current associated with the particle's orbital motion. Since the orbital angular momentum is h in order of magnitude, it is convenient to rewrite (6.29) as vi = -gi jb (L/h) . (6.30)
Here gl is a dimensionless number called the Landé g-factor, equal to unity in this case. The quantity jb is the Bohr magneton, given by eh to = -
The spin contribution fj,s obeys an equation analogous to (6.30), but with a value for gs of 2.0.
Given fxl and (j,s, we may use Equations (6.27) and (6.28) to evaluate Emag. However, in the usual representation of the molecular wavefunction, neither the component of L nor of S in the direction of B has an associated quantum number. As we saw in § 5.5, both L and S precess rapidly about the internuclear axis because of the torque exerted on the electron by the magnetic field of the nuclei and remaining, paired electrons. Within each A-doubled rotational state shown in Figure 5.15, the precise energy depends on the relative orientation of B and the grand total angular momentum F. Let us denote by AEmag the perturbation of the state energy from its value at B = 0. Then AEmag depends on all the quantum numbers characterizing the state (A, Q, S, J, I, and F), and on the additional quantum number MF. The latter represents the projection of F along B and can take on 2F +1 integer values, ranging from MF = —F to +F. The final expression for the energy perturbation is then
Here the full Landé g-factor is again of order unity and is a function of all the quantum numbers except Mf .
Dividing equation (6.32) by h, we obtain the splitting in terms of frequency. We may write the result as
The factor 1/2 is convenient because the frequency splitting is symmetric with respect to the unperturbed line. For the dominant 1665 and 1667 MHz transitions, the constant b has the values 3.27 and 1.96 Hz |G_1, respectively. The actual Avmag for typical cloud field strengths is therefore relatively small (Avmag/v0 ~ 10~8). Nevertheless, it is much greater than in molecules lacking an unpaired electron. Here, A-doubling is absent, and the magnetic interaction that splits each rotational state involves the spins of two nucleons, rather than an electron and a nucleon. From equation (6.31), the nuclear magneton ¡j>n is smaller than ¡j,b by the electron-nucleon mass ratio of 1/1836. The b-values of NH3 and H2O, for example, are only 7.2 x 10~4 and 2.3 x 10~3 Hz |G_1, respectively.
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