On Dielectric Constants and Magnetic Susceptibilities in the New Quantum Mechanics. Part II—Application to Dielectric Constants

Abstract

1,2. The proof of the Langevin-Debye formula given in part I with the new quantum mechanics is rather abstract because of its very generality, and so the results are made more concrete and non-mathematical by discussing some of the models that are included as special cases. This general derivation shows that the Debye formula applies to asymmetrical models with three unequal moments of inertia quite as well as to the symmetrical molecules previously studied by various writers with the aid of special models. Unlike the old quantum theory, there is no abrupt change of dielectric constant with pressure or field strength due to the passage from "weak" to "strong" spacial quantization. 3. Influence of a magnetic field. The mathematical theory of part I shows that very generally a magnetic field $H$ should be without effect on the dielectric constant (or, 4, on the refractive index, except for the Faraday and Cotton-Mouton effects) unless we consider very small terms in $H^2$. Recent spectroscopic data show that apparently the only feasible explanation of the absence of a "magneto-electric directive effect" in NO is Piccard's and de Haas' suggestion of equal numbers of "left" and "right-handed" molecules having mutually opposite senses of electronic rotation relative to the molecular axis. 4. Refractive index—effect of vibration bands. A general formula is given for the refractive index which shows that the refractivity per molecule should not vary with temperature despite "temperature rotation" of the nuclei about the center of ${}_{\mathrm{g}}{}^{}{}_{}{}^{}\mathrm{ravity}$. The experimental confirmation of this fact may be regarded as verifying the sum-rules" characteristic of the new quantum mechanics. The suggestion of Debye "nd Ebert that the experimental discrepancy between ${n_0}^2-1\mathrm{}$ and $4\pi N\alpha$ is due to infra-red nuclear vibration bands is shown to be untenable in molecules such as HCl. Here $n_0$ denotes the extrapolation of the index of refraction to zero frequency from visible dispersion curves, and $N\alpha$ is the part of the static dielectric susceptibility due to "induced polarization." Measurements by Bourgin and others of the intensity of infra-red absorption bands furnish values of the "effective charge" associated with the nuclear oscillations and so enable one to calculate the contribution of the vibration bands to the polarization. In HCl this contribution proves to be about $\frac{1}{100}$ of that necessary to account for the discrepancy mentioned above. 5. Limit of accuracy of the Debye formula. Because a fraction, usually small, of the molecules have sufficiently large rotational quantum numbers to make their frequencies of rotation comparable with $\frac{\mathrm{kT}}{h}$, the Debye formula holds only asymptotically at high temperatures in the new quantum mechanics, and at ordinary temperatures the correction for departures from the asymptotic value consists approximately in adding a very small term $-\frac{\mathrm{NC}}{T^2}$ to the ordinary Debye expression $N\alpha +\frac{N{\mu }^2}{3kT}$ for $\frac{(\epsilon -1)}{4\pi }$. The value of $C$ is calculated for the most general rigid asymmetrical molecule, and reduces for symmetrical molecules to the value previously obtained for them by Kronig and Manneback by an entirely different method. 6. General classical derivation of the Langevin-Debye formula. It is shown that according to classical mechanics in any multiply periodic dynamical system amenable to statistical theory the susceptibility equals $\frac{N\overline{{\mathrm{M}}^2}}{3kT}$, where $\overline{{\mathrm{M}}^2}$ is the statistical mean square of the vector moment of the molecule; i.e., the average over the phase space weighted according to Boltzmann factor, which is not to be confused with the time average for an individual molecule. This formula for the susceptibility is a generalization of the Langevin-Debye formula, and reduces to the ordinary Debye expression $A+\frac{B}{T}$ if the dynamical system consists of a rotating elastic polar molecule executing small vibrations about an equilibrium configuration. DOI: https://doi.org/10.1103/PhysRev.30.31 ©1927 American Physical Society