2 . (5'15)
Here J = 1, 2, 3, etc. and AEvv•• is the energy difference between two J = 0 states. Note that J = 0 ^ 0 dipole transitions do not exist, so that the band contains a gap at the corresponding frequency. Since Bv•• is slightly greater than Bv>, equation (5.15) shows that the frequencies of the J ^ J — 1 lines, known collectively as the R-branch of the band, increase, reach a maximum, and then begin to decline with higher J. For the fundamental v = 1 ^ 0 band, this maximum is reached at A = 4.30 pm, while it occurs at 2.29 pm in the first overtone v = 2 ^ 0 band. Figure 5.9 shows the v = 2 ^ 0 band head in SSV 13, an embedded infrared source driving a molecular outflow.
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