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Physical Review

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

John Hasbrouck Van Vleck

Published Apr 1, 1928• 110 citations
Publisher

Abstract

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 Schr\"odinger 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 $\ensuremath{\chi}=N\frac{[4s(s+1)+k(k+1)]{\ensuremath{\beta}}^{2}}{3kT}$ and $\ensuremath{\chi}=[\frac{N{g}^{2}j(j+1){\ensuremath{\beta}}^{2}}{3kT}]+N\ensuremath{\alpha}$ where $\ensuremath{\beta}$ is the Bohr magneton and $\ensuremath{\alpha}$ 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 ${\mathrm{cm}}^{\ensuremath{-}1}$. 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.

DOI
10.1103/physrev.31.587

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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 SSS 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 Schr"odinger wave function has nearly as much symmetry as in an atomic SSS state.3.Paramagnetism of atoms.---Limiting values for the paramagnetic susceptibilities of atoms are \ensuremathχ=N[4s(s+1)+k(k+1)]\ensuremathβ23kT\ensuremath{\chi}=N\frac{[4s(s+1)+k(k+1)]{\ensuremath{\beta}}^{2}}{3kT}\ensuremathχ=N3kT[4s(s+1)+k(k+1)]\ensuremathβ2​ and \ensuremathχ=[Ng2j(j+1)\ensuremathβ23kT]+N\ensuremathα\ensuremath{\chi}=[\frac{N{g}^{2}j(j+1){\ensuremath{\beta}}^{2}}{3kT}]+N\ensuremath{\alpha}\ensuremathχ=[3kTNg2j(j+1)\ensuremathβ2​]+N\ensuremathα where \ensuremathβ\ensuremath{\beta}\ensuremathβ is the Bohr magneton and \ensuremathα\ensuremath{\alpha}\ensuremathα is a constant. These two formulas are rigorously applicable when the multiplet intervals are respectively very small or very large compared to kT\mathrm{kT}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 kT\mathrm{kT}kT. The experimental susceptibilities for O2{\mathrm{O}}_{2}O2​ and ClO2{\mathrm{O}}_{2}O2​ are in accord with the assumption that the normal states are respectively 3S^{3}S3S and 2S^{2}S2S terms. In the particular case of SSS 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 2P^{2}P2P terms separated by 120.9 cm\ensuremath−1{\mathrm{cm}}^{\ensuremath{-}1}cm\ensuremath−1. 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 kT\mathrm{kT}kT. These deviations should be detectable experimentally if the susceptibility of NO could be measured over a wide temperature range.
1. In a recent paper Landau has shown that, when electrons are moving freely in a magnetic field, they exhibit, in addition to the paramagnetic effect of their spin, a diamagnetic effect due to their motion. This result is rather unexpected, since it is quite contrary to the classical case. There it might appear as though the circles described by the electrons must produce a magnetic moment, but the error was long ago pointed out by Bohr. The motion of the electrons must be confined to some region by means of a boundary wall, and the electrons near the wall describe a succession of circular arcs, repeatedly bouncing on the wall, and slowly creeping round it in the direction opposite to that of the uninterrupted circles; when the moment of these electrons is taken into account, it exactly cancels out that due to the free circles. In Landau's work it is of course necessary to consider the boundary, but he shows how allowance is to be made for it by an appropriate process. The complete justification is rather subtle, and so it may be worth considering a special case, admitting of exact solution, which takes the boundary into account, and so makes it possible to follow more closely the analogy between the classical and quantum problems. With regard to the general case with a boundary wall of any type, we shall only observe that the different results arise, because in the wave problem ψ must vanish at the bounding “potential wall,” and so will be small near it; this upsets the balance of the electric current near the wall, and yields the magnetic moment.
How does one explain magnetic effects in matter when one views matter as a collection of classical charges in motion? The answer is: not at all! This is one of the points that Niels Bohr made in his doctoral dissertation in 1911, two years before addressing the issue of the stability of the hydrogen atom. The result, later rediscovered by H.J. van Leeuwen was amplified and formalized in Van Vleck’s 1932 text on electric and magnetic susceptibilities and it is currently known as the Bohr-van Leeuwen theorem. We will review Bohr’s two derivations, one statistical and one based on the motion of individual electrons. We will then propose reasons why this result, unlike that on the stability of hydrogen, did not lead to a major development in quantum theory but, instead, had to wait until after the introduction of spin and exchange forces in quantum mechanics to become generally known. The classical-quantum discontinuity • Classical Theory (mechanics, field theory, thermal, ...) 1. Critical discoveries: x, α, β, γ, e (e/m), nucleus 2. h-bar effects: Planck, Einstein, Compton, ... 3. Atomic models and stability of matter (Old quantum theory: Bohr & Sommerfeld) • Quantum Mechanics (Schrodinger, Heisenberg, Dirac, ...) Did we miss something? Classical Magnetism (1) • B fields in vacuum: Maxwell’s equation • Lorentz’ force: F = q ( E + v x B ) • Magnetism in matter susceptibilities (~T) Curie 1895 OK Zeeman effect Lorentz 1896 OK diamagnetism Langevin 1905 OK paramagnetism Langevin 1905 ? & permanent magnets Weiss 1907 ? Classical Magnetism (2) Failures... Anomalous Zeeman effect Anomalously large molecular fields ... and demise! Bohr-Van Leeuwen theorem: Lack of relevance of classical theory & Need for quantum theory!
Second-order contributions of diamagnetic interaction to the energy shift and splitting are studied in the basis of the angular momentum eigenfunctions for two kinds of Rydberg states in atoms: (i) hydrogen-like degenerate states and (ii) nondegenerate Rydberg states with small magnetic quantum number, m ? 3, in many-electron atoms. General formulae are presented for calculations of higher order energy corrections in degenerate states with the use of reduced Green's function. The analytical expressions are derived for the second-order radial matrix elements of operator r2 and for the second-order diamagnetic susceptibilities of hydrogen-like states with m ? n-5. The quasiclassical quantum defect method is used to calculate the irreducible components ?nl(p) of the diamagnetic susceptibility ?nlm(2) for Rydberg states of alkali atoms. The numerical results are presented for susceptibilities of degenerate hydrogen substates, and for susceptibilities of alkali atoms in Rydberg s-, p-, d-states. The asymptotic dependence of susceptibilities in alkali atoms on the effective principal quantum number ? is determined numerically and the deviation from that of the hydrogen-like states is discovered. The data is also presented for the scaling parameters cl(p), determining the irreducible parts of the diamagnetic susceptibilities in states with high ?, according to the asymptotic formula ?nl(p) = cl(p)?11.
By combining methods from thermal field theory and statistical mechanics, we reexamine the spin polarization caused by the relativistic Barnett effect in a rigidly rotating Fermi gas. We determine the pressure of this medium and show that it depends on an effective chemical potential, which includes contributions from orbital angular momentum-rotation and spin-rotation coupling. We introduce a specific regularization scheme to sum over the angular momentum quantum numbers. As a result, the thermal pressure and all thermodynamic quantities are separated into two parts that differ only in the spin fugacities of spin-up and spin-down fermions. We calculate the Fermi energy for both components and show that the Fermi energy of the spin-down fermions is lower than that of the spin-up ones. This difference arises from the spin-rotation coupling and leads to a spin polarization consistent with the Barnett effect. In particular, we introduce the spin-chemicorotational ratio η≡Ω(0)/2μ(0)η\equiv Ω^{(0)}/2μ^{(0)}η≡Ω(0)/2μ(0), which adjusts the spin polarization of the Fermi gas. Here, Ω(0)Ω^{(0)}Ω(0) and μ(0)μ^{(0)}μ(0) represent the angular velocity and chemical potential at zero temperature, respectively. The factor 1/21/21/2 accounts for the fermion's spin. We explore the temperature dependence of μμμ and ΩΩΩ, while assuming that the number of spin-up and spin-down fermions remains temperature independent. Our findings indicate that the spin-down component of the rotating Fermi gas dilutes at lower temperatures compared to the spin-up component. Additionally, we calculate the magnetic susceptibility arising from the Barnett magnetization and demonstrate that it is proportional to the moment of inertia III of the rotating Fermi gas. Finally, we prove that III exhibits a 1/T1/T1/T behavior in the high-temperature limit, similar to the Curie law of paramagnetism.