We saw in Chapter 3 how molecular clouds are threaded by the magnetic field lines that permeate interstellar space. Compression of this field creates an effective pressure that partially supports clouds against gravitational collapse. To measure B accurately, a molecular probe should have a relatively large magnetic moment. Especially important in this regard are molecules with one unpaired electron and therefore a nonzero electronic angular momentum. Classified chemically as free radicals, such compounds are violently reactive in the laboratory, but can survive for long periods in the rarefied environments of molecular clouds. The most widely used species of this kind is OH.
The rotational levels of OH again have energies given by equation (5.6). However, it is the more finely spaced hyperfine transitions that have found application in magnetic field measurements. A given rotational state, again labeled by the quantum number J, is split into two sublevels of nearly equal energy by a phenomenon known as A-doubling. Here, Ah is the projection of the orbital angular momentum of the unpaired electron along the molecule's internuclear axis. The molecule is rotationally symmetric about this axis, so there are no torques to alter the corresponding component of the angular momentum. It follows that A is a valid quantum number. Since a state with -A, i. e., with the orbital motion reversed, has almost the same energy as the +A state, the label A is conventionally restricted to nonnegative integer values. The projection of the electronic spin angular momentum along the internuclear axis is another good quantum number, denoted E. Since OH has only one unpaired electron, E is restricted in value to ±1/2. The projection of the electron's total angular momentum, a quantity denoted 0, is given by |A + E|.
We emphasize that it is only the axial projections of the electronic angular momenta that are constants of the motion. In a semi-classical description, the spin and orbital angular momentum vectors, denoted S and L respectively, are not fixed in space, but undergo a complex motion (Figure 5.17). First, the unpaired electron is attracted toward the internuclear axis by the powerful electrostatic force of the chemical bond. The resulting torque causes L to precess rapidly about this axis. Second, the electron sees, in its own reference frame, a magnetic field
Figure 5.17 Torque-free motion of OH. The vectors L and S, representing respectively the unpaired electron's orbital and spin angular momentum, have projections A and £ along the internuclear axis. Both L and S precess rapidly about this axis. Meanwhile, the internuclear axis itself, and the associated nuclear angular momentum O, precess slowly about the total angular momentum J.
from the motion of the nuclei and remaining electrons. The field creates a torque on the magnetic moment associated with S. (Recall the discussion of the hydrogen atom in § 2.1.) As a result of this spin-orbit coupling, S also precesses quickly about the internuclear axis. Finally, the axis itself tumbles slowly end over end, through rotation of the O and H nuclei. Note that the angular momentum associated with this rotation, denoted O, lies perpendicular to the axis, since the atomic nuclei have negligible moments of inertia about the line joining them.
Regardless of these internal torques, the angular momentum J, formed by adding vectorially S, L, and O, is very nearly constant in magnitude and direction. We may picture the vector O and the projection of S + L along the internuclear axis both precessing about the fixed J (Figure 5.17). The motion is analogous to that of a symmetric top. (Recall Figure 5.11.)3 Thus, J is another good quantum number, whose possible values are given by 0, 0 + 1, 0 + 2, etc. This sequence is a generalization of J = 0,1, 2, etc. for molecules without electronic angular momentum.
The rotational states of the molecule are thus labeled, not only by J, but also by the quantum numbers A and 0. In addition, one must specify S, the magnitude of the electronic spin. This number, fixed for all rotational states, is here equal to 1/2. In spectroscopic notation, the ground state for OH is symbolized 2n3/2 and has J = 3/2. The n denotes A = 1, while the subscript is 0. Since E = +1/2 for this state, 0 = |1 + 1/2| = 3/2. Figure 5.18 shows the rotational ladder of 2n3/2 levels with J = 3/2, 5/2, 7/2, etc. The superscript in the spectroscopic symbol is the multiplicity, equal to 2S +1. In this case, the multiplicity of 2 indicates that there exists another state with the same A, but with the unpaired electron's spin oriented oppositely, i. e., with E = -1/2 and therefore 0 = 1/2. Any state with these values of A and 0 is denoted
3 The precession of angular momentum vectors described here applies only to the lower rotational states of OH. In states of higher J, the spin-orbit coupling can no longer effectively lock S to L directly.
Figure 5.18 Rotational states of OH. The two ladders correspond to opposite orientations of the unpaired electron's spin. The splitting of the levels due to both A-doubling and the magnetic hyperfine interaction is shown schematically. Also indicated are the allowed transitions within the ground rotational state.
2n^2; the lowest one has J = 1/2. Figure 5.18 includes the separate ladder of 2n^2 states with J =1/2, 3/2, 5/2, etc.
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