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Journal of Physics A Mathematical and Theoretical

On the Biot–Savart law of electromagnetism applied to the atomic circulation current *

J D Hey

Published Feb 19, 2021• 2 citations
Publisher

Abstract

Abstract The law of Biot and Savart, derived by observation of the magnetic field produced by electric current flowing in a macroscopic conductor, is shown to yield the correct expression for the nuclear hyperfine interaction, when applied to the electron circulation current derived for hydrogenic atoms from the theories of Pauli and Dirac, as further developed independently by Darwin, Gordon and Hartree. The law thus applies also on the microscopic scale to the magnetic field generated by the electron probability flux of the various atomic eigenstates, and may clearly be generalised formally for application to many-electron atoms, as outlined by Hartree. The present, fully relativistic, treatment is advantageous in avoiding the usual artificial splitting of the hyperfine interaction into a magnetic dipole–dipole contribution and a separate contribution arising from electron orbital motion. The formula obtained forms the basis for the inclusion of higher order (quantum electrodynamic and nuclear recoil) corrections. Interest in this subject is enhanced by observations of ‘strongly forbidden’ (magnetic dipole) atomic transitions arising from the nuclear magnetic interaction, notably the 21 cm line of hydrogen, which has played an important role in astrophysics, in both measurements of galactic rotation and studies of the early Universe.

DOI
10.1088/1751-8121/abe832

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physics.atom-phCross-listedarXiv:2602.18144•Feb 23, 2026
Abstract The law of Biot and Savart, derived by observation of the magnetic field produced by electric current flowing in a macroscopic conductor, is shown to yield the correct expression for the nuclear hyperfine interaction, when applied to the electron circulation current derived for hydrogenic atoms from the theories of Pauli and Dirac, as further developed independently by Darwin, Gordon and Hartree. The law thus applies also on the microscopic scale to the magnetic field generated by the electron probability flux of the various atomic eigenstates, and may clearly be generalised formally for application to many-electron atoms, as outlined by Hartree. The present, fully relativistic, treatment is advantageous in avoiding the usual artificial splitting of the hyperfine interaction into a magnetic dipole–dipole contribution and a separate contribution arising from electron orbital motion. The formula obtained forms the basis for the inclusion of higher order (quantum electrodynamic and nuclear recoil) corrections. Interest in this subject is enhanced by observations of ‘strongly forbidden’ (magnetic dipole) atomic transitions arising from the nuclear magnetic interaction, notably the 21 cm line of hydrogen, which has played an important role in astrophysics, in both measurements of galactic rotation and studies of the early Universe.
The mechanical angular momentum and magnetic moment of the electron and proton spin have been calculated semiclassically with the aid of the uncertainty principle for energy and time.The spin effects of both kinds of the elementary particles can be expressed in terms of similar formulae.The quantization of the spin motion has been done on the basis of the old quantum theory.It gives a quantum number n = 1/2 as the index of the spin state acceptable for both the electron and proton particle.In effect of the spin existence the electron motion in the hydrogen atom can be represented as a drift motion accomplished in a combined electric and magnetic field.More than 18,000 spin oscillations accompany one drift circulation performed along the lowest orbit of the Bohr atom.The semiclassical theory developed in the paper has been applied to calculate the doublet separation of the experimentally well-examined D line entering the spectrum of the sodium atom.This separation is found to be much similar to that obtained according to the relativistic old quantum theory.
Introduction The selection rules discussed in Sec. 3.5 forbid many kinds of transitions that are frequently observed or exploited. It must be remembered that these “rules” are based on the approximations of Sec. 3.2, but of course, those can be extended. Such expansions to higher orders of approximation may seem to be little more than a tedious exercise, but this chapter will show that the results are extremely important, because they cause qualitatively different kinds of transitions. Since they involve both the electric and magnetic components of the light, their atomic dipole moments will be identified with the superscript E and B for the remainder of this chapter. The magnetic dipole transitions, called M1, discussed in Sec. 4.2.1 that arise from extension of the electric dipole approximation encompass all of magnetic resonance, including atomic clocks, forced evaporative cooling in Bose–Einstein Condensation (BEC), MRI, and astrophysical processes such as generation of the all-important 21 cm radio frequency line. The electric quadrupole transitions, called E2, discussed in Sec. 4.2.2 describe myriad stellar spectral features as well as the vast majority of nuclear γ-ray transitions. Even with these extensions there are many cases where the relaxation of the electric dipole (E1) selection rules to include M1 and E2 transitions as in Sec. 4.2 still do not account for observed processes. Thus other approximations have to be extended so that their limitations can be relaxed as well. Extension of the perturbation approximation (see Sec. 4.3.1) provides an analytical description of most of non-linear optics, including second harmonic generation, four wave mixing, Raman spectroscopy, and the fabled decay of the hydrogen 2S state. A semi-classical description of the interaction between radiation and atoms can begin with a multipole expansion of the Hamiltonian. The lowest-order term is just from the point charges α located at, and vanishes for neutral atoms. The next-order term is the electric dipole interaction that arises from the non-isotropic distribution of charges induced by the electric field of the light. This term produces the electric dipole transitions (E1) that have been described in some detail in Chap. 3.
Niels Bohr’s model of the hydrogen and hydrogen-like atoms, in which the electron is allowed to revolve in certain circular orbits around the nucleus, was extended by Arnold Sommerfeld to allow for the possibility of elliptical orbits. Each elliptical orbit was constrained by two quantization conditions specified by a positive integer nφ and a non-negative integer nr. If, in the presence of the attractive Coulomb force between the electron and the nucleus, the dynamics of the atom were taken to be governed by the Newtonian laws of motion, the energy Ɛ of the two-particle system was found to be a function of the positive integer n = nr + nφ. The atomic energy levels thus exhibited an n-fold degeneracy, which would be lifted in the presence of external electric or magnetic fields, thus accounting for the fine-structure of the spectral lines emitted by the atom—phenomena observed, respectively, in the Stark effect and the Zeeman effect. Sommerfeld also examined the behavior of his elliptical orbits by allowing the revolving electron to obey Einstein’s relativistic dynamics. In this case, the energy-level degeneracies were lifted (even in the absence of external perturbations), and the elliptical orbits were found to precess around the nucleus—similarly to the precession of, say, the orbit of the planet Mercury around the Sun. What is astonishing about Sommerfeld’s relativistic expression of the energy Ɛ as a function of nφ and nr is that it is precisely the same expression as obtained for hydrogen and hydrogen-like atoms from Dirac’s equation, which came more than a decade later. Recall that the spin of the electron was not yet discovered when Sommerfeld developed his model, and that in Dirac’s fully-relativistic quantum mechanics, the spectral fine-structure arises from the coupling of the electron’s spin to its orbital motion. The present paper provides a streamlined yet detailed account of Sommerfeld’s elliptical orbits in hopes of bringing about a deeper appreciation for the miraculous coincidence of Sommerfeld’s result with that derived from Dirac’s equation.
Contrary to popular belief, the Zeeman effect can be treated exactly in single-electron systems, for arbitrary magnetic field strengths, as long as the term quadratic in the magnetic field can be ignored. These formulas were actually derived already around 1927 by Darwin, using the classical picture of angular momentum, and presented in their proper quantum-mechanical form in 1933 by Bethe, although without any proof. The expressions have since been more or less lost from the literature; instead, the conventional treatment nowadays is to present only the approximations for weak and strong fields, respectively. However, in fusion research and other plasma physics applications, the magnetic fields applied to control the shape and position of the plasma span the entire region from weak to strong fields, and there is a need for a unified treatment. In this paper we present the detailed quantum-mechanical derivation of the exact eigenenergies and eigenstates of hydrogen-like atoms and ions in a static magnetic field. Notably, these formulas are not much more complicated than the better-known approximations. Moreover, the derivation allows the value of the electron spin gyromagnetic ratio g s to be different from 2. For completeness, we then review the details of dipole transitions between two hydrogenic levels, and calculate the corresponding Zeeman spectrum. The various approximations made in the derivation are also discussed in details.
The study of highly charged electronic and muonic hydrogen-like ions, provides an intriguing way to probe the internal structure of their atomic nuclei. In this work, we use nuclear structure calculations to accurately calculate the hyperfine splitting of electronic and muonic hydrogen-like ions, focusing in particular on the incorporation of finite-volume corrections, such as Bohr-Weisskopf and Breit-Rosenthal, due to the penetration of the electron and muon wavefunction into the nuclear electric charge and magnetic dipole densities. These corrections are essential for refining our understanding of the nuclear magnetic dipole and electric quadrupole moments. Our simulations use a Skyrme-Hartree-Fock-BCS model known for its effectiveness in modeling well-deformed nuclei such as 159Tb64+{}^{159}\mathrm{Tb}^{64+}159Tb64+ and 165Ho66+{}^{165}\mathrm{Ho}^{66+}165Ho66+, with particular emphasis on 161,163Dy65+{}^{161,163}\mathrm{Dy}^{65+}161,163Dy65+ isotopes. It can also be generalised to multi-electron ions by studying the hyperfine anomaly between two isotopes.