The liquid that will not slow down

Every liquid this collection has discussed resists being sheared, and the resistance is momentum diffusing across the flow. One liquid does not. Below 2.17 kelvin, helium-4 flows through channels a few tens of nanometres wide with no measurable pressure difference, and a ring of it set circulating shows no detectable slowing after a year.

Getting to this point requires two separate pieces of physics, and the first is the one usually skipped.

Why there is a liquid there at all #

Everything else freezes. Cool any other substance far enough and its atoms settle into a lattice, because the attraction between them eventually overwhelms whatever motion is left. Helium does not solidify at any temperature under its own vapour pressure, and it takes about 25 atmospheres to force it.

The reason is zero-point energy. Confining an atom to a region costs kinetic energy — the uncertainty relation requires it — and that cost is inversely proportional to the atom’s mass and to the square of the confinement. Helium is very light and its interatomic attraction is exceptionally weak, being a closed-shell atom with only van der Waals forces available.

Put numbers to it and the zero-point energy of an atom confined to a lattice site exceeds the binding energy that site would provide. The lattice is not a bound state; the atom would rather wander. So helium remains liquid down to absolute zero, and it is the only substance that does.

This is not a footnote. It is the reason any of the rest is observable: quantum statistics govern every substance, and in every other one the effects are pre-empted by freezing long before the temperature is low enough for them to appear.

A particle confined to a region has a lowest energy that is not zero, and the lighter the particle and the tighter the box the larger it is. Helium is the extreme case: light atoms, weak attraction, and a zero-point energy comparable with the binding — so the atoms cannot settle into a lattice at any temperature at atmospheric pressure. That is why there is a liquid there at all, and it is the one substance for which that is true.

What happens at 2.17 kelvin #

Cooling liquid helium through 2.17 K produces a visible change. Above it the liquid boils vigorously, bubbling throughout. Below it the bubbling stops abruptly and the surface goes glassy still — because the thermal conductivity has become enormous and no local hot spot can survive long enough to nucleate a bubble.

The specific heat has a sharp peak at that temperature whose shape gives the transition its name: the lambda point. Below it the liquid is called helium II and has properties that no classical liquid has.

It flows without measurable viscosity through channels too fine for ordinary helium to enter — superleaks, packed powders with gaps of tens of nanometres.

It conducts heat by a mechanism that is not conduction, several hundred times better than copper, by a counterflow that carries entropy bodily.

It climbs the walls of its container as a film about 30 nanometres thick and siphons itself out over the rim, until the vessel is empty.

It shows a fountain effect: heat one side of a superleak and the liquid rises on that side, producing a pressure from a temperature difference with nothing mechanical involved.

Bose statistics, and why a transition must exist #

The underlying reason is indistinguishability, taken the other way round from the exclusion essay.

A helium-4 atom contains two protons, two neutrons and two electrons — six fermions, so it is a boson. Bosons carry the opposite symmetry from fermions: instead of being forbidden to share a state, they are more likely to occupy a state the more of them are already in it.

For an ideal Bose gas that has a dramatic consequence. Below a critical temperature the excited states cannot hold all the particles, and the surplus accumulates in the single lowest state — not a small excess but a macroscopic fraction, growing to everything at absolute zero:

$$\frac{N_0}{N} = 1 - \left(\frac{T}{T_c}\right)^{3/2}.$$

Putting helium’s density and atomic mass into the formula for $T_c$ gives about 3.1 K, against an observed 2.17 K. Same order, wrong number — which is the appropriate outcome for a calculation that assumes the atoms do not interact, applied to a liquid where they certainly do.

Where the ideal calculation fails #

The disagreement is worth being precise about, because “helium is a Bose–Einstein condensate” is repeated often and is at best a half-truth.

The transition temperature is 30 per cent out, as above.

The shape of the curve is wrong. The ideal fraction goes as $1 - (T/T_c)^{3/2}$; helium’s superfluid fraction goes as roughly $1 - (T/T_\lambda)^{5.6}$. The figure draws both, and they part company immediately.

The condensate fraction is not the superfluid fraction. At absolute zero, all of helium is superfluid — but only about ten per cent of its atoms are in the zero-momentum state, because interactions scatter them out of it. The two quantities are different, and confusing them is the standard error.

Interactions are essential rather than a correction. A genuinely ideal Bose gas would condense and would not be a superfluid: Landau’s criterion shows that flow is only protected against dissipation if the excitation spectrum has the right shape at small momentum, and an ideal gas’s does not. Helium’s does, because its atoms repel each other, and the resulting phonon-like spectrum sets a critical velocity below which the flow cannot lose energy at all.

So the honest statement is: Bose statistics explain why a transition has to happen; interactions explain why the resulting state is a superfluid; and neither alone is sufficient.

The two-fluid model, which is a decomposition rather than a mixture #

Below the lambda point, helium behaves as though it were two interpenetrating fluids: a normal component with ordinary viscosity carrying all the entropy, and a superfluid component with no viscosity and no entropy. The proportions shift with temperature — all normal at $T_\lambda$, all super at zero.

It predicts a great deal correctly. The fountain effect, the film flow, the enormous thermal conductivity and the existence of second sound — a wave in which temperature oscillates while density does not — all fall out of it.

And there is no such mixture. No experiment separates the two components, no atom belongs to one or the other, and the liquid is one substance throughout. The split is a decomposition of the response: helium reacts to a slow shear as if part of it were absent and to a temperature gradient as if part of it carried everything. Two responses, one fluid.

That distinction matters because the picture is so vivid that it is easy to start believing it. It is the same caution field lines earn — a representation that computes correctly and describes nothing, whose usefulness is exactly what makes the reification tempting.

An ordinary liquid sustains a velocity gradient under a stress, with momentum diffusing across it. The superfluid component supports no such gradient — it has no viscosity to transmit one — so a two-fluid description is a decomposition rather than a mixture: the same atoms participate in both, and the fractions are properties of the temperature rather than of any separable substance.

Two viscosities from one liquid #

The cleanest evidence that the two-fluid picture is a decomposition rather than a mixture comes from two measurements that disagree, on purpose.

Push helium II through a fine capillary and measure the flow: the viscosity comes out at zero, because the superfluid component passes through unimpeded and the normal component is held back by the narrow channel.

Now spin a disc in a bath of helium II and measure the drag: the viscosity comes out non-zero, and close to that of the normal component alone, because the disc couples to the normal fluid and the superfluid slips past.

Two experiments on the same liquid at the same temperature, giving zero and not-zero. Neither is wrong. What they show is that “the viscosity of helium II” is not a well-defined quantity — which is the same lesson a shear-thinning fluid teaches in a much less extreme form, and the reason both belong in a collection about where models stop.

Heat that travels as a wave #

The strangest of helium II’s properties, and the one that most clearly marks it as something other than a liquid with a small viscosity, is that heat propagates through it as a wave.

In every ordinary material heat diffuses: a hot spot spreads out, its width growing as the square root of time, by the same equation that governs ink and momentum. In helium II a temperature disturbance instead travels at a definite speed — around 20 m/s over much of the range — and can be reflected, and can form standing waves in a resonator.

The two-fluid model explains it directly. Ordinary sound is a wave in which the two components move together and the density oscillates. Second sound is a wave in which they move in opposite directions, so the total density stays constant while the proportion of normal to superfluid oscillates — and since the normal component carries all the entropy, a wave in that proportion is a wave in temperature.

So helium II supports two sound speeds, from one liquid, and measuring the second is one of the standard ways of determining the superfluid fraction that the figure at the top of this page plots. The curve is not a theoretical construct; it is what a resonator returns.

It is worth noticing that a wave equation for temperature is impossible in ordinary physics. The diffusion equation runs one way only and has no wave solutions; a wave equation is reversible and has no arrow. That helium supports the second is a sign that the entropy is being carried bodily by a component that can move without dissipating, which is not something a classical fluid has available.

Second sound forms standing patterns in a cavity exactly as ordinary sound does, and the frequencies selected are the ordinary ones for a wave of its speed. That is the cleanest demonstration that it is a wave rather than a diffusive spreading: diffusion has no resonances, and this has the full set.

What circulation does instead #

A superfluid cannot rotate the way a normal liquid does. Spin a bucket of water and the whole body eventually turns as a solid; spin a bucket of helium II slowly and the superfluid component simply does not.

Spin it faster and it does rotate, but only by admitting quantised vortices — thin lines around which the circulation takes one of a discrete set of values,

$$\oint \mathbf{v}\cdot\mathrm{d}\boldsymbol{\ell} = n\frac{h}{m},$$

with $n$ an integer. Circulation is quantised, in units built from Planck’s constant and the atomic mass, and the vortices arrange themselves into a regular lattice whose spacing depends on the rotation rate. They have been photographed. The lattice is regular for the same reason any repelling set of like objects arranges itself regularly, and the spacing is set by the rotation rather than by anything about helium.

That quantisation is the clearest signature that this is one quantum state on a macroscopic scale rather than a peculiar liquid. It is the same kind of statement as a bound electron having only certain energies — a continuous-looking system returning a whole-number answer — scaled up until it can be seen in a bucket.

Why it was not predicted #

The discovery order here is worth recording, because it is the reverse of the usual telling.

Kapitza in Moscow and Allen and Misener in Cambridge independently reported the vanishing viscosity in 1938, in back-to-back papers. Bose–Einstein condensation had been predicted in 1924–25 and then largely set aside as a curiosity of an idealised gas that nothing real would exhibit. Einstein himself was unsure it described anything physical.

So the phenomenon was found first and connected to the statistics afterwards — by London, within months, who suggested that the lambda transition was condensation modified by interactions and was met with considerable scepticism. Landau built the two-fluid theory in 1941 while explicitly rejecting the condensation interpretation, and it worked anyway, which is a good demonstration that a correct phenomenology does not require a correct microscopic story.

Both turned out to be partly right. Landau’s spectrum is what protects the flow; London’s statistics are why there is anything to protect. That neither is sufficient alone is the point the section above makes about the ideal-gas curve, and it is why this page states the two ingredients separately rather than collapsing them into one sentence.

The dilute trapped gases of 1995 closed the loop. There the atoms genuinely barely interact, the ideal-gas prediction applies, and the measured condensate fraction follows the three-halves curve accurately — which is the cleanest possible evidence that helium’s departure from it is caused by interactions rather than by the theory being wrong.

Interactions split and shift what isolated systems would have shared, and that is exactly what makes the transition hard to predict from first principles. Helium’s superfluidity was not anticipated because the ideal Bose gas condensation temperature and the observed lambda point differ, and the interactions that account for the difference are strong enough that no perturbative treatment reaches them. It was found before it was explained.

The stack of discs that weighed the normal component #

The curve at the top of this page is a measurement, and the experiment that first produced it is worth describing because it is the two-fluid decomposition made into an apparatus.

Andronikashvili, in 1946, hung a stack of closely spaced metal discs from a torsion fibre and immersed the whole thing in helium II. The gaps between the discs were made smaller than the distance a normal fluid’s momentum diffuses in one oscillation, so any normal component in those gaps is dragged bodily round with the stack and adds to its moment of inertia. The superfluid component, having no viscosity, is not dragged at all and adds nothing.

Measuring the period of the torsional oscillation therefore weighs the normal fraction directly. Above the lambda point the whole liquid is entrained and the period is long; cooling the bath shortens it, smoothly, until near absolute zero the discs oscillate as though swinging in a vacuum with only their own inertia.

What makes it a good experiment rather than a good demonstration is that it measures a ratio of densities, on an apparatus whose calibration is its own empty period, with no model of the liquid needed anywhere. The number that comes out is $\rho_n/\rho$, and the curve plotted against the ideal Bose gas’s in the first figure is that measurement — which is why the disagreement between them is a fact about helium and not an argument about theories.

The technique did not retire. A torsional oscillator is still the standard probe for superfluidity in helium confined in small pores, and it is what the claims about a possible supersolid phase in solid helium-4 rested on — and what eventually undid them, when the anomalous period shift turned out to come from the solid’s own stiffness rather than from anything flowing.

What it is for #

Superfluid helium is not only a demonstration. It is the working coolant of the largest cryogenic system ever built, and the reasons are the properties on this page rather than merely its low temperature.

The Large Hadron Collider’s magnets sit in about a hundred and twenty tonnes of helium at 1.9 kelvin — below the lambda point on purpose. Two of this essay’s properties are being bought. The enormous effective thermal conductivity carries heat out of the middle of a magnet winding faster than any solid could, so a local deposit of energy from a stray beam particle is spread before it can drive the conductor normal. And the vanishing viscosity lets the liquid penetrate the gaps in the cable insulation, so the coolant reaches the wire rather than the outside of the coil. A bath at 4.2 kelvin would do neither.

The film flow, meanwhile, sets a limit that shaped the whole of low-temperature practice. Pumping on a bath of helium-4 to cool it works until about 0.7 kelvin, at which point the creeping film climbs the pumping line, reaches somewhere warm, evaporates and returns as a gas load the pump cannot beat. That floor is why helium-3 — which has no such film at those temperatures — and then the dilution refrigerator had to be invented, and it is a rare case of a quantum property being the practical obstacle rather than the goal.

And the fountain effect has been flown. A cryogenic telescope in orbit cannot let its helium separate from its vapour by gravity, because there is none, so its tank is vented through a porous plug that exploits the thermomechanical effect directly: the superfluid moves toward the warmer side and holds itself in while the vapour escapes. Several infrared observatories stayed cold for years on a phase separator with no moving parts, whose operating principle is that a temperature difference across a superleak produces a pressure.

Where the model stops #

Helium-3 is not covered. It is a fermion, so none of the Bose argument applies, and it nevertheless becomes superfluid — at about 2.5 millikelvin, three orders of magnitude colder, by atoms pairing up in a manner analogous to superconductivity. The pairs are bosons and condense; the mechanism is entirely different from anything on this page.

Landau’s criterion is quoted, not derived. The critical velocity it predicts is often not observed, because vortices are nucleated at surfaces well below it, and the practical critical velocity is set by geometry rather than by the spectrum.

The transition is not fully solved. The lambda transition’s critical exponents are known to high precision — the specific-heat exponent was measured on the Space Shuttle to avoid gravity broadening the transition — and they are a benchmark for the theory of critical phenomena rather than a settled consequence of it.

The two-fluid model has limits. It fails at high flow rates, where vortices proliferate and produce quantum turbulence, and near the transition where the components are strongly coupled.

The film flow has its own mechanism. The creeping film is held to the wall by van der Waals attraction and is only able to move because it is superfluid; its thickness follows from a balance this page does not set up, and it is closer to a wetting problem than to anything about condensation.

Nothing here explains the ten per cent. Why the condensate fraction is so small while the superfluid fraction is total requires many-body theory that this page does not attempt.

The ladder from here #

Later rungs on this anchor: Landau’s excitation spectrum, with its phonon and roton branches, and the critical velocity it implies. Quantised vortices and their lattice. Second sound. Helium-3 and its pairing. Bose–Einstein condensation in dilute trapped gases, where the ideal-gas calculation does apply because the atoms genuinely barely interact — and where the predicted curve is reproduced accurately, which is the cleanest possible demonstration that helium’s disagreement is caused by its interactions. And superconductivity, which is this phenomenon with charged pairs and a magnetic field to exclude.