On Dielectric Constants and Magnetic Susceptibilities in the New Quantum Mechanics Part III—Application to Dia- and Paramagnetism

1.General mathematical theory.—Modifications are given in the general derivation of the Langevin-Debye formula by means of quantum mechanics published in part I which are required by the moment now being magnetic rather than electric. The main additions are the appearance of a diamagnetic term, and allowance for the fact that the magnetic moment consists of two parts arising respectively from orbital electronic motions and from the electrons' internal spins. These parts cannot always be treated as rigidly coupled to form a "permanent" resultant moment. The Hamiltonian function used for the internal spin is that of a spherical top.

2.Diamagnetism.—The writer's previous claim that Pauli's formula for the diamagnetic susceptibility can be applied to molecules as well as atoms is shown to be invalid because of the anomalous fact that the square of the angular momentum, unlike the average angular momentum, does not vanish even in $S$ states when there is more than one nucleus. Pauli's formula is instead an upper limit to the diamagnetism in non-monatomic molecules and is a good approximation when the Schrödinger wave function has nearly as much symmetry as in an atomic $S$ state.

3.Paramagnetism of atoms.—Limiting values for the paramagnetic susceptibilities of atoms are $\chi =N\frac{\mathrm{}\left[4s\right(s+1)+k(k+1\mathrm{}\left)\right]{\beta }^2}{3kT}$ and $\chi =\left[\frac{N{g}^{2}j(j+1\mathrm{})\mathrm{}{\beta }^2}{3kT}\right]+N\alpha$ where $\beta$ is the Bohr magneton and $\alpha$ is a constant. These two formulas are rigorously applicable when the multiplet intervals are respectively very small or very large compared to $\mathrm{kT}$, and are valid regardless of whether the magnetic field is strong enough to change the quantization by producing a Paschen-Back effect. The formula for small multiplets yields susceptibilities slightly different from those given by Laporte and Sommerfeld's expression for this case, and is much simpler, as they overlooked the contribution of the portion of the magnetic moment which is perpendicular to the invariable axis.

4.Paramagnetism of molecules.—The susceptibility is calculated on the basis of the Hund theory of molecular quantization. Formulas are given applicable to his couplings of type (a) and type (b) provided in the former the multiplet intervals are either very large or very small compared to $\mathrm{kT}$. The experimental susceptibilities for ${\mathrm{O}}_2$ and Cl${\mathrm{O}}_2$ are in accord with the assumption that the normal states are respectively ${}_{}{}^3{}_{}{}^{}S$ and ${}_{}{}^2{}_{}{}^{}S$ terms. In the particular case of $S$ terms the numerical results are the same in the atomic and molecular formulas, but, unlike previous theories, it is not necessary to suppose the orbits are as freely oriented in molecules as in atoms. Polyatomic molecules may have lower paramagnetic susceptibilities than diatomic ones because the dissymmetry causes large fluctuations in electronic angular momentum.

5.Paramagnetism of nitic oxide.—Spectroscopists have recently found that the normal states of the NO molecule are ${}_{}{}^2{}_{}{}^{}P$ terms separated by 120.9 ${\mathrm{cm}}^{-1\mathrm{}}$. This permits an unambigous calculation of the susceptibility of NO which agrees with the experimental value within 1.5 per cent. Deviations from Curie's law are calculated which result from the doublet interval being comparable with $\mathrm{kT}$. These deviations should be detectable experimentally if the susceptibility of NO could be measured over a wide temperature range.