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Wolfgang Pauli introduced the Bohr magneton as a fundamental unit of magnetic moment during an effort to find a quantum basis for magnetism, as Davide Castelvecchi recounts.

The year was 1920, and Austrian-born Pauli (pictured) was a doctoral student in Munich, Germany, under the supervision of Arnold Sommerfeld. At that time, the leading model of the atom was the ‘old quantum theory’, a semiclassical model of the atom developed by Niels Bohr and Sommerfeld that was eventually superseded by quantum mechanics. Given the focus of his doctoral advisor, it was unsurprising that Pauli would try to find an old-quantum interpretation of magnetic phenomena, beginning with the surprisingly strong diamagnetism of helium.

Credit: Emilio Segre Visual Archives / American Institute of Physics / Science Photo Library

In Bohr–Sommerfeld theory, electrons orbit in elliptical trajectories around the nucleus. The resulting current produces a discrete magnetic moment, which is proportional to the orbit’s quantum number and the electron’s elementary charge, and inversely proportional to the mass of the electron. When Pauli looked at magnetization, it was thus natural to conclude that the corresponding fundamental quantum should be the one produced by an electron in the ground state, which equals *eħ*/2*m* — a quantity that he dubbed the Bohr magneton, *μ*_{B}. This reflected the idea of the old quantum theory that an electron in the ground state had nonzero angular momentum. However, quantum mechanics showed this assumption to be false. The orbital angular momentum of electrons in atoms — and therefore their magnetic moment — is complicated by quantum mechanical and relativistic effects. As Pauli was already well aware in 1920, the old quantum theory failed to account for many of helium’s properties, such as its spectrum, and magnetism was no exception.

But the Bohr magneton stuck nevertheless, and is now known to be 9.2740100783 × 10^{–24} J T^{–1} (ref. ^{1}). The reason for the persistence of the Bohr magneton might lie in a development that took place in 1928. After the discovery of the electron’s spin, Paul Dirac calculated that the electron’s magnetic moment should be equal to exactly one Bohr magneton. More generally, an elementary particle with charge *q*, mass *m* and spin *s* has a magnetic moment *μ* of (g {q over {2m}}s), where *g* is a dimensionless constant and *s* is typically ½*ħ*. For the electron, this reduces to *μ* = ({textstyle{g over 2}})*μ*_{B}. But as it turns out, *g* is not exactly 2 due to higher-order corrections.

Experiments with single electrons held in Penning traps^{2} have found half of the electron’s *g*-factor to be 1.00115965218073, with an uncertainty of less than three parts per 10 trillion. This is one of the most precise experimental measurements in physics, and has provided the most stringent test of quantum electrodynamics: it agrees with theoretical predictions (calculations that themselves required Herculean efforts) to 1.1 parts per trillion. But for the muon, there seems to be a contradiction with theory as Thomas Teubner explained in another Measure for Measure^{3}. An experiment currently underway at Fermilab in the United States and a future project at J-PARC in Japan are trying to improve those earlier measurements, but in the meantime more recent calculations suggest that the tension might not be there after all^{4}.

For hadrons, the *g*-factors are further complicated by the fact that hadrons are not elementary particles. But they have been measured, for example for the proton and the neutron, whose magnetic moments are nonzero despite the particles having no net electric charge.

As for Pauli’s favourite particles — neutrinos — their *g*-factors are still unknown, like many of their other properties. Although neutrinos are electrically neutral and — contrary to the neutron — elementary, theorists have calculated that in the Standard Model of particle physics, they should have a tiny but finite magnetic moment of around 10^{–20}*μ*_{B} (ref. ^{5}), which is unobservably small. Experimental limits come indirectly from astrophysics, which are of the order of 10^{–12}*μ*_{B}, and the most stringent direct limit is an order of magnitude larger^{5}. However, a significantly larger magnetic moment of the neutrino could hint at physics beyond the Standard Model and the observation of a magnetic moment exceeding roughly 10^{–15}*μ*_{B} would mean that neutrinos are Majorana rather than Dirac fermions. In a recent preprint, the dark matter experiment XENON1T reported an excess over their expected background, which could be interpreted as a hint for such an anomalous magnetic moment of the neutrino^{5}. If this result turns out to be correct, it could potentially not only point to an unexpectedly large neutrino magnetic moment, but also to a striking violation of the standard model.

## References

- 1.
*Bohr Magneton*(NIST, 2019); https://go.nature.com/3gAbPsL - 2.
Hanneke, D., Fogwell, S. & Gabrielse, G.

*Phys. Rev. Lett.***100**, 120801 (2008). - 3.
Teubner, T.

*Nat. Phys.***14**, 1148 (2018). - 4.
Borsanyi, Sz. et al. Preprint at https://arxiv.org/abs/2002.12347 (2020).

- 5.
Aprile, E. et al. Preprint at https://arxiv.org/abs/2006.09721 (2020).

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Castelvecchi, D. Just a moment.

*Nat. Phys.* (2020). https://doi.org/10.1038/s41567-020-1022-6

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