The melting curve that has to arrive flat

A law about spectra, not about heat restates the third law as a condition on the bottom of a spectrum: the entropy of a system in its ground state is the logarithm of how many ground states it has, and it vanishes only when there is one. That form of the law has a consequence that is easy to state and not obvious at all. It constrains the shape of every phase diagram.

The melting curve that leans the wrong way derives the slope of any line on which two phases coexist:

$$\frac{\mathrm{d}P}{\mathrm{d}T} = \frac{\Delta S}{\Delta V},$$

the difference in entropy between the phases over their difference in volume. Water’s melting line slopes backwards because the volume difference is negative — ice is less dense than water — while the entropy difference has its usual sign. Now take the line down towards absolute zero. The volume difference between two phases stays finite; the third law says the entropy difference must vanish. So every coexistence line in nature must arrive at absolute zero with zero slope. Most substances freeze long before the question arises. Helium does not: it stays liquid under its own vapour pressure all the way down, and has to be squeezed to more than twenty-five atmospheres to freeze, so its melting line is the one place where the constraint can be watched in action. The lighter isotope, helium-3, shows it in the most surprising form possible.

Why helium is still liquid at the bottom #

Every other substance freezes on cooling long before the third law’s constraint on its melting line could be seen, and helium does not, for a reason that is itself a piece of quantum mechanics. Helium atoms attract one another only weakly — a closed shell of two electrons has no chemistry and very little polarisability — and they are the lightest atoms that are not hydrogen. Confining a light particle costs kinetic energy, and the motion that cannot be stopped finds that the cost of localising a helium atom on a lattice site is larger than the attraction it would gain there. The zero-point motion alone keeps the atoms from settling into a crystal, and helium stays liquid under its own vapour pressure down to absolute zero.

Pressure changes the balance. Squeezing the liquid brings the atoms close enough that the attraction and the packing win, and helium-4 freezes at about 25 atmospheres, helium-3 at about 34. The melting line is therefore reachable at every temperature, however low, simply by turning up the pressure — which is why helium is the only substance whose melting curve can be followed all the way down, and why it is the natural place to watch the third law flatten one.

The melting curve that turns back, then lies flat #

Above about a third of a kelvin, helium-3’s melting curve behaves like any other: warmer liquid needs more pressure to freeze. At 315 millikelvin it reaches a minimum, and below that it turns round. Colder liquid needs more pressure to freeze, not less, and the curve climbs by half a megapascal as the temperature falls towards a millikelvin, before levelling off at a little under 34.4 atmospheres.

The slope of the curve is the entropy difference over the volume difference, and the volume difference does not change sign — the solid is denser than the liquid throughout, by about 1.3 cubic centimetres per mole. So below 315 millikelvin the entropy difference has changed sign. The solid has more entropy than the liquid it freezes from. Freezing helium-3 there makes it more disordered.

This curve is now more than a curiosity. Between 0.9 millikelvin and one kelvin, the International Temperature Scale gives way to a provisional scale, PLTS-2000, and PLTS-2000 is this curve: a polynomial in temperature, fitted to the best measurements, whose value at any measured melting pressure defines the temperature. The fixed points in the drawings — the minimum, the superfluid transitions, the magnetic ordering of the solid — are fixed points of the scale, and the polynomial the drawings use reproduces all four to five decimal places. Below one kelvin, the unit of temperature is realised by squeezing helium-3.

The entropy the melting curve says the solid is holding #

Reading Clausius and Clapeyron backwards turns the melting curve into an entropy meter, and the drawing does it. The entropy difference between solid and liquid rises from zero at the minimum to about 0.65 of the gas constant at a few millikelvin — very nearly $R\ln 2$, and the reason is the nucleus.

A helium-3 nucleus has a spin of one half, which can point up or down. In the solid, the atoms sit on a lattice and their nuclear spins interact only through a weak exchange coupling, worth about a millikelvin; above a few millikelvin they point every which way, and each spin contributes $k\ln 2$ of entropy — $R\ln 2$ per mole, exactly the count a law about spectra makes for a doubly degenerate ground state. The liquid is different. Helium-3 atoms are fermions, and the liquid is a degenerate Fermi liquid, like the electrons in a metal that are at eighty thousand kelvin in a room-temperature wire: only the atoms near the top of the Fermi sea can change their state, so the liquid’s entropy is small and proportional to temperature. The drawing’s reading at twenty millikelvin gives a liquid entropy of 4.5 $RT$, close to the measured heat capacity of the liquid at that pressure.

So at low temperature the ordered phase is the liquid and the disordered phase is the solid, because the liquid’s order is quantum-statistical and the solid’s disorder is magnetic. Below a few millikelvin the solid’s spins begin to feel their exchange coupling and line up, its entropy falls, the difference falls, and the slope of the curve falls with it. At 0.902 millikelvin the solid orders antiferromagnetically; below that its entropy drops rapidly towards zero, and the curve lies flat. The third law is obeyed, but only after the spins have been dealt with — which is the same observation a law about spectra makes about copper’s nuclei carrying entropy down to microkelvin temperatures.

Cooling a liquid by squeezing it into a solid #

In 1950 Isaak Pomeranchuk saw what the backward slope implies. Freezing is ordinarily exothermic: a liquid gives up its latent heat as it becomes the more ordered solid. For helium-3 below the minimum the solid is the less ordered phase, so freezing takes in heat — $T\Delta S$ per mole — and a mixture of liquid and solid that is compressed with no heat supplied from outside must supply that heat from itself. It cools.

The drawing runs the process. Start with liquid on the melting curve at twenty millikelvin and compress it slowly in an insulated cell. Each mole that freezes absorbs the entropy difference, so the rest of the mixture must lose entropy, which a Fermi liquid does by getting colder. Freezing a tenth of the liquid takes the cell to under six millikelvin, and freezing a fifth reaches the lower end of the scale, below a millikelvin. The liquid’s small heat capacity is what makes this efficient: it has so little entropy to lose that a modest amount of freezing drains it.

Pomeranchuk’s method is a cousin of the staircase that never reaches the floor, which cools by magnetising a salt and then letting its spins disorder. Both use a reservoir of spin entropy as a heat sink. The salt’s spins are in a crystal and are ordered by a field; helium-3’s are in a solid that is made on demand from the liquid being cooled, and ordered by nothing — they are simply disordered, and freezing puts more of the sample into that disordered state. Neither can reach absolute zero, and for the same reason: the entropy reservoir itself empties as its spins begin to order.

The last few millikelvin #

In late 1971 Douglas Osheroff, a graduate student at Cornell working with Robert Richardson and David Lee, was compressing helium-3 in a Pomeranchuk cell and recording its pressure as it cooled. The record showed two small kinks, reproducible, near 2.6 and 1.8 millikelvin: changes in the rate at which the pressure rose, as though something in the cell had changed its heat capacity. They first attributed them to a magnetic transition in the solid. Within months, experiments that could tell the phases apart showed that the changes were in the liquid. Helium-3 had become a superfluid — its atoms, fermions, pairing up as electrons do in a superconductor that has nothing to order but pairs, but with the pairs carrying spin and orbital angular momentum, so that the superfluid has two distinct phases, A and B, with different internal structures. The discovery earned the three the Nobel Prize in 1996.

The drawing shows where they sit: on the flat end of the curve, where the third law has almost finished its work. The pressure differences involved are tiny — the transitions are fractions of a kilopascal apart — and it is a measure of how precisely melting pressure can be measured that these features are now among the fixed points by which the world’s millikelvin thermometers are calibrated.

Helium-4, which has no spin to spend #

The comparison with the other isotope isolates what is special about helium-3. A helium-4 nucleus has no spin, and helium-4 atoms are bosons. Its liquid becomes a superfluid at 2.17 kelvin, and below about a kelvin both liquid and solid carry entropy only in their sound waves, which is tiny and falls as the cube of the temperature in both phases. The entropy difference between them is therefore tiny too, and helium-4’s melting line is flat below about a kelvin, at 25.3 atmospheres, with no backward excursion worth speaking of: the third law’s flattening, arriving early and quietly.

Helium-3 differs in two ways, and the backward curve needs both. Its nuclei have spin, which gives the solid a large reservoir of entropy that survives down to millikelvins; and its atoms are fermions, which makes the liquid’s entropy small and linear in temperature. Remove the spin and the solid has nothing to hold; make the atoms bosons and the liquid condenses instead. The Pomeranchuk effect is the signature of a fermion liquid in contact with a paramagnetic solid, and helium-3 is the only substance that offers both.

A pressure measured to a fraction of a pascal #

Using a melting curve as a thermometer means measuring pressure extraordinarily well inside a cell that is a few millikelvin cold. The obvious method — a tube leading up to a gauge at room temperature — fails at once, because helium in the tube freezes and plugs it. The pressure is measured in the cell itself, with a gauge made of a thin flexible diaphragm forming one plate of a capacitor: the helium’s pressure bends the diaphragm, and the capacitance, read by a bridge, changes accordingly. Gauges of this kind resolve changes of a fraction of a pascal against a pressure of 3.4 million.

That resolution is what makes the flat end of the curve usable. Near a millikelvin the melting pressure changes by only a few kilopascals per millikelvin, so a resolution of a tenth of a pascal is a temperature resolution of tens of nanokelvin. The flatness the third law imposes makes the curve a poor thermometer at the very bottom and forces the measurement to be that much better; the superfluid transitions and the solid’s ordering, marked on the flat part, give fixed points that do not depend on anyone’s calibration of the gauge.

The same flattening on a magnetic phase boundary #

The rule is not about helium or about melting. A superconductor in a magnetic field has two phases, normal and superconducting, divided by a critical field that depends on temperature, and the boundary between them obeys its own Clausius–Clapeyron equation, with the magnetic field playing the role of pressure and the magnetisation that of volume. The entropy difference between the phases is set by the slope of the critical field. The drawing reads it off for lead, a classic superconductor, and finds it vanishing at the two ends: at the critical temperature, where the two phases become one, and at absolute zero, where the third law demands it. The parabolic critical field that experiments find is exactly a curve that reaches zero temperature with zero slope.

The same argument applies to every boundary in every phase diagram that reaches absolute zero: the line between a ferromagnet’s two directions, between a crystal’s structures, between the liquid and the gas of any substance that could be kept fluid. All must be flat at the bottom. Helium-3’s melting curve is the one where the flattening is watched through a whole sequence of stages — solid disorder beating liquid disorder, then losing, then vanishing — and measured to a few pascals.

The boiling line, which flattens for another reason #

One coexistence line seems to escape the argument. A liquid and its vapour also coexist along a line, and the entropy difference between them does not vanish at low temperature: a mole of vapour, spread thin, has an entropy that grows without limit as its density falls, and the latent heat of evaporation stays finite. By the same equation the slope of the boiling line is the latent heat over the temperature and the volume difference, and nothing in that forces it to zero.

What forces it is the vapour’s disappearance. The volume difference is essentially the volume of a mole of vapour, which grows as the vapour pressure falls, and the vapour pressure falls exponentially, as $e^{-L/RT}$. Put the two together and the slope of the boiling line is the pressure times the latent heat over $RT^2$, which goes to zero because the pressure does — faster than any power of the temperature. A boiling point is a pressure, and near absolute zero that pressure is effectively nothing.

The distinction matters because it shows which statement the third law makes. It is a statement about systems that remain in equilibrium with finite density — condensed phases, which have ground states to count. A gas at vanishing density is not approaching a ground state at all; it is disappearing. The melting curve is the clean test, because both of its phases are condensed and both must reach the bottom of their spectra together.

Where the Clapeyron reading stops #

A constant volume difference. The entropy figure takes the difference in molar volume between liquid and solid as 1.31 cubic centimetres per mole all along the curve. In reality it varies by a few per cent across the range, and the entropy read off the curve inherits that uncertainty.

Equilibrium along the curve. Clausius and Clapeyron apply to two phases in equilibrium. At the lowest temperatures the solid and liquid take a long time to exchange heat — a thermal boundary resistance between them grows as the temperature falls — and a real Pomeranchuk cell has temperature differences within it that the drawing does not.

The cooling model. The cooling curve takes the liquid’s entropy as linear in temperature, with its coefficient fitted at twenty millikelvin, and the solid’s as the liquid’s plus the difference the curve implies. That ignores the superfluid transitions, where the liquid’s entropy changes its form, and it is least trustworthy at the lowest temperatures drawn.

What the curves cannot show #

The drawings show pressure and entropy, averages over enormous numbers of atoms. They cannot show what makes helium-3 behave this way, which is quantum mechanics at the scale of the atoms themselves: atoms so light, and so weakly attracted to one another, that their zero-point motion keeps them liquid at absolute zero unless they are squeezed, and whose nuclear spins, invisible to every chemical property, carry more entropy than all their motion. None of the curves looks quantum. Every feature of them is.

Still open: what the solid’s spins do #

Solid helium-3’s magnetism was expected to be simple — a lattice of spins with a nearest-neighbour exchange coupling — and it is not. Its ordered phase below 0.9 millikelvin has an unusual structure, with spins in pairs of planes pointing up and pairs pointing down, and explaining it needed exchange processes in which three, four and more atoms swap places at once, a consequence of how easily the light atoms tunnel through the lattice. Two-dimensional helium-3, a film a single atom thick on graphite, shows a spin state that does not order at all down to the lowest temperatures reached, and whether it is a quantum spin liquid — a state whose disorder is a property of its ground state, the case a law about spectra leaves open — is under active investigation. The third law will be obeyed there too; how is the question.

The habit worth carrying away is to read a phase diagram’s slopes as entropy. Every coexistence line’s slope is an entropy difference over a volume difference, so the third law, which forbids entropy differences at absolute zero, forces every such line to arrive flat — and a line that runs backwards before it does is announcing that the phase everyone assumes is more ordered is, at that temperature, the more disordered one.