The magnetism classical physics forbids

A bar magnet is the most ordinary object in physics. It is also, on the strictest reading of classical mechanics, impossible.

The claim is not that classical physics gets magnetism wrong by a factor, or that it works at high temperature and fails at low. It is that classical statistical mechanics predicts exactly zero magnetisation, at every field, at every temperature, for every arrangement of charges — and the proof takes two lines.

The two lines #

The energy of a classical charge in a magnetic field is

$$H = \frac{(\mathbf{p} - q\mathbf{A})^2}{2m} + V(\mathbf{r}),$$

with the vector potential $\mathbf{A}$ carrying the field and the potential $V$ carrying everything else — the nucleus, the neighbours, the walls of the container. The partition function is the integral of $e^{-\beta H}$ over every position and every momentum.

Hold the position fixed and do the momentum integral. The substitution $\mathbf{p}' = \mathbf{p} - q\mathbf{A}(\mathbf{r})$ shifts the variable by a constant, and the domain of integration is all of momentum space, which a shift maps onto itself. So the momentum integral returns the same number it would have returned with no field at all, whatever $\mathbf{A}$ is, and the position integral never contained $\mathbf{A}$ in the first place.

The free energy therefore has no $B$ in it. The magnetisation is $-\partial F/\partial B$. It is zero.

That is the theorem Niels Bohr proved in his doctoral thesis in 1911 and Hendrika van Leeuwen proved again, independently, in hers in 1919 — and which neither of them could get anybody to care about for twenty years, because by the time it mattered the subject had moved on to explaining magnetism with an assumption the theorem forbids.

Its reception is worth a sentence, because it is the ordinary fate of a negative result. Bohr’s thesis was written in Danish and never translated in his lifetime; van Leeuwen’s was published in a journal, in French, and cited for a decade mostly as a curiosity about a cancellation. Neither reads as a crisis, and neither was treated as one. A theorem that says a whole class of explanation cannot work does not tell anybody what to do next, so it waits until somebody arrives with the thing it was clearing space for — which in this case was an angular momentum with a fixed length, fifteen years later.

What it disposes of #

The result is more destructive than it first looks, because the standard classical stories about magnetism come in two flavours and it kills both.

The paramagnetic story. Each molecule carries a little magnet; the field lines them up; thermal agitation knocks them about; the competition gives a magnetisation that rises with field and falls with temperature. That story is Langevin’s, from 1905, and it produces the observed Curie law. Its hidden assumption is a moment of fixed magnitude. A classical charge distribution has no such thing: its moment is $\tfrac{q}{2}\,\mathbf{r} \times \mathbf{v}$, and switching on a field changes $\mathbf{v}$ — that is the whole of what a field does to a charge.

The diamagnetic story. Switching the field on drives a current round each orbit which opposes the change, by the rule that a circuit answers a changing field by opposing it; the induced moments are negative; every material should therefore be weakly repelled. That story is also wrong, and it is wrong in a way worth seeing rather than merely being told.

A charge in a uniform field goes round a circle of radius $mv/qB$, and each such orbit does carry a magnetic moment. What the theorem disposes of is the idea that those moments add up to anything: the orbits in the interior carry one sign and the skipping orbits along the boundary carry the other, and the two cancel exactly. Not approximately, and not to leading order — exactly, for any container and any field.

The boundary term is the part every intuitive account leaves out, and it is not small: it is the whole of the interior term with the sign reversed. A calculation of the induced moments that stops at the closed orbits gets a diamagnetic answer of exactly the right magnitude and the wrong conclusion, because it has integrated over part of the phase space and called the result a property of the substance.

The measurement the theorem has to explain away #

None of this would matter if magnetism were subtle. It is not. A paramagnetic salt in a strong field at a low temperature magnetises almost completely, and the curve it follows is one of the best-measured things in low-temperature physics.

The quantum calculation differs from Langevin’s in exactly one respect. Langevin averages the moment’s component along the field over all directions on a sphere, which is an integral; the quantum calculation averages over $2J+1$ allowed values, which is a sum. Everything else — the Boltzmann weighting, the competition with temperature, the shape of the answer — is identical.

A moment sent through a field gradient does not spread into a band; it splits into a discrete set. That is the measurement the theorem has to survive and does not: the integral over orientations that produces the cancellation assumes a continuum of them, and there is no continuum. Replace the integral by a sum over two values and the cancellation fails.

And the sum is what supplies the moment of fixed magnitude Langevin needed. A quantum angular momentum has a length that is a property of the state and not of the field, so it is the object classical mechanics could not provide.

Why the classical answer looked right for forty years #

Because the two theories agree on the one thing a nineteenth-century laboratory could measure.

The collapse onto a function of $B/T$ is Curie’s law in its raw form, and it holds for both theories because both are a Boltzmann average of a moment energy $-\mu B$ over a temperature $T$, and the ratio is the only place either quantity appears. So Curie’s measurement confirmed the structure of Langevin’s argument without touching its assumption.

A state costing an energy above the ground state is occupied in proportion to $e^{-E/kT}$, and that same exponential does the work in both accounts. The classical answer looked right because the shape of the resulting curve is the same — a susceptibility falling as one over the temperature — so Curie’s law was reproduced by a theory that predicts no magnetism at all. Getting the shape right is not evidence.

The instrument that comes out of it #

The weak-field limit of any of those curves is a straight line, and the reciprocal of the susceptibility is straighter still.

That is the standard method for finding how many unpaired electrons an ion has, and it is worth noticing what it depends on. The slope contains $J(J+1)$ rather than $J^2$, which is the quantum expression for the square of an angular momentum and not the classical one. A measurement of the Curie constant is therefore a measurement of $\hbar$ in disguise, in the same way that a heat capacity that comes out wrong for hydrogen is a thermometer measuring Planck’s constant.

The departures from the straight line are as informative as the line. A material whose reciprocal susceptibility hits zero at some positive temperature rather than at the origin has moments that have begun to notice each other, and the intercept is roughly the temperature at which they will order.

Where the moments come from #

Nothing above says what carries the moment. Two things do, and they contribute differently.

A spectrum is a difference of levels, and the same discreteness that produces spectral lines produces the moments. That is where they come from: an atom’s electrons occupy states with definite angular momenta, each carrying a moment fixed by the Bohr magneton, and the available orientations are counted rather than continuous.

The orbital motion of an electron contributes a moment proportional to its orbital angular momentum. Its spin contributes another, twice as large per unit of angular momentum, which is an angular momentum that is not the angular momentum of anything going round. In a free ion the two combine into a total $J$ with a $g$-factor between one and two; in a solid the orbital part is often largely quenched by the electric field of the neighbours, which is why so many transition-metal salts behave as though their moments were pure spin.

A magnetic field has no ends, and a current loop’s field has the right shape for a moment — so the classical picture is right about the geometry and wrong about everything else. It cannot say why the moment has the size it has, why it has a discrete set of orientations, or why the electron’s gyromagnetic ratio is twice what a rotating charge would give.

Conduction electrons in a metal behave differently again, and the difference is the second place the theorem’s shadow falls.

Only the fraction $T/T_F$ of the electrons near the top of the distribution can change state at all, which is why a metal is a feeble paramagnet. Every deeper electron has nowhere to be promoted to, so the susceptibility is smaller than Curie’s law would give by a factor of about a hundred at room temperature — and it is nearly independent of temperature, which Curie’s law is not.

What it costs #

The theorem is exact, and its exactness is bought with three conditions, each of which is worth naming because each is a place a real system leaves the argument.

The charges must be classical. That is the point, and the escape.

The momentum integral must run over all momenta. In a strictly two-dimensional system with a boundary, the argument still holds — the boundary orbits are what cancel the interior — but the cancellation is between two large terms rather than a term and nothing, which is why a finite quantum system can have a large orbital response that a naive bulk calculation misses entirely.

The statistics must be Boltzmann. Substituting Fermi–Dirac statistics changes the counting, and Landau’s diamagnetism of a free electron gas — exactly one third of the Pauli paramagnetism, with the opposite sign — has no classical limit at all. That factor of three is a good measure of how far the quantum answer is from nothing: the two effects come from the same electrons, in the same metal, at the same field, and they are the orbital motion and the spin of the same particle. Classically both are zero and their ratio is undefined. Quantum-mechanically both are finite, of opposite sign, and their ratio is a pure number that contains neither the field, the temperature nor the density of the metal.

The oscillation classical physics has no room for #

The theorem forbids a magnetisation. The sharpest refutation of it is not a magnetisation that is merely non-zero but one that oscillates — going up and down as the field is raised, many times, with a regularity that nothing continuous could produce.

Put a pure metal at a low temperature in a strong field and measure its magnetisation as the field is swept. It is not a smooth curve. It wobbles, periodically — not in the field itself but in the field’s reciprocal — and the wobbles can be counted.

The cause is that a magnetic field does not leave an electron’s allowed states alone. The circular orbits the classical picture drew are quantised, so their energies form a ladder whose rungs are spaced by the cyclotron energy, and raising the field spreads the ladder out. Each time a rung passes up through the Fermi level, a set of states empties, and every quantity that depends on the states near the Fermi level jumps a little.

Since the rungs are evenly spaced in energy and the spacing is proportional to the field, the passages are evenly spaced in $1/B$ — which is why the oscillation is periodic in the reciprocal.

What makes the effect an instrument rather than a curiosity is what the period contains. It is fixed by the area of the Fermi surface’s cross-section perpendicular to the field, at the place where that area is extremal. So measuring a period gives an area; turning the sample and measuring again gives another area in another direction; and enough directions reconstruct the shape of the Fermi surface — a three-dimensional surface in momentum space, mapped by counting wobbles in a magnetisation.

That is the de Haas–van Alphen effect, found in bismuth in 1930 and now the standard method for the job. It demands a pure sample and a low temperature — the electron must complete an orbit before it scatters, and the ladder’s rungs must be sharper than $k_BT$ — and where those are met it is the most direct measurement of a metal’s electronic structure there is.

Set against a theorem that predicts exactly zero, an oscillating magnetisation whose period measures a geometry in momentum space is about as far from a small correction as an experimental result can be.

The perfect diamagnet #

The theorem’s other spectacular failure is a material that expels a field entirely, and it is worth adding because it fails the classical argument in a second, independent way.

A superconductor below its transition temperature has a magnetic susceptibility of exactly $-1$: the field inside it is zero, and it achieves that by carrying surface currents that cancel the applied field throughout its interior. Against the parts-per-million susceptibilities of ordinary diamagnets, that is a response larger by five orders of magnitude, in a direction the classical theory says is unavailable at any size.

The detail that makes it decisive is not the expulsion itself but when it happens. A merely perfect conductor would, by induction, oppose any change of field — so a sample cooled in a field would keep that field frozen inside it, and a sample cooled in zero field and then placed in one would exclude it. The two histories would give different final states.

They do not. A superconductor cooled in a field expels it as it crosses the transition, arriving at the same state as one cooled first and magnetised afterwards. That is the Meissner effect, found in 1933, and it establishes that the state is a thermodynamic equilibrium rather than a consequence of a history — which is exactly what a partition function is supposed to describe, and exactly what the classical partition function says cannot happen.

The mechanism is the same one as everywhere else in this essay: a macroscopic number of electrons share a single quantum state with one phase, and the current is fixed by that phase’s gradient rather than by anything a classical charge could do. The theorem’s assumption fails at the first word.

Where the model stops #

Interaction is left out entirely. Everything above treats the moments as independent, and a ferromagnet is what happens when they are not. The exchange interaction that lines neighbouring spins up is not magnetic in origin at all: it is electrostatic repulsion plus the requirement that the total wavefunction change sign under exchange, which is the rule that gives matter its volume. A mean-field treatment of it turns the Curie law into the Curie–Weiss law and predicts an ordering temperature, and its failures near that temperature are a subject of their own.

Saturation needs conditions no ordinary experiment reaches. The curves part company where $g J \mu_B B$ approaches $k_B T$, which at a laboratory field of one tesla is a temperature below about one kelvin. Above that, every one of the Brillouin curves and the Langevin curve are straight lines through the origin differing only in slope, and a slope alone cannot say which is which without an independent measurement of the number of moments.

The $g$-factor has been treated as a number. In a crystal it is a tensor, it depends on direction, and the anisotropy it encodes is what makes a magnet have an easy axis at all — which is where the shape of a magnet decides its working point and where hysteresis comes from.

None of the curves in this essay has any memory in it. A real permanent magnet’s response depends on where it has been, and the area of its loop is an energy paid per cycle — so the whole treatment here describes a paramagnet and says nothing about the magnets anybody owns. That is where the model stops, and it stops well short of the phenomenon most people mean by magnetism.

And the theorem says nothing about what magnetism is. It is a prohibition, not a mechanism. It rules out a whole family of explanations and leaves the field clear; what actually fills the gap is angular momentum that comes in units, and the reason it comes in units is not magnetic.

What the picture cannot show #

The hero figure draws a displacement in momentum space and the numbers beside it are the integral’s answer, and neither is a picture of a magnet. Nothing in it is an object; the circles are contours of a Boltzmann weight over a variable that is not a position, and the displacement $q\mathbf{A}$ is not gauge-invariant — a different choice of vector potential moves it somewhere else and leaves every physical quantity alone. That the answer is independent of the choice is the deeper statement, and a drawing of one gauge cannot make it.

Nor can the Brillouin curves show what is discrete. They are smooth functions of a continuous field, and the discreteness lives in the sum that produced them. A figure of a smooth curve is a poor advertisement for quantisation, and the only honest signature in the drawing is the initial slope: the number $(J+1)/3J$ is a fingerprint of how many terms were in the sum.

Where this ladder goes next #

Two rungs stand on magnetisation now. The first asked what a magnet’s own field does to it and found the answer in its shape; this one asks what a magnet is and finds that the question has no classical answer at all. The pair have a shape worth noticing — one is about a body and one is about a constituent, and neither could have been reached from the other.

The habit to carry away is about the domain of integration. Both wrong stories above, the paramagnetic and the diamagnetic, come from averaging over part of a phase space and reporting the result as a property of matter. Langevin averaged over a sphere of orientations that is not there; the induced-moment argument averaged over closed orbits and left out the ones that hit the wall. When an argument produces a physical property by averaging, the first question is what the average runs over and whether anything has been left outside it.

What is left on this ladder is the interaction. Two moments a lattice spacing apart influence each other by an energy a thousand times larger than their magnetic interaction, for reasons that are not magnetic; the ordering that follows has a temperature, a critical exponent and a correlation length, and the mean-field treatment that predicts the first gets the other two wrong in a way that took a century to fix.