Doubling

Focusing again on an individual rotational state, we saw how the motion of the internuclear axis causes the axial projection of S + L to precess about J. Such precession by itself would not affect the molecule's energy. However, the nuclear rotation also distorts slightly the electron's orbital motion. Figure 5.19 depicts two orthogonal probability distributions for the electron in the plane perpendicular to the internuclear axis. Because of the molecule's symmetry about the axis, the two cases (a) and (b) might be expected to have identical energy. However, the molecule as a whole rotates about the perpendicular axis indicated in the figure. This rotation induces a centrifugal force, so that the distribution in (a) has higher energy than in (b), where the electron is, on average, farther from the rotation axis.

This splitting of the previously degenerate ±A levels constitutes A-doubling. Note that the actual wavefunctions for the two eigenstates (known as the + and - states) are linear combinations of those corresponding to orthogonal directions of electronic rotation. Since the nuclear rotation of the molecule is very slow compared to the electron's orbital speed, the resulting perturbations to the energy are slight. Figure 5.18 shows schematically that the energy split increases going up the rotational ladder to states of higher J, i. e., to faster molecular rotation. For the 2n3/2 (J = 3/2) state, the temperature equivalent of the energy difference is 8.0 x 10~2 K. A photon emitted during the transition between sublevels has a frequency of about 1700 MHz and a wavelength of 18 cm.

Figure 5.19 Physical origin of A-doubling in OH. The molecule's energy depends on whether the symmetry axis of the unpaired electron's orbital motion (a) coincides with the internuclear axis, or (b) lies orthogonal to that axis.

5.6.3 Magnetic Hyperfine Splitting

Each of the sublevels of the 2 n states is further split by an interaction between the spins of the unpaired electron and the hydrogen nucleus. With reference to Figure 5.17, the result is that even J is not strictly constant, but precesses with I, the proton angular momentum, about their sum, the grand total angular momentum F. The new interaction arises because the magnetic field from the spinning electron at the position of the proton depends on the relative orientation of the two spin axes. Thus, in Figure 5.19, the electronic angular momentum S appears parallel to I in both panels. However, the magnetic field from the electron is actually parallel to I in (a) and antiparallel in (b). Reversing the direction of I would result in two other distinct states.

The net effect of this magnetic hyperfine splitting is quantitatively small; each sublevel of the ground 2n3/2 state is split in frequency by only about 60 MHz. The F-values of the final states are given in Figure 5.18. Transitions that connect states of the same F are said to produce main lines, while the others emit satellite lines. The reader may recall from Chapter 2 that the same magnetic interaction is responsible for the famous 21 cm line of atomic hydrogen. The split in energy, corresponding in that case to a frequency of 1420 MHz, is larger than in OH because the average separation of the electron and proton is less.

The short vertical line segments in Figure 5.18 indicate the allowed radiative transitions within the 2n3/2 (J = 3/2) ground state of OH. Historically, the 1963 detection of the four lines at 1612, 1665, 1667, and 1720 MHz constituted the first radio identification of an interstellar

molecule. In this case, the lines appeared in absorption against the radio source Sgr A*. Their relative intensities were close to the theoretically predicted values in an optically-thin medium, with the strongest lines being the main ones at 1665 and 1667 MHz. Soon after, observers found three of the four lines in emission, but the one at 1665 MHz was far brighter than the others. This line, initially dubbed "mysterium," was also much narrower than expected and strongly variable. What was seen, in fact, was the first instance of an interstellar maser.

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