On the Saturation of the Electric and Magnetic Polarization of Gases in Quantum Mechanics

The calculations of dielectric and magnetic susceptibility in quantum mechanics previously made by Van Vleck are extended to include higher powers of the field strength. This is necessary in fields so strong that the moment is not linear in the field strength.Electric polarization. As noted by Debye the electric polarization of a gas results from two effects: (a) a change in the spacial orientation of the rigid or permanent moment of the molecule, (b) the inducing of an elastic polarization or deformation of the molecule.1. Rigid molecules. The effect of (a) is calculated in \textsection{}2 to all powers of the field strength and yields exactly the classical Langevin function $L(\ensuremath{\mu}, F, T)=\ensuremath{\mu}[\mathrm{cotgh}(\frac{\ensuremath{\mu}F}{\mathrm{kT}})\ensuremath{-}\frac{\mathrm{kT}}{\ensuremath{\mu}F}]$ provided only the (rotational) energy changes in "allowed" transitions are small compared to $\mathrm{kT}$ Here $\ensuremath{\mu}$ is the permanent moment of the molecule.2. Deformable molecules. The part of the moment resulting from the effect (b) (induced polarization, described with matrix elements whose frequencies are large compared to $\frac{\mathrm{kT}}{h}$ and the part arising from the superposition of (a) and (b) is calculated in \textsection{}3 to terms of the third order in the field $F$ The complete formula for the moment is to this order: ${m}_{F}=\left(\frac{{\ensuremath{\mu}}^{2}}{3kT}+{p}_{0}\right)F+\left({q}_{0}+\frac{{q}_{1}}{\mathrm{kT}}+\frac{{q}_{2}}{{k}^{2}{T}^{2}}\ensuremath{-}\frac{{\ensuremath{\mu}}^{4}}{45{k}^{3}{T}^{3}}\right){F}^{3}$ where ${p}_{0}$, ${q}_{0}$, ${q}_{1}$, ${q}_{2}$ are constants whose explicit expressions in terms of the individual matrix elements are rather complicated. The expression is of the same type form as that of Debye's, except for addition of the term ${q}_{0}$ and reduces indentically to his type form if, following Debye, we specialize the model by supposing the restoring forces to be simple harmonic.Magnetic polarization of atoms. Brillouin showed that if we neglect the "spin," the magnetic polarization per atom in a monatomic gas is ${m}_{H}=\ensuremath{\beta}\left[\frac{l{e}^{\frac{l\ensuremath{\beta}H}{\mathrm{kT}}}+(l+1){e}^{\ensuremath{-}\frac{(l+1)\ensuremath{\beta}H}{\mathrm{kT}}}}{{e}^{\frac{l\ensuremath{\beta}H}{\mathrm{kT}}}\ensuremath{-}{e}^{\ensuremath{-}\frac{(l+1)\ensuremath{\beta}H}{\mathrm{kT}}}}\ensuremath{-}\frac{1}{{e}^{\frac{\ensuremath{\beta}H}{\mathrm{kT}}}\ensuremath{-}1}\right]=B(l, \ensuremath{\beta}, H, T)$ where $l+1$ is the azimuthal quantum number of the old quantum theory and $\ensuremath{\beta}$ the Bohr magneton. We show that the reason this "Brillouin" function differs from that of Langevin is because the various Cartesian components of the angular momentum matrix do not commute in multiplication, a complication not found in the electric case. The appearance of a Brillouin function (which is expressible as the difference of two Langevin functions) instead of a single $L$-function obviates Debye's objection that the classical Langevin theory yields infinite entropy at $T=0$ in contradiction to the Nernst heat theorem.Magnetic polarization of atoms with spin. When now the spin is included two limiting cases can conveniently be considered; viz. those in which the spin multiplets are very small or very large compared to $\mathrm{kT}$. In the first case the magnetic polarization is the sum $B(l, \ensuremath{\beta}, H, T)+B(s, 2\ensuremath{\beta}, H, T)$ of two Brillouin functions representing the orbital and spin effects respectively. In the second case we need consider only the component of the magnetic moment parallel to the total angular momentum associated with the inner quantum number $j$. The polarization per atom then becomes $B(j, g\ensuremath{\beta}, H, T)$ where $g$ is the Land\'e-factor.Magnetic polarization of diatomic gases. With narrow multiplets $\ensuremath{\Delta}\ensuremath{\nu}\ensuremath{\ll}\frac{\mathrm{kT}}{h}$ the polarization is $L({\ensuremath{\sigma}}_{l}\ensuremath{\beta}, H, T)+B(s, 2\ensuremath{\beta}, H, T)$ while with wide multiplets the formula is instead $L(({\ensuremath{\sigma}}_{l}+2{\ensuremath{\sigma}}_{s})\ensuremath{\beta}, H, T)$ where ${\ensuremath{\sigma}}_{l}$ and ${\ensuremath{\sigma}}_{s}$ have their usual spectroscopic meaning. The reason that the Langevin function appears whereas the Brillouin function alone was encountered in the atomic case, is that now only the component of orbital angular momentum parallel to the axis of figure is of the important low frequency type and with only one effective Cartesian component no questions of non-commutability can arise.