The fountain a lamp can drive

Helium below 2.17 kelvin flows through channels that stop every other liquid, and it also drags on a rotating disc in the ordinary way. Both are true at once, of the same liquid, and the resolution is the subject of this essay.

The height a 1 mK difference lifts helium. The head of liquid that a temperature difference of 1 millikelvin can support across a superleak — a plug fine enough that the normal fluid cannot pass and the superfluid can — against the temperature it is done at. Only the normal component carries entropy, so warming one side makes the superfluid flow toward the warm side until the pressure difference balances, and equilibrium is at ΔP = ρSΔT. The height that supports is SΔT/g, in which the density cancels exactly; computed both ways here the two agree to machine precision. The numbers are the striking part: 2.0 mm at 1.2 K, 4.6 mm at 1.4 K, 9.2 mm at 1.6 K, 19.4 mm at 1.8 K, 46.9 mm at 2 K, from a temperature step a thousand times smaller than anything a hand could feel. Aim a light at the warm side and the liquid does not merely rise but jets out of the tube, which is the fountain effect Allen and Jones found in 1938 and the most direct demonstration that helium II is two fluids rather than one. The entropies used are measured values; everything else on this chart is computed from them. — Fig. 1 The head of liquid helium that a temperature difference of one millikelvin supports across a plug too fine for the normal fluid to pass: 2 mm at 1.2 K rising to 47 mm at 2.0 K. The relation is ΔP = ρSΔT, so the height is SΔT/g and the density cancels exactly.

Two velocity fields, not two substances

Tisza proposed in 1938, and Landau put on a proper footing in 1941, that the motion of helium II be described by two velocity fields at once:

\rho = \rho_s + \rho_n, \qquad \rho\mathbf{v} = \rho_s\mathbf{v}_s + \rho_n\mathbf{v}_n

with the normal component carrying all of the liquid’s entropy and all of its viscosity, and the superfluid component carrying neither.

It has to be said immediately what this is not. There are no two kinds of helium atom; nothing can be filtered, decanted or separated; and asking which component a particular atom belongs to has no answer. The decomposition is of the motion, and it is legitimate because the two fields have independent equations and can move in opposite directions at the same place.

Two transitions, one shape and one not. The condensate fraction of an ideal Bose gas, 1 − (T/Tc)^3/2, drawn against the superfluid fraction of liquid helium-4, which goes as roughly 1 − (T/Tλ)^5.6. Both reach one at absolute zero and zero at their transition, and in between they disagree everywhere. The ideal calculation says why a transition has to exist; it does not describe the one that does, because its atoms do not interact and helium's do. — Fig. 2 The superfluid fraction against temperature for helium and for an ideal Bose gas. Both go to one at absolute zero and to nothing at the transition, and they disagree everywhere between — the ideal calculation says why a transition must exist and does not describe this one, because its atoms do not interact and helium’s do.

The justification is the ordinary one for a model: it predicts. A disc oscillating in helium II feels a drag proportional to $\rho_n$ and not to $\rho$ , so the normal fraction can be measured directly by a torsion pendulum; the same fraction then appears in the fountain effect, in second sound, and in the critical velocity, with no further adjustment. That is the whole argument, and it is why “two fluids” survived being obviously impossible.

How the fraction is measured

The model is only worth anything if $\rho_n/\rho$ can be measured independently of the effects it is used to explain, and Andronikashvili did it in 1946 with a device of remarkable directness.

He hung a stack of closely spaced discs on a torsion fibre and set it oscillating in the liquid. The gaps between the discs are far smaller than the viscous penetration depth of the normal fluid, so the normal component is dragged round with the stack and adds its mass to the pendulum’s; the superfluid, having no viscosity, is not dragged and does not. The period of the pendulum therefore measures the normal density directly.

A torsion oscillator ringing down measures both components at once: its period reports how much mass is moving with it and its decay reports how much is being dissipated. In helium II the first of those falls as the temperature drops while the second falls faster — so the superfluid fraction is read off the period and the normal fraction off the damping, which is how the two-fluid picture was made quantitative rather than merely suggestive.

The result is the curve on the previous figure: the normal fraction goes to zero as the temperature falls and reaches one at the lambda point. Everything else in this essay is then a prediction rather than a fit, which is the difference between a model and a description.

The pressure that a temperature is

Take two vessels of helium II joined by a plug of packed powder, with channels so fine that viscosity stops the normal fluid completely while the superfluid passes freely. Warm one side.

The superfluid carries no entropy, so it can flow through the plug without carrying heat, and it does — toward the warm side, because that is where the chemical potential is lower. Liquid piles up on the warm side until the pressure difference stops the flow. Equilibrium is at

\Delta P = \rho S\,\Delta T

which is the thermomechanical effect. The head that pressure supports is $S\Delta T/g$ , in which the density cancels, so the height depends on the entropy and the temperature step and on nothing else about the liquid.

At 1.8 K a difference of one millikelvin supports nineteen millimetres of helium. Warm the tube with a small lamp instead of a millikelvin and the equilibrium head exceeds the length of the tube, so the liquid does not rise but jets: Allen and Jones photographed a fountain forty centimetres high in 1938, driven by a torch.

The fountain works because the entropy is all in one of the two components. That is the unusual statement: entropy here is something that can be located and moved rather than merely accounted for, so a temperature difference across a narrow channel drives a flow of the component that carries no entropy, and the pressure it develops is a thermodynamic quantity with a mechanical effect.

The effect runs both ways, which is the part that convinced people. Force superfluid through a superleak mechanically and the far side cools, because the arriving liquid brings no entropy with it and dilutes what is there. The mechanocaloric effect and the fountain effect are one relation read in two directions, and their ratio is fixed by thermodynamics with nothing left to fit.

Heat that is carried rather than conducted

The two-fluid picture immediately changes what “conduction” means in this liquid.

The speed the normal fluid must run to carry the heat. Helium II does not conduct heat; it carries it. All the entropy is in the normal component, so a heat flux q is transported by that component moving at v = q/ρST, with the superfluid flowing back underneath it carrying none — two fluids passing through each other in opposite directions in the same tube. The curves are that speed against temperature for fluxes of 0.1 W/cm², 0.5 W/cm², 1 W/cm², 2 W/cm². At 1.8 K and 1 W/cm² the normal fluid runs at 0.202 m/s and the superfluid returns at 0.108 m/s. The consequence is the one that made helium II famous before anybody had a theory of it: the same flux through a metre of copper would need a temperature difference of 25 K, and helium II moves it on a difference of millikelvin — a thermal conductance four orders of magnitude better than the best metal. That is why helium II does not boil — no bubbles form, because no part of it is hotter than any other for long enough — and it is why superconducting magnets are cooled with it. The speed rises steeply at low temperature because the entropy falls: the colder the liquid, the less each kilogram of normal fluid can carry, and the faster it has to go. — Fig. 3 The speed at which the normal fluid must run to carry a given heat flux, against temperature. At 1.8 K and one watt per square centimetre it runs at 0.202 m/s and the superfluid returns at 0.108 m/s in the opposite direction through the same tube. The same flux in a metre of copper would need a 25 K temperature difference.

Apply heat at one end of a tube of helium II and nothing diffuses. Instead the normal component flows bodily toward the cold end carrying its entropy, and the superfluid flows the other way carrying none, with no net transfer of mass. This is counterflow, and the heat is convected.

The numbers are extreme. Helium II moves a heat flux on a temperature difference four orders of magnitude smaller than the best metal would need, which is why the liquid does not boil: bubbles need a local hot spot to nucleate in, and no part of the liquid stays hotter than the rest for long enough. The visible transition at 2.17 K — a vigorously boiling liquid that goes abruptly still — is the moment counterflow switches on, and it is one of the few phase transitions whose arrival is announced by something stopping rather than starting.

Helium’s lambda transition is named for the shape of its heat-capacity curve, which rises to a sharp spike at 2.172 K and looks like the Greek letter. That shape is the signature of a continuous transition rather than a latent-heat one — nothing is absorbed at a fixed temperature, and the divergence is in the derivative, which is why the two fluids are not two substances that could be separated.

It is worth being exact about what fails here. The transport is not conduction, so it has no conductivity: the effective one depends on the heat flux, on the tube’s width, and on the temperature, and above a critical counterflow velocity it collapses because quantised vortices appear and couple the two fluids together. Vortices in a superfluid come in one size, and the mutual friction between a tangle of them and the normal fluid is what limits how much heat a real cryostat can move.

The wave that is a temperature

If the two components can move in opposite directions, there should be a wave in which they do so oscillating — and there is.

Ordinary sound is the two components moving together, so it is a density wave of the ordinary kind — a shape travelling through a medium whose speed the medium decides. The other mode is the two moving oppositely with the total density fixed, which makes it a wave of the ratio of normal to superfluid — that is, a wave of entropy, and therefore of temperature.

Heat with an arrival time, and heat without one. A heat pulse released at one end of a 20 mm channel, and what a thermometer at the far end reads. In helium II the disturbance is a wave: it arrives at 1.00 milliseconds, which is the path divided by the second-sound speed of 20 m/s near 1.6 K, and it arrives as a pulse with a front. In an ordinary liquid the same disturbance diffuses, and the curve for helium I's thermal diffusivity peaks at 0 milliseconds and has no front at all — a diffusive signal is nonzero at the far end immediately and merely very small, so there is no arrival to time. That is the operational difference between the two, and it is how second sound was found: Peshkov released heat pulses in 1944 and timed them. Measuring the speed then measures the ratio of superfluid to normal density, so the model's one parameter is read off one of its own predictions. — Fig. 4 The difference between a wave and a diffusion, put to a thermometer. A heat pulse is released at one end of a twenty-millimetre channel and read at the other. In helium II it *arrives* — at 1.00 milliseconds, which is the path divided by the second-sound speed near 1.6 K — with a front, and before it nothing has happened at the far end. In an ordinary liquid the same disturbance diffuses: the reading begins to rise immediately, peaks at no particular time, and has no front to time. Heat in helium II has an arrival time, which is the whole of what makes it a wave.

Helium II carries a second wave alongside ordinary sound, at a quite different speed — around 20 m/s over much of the range against 240 for the first. What propagates in it is not a pressure variation but a temperature one: the two components move in antiphase, the density stays put, and the oscillation is in how the entropy is distributed.

Second sound is a temperature wave that propagates rather than diffuses, with a definite speed, reflections and standing modes in a resonator. It has no analogue in an ordinary liquid, where a temperature disturbance spreads diffusively and has no speed at all. Measuring its velocity is now the standard way of determining the ratio $\rho_s/\rho_n$ , which closes the loop: the model’s parameter is measured by one of its own predictions.

A second-sound resonator is a standing-wave cavity with a heater at one end and a thermometer at the other, and the modes it holds are modes of a temperature wave. That is the cleanest demonstration that the second wave is real: it obeys the same boundary conditions and the same mode counting as any other wave, and only the quantity oscillating is unfamiliar.

The film that climbs out of the beaker

One consequence is visible without any apparatus and is worth including because it is the most-photographed and the least-explained.

Helium II wets its container completely, and van der Waals attraction to the wall holds a film of it about thirty nanometres thick over every surface above the liquid — that much is true of ordinary liquid helium too. What is different below the lambda point is that the film has no viscosity, so it is not held back by the walls it is crawling over: it flows at a speed set by gravity and by the critical velocity rather than by drag, which is around twenty centimetres a second.

A helium film is ordinary wetting physics with an ordinary origin in molecular attraction. What makes it drain a beaker is not that the film is unusual but that it is frictionless: an ordinary liquid climbs a wall the same way and stops, because viscosity holds it. Remove the viscosity and the same film flows over the rim and away.

So a beaker lifted out of a bath empties itself: the film creeps up the inside, over the rim and down the outside, and drips back. Lowered into the bath, it fills. The rate is proportional to the perimeter of the rim rather than to the area of the beaker, which is a testable statement and holds.

What the superfluid cannot do

The component with no viscosity also has no ordinary rotation: its flow is irrotational, and the only way it can carry angular momentum is in quantised vortices.

A superfluid cannot rotate as a body, so it imitates rotation with a regular array of line defects each carrying a quantised circulation. That is the sharpest statement of what the superfluid component is not allowed to do — its flow is irrotational everywhere it exists, and the only way to carry angular momentum is to exclude itself from an array of cores.

And it has a critical velocity: below some speed the superfluid flows without dissipation, and above it the flow generates vortices and dissipates. Landau’s argument gives an upper bound from the shape of the excitation spectrum — a flow slower than the minimum of $\varepsilon(p)/p$ cannot create any excitation and must be dissipationless — and the measured values are far below that bound, because vortices are nucleated at walls long before rotons can be made in the bulk.

That gap between the bound and the observation is a good example of the difference between a proof of impossibility and a prediction: Landau’s criterion says what cannot dissipate, and the real limit is set by a mechanism his argument did not cover.

Ordinary viscous drag is momentum diffusing sideways through a liquid, and helium II shows exactly that behaviour with a viscosity measuring the normal component alone. So the same sample gives a finite viscosity in one experiment and zero in another, and both readings are correct — which is what forced the two-fluid picture rather than a single fluid with strange properties.

The pump with no moving parts

The fountain effect is a laboratory demonstration and it is also a component, and the component’s virtue is what it lacks.

A fountain pump is a superleak with a heater on one side. Heat the far side and superfluid is drawn through the plug toward it; the pressure difference is the thermomechanical one, and it circulates liquid round a loop with no impeller, no bearing, no seal and no moving part of any kind. The only thing that consumes power is a resistor, and the only thing that can wear out is a resistor.

That combination is worth a great deal in a place nobody can visit. Space telescopes cooled by superfluid helium use fountain pumps to circulate the coolant and to control where the liquid sits in a tank under no gravity — a problem with no obvious other solution, since without gravity the liquid does not know which end of the tank is the bottom. A porous plug at the vent also serves as a phase separator, letting vapour out while the thermomechanical pressure holds the liquid in.

The larger use is not in space. The superconducting magnets of a large particle accelerator run at 1.9 kelvin rather than at 4.2, which is below the lambda point, and the reason is this essay’s second half: the superfluid penetrates the windings where an ordinary liquid could not, and carries heat out of them by counterflow at a rate no conduction path could match. A hundred tonnes or so of helium II is used that way, as a coolant chosen for a transport mechanism rather than for a temperature.

Both cases share a design logic worth naming. Superfluid helium is not being used because it is cold — liquid helium at 4.2 K is easier to make and easier to handle — but because at 1.9 K it acquires two properties, zero viscosity and convective heat transport, that solve problems ordinary cryogenics cannot. Going further down in temperature to get better heat transport is a peculiar-looking decision until those two properties are on the table.

The transition measured in orbit

The lambda point has a feature worth recording, because measuring it properly required leaving the planet.

The heat capacity of helium diverges as the transition is approached, and how it diverges — the exponent of the divergence — is a prediction of the theory of critical phenomena that can be computed and compared. The comparison needs the heat capacity measured extremely close to the transition, within nanokelvins, because the exponent only reveals itself in the last few decades of approach.

On Earth that is impossible for a reason that has nothing to do with thermometry. Liquid helium under its own weight has a pressure gradient, and the lambda temperature depends slightly on pressure, so the transition temperature varies from the top of a sample to the bottom. A centimetre of liquid smears the transition over about a microkelvin, and no amount of care with the cryostat removes it: the sample is transitioning at different temperatures in different places.

The answer was to fly the experiment. A calorimeter of helium flown on a Space Shuttle mission in 1992 had no hydrostatic gradient to smear it, and measured the heat capacity to within a couple of nanokelvins of the transition — three decades closer than any ground measurement.

What came out is one of the sharpest tests of critical-phenomena theory there is, and it is a test of a prediction about a universality class rather than about helium: the same exponent is supposed to govern any system whose order parameter has two components, which includes helium’s superfluid transition and a number of magnetic systems that have nothing else in common with it. The measured value and the calculated one agree to a fraction of a per cent, and the disagreement that remains is a live question.

Nine per cent and a hundred

The caution about condensate fraction and superfluid fraction deserves expanding, because it is the single most common confusion about this subject and the numbers are startling.

At absolute zero the superfluid fraction of helium II is exactly one: the whole liquid participates in frictionless flow, the normal density vanishes, and a torsion pendulum in it feels no drag at all. The condensate fraction — the proportion of atoms actually occupying the single lowest-momentum state — is about nine per cent.

Both numbers are measured. The first from a pendulum, the second from neutron scattering, which reads the momentum distribution directly and finds a sharp peak at zero momentum sitting on a broad background. Nine per cent of the atoms are in the peak.

They differ because the atoms interact. In a non-interacting Bose gas the two fractions coincide, and every introductory account of superfluidity uses that gas — which is why the confusion is so widespread. Turn the interactions on and the condensate is depleted: an atom in the condensate is continually scattered out of it and back by its neighbours, so the occupancy of the zero-momentum state falls well below one even at zero temperature. What does not fall is the coherence: the whole liquid still moves with one phase, and the superfluid density is a property of that coherence rather than a count of atoms.

The distinction has a practical edge. A dilute atomic gas can be made with weak interactions, and there the two fractions are nearly equal; helium is strongly interacting and they differ by a factor of eleven. Comparing an experiment on one with a theory built for the other, without noticing which fraction is being quoted, is a mistake that has been made repeatedly.

Where the same bookkeeping is used elsewhere

The two-fluid decomposition is not confined to helium, and its reappearances are worth naming because they show what kind of idea it is.

Superconductors were described this way first. The London brothers wrote the electrons of a superconductor as a superfluid component carrying a dissipationless current and a normal component carrying an ordinary one, in exactly this form, in 1935 — before helium II had a two-fluid model and long before the microscopic theory. The normal component is why a superconductor still absorbs at high frequency and why a field still diffuses into it over a depth rather than stopping at the surface.

A field decaying into a conductor has a penetration depth, and the superconducting version of that figure is computed from a two-fluid model of exactly this shape. That is worth ending on: the bookkeeping invented for helium was carried over to superconductivity almost unchanged, with the superfluid fraction becoming the condensate and the normal fraction the unpaired electrons.

Helium-3 needs it twice over, once for each of its superfluid phases, and the same shape of description works for a Bose–Einstein condensate of dilute atoms, where the condensate fraction and the superfluid fraction can be measured separately and differ.

And it is the standard shape for any system with a condensate. What is common is that a macroscopic number of particles share one quantum state, so their motion has one phase and one velocity field, while everything else in the system behaves ordinarily. The decomposition into “the coherent part and the rest” is what gets written down, whatever the substance.

What the pictures cannot show

The entropies are measured, not computed. Every number in this essay rests on a table of published specific entropies for helium II between 1.0 and 2.15 K, interpolated logarithmically. What is computed from them — fountain heights, counterflow speeds — is the two-fluid model’s own arithmetic, and that is the part being tested.

The two-fluid model is a description and not a mechanism. It says nothing about why a fraction of the liquid should have no entropy. The underlying account is Bose–Einstein condensation modified by strong interactions, and the condensate fraction in helium II is only about nine per cent even at absolute zero, while the superfluid fraction is a hundred. Those two numbers are different quantities and confusing them is the standard error.

The critical velocity is not one number. It depends on the channel width, on the surface, on the temperature and on how the flow was started, and reported values for the same geometry vary by an order of magnitude between laboratories. What is reproducible is the existence of a threshold and the quantisation of what appears above it, which is why a quantised vortex is the object worth measuring rather than the speed at which the first one arrives.

Nothing here is above the lambda point or below one kelvin. Above 2.17 K there is only one fluid. Below 1 K the entropy is dominated by phonons and the table used here stops; the fountain effect continues and gets weaker, following the entropy down as $T^3$ .

And helium-3 is a different subject. It becomes superfluid too, at about two millikelvin, but its atoms are fermions and must pair first — so its two-fluid description has the same shape and an entirely different microscopic origin, closer to superconductivity than to what is described here.

The ladder from here

Later rungs on this anchor: Landau’s excitation spectrum, with the roton minimum and the critical velocity it bounds; second sound derived from the two-fluid equations rather than described; the mutual friction between vortices and the normal fluid, which is what limits heat transport in practice; and the relation between the superfluid fraction and the condensate fraction, which are not the same number and are routinely confused.

The neighbouring ladders are the liquid that will not slow down, which is the phenomenon this model was built to describe, the whirlpool that comes in one size, which is what the superfluid does instead of rotating, and the viscosity that does not care how much gas there is, which is the ordinary mechanism the normal component still obeys.