The mixture that stays mixed at absolute zero

A law about spectra, not about heat put the third law of thermodynamics in its statistical form: as the temperature falls, a system settles into its lowest energy states, and its entropy goes to zero if the lowest state is unique. The two laws absolute zero will not allow drew one consequence for equations of state: anything whose entropy would stay finite at zero, as an ideal gas’s or a free magnet’s would, must change its behaviour on the way down. Mixtures are a third case, and the rule they seem to imply is severe.

Mix two kinds of atom at random and the number of arrangements rises by a factor that the entropy of mixing records — the quantity mixing what is already mixed derives — and that entropy does not depend on temperature. If the third law is to hold, a mixture cooled towards absolute zero must lose it, and the only ways to lose it are to separate completely into the pure components or to order into a regular compound in which every atom’s place is fixed. Solid alloys do exactly this: brass, cooled slowly enough, orders, and many metal mixtures that are uniform at high temperature separate into two phases as they cool, each phase becoming purer as the temperature falls. Classical solution theory says the separation must be complete at absolute zero.

One liquid mixture refuses, and it is the basis of the most important refrigerator in modern physics.

Two liquids, one of which keeps its guests #

Helium has two stable isotopes. Helium-4, with two protons and two neutrons, is a boson, and below 2.17 kelvin its liquid becomes a superfluid, the state the liquid that will not slow down describes. Helium-3, with one neutron fewer, is a fermion, and its liquid remains an ordinary — though strongly quantum — liquid down to a few millikelvin. The two mix freely at higher temperatures. Below 0.87 kelvin, a mixture with enough helium-3 separates into two liquid layers: a lighter one, rich in helium-3, which floats on top, and a heavier one, mostly helium-4.

As the temperature falls further, the upper layer purifies exactly as classical theory expects. Its helium-4 content falls exponentially, and at a tenth of a kelvin it is pure helium-3 to one part in ten thousand. The lower layer does not. Its helium-3 content falls to 7.1 per cent at a tenth of a kelvin and then stops falling, levelling off at 6.6 per cent at absolute zero. However cold the mixture is made, a mole of the dilute phase contains about a fifteenth of a mole of helium-3, dissolved and uniformly spread through the superfluid.

The reason it stays has two parts, and both are quantum. A helium-3 atom is lighter than a helium-4 atom, so its zero-point motion — the jiggling that the motion that cannot be stopped shows no liquid can be cooled out of — is larger, and helium-3 atoms in their own liquid keep each other further apart. Surrounded by helium-4 instead, a helium-3 atom sits in a denser liquid and is bound more strongly than it would be among its own kind, by about three-tenths of a kelvin per atom. Dissolving is favourable. What stops it going on indefinitely is that helium-3 atoms are fermions: no two can be in the same state, so each additional dissolved atom must take a higher-momentum state than the last, and the dissolved atoms fill a Fermi sea whose top energy rises with their concentration. At 6.6 per cent the energy of the top state has risen by the full binding advantage, and the next atom gains nothing by dissolving. The solubility limit is where the Fermi energy of the dissolved gas equals the binding advantage, and since neither depends on temperature, the limit survives to absolute zero.

Why the third law allows it #

The entropy of a dissolved helium-3 atom is not the classical entropy of mixing. That entropy counts the ways of placing distinguishable solute atoms among the sites of a solvent, and for identical quantum particles the count is meaningless: the dissolved helium-3 atoms are not on sites, they are a gas of indistinguishable fermions occupying momentum states. At absolute zero that gas is in its ground state — every momentum state below the Fermi level filled, every one above it empty — and there is exactly one such state. Its entropy is zero. At low temperature a thin shell of states near the Fermi level is thermally blurred, and the entropy is proportional to the temperature, falling smoothly to nothing.

So the dilute phase satisfies the third law while remaining a mixture. What the law forbids is a mixture that keeps an entropy of mixing at zero, and a degenerate Fermi gas carries none. The classical argument for complete separation silently assumes that the solute’s atoms are distinguishable and localised; when they are neither, the mixture can stay mixed. The same reasoning makes the electrons in a metal, which are a degenerate Fermi gas spread through a lattice of ions, compatible with the third law. The helium mixture is the case where the solute is an entire species of atom.

A vapour that never thins out #

The dilute phase’s fixed helium-3 content is what makes a refrigerator possible, and the idea, due to Heinz London in 1951, is an analogy with evaporation. A liquid cools when it evaporates because the molecules that leave take their latent heat with them; pumping on a bath of liquid helium-4 cools it to about a kelvin, and on helium-3 to about a quarter of a kelvin. In the dilution refrigerator, the concentrated helium-3 phase plays the liquid, the dilute phase plays the vapour, and the superfluid helium-4, which at these temperatures carries almost no excitations, plays the part of empty space through which the “vapour” moves. Helium-3 atoms crossing from the concentrated phase into the dilute one absorb heat as an evaporating molecule does.

The heat absorbed is the difference between the enthalpies of helium-3 in the two phases, and both are Fermi liquids, whose heat capacities are proportional to temperature and whose enthalpies therefore go as its square. The dilute one is larger, because its atoms are spread out: their Fermi temperature is lower, so a greater fraction of them are thermally excited at any given temperature and more heat is stored per atom. Per mole of helium-3 the difference is $84T^2$ joules, with $T$ in kelvin — eighty-four hundredths of a joule at a tenth of a kelvin, eight thousandths at ten millikelvin.

That number falling as $T^2$ is the third law in practical form, and the comparison with evaporation shows why it is nevertheless a triumph.

The cooling an evaporating bath delivers is proportional to the rate at which molecules leave it, and that is proportional to its vapour pressure, which falls as $e^{-L/kT}$, where $L$ is the energy binding a molecule to the liquid. For helium-4, $L/k$ is about seven kelvin, and below a kelvin the vapour pressure collapses so fast that no pump can draw enough gas; for helium-3 it is about two and a half kelvin, which buys a factor of four in temperature and then fails in the same way. With a pump of a hundred litres a second, the helium-3 bath that delivers a tenth of a milliwatt at 0.3 kelvin delivers $10^{-24}$ watts at 50 millikelvin — an exponential collapse, the vapour thinning out to nothing.

The dilute phase does not thin out. Its helium-3 density is pinned at 6.6 per cent, so the rate at which atoms can cross into it is set by how fast they are circulated, not by an exponential, and the cooling falls only as the square of the temperature. A refrigerator circulating a hundred micromoles of helium-3 a second gives twenty microwatts at 50 millikelvin and nearly a microwatt at ten. On the logarithmic axes the dilution line is straight and the evaporation curves plunge; below about 0.3 kelvin there is no contest.

How the cycle is closed #

A refrigerator must run continuously, so the helium-3 that has crossed into the dilute phase must be extracted and returned. The dilute phase is connected by a tube to a chamber called the still, held at about 0.7 kelvin. There the mixture’s vapour is almost pure helium-3, because helium-3’s vapour pressure is so much higher than helium-4’s at that temperature, and pumping on the still removes helium-3 preferentially. Removing it lowers the helium-3 concentration in the still, and helium-3 diffuses up from the mixing chamber through the stationary superfluid to replace it, driven by the difference in concentration — an osmotic flow through a solvent that does not move, the same physics as the fountain a lamp can drive with the roles of the two isotopes reversed.

The pumped helium-3 gas goes out to room temperature, is compressed, and comes back down the cryostat, where it is condensed at about 1.5 kelvin on a plate cooled by pumped helium-4, then cooled further at the still, and finally passed through a series of heat exchangers in which it gives up its heat to the cold dilute stream flowing the other way. It arrives in the mixing chamber as liquid helium-3 at nearly the chamber’s own temperature, crosses the boundary, absorbs its $84T^2$ joules a mole, and starts round again. The first such machine, built at Leiden in 1965 by Pieter Das, Reinier de Bruyn Ouboter and Krijn Taconis, reached 0.22 kelvin; within a year Boris Neganov’s machine at Dubna had reached 25 millikelvin, and the best have since gone below 2 millikelvin.

The incoming helium-3 is the refrigerator’s main internal burden. If it reaches the mixing chamber warmer than the chamber, its own enthalpy must be removed first, and the cooling becomes $\dot n(95T^2 - 11T_{in}^2)$; an exchanger that delivers the returning helium at twice the chamber’s temperature costs about half the cooling. The exchangers are therefore the parts of a dilution refrigerator that decide its base temperature, and the reason they are hard to make is the subject of the last figure.

The floor a leak sets #

The cooling power going as $T^2$ means that a dilution refrigerator, like every refrigerator, can only hold a temperature at which its cooling balances whatever heat leaks in.

Setting $84\dot n T^2$ equal to a leak $Q$ gives a floor at $T = \sqrt{Q/84\dot n}$, rising as the square root of the leak. A hundred nanowatts — a modest leak, from the vibration of the pumps, from radio-frequency pickup on the wires that run down to the experiment, or from the slow heat release of hydrogen molecules trapped in the copper converting from their ortho to their para form — holds a hundred-micromole refrigerator at 3.5 millikelvin. Quadrupling the circulation halves that; so does cutting the leak by a factor of four. The same leak against a pumped helium-3 bath holds it at about 175 millikelvin. Neither curve touches zero, and no change of circulation can make one: this is the staircase that never reaches the floor restated as a budget, the unattainability of absolute zero expressed as the cost of each halving of temperature — four times the flow, or a quarter of the leak.

Modern refrigerators are mostly “dry”: a mechanical pulse-tube cooler replaces the bath of liquid helium-4 that used to precool them, and the whole instrument runs for months from an electrical supply. Their mixing chambers reach about ten millikelvin with a few hundred microwatts of cooling at a hundred millikelvin, and that is the environment every superconducting quantum processor, most single-electron devices of the kind in the island that takes electrons one at a time, and the cryogenic detectors of the bar a wave of gravity barely touches have been built around.

Below the dilution refrigerator #

Colder temperatures are reached, but not continuously. The method is the one the staircase that never reaches the floor draws as a sequence of steps: magnetise a material at a fixed temperature, so that its magnetic moments line up and its entropy falls, isolate it, then remove the field slowly so that the moments disorder again at the expense of the material’s own heat. Electronic paramagnetic salts demagnetised this way reach a few millikelvin; the nuclear moments of copper, whose interactions with one another are a thousand times weaker, reach microkelvin, and with the copper’s nuclei alone, briefly, nanokelvin. But a demagnetisation is a single stroke. Once the field is gone the stage warms under whatever leak reaches it, and it must be remagnetised and its heat dumped again.

Every nuclear demagnetisation stage therefore hangs below a dilution refrigerator, which precools it to ten millikelvin or less in a strong field and absorbs the heat of magnetisation — joules, removed over days — before the stroke begins. The division of labour follows from the third law in two different forms. Demagnetisation exploits an entropy that a field can switch, and runs out when that entropy is spent; dilution exploits an entropy that falls only as the temperature, the Fermi liquid’s, and can be drawn on indefinitely because the helium-3 that carries it is circulated rather than spent. One reaches lower, the other holds still, and the coldest experiments use both, the second as the floor the first steps down from.

Getting the heat out #

Cooling the helium is only half the job; the heat must also get from the experiment, and from the returning helium-3, into the cold helium. At millikelvin temperatures this is surprisingly hard. Heat crosses a boundary between liquid helium and a metal as phonons, sound quanta, and the two materials are so mismatched — helium is light, soft and slow, a metal dense, stiff and fast — that a phonon arriving at the boundary is almost always reflected. The result is a thermal boundary resistance, discovered by Pyotr Kapitza in 1941, which grows as the inverse cube of the temperature.

To pass a microwatt across the boundary with a temperature step of a tenth of the temperature takes twenty square centimetres of contact at a hundred millikelvin, twenty square metres at ten, and a couple of thousand square metres at three. No polished surface offers that, so the exchangers and the mixing chamber are lined with sintered silver: powder of grains a tenth of a micrometre across, pressed and heated until the grains stick together into a sponge offering about a square metre of surface per gram. A few hundred grams of it make the area that millikelvin heat transfer needs. Below a few millikelvin even that runs out, and the limit on how cold a sample becomes is no longer the refrigerator’s cooling but the resistance between the sample and the helium — a second wall, set by the same physics of phonons that makes the cooling fall.

Fits that hold only in the cold #

The phase-separation curve uses the standard low-temperature fits for the two branches, which hold below about 0.3 kelvin; above that the curves bend towards each other and meet at 0.87 kelvin, where the separation begins and where the line marking the onset of superfluidity in the mixture joins it. The enthalpies $95T^2$ and $11T^2$ are the low-temperature forms and fail above about a tenth of a kelvin, where the dashed continuation of the cooling curve stops meaning anything. The vapour pressures come from an ideal-vapour Clausius–Clapeyron law with each isotope’s binding energy at absolute zero, anchored at its normal boiling point; this reproduces helium-4’s measured pressure at one kelvin within about a quarter, which is ample for a curve falling by twenty orders of magnitude, but it is not a thermometry table. The boundary resistance used is a typical value for helium against a metal; real values vary by a factor of several between surfaces, and below a few millikelvin, against silver, a magnetic coupling between helium-3 and the metal’s surface lowers it.

Still open: what limits the base temperature now #

Laboratory refrigerators have hovered between about two and ten millikelvin for decades, and no machine has gone much below the record of under two millikelvin set in Leiden in the 1980s. The question of what stops them is partly engineering — heat exchanger area, the viscous heating of helium-3 flowing through narrow channels, which grows as the temperature falls — and partly physics, since the properties of the dilute phase and the boundary resistance below a millikelvin are not measured as well as above it. The quantum computing industry’s demand for very large mixing chambers, with thousands of wires to cool and microwatts of heat to remove at ten millikelvin, has made cooling power at a fixed temperature, rather than base temperature, the quantity that matters. Whether the dilution principle can be scaled to the kilowatt-class refrigerators that million-qubit machines would need at their warmer stages, and whether the world’s helium-3 supply, which comes almost entirely from the decay of tritium in nuclear weapons stockpiles, can support it, are questions being argued now.

The habit worth carrying away is to ask what the third law actually forbids. It forbids entropy at absolute zero, not mixtures — and a degenerate Fermi gas has a single ground state at any density, so helium-3 can stay dissolved in helium-4 at 6.6 per cent forever, and moving it across that boundary cools like evaporation from a vapour that never thins out: $84\dot nT^2$ watts instead of an exponential collapse, three and a half millikelvin against a hundred-nanowatt leak. The law still collects its due as the $T^2$, and as the square-metre surfaces needed to get the heat out.