Physical · a theorem you can watch · Bohr 1911, van Leeuwen 1919

The Crowd at the Wall

Put electrons in a box and switch on a magnetic field. Every one of them curls into a little circle, and every circle is a loop of current that pushes back against the field. So a box of electrons ought to be a magnet. It is not, and the reason is the electrons at the walls: they skip along the edge the other way round, and their one big loop cancels all the small ones. In 1911 Niels Bohr showed that classical physics, applied consistently, leaves no magnetism at all. Below, 20,000 electrons show you why, and then a famous experiment of 1915 measures what it expected to find.

Every circle pushes back

A charge moving through a magnetic field is pushed sideways, at right angles to its motion, so it never speeds up or slows down: it just turns, and keeps turning, in a circle. A charge going round a circle is a small loop of current, and a loop of current is a small magnet. Work out which way that magnet points and you find it always opposes the field that made it. A negative electron circles one way and a positive charge the other, but the current and the magnet come out the same: against the field, every time. Its strength is the particle's kinetic energy divided by the field.

So here is the prediction the classical picture makes. Every metal is full of free electrons. Put the metal in a field and each electron turns into one of these small back-pushing magnets. The metal should be repelled by magnets, and strongly: at room temperature the electrons are fast, and the magnet each one makes grows with its energy. Test it. The box below holds 20,000 electrons in a square of side 60 in the page's units, moving as fast as heat makes them, not touching each other. A field points out of the screen.

The blue bar is the prediction coming true. The blue electrons are the ones whose circles fit inside the box without touching a wall, and together they make a magnet pointing against the field, exactly as strong as the formula says (the note under the bars works it out from their speeds, with no simulation, and the running average agrees). At the page's starting field, 92 per cent of the electrons are blue.

The other eight per cent are the orange ones. Their circles meet a wall, so they never finish one. They bounce, start a new arc, bounce again, and hop along the wall in a string of scallops, and because every one of them turns the same way, they all hop round the box in the same direction: the opposite direction to the way each blue electron circles. Together they are one enormous loop of current running round the inside of the walls, and a loop's magnet grows with the area it encloses. There are few of them, but they go round everything. Watch the bars for a while: the orange bar grows to the length of the blue one, pointing the other way, and the whole box settles at zero, give or take the scatter you would expect from a box of 20,000 (the ±).

This picture is more than a century old. Hendrika van Leeuwen, whose proof of the theorem is the second half of its name, credited it in 1921 to the lectures of H. A. Lorentz at Leiden, and noted who had missed it:

“Lorentz a démontré dans ses leçons professées à l'université de Leyde, que les électrons intérieurs, dont les projections sur un plan perpendiculaire aux lignes de force parcourent des cercles complets. correspondent à un moment négatif, mais les électrons près de la périphérie du métal correspondent à un moment positif de la même grandeur [...]. Les autres auteurs cités n'ont pas fait attention à ce dernier moment.”

H.-J. van Leeuwen, “Problèmes de la théorie électronique du magnétisme”, J. Phys. Radium 2 (1921), 361 to 377, p. 376; transcribed from the page image. In English (our translation): Lorentz showed in his lectures at the University of Leiden that the inner electrons, which go round complete circles, correspond to a negative moment, but the electrons near the edge of the metal correspond to a positive moment of the same size. The other authors cited paid no attention to this last moment.

John Van Vleck drew the same picture in his 1932 textbook on magnetism and put it more bluntly: “These boundary electrons are very vital, as without them there would be diamagnetism.”

Not just this box: every field

Slide the field and the balance changes completely. A weak field makes big circles, and most of them touch a wall: at the slider's weakest setting only 6 per cent of the electrons circle clear. A strong field makes tiny circles and almost everyone is blue. The strength of the blue magnet rises and falls with the field, largest near B = 0.17, where the circling electrons would give 2.45 per electron against the field if they were alone. The curve below is that prediction, worked out as a one-line integral over the speeds heat gives the electrons. The orange curve is its mirror image. Each field you visit leaves its measured points on the chart.

The orange crowd matches the blue one at every field, not because the wall electrons were tuned to, but because they cannot help it. Here is why, in the form the textbooks give it. In thermal equilibrium, how likely any arrangement of the electrons is depends only on its energy. A magnetic force never speeds anything up or slows it down, so it never changes anyone's energy. The field changes which way every electron goes, but not how likely any position or speed is; and if nothing about the likelihoods depends on the field, then nothing measured by averaging over them can depend on it either, magnetism included. In the notation of a lecture course: an electron's energy in a field is written as (p + eA)²/2m, where A is the field's vector potential, and the likelihood of every arrangement is summed over all momenta p. Rename p + eA as a new momentum p′, and the field has vanished from the sum. David Tong's Cambridge notes give the name:

“This result [...] is known as the Bohr-van Leeuwen theorem: it states that there can be no classical magnetism.”

David Tong, Statistical Physics, University of Cambridge Part II lecture notes, section 3.6.7, “Landau Diamagnetism”.

Nothing in that argument cares about the shape of the box, how many electrons there are, or what forces they exert on each other, provided those forces depend only on where the electrons are. A box of any shape, full of any charges pushing on each other in that way, settles to no magnetism at all. The simulation is the same argument done by counting: the blue electrons are where the classical picture of a magnet lives, and the orange ones are the part of the classical picture nobody draws.

One honest limit on that reach. The argument uses only the field applied from outside. It leaves out the magnetic fields the moving electrons make themselves, and the forces those fields exert on the other electrons. Hanno Essén and Miguel Fiolhais, in a review published in the American Journal of Physics in 2012, argue that once those are kept (in a form of the energy due to Charles Galton Darwin), the theorem no longer holds and classical physics does give diamagnetism; they count “There is no classical diamagnetism” among the myths that textbooks repeat. The simulation here cannot settle that either way: its electrons ignore each other completely. What it shows is the theorem as it is usually stated, and why it is true on its own terms.

The most deflationary thesis

Niels Bohr worked this out in his 1911 doctoral dissertation on the electron theory of metals (Studier over Metallernes Elektrontheori), two years before his theory of the hydrogen spectrum. Hendrika Johanna van Leeuwen proved it independently, by a slightly different method, in her Leiden dissertation of 1919, and published a summary in French in 1921; a footnote there says that only after her thesis was published did she learn of “une remarque analogue” in Bohr's. The same paper thanks Paul Langevin for checking her French, which is a quiet irony: the theorem shows that Langevin's own theory of magnetism was not the classical theory it seemed, as the next paragraphs explain. John Van Vleck, in his Nobel lecture of 1977:

“If one applies classical dynamics and statistical mechanics consistently, a very simple calculation, which can be made in only a few lines but I shall not reproduce it here, shows that the diamagnetic and paramagnetic contributions to the susceptibility exactly cancel. Thus there should be no magnetism at all. This appears to have been first pointed out by Niels Bohr (4) in his doctor’s dissertation in 1911, perhaps the most deflationary publication of all time in physics.”

J. H. Van Vleck, “Quantum mechanics: the key to understanding magnetism”, Nobel lecture, 8 December 1977; his note 4 credits “Miss J. H. van Leeuwen, Dissertation, Leiden 1919, J. de Physique 2, 361 (1921)”.

And yet magnets exist: iron holds up the shopping list on the fridge. Something classical physics does not have is doing it. Two things, it turned out, and both are quantum.

The first is that the circles themselves come in steps. In quantum mechanics an electron in a field cannot circle with any energy it likes, only in a ladder of allowed levels, and once the circles are counted that way the cancellation between the circling electrons and the wall electrons is no longer perfect. Lev Landau worked it out in 1930; in Van Vleck's words, he “showed that spinless free electrons had a small susceptibility of diamagnetic sign, in contrast to the zero result of classical mechanics.” A small, real repulsion by magnets, now called Landau diamagnetism.

The second is much stronger, and it is the one your fridge magnet runs on. The electron is a magnet in itself: it has a magnetic moment that is not a charge going round anything. That property is called spin, and it is exactly what the proof above has no room for, because the proof only knows about charges and where they move. Van Vleck said the same of the classical theory of paramagnetism, in which Paul Langevin had given each atom a fixed magnetic moment in 1905: “When Langevin assumed that the magnetic moment of the atom or molecule had a fixed value µ, he was quantizing the system without realizing it”. A fixed moment is a quantum object smuggled into a classical calculation. In iron, the spins of neighbouring atoms line up with each other, and what lines them up is not magnetism either: that part of the story, and the number that proves it, are on What Holds a Magnet Together.

The rod that turned half as much

In 1915, four years after Bohr's thesis, Albert Einstein and Wander de Haas set out to show that the magnetism of iron is made of circulating electrons. If magnetism is electrons going round inside atoms (the molecular currents André-Marie Ampère had proposed at the beginning of the nineteenth century), then every one of those circulating electrons carries angular momentum as well as magnetism, in a fixed ratio. Flip the magnetization of an iron rod and you flip all that angular momentum, and since angular momentum is conserved, the rod as a whole has to turn the other way. How much it turns tells you the ratio. For an electron going round a circle the ratio is fixed at 2m/e, twice its mass over its charge, which they put at 1.13 × 10⁻⁷ in the units of the day.

They hung a soft-iron cylinder 1.7 mm thick on a glass wire, in the Physikalisch-Technische Reichsanstalt, reversed its magnetization back and forth in resonance with its natural twisting swing so that the tiny kicks built up, and reported in the Proceedings of the Amsterdam Academy, under the title “Experimental proof of the existence of Ampère's molecular currents”:

“λ = 1,1 . 10⁻⁷, which agrees very well with the theoretical one 1,13. 10⁻⁷. We must observe, however, that we cannot assign to our measurements a greater precision than of 10%.”

A. Einstein and W. J. de Haas, “Experimental proof of the existence of Ampère's molecular currents”, KNAW Proceedings 18 I (1915), 696 to 711, p. 711; transcribed from the page image.

The measurements that followed, by physicists in Europe and America, most of them using the same resonance method, did not agree. V. Ya. Frenkel', in a history of the episode published in 1979:

“The subsequent studies [...] have unambiguously shown the value of λ to be 0.57×10⁻⁷. This is half the value found by Einstein and de Haas (1.11×10⁻⁷), which corresponded to their simple theory that seemed unconditionally correct.”

V. Ya. Frenkel', “On the history of the Einstein-de Haas effect”, Soviet Physics Uspekhi 22 (1979), 580 to 587, p. 585; transcribed from the page image.

Frenkel is careful about why. Einstein and de Haas, he notes, paid special attention to possible errors; but “Evidently, when the value of λ that they had calculated from the experimental data approached that expected from Eq. (1), they considered their work complete.” The half has an explanation that did not exist in 1915. In iron and the other ferromagnets, as Frenkel puts it, “the spin component of the magnetic moment plays the dominant role”, and for spin the ratio is not 2m/e but half of it: the electron's spin carries twice as much magnetism for its angular momentum as an orbiting charge does (in the physicists' shorthand, g = 2 rather than g = 1). Spin was proposed by George Uhlenbeck and Samuel Goudsmit in 1925, ten years after the rod turned.

Which leaves the title. The experiment was called a proof of Ampère's molecular currents, circulating charges, the very thing Bohr's theorem says cannot make a magnet. Its correct result shows the opposite. Frenkel again: “the experiments of the Einstein-de Haas effect have demonstrated the absence of a contribution to this effect from the orbital motion of the electrons. Hence, formally these experiments cannot be considered as a "proof of the existence of molecular Ampère currents!"” The rod did turn: magnetism really is angular momentum. It turned half as much as circulating charges would make it, because the angular momentum is spin.

Sources

1. D. Tong, Statistical Physics, lecture notes, University of Cambridge, section 3.6.7, damtp.cam.ac.uk/user/tong/statphys.html: the proof by a shift of variables, and Landau diamagnetism.
2. J. H. Van Vleck, “Quantum mechanics: the key to understanding magnetism”, Nobel lecture, 1977, nobelprize.org.
3. N. Bohr, Studier over Metallernes Elektrontheori, doctoral dissertation, Copenhagen, 1911; H. J. van Leeuwen, dissertation, Leiden, 1919, and “Problèmes de la théorie électronique du magnétisme”, J. Phys. Radium 2 (1921), 361, doi:10.1051/jphysrad:01921002012036100. (The 1921 paper was read for this page, from the scan; Bohr's thesis and the 1919 dissertation were not, and are cited as Van Vleck cites them.)
4. J. H. Van Vleck, The Theory of Electric and Magnetic Susceptibilities (Oxford, Clarendon Press, 1932), archive.org scan.
5. L. Landau, “Diamagnetismus der Metalle”, Z. Phys. 64 (1930), 629, doi:10.1007/BF01397213.
6. H. Essén and M. C. N. Fiolhais, “Meissner effect, diamagnetism, and classical physics: a review”, Am. J. Phys. 80 (2012), 164 to 169, doi:10.1119/1.3662027; preprint arXiv:1109.1968.
7. A. Einstein and W. J. de Haas, “Experimental proof of the existence of Ampère's molecular currents”, KNAW Proceedings 18 I (1915), 696 to 711, dwc.knaw.nl.
8. V. Ya. Frenkel', “On the history of the Einstein–de Haas effect”, Sov. Phys. Usp. 22 (1979), 580, doi:10.1070/PU1979v022n07ABEH005587.
9. G. E. Uhlenbeck and S. Goudsmit, “Ersetzung der Hypothese vom unmechanischen Zwang durch eine Forderung bezüglich des inneren Verhaltens jedes einzelnen Elektrons”, Naturwissenschaften 13 (1925), 953, doi:10.1007/BF01558878.