0 1 2 3 4 5 6 Angular Momentum K
Figure 5.12 Rotational levels of NH3. Those levels lying on the lower border constitute the rotational backbone.
Here, A and B are the two rotational constants, while J and K are quantum numbers that measure, respectively, the magnitude of the total angular momentum and its component along the symmetry axis. For a given value of J, the possible K-values range from — J to + J. Since, according to (5.19), states with ±K have the same energy, it is conventional, when labeling states, to restrict K to values greater than or equal to 0.
The total set of rotational levels is conveniently arranged into columns of fixed K-value (Figure 5.12). Within a given column, the state of lowest energy has J = K. Downward transitions from (J, K) to (J — 1, K) occur very rapidly, with typical A-values from 10"2 to 10"1 s"1. By symmetry, the molecule's electric dipole moment vector ¡i lies along the central axis. Classically, rotation about this axis can therefore produce no dipole radiation. Correspondingly, quantum-mechanical dipole transitions with nonzero AK are forbidden. States along the lower border in the diagram consitute the rotational backbone. Downward quadrupole transitions along this border, (J, K) ^ (J — 1, K— 1), do occur, but with A-values of order 10"9 s"1. Thus, the states along the backbone are metastable.
The most useful spectral lines from NH3, as we shall demonstrate in Chapter 6, arise from inversion, the oscillation of the nitrogen atom through the hydrogen plane. In most molecules, vibrational transitions yield infrared photons of much higher frequency than those from rotational modes. The inversion transition of NH3, however, produces microwave photons, in contrast to the far-infrared rotational lines. The reason is that, from the classical viewpoint, the nitrogen atom does not have sufficient energy to cross the central plane, i.e., there exists a barrier in the potential well (Figure 5.13). In quantum mechanics, the atom's wavefunction can tunnel through this barrier, given sufficient time. Oscillation thus occurs, but at a far lower rate than in a simple parabolic potential well. To produce the low-frequency emission, each rotational level (J, K) with K > 0 is split into two sublevels with an energy separation of order 10~4 eV (Figure 5.14). The transition from the upper to the lower sublevel yields the main line of the NH3 microwave spectrum. For the (1,1) state, this line has a wavelength of 1.27 cm.
Additional effects further split the two inversion states. The nitrogen nucleus has a non-spherical charge distribution and an electric quadrupole moment. It can therefore be torqued in the presence of an electric field gradient. The system energy depends on the relative orientation of the nuclear spin and the total angular momentum vector of the electrons, which in turn varies with the rotational state of the molecule. Consequently, each inversion state splits into three sublevels, as illustrated in Figure 5.14. When the appropriate selection rules are enforced, the allowed transitions between the upper and lower levels give rise to a total of five lines - the original main line and two pairs of satellite lines, separated from the main line by about 1 MHz.
Finally, even weaker, magnetic interactions between the spins of the various nuclei split the lines again, with typical separations of 40 KHz. The net result is that the observationally important (1,1) and (2,2) rotational states each produce a total of 18 lines. The original main line, now split into 8 closely spaced components, has about half the total intensity of the inversion transition, while each cluster of satellite lines carries roughly equal intensity.
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