The Theory of the Faraday Effect in Molecules

In treating the Faraday effect two cases may be distinguished, depending upon whether the frequency of the incident light is near resonance or well removed from resonance with absorption lines of the molecule.Frequency of incident light well removed from resonance with any absorption lines. In this case it is imperative to include the perturbation of the intensities by the magnetic field, as well as the perturbation of the energies. A general expression is obtained for the rotation by molecules (poly-, di-, or monatomic), of the form $V=\ensuremath{\Sigma}\stackrel{}{{n}^{\ensuremath{'}}}\left\{\frac{{\ensuremath{\nu}}^{2}A(n{n}^{\ensuremath{'}})}{{(\ensuremath{\nu}{({n}^{\ensuremath{'}}n)}^{2}\ensuremath{-}{\ensuremath{\nu}}^{2})}^{2}}+\frac{{\ensuremath{\nu}}^{2}B(n{n}^{\ensuremath{'}})}{\ensuremath{\nu}{({n}^{\ensuremath{'}}n)}^{2}\ensuremath{-}{\ensuremath{\nu}}^{2}}+\frac{{\ensuremath{\nu}}^{2}C(n{n}^{\ensuremath{'}})}{T(\ensuremath{\nu}{({n}^{\ensuremath{'}}n)}^{2}\ensuremath{-}{\ensuremath{\nu}}^{2})}\right\},$ where $V$ is the Verdet constant. This formula contains dia- and paramagnetic terms of the usual type, but augmented by terms arising from perturbation of the intensities. It contains, in addition, other diamagnetic terms which have the same frequency dependence as the paramagnetic terms. For atoms this expression reduces to that given by Rosenfeld. However for diatomic molecules our results differ from Kronig's, since we include the effects of the components of magnetic moment perpendicular to the axis of figure. The terms arising in this way were omitted by Kronig, although generally they are of the same order of magnitude as the contribution of the parallel component of the moment.Independence of spin. When the over-all spin-multiplet width is small compared to $\frac{\mathrm{kT}}{h}$ the rotation is completely independent of spin. As a consequence the paramagnetic terms vanish for nonlinear polyatomic molecules, and for linear polyatomic and diatomic molecules in $\ensuremath{\Sigma}$ states.Magnitude of the rotation and comparison with experiment. The classical Becquerel formula for the Verdet constant is $V=\frac{\ensuremath{\gamma}(\frac{e}{2m{c}^{2}})\ensuremath{\nu}\ensuremath{\partial}n}{\ensuremath{\partial}\ensuremath{\nu}}$, with $\ensuremath{\gamma}=1$. It is shown that the rotations, in the visible and near ultraviolet, of the gases for which data are available should be approximately representable by a formula of this form, provided $\ensuremath{\gamma}$ is given the proper value. The $\ensuremath{\gamma}$ value should lie between zero and one. This conclusion, in all cases but one, agrees with the known facts. The exception is oxygen, but the data are probably in error because of polymerization effects which seriously alter the absorption at high pressures.Frequency of incident light near resonance with an absorption line. Here only the perturbation of the energies by the magnetic field need be considered. It is shown that the rotation in iodine vapour, observed by Wood, is due to rotational distortion of the excited $^{3}\ensuremath{\Pi}_{0}$ level, which partially uncouples the spin moment from the figure axis. The magnetic rotation spectrum of the alkalis, observed by Wood and Loomis, is also explained, in particular the quenching of lines of large rotational quantum number.