What holds a magnet together is not magnetism

A bar magnet is made of atoms that are themselves tiny magnets, and the obvious story about why iron is magnetic is that the tiny magnets have lined each other up. It is the story every diagram of a ferromagnet tells, with its rows of little arrows all pointing the same way, and it has the great virtue of needing nothing that has not already been introduced.

It is also arithmetically impossible, and the arithmetic takes one line.

Two magnetic moments a distance $a$ apart interact with an energy of order $\mu_0\mu^2/4\pi a^3$. Iron’s atoms carry 2.22 Bohr magnetons each and sit 2.48 ångströms apart, and putting those numbers in gives $2.8\times10^{-24}$ joules. Divided by Boltzmann’s constant that is a fifth of a kelvin — which is to say that an arrangement of moments held in line by their own magnetic fields would be shaken apart by any temperature above that of liquid helium.

Iron is magnetic at room temperature. It is magnetic in a furnace. It stays magnetic to 1,043 K, above which it stops being magnetic all at once, and 1,043 K is five thousand times the only energy the moments have available to hold each other with.

So the picture of little magnets lining each other up is not approximately right, or right in spirit, or right for the wrong reason. Whatever holds a ferromagnet together is a different interaction, it is thousands of times stronger than magnetism, and finding out what it is took the better part of thirty years after the problem was posed.

The material that passes the test #

A comparison across five materials that all fail the same way is weaker evidence than it looks, because it could be a mistake in the comparison rather than a fact about the materials. So the figure carries a sixth.

Lithium holmium fluoride orders at 1.53 K. Its dipolar scale, computed exactly as the others were — holmium’s ten Bohr magnetons, the same expression, the same constant — is 1.23 K. The two agree within a quarter, and they agree because in that material the dipole interaction really is what does the ordering. Holmium’s magnetic electrons are buried in the 4f shell, they overlap almost not at all with their neighbours’, and the strong interaction the other five have is simply absent. What is left is magnetism, and magnetism orders it at the temperature magnetism says it should.

That is the control the argument needs. The arithmetic is not systematically wrong, the constant is not misplaced, and a material whose ordering is genuinely magnetic lands on the line. The five that miss by decades miss because something else is holding them.

It is worth putting the size of that something else into the unit a magnet is usually described in. To align iron’s moments against thermal agitation at 1,043 K by an applied field alone would take a field of about a thousand tesla — a hundred times the strongest continuous field ever produced in a laboratory, and ten times the strongest achieved by blowing up a coil.

Weiss wrote that field down in 1907 and called it the molecular field. He could fit the whole of ferromagnetism with it, he could get the Curie temperature right, and he had no idea what it was — only that it was enormous and that nothing in electromagnetism could supply it.

A spin energy out of a Hamiltonian with no spin in it #

What supplies it was found by Heisenberg and Dirac in 1926, within months of the exclusion principle being understood, and the cleanest way to see it is in the smallest system that has it: two sites, two electrons.

The model has two ingredients. An electron on one site can hop to the other, with amplitude $t$. Two electrons on the same site repel, with energy $U$. There is no magnetic field, no moment, no dipole and no interaction between spins anywhere in it.

Now count the states. If the two electrons have opposite spins, one may hop onto the other’s site — paying $U$, but gaining the energy that comes from being able to move at all. If they have the same spin, that hop is forbidden, because the doubly occupied state would have two identical electrons in one orbital and no two electrons may be in the same state. The parallel arrangement is stuck.

Being able to move lowers an energy. The antiparallel arrangement can borrow the doubly occupied configuration virtually — hop across and back — and that second-order process lowers it by $4t^2/U$, which is what the figure computes and confirms. The parallel arrangement cannot, so it is left where it was.

So a Hamiltonian with no spin in it produces an energy difference between two spin arrangements, and it does so because the spins decide which orbital motions are permitted rather than because they pull on one another. The energy being decided between is the kinetic and electrostatic energy of the electrons, which is a chemical energy, and that is why the answer comes out thousands of times larger than anything magnetic.

Dirac’s way of writing the result is the one the subject has used ever since. Whatever the actual mechanism, an energy that depends on whether two spins are parallel can be written as $-2J\,\mathbf{S}_1\!\cdot\!\mathbf{S}_2$, which looks exactly like two little magnets interacting and is nothing of the kind. The form is forced by symmetry — it is the only rotationally invariant scalar two spins can make — and the constant $J$ in front of it is a number about orbitals and Coulomb integrals with no magnetic content whatever. A great deal of confusion about magnetism comes from reading that expression as the thing it resembles.

The same statement in space #

The spectrum makes the point with numbers; the wavefunction makes it with a picture, and the picture is worth having because it says where the energy difference physically is.

A two-electron state must change sign when the two electrons are exchanged. The spin part of a triplet is symmetric under that exchange, so its spatial part must be antisymmetric; the singlet’s spin part is antisymmetric, so its spatial part is symmetric. That is the whole of the connection between spin and space, and it is the same bookkeeping that makes one of a pair of spins odd.

An antisymmetric function of two coordinates vanishes when the coordinates are equal. So two electrons in the triplet state are never found at the same place, and — because the function is smooth — are unlikely to be found near each other either. The dip around the other electron is the Fermi hole, and it is not a force keeping them apart. It is a property of the only functions the exclusion principle permits.

Electrons that stay apart pay less repulsion. The figure integrates $1/r$, softened so that it is integrable in one dimension, over both states and finds the triplet cheaper by two and a half per cent of a large number. That difference is the exchange energy: an electrostatic quantity, of electrostatic size, that appears to depend on spin because spin decides which spatial function is legal.

The general habit this establishes is worth more than the result. When a quantity appears to depend on something it has no interaction with, look for a constraint rather than a force. Here spin does not couple to anything; it restricts which states are available, and a restriction on the available states is as good as an interaction for producing an energy difference. The same move explains why matter has volume without any repulsive force between electrons, and it is why the exclusion principle is spoken of as a force by people who know perfectly well that it is not one. Spin itself is the same sort of object — an angular momentum that is not the rotation of anything, and whose whole physical work here is to label which spatial state is permitted.

Two mechanisms, two signs, and the case iron is #

The two-site model gives an antiferromagnet. Its singlet is the ground state at every value of $U$, which means two electrons sharing a bond prefer opposite spins — and most insulating magnetic compounds are indeed antiferromagnets for exactly that reason.

Iron is not. So the model that explains why the interaction is large does not, by itself, explain the sign of the case everybody starts from, and it is worth being explicit about that rather than letting the argument run past it.

There is a second mechanism with the opposite sign. Within a single atom, two electrons in different orbitals are not competing for the same place, so the kinetic term above does not arise; what is left is the bare exchange integral, which always favours the parallel arrangement because the Fermi hole reduces the repulsion. That is Hund’s first rule, it is why an isolated iron atom has a large moment at all, and it is a genuinely ferromagnetic exchange.

In a solid both are present. Electrons on neighbouring atoms can hop, which favours antiparallel; electrons in different orbitals on the same atom favour parallel; and which wins is a quantitative question that depends on the overlaps, the orbital energies and the filling. Iron, cobalt and nickel win it on the ferromagnetic side and their immediate neighbours in the periodic table do not, which is why there are three elemental ferromagnets rather than thirty. The competition is a quantitative one of exactly the kind a classical statistical mechanics that forbids magnetism altogether cannot even begin, since it has no exclusion principle to do the deciding.

It has to be said that the localised picture used throughout this essay is not quite what iron is. Its moment is 2.22 Bohr magnetons per atom, and 2.22 is not a number of electrons. The magnetic electrons in a metal are in bands rather than on atoms, the moment is a difference between two slightly unequal band fillings, and a description that assigns a fixed spin to each site is a caricature of that. What survives the change of description is the size and the origin of the energy: it is electrostatic, it is chemical in magnitude, and the exclusion principle is what makes it depend on spin. The caricature gets those right, which is why it is still the standard language.

What a Curie point is worth in millielectronvolts #

With the mechanism identified, the measured ordering temperatures can be turned round and asked what exchange constant they imply.

The mean-field answer is that each spin sits in the average field of its neighbours, which gives $k_BT_c = \tfrac{2}{3}zJS(S+1)$ and a Curie point proportional to $J$ and to the number of neighbours. Read backwards, iron’s 1,043 K needs about eight millielectronvolts.

Eight millielectronvolts is a small chemical energy — a hundredth of a typical covalent bond, a third of $k_BT$ at room temperature. It is also five hundred times the 0.017 meV of the dipolar coupling between the same two moments. That ratio is the essay’s whole claim, arrived at from the opposite direction: rather than computing the magnetic interaction and finding it too small, one takes the measured ordering temperature and asks what interaction it implies, and the answer lands in the range of chemistry rather than the range of magnetostatics.

The exercise also shows how weakly the ordering temperature constrains anything. Changing the coordination number from four to twelve changes the implied $J$ by a factor of three, and real lattices differ by that much; the mean-field relation is a way of putting a measurement into an energy unit rather than a determination of an exchange constant. Better determinations come from the shape of the magnetisation at low temperature and from neutron scattering, and they disagree with the mean-field value by tens of per cent.

Where the mean field is easiest to catch #

Mean-field theory has a reputation for getting the transition temperature roughly right and the critical behaviour wrong, and that is true. It is more useful to know that it is also wrong at the other end of the temperature range, where no critical exponent is involved and the failure is enormous.

The mean field allows a magnet one kind of excitation: a spin turns over, against the average field of all its neighbours, at a cost of order $k_BT_c$. That is an enormous energy at low temperature, so the predicted number of overturned spins is $e^{-1565/T}$ — utterly negligible at 20 K, and negligible at 100 K, and the predicted magnetisation is flat across the whole low-temperature range.

Measured magnetisation is not flat. It falls as $T^{3/2}$ from the lowest temperatures anybody can reach, and it does so because the cheapest excitation is not a flipped spin at all. It is a long wave: every spin tilted slightly, with the tilt precessing round the lattice, and the energy of such a wave goes to zero as its wavelength grows. There is no gap, so there is no temperature low enough for the excitations to be frozen out, and the deficit is a power of the temperature rather than an exponential.

The figure computes that by summing the thermal occupation of every spin wave in the Brillouin zone, with the exact lattice dispersion, and recovers the exponent 1.51 rather than assuming it. Two things are worth taking from it. The first is the size of the discrepancy: thirty-one decades is not a correction, and any experiment that measures the low-temperature magnetisation at all distinguishes the two accounts immediately. The second is why the mean field misses it. Averaging over a spin’s neighbours destroys precisely the correlations that let a disturbance spread as a wave, and a theory that cannot represent a collective excitation will always predict a gap where there is none.

That failure is the same one, seen from a different side, as the theory’s failure at the transition. Mean field replaces the fluctuating environment of each spin by its average, so it has no long-wavelength physics at either end — no spin waves at the bottom and no diverging correlation length at the top. Its successes are the quantities that do not depend on either, and the transition itself is not one of them.

The cheapness of a long wave is not peculiar to magnets. It is the same statement as a lattice of masses having modes that cost arbitrarily little as their wavelength grows, and it is why a solid’s heat capacity rises as a power of the temperature rather than switching on at a threshold: whenever the excitations of an ordered state are the slow distortions of the order itself, there is no gap, and counting them is what a heat capacity is.

The energy that also decides a shape #

There is a consequence of all this for everything built on top of it, and it is the reason the exchange interaction is not merely a footnote about origins.

A magnet’s own field opposes its magnetisation and is what makes a short fat magnet weak, and that field is genuinely magnetic — it is the dipolar interaction, summed over the whole body. So a ferromagnet contains two interactions of wildly different strengths: an exchange interaction that is enormous and acts only between neighbours, and a dipolar interaction that is feeble and acts across the whole sample. Both matter, because the second is summed over an astronomical number of pairs and the first is not.

That competition is where domains come from, and it is where this subject goes next. It also sets the internal structure of the boundary between them: the exchange interaction wants a reversal spread out over as many atoms as possible, because it penalises any angle between neighbours, and something else wants it compressed. The length where those balance is the first length in magnetism that belongs to the substance rather than to the shape of the sample — and the avalanches a magnetisation curve is made of are that boundary moving.

Two sites are not a metal #

The two-site model is not iron. It has two orbitals, two electrons and one hopping amplitude, and it predicts an antiferromagnet. What it establishes is the origin and the order of magnitude of a spin-dependent energy; it does not predict the sign for any real material, and getting the sign right for the three elemental ferromagnets is a calculation that still needs care.

The moments are treated as localised and in iron they are not. A metal’s magnetic electrons are shared, the moment per atom is not an integer, and the excitations at high temperature are not the ones a lattice of fixed spins would have. The Stoner picture, in which ferromagnetism is a rearrangement of the bands themselves, gets the moment right and the Curie temperature badly wrong; the localised picture does the reverse. Neither is the whole answer and the disagreement between them is an active subject.

The dipolar arithmetic is for one pair. The figure computes the interaction between two neighbours, which is the right comparison for whether the interaction can order the material. It is not the right comparison for everything: summed over a whole sample the dipolar energy is comparable to the exchange energy, because there are so many pairs, and that sum is what shapes a magnet.

And the mean-field readback uses $S = 1$. The spin per atom in iron is not one, the coordination number depends on which neighbours are counted, and the relation between them is a mean-field one that is known to be inaccurate. The eight millielectronvolts should be read as “of order ten”, and the comparison with the dipolar energy survives any plausible revision because the two differ by a factor of five hundred.

A node in six dimensions, drawn along one line #

The Fermi hole figure draws a conditional density in one dimension, and the object it is about lives in six. What matters physically is the pair density — the probability of finding one electron here and the other there — and that is a function on a six-dimensional space of which the figure shows one line. The node the whole argument turns on is a five-dimensional surface in that space, and the drawing can only show where it crosses one line through it.

The comparison figure draws two temperatures per material and cannot show how they were obtained. The ordering temperatures are measured; the dipolar temperatures are computed from a moment and a separation, both of which are themselves measured, and one of which — the moment per atom — is a quantity a localised picture does not strictly possess. The figure presents a computed number beside a measured one, in the same units, and the honesty of the comparison rests on an assumption the picture cannot display.

And the spin-wave figure draws a deficit, not a magnet. What is actually happening at 20 K is that a tiny fraction of the moment has been taken by waves of enormous wavelength — tens of thousands of lattice spacings — each one a barely perceptible tilt spread over an enormous number of atoms. Nothing about that is visible in a curve of a scalar against temperature, and it is the whole content of the result.

Still open: why three elements and not thirty #

The question this essay leaves is the one it was careful not to answer. Both mechanisms above are real, they have opposite signs, and which wins is decided by quantities — orbital overlaps, on-site repulsions, band fillings — that vary smoothly across the periodic table. Yet the outcome does not vary smoothly: iron, cobalt and nickel order ferromagnetically at high temperatures, manganese and chromium order antiferromagnetically, and copper does not order at all.

Working out where the boundary is from first principles remains hard. Calculations that start from the electronic structure get the ground-state moments of the three ferromagnets to within a few per cent and their Curie temperatures wrong by factors approaching two, because the temperature is set by the excitations rather than by the ground state and the excitations are collective. Whether the right description of a hot ferromagnet is a lattice of moments that have lost their alignment or a band structure that has lost its splitting is a question with experimental evidence on both sides.

The habit worth carrying away is the one the first figure is built on. When a mechanism is proposed for something, compute its size before asking whether it is right. The little-magnets story fails on a single line of arithmetic performed with quantities everybody already knew — a moment, a spacing, a constant — and it failed on that line for twenty years while people looked for reasons it might work. One estimate, made early, would have said that the interaction was the wrong one by three orders of magnitude and that the search should be for something else entirely.