Illustrated Physics
physics, drawn
Concept

Superfluidity — where it appears

Flow with no viscosity at all, appearing below a transition temperature and carrying no entropy, so that it cannot be stirred or slowed. Its existence follows from a gap in the excitation spectrum: below a critical speed there is no excitation the flow can create, so there is no way to lose energy.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

The liquid that will not slow down

Cool helium below 2.17 kelvin and it starts flowing through gaps no ordinary liquid could enter, climbs out of its own container, and circulates for as long as anyone has been willing to watch. The viscosity is not small. As far as any measurement can tell, it is zero.

quantum · Superfluidity Two costs, and the width that balances them. The energy of a particle in a harmonic well against how tightly its wavefunction is squeezed, in units of ħω and of the width that minimises the total. Two terms compete. Squeezing the particle into a smaller region raises its kinetic energy, because the uncertainty relation makes a narrow position spread a wide momentum spread and momentum is squared in the energy; that term rises as the inverse square of the width and goes to infinity as the particle is localised. Letting it spread out raises its potential energy, since the well gets steeper away from the bottom; that term rises as the square of the width. The sum has a minimum at a width of 1.0000 in these units, where the total is 0.5000 ħω and the two terms are equal at a quarter each. That number is exactly the true ground-state energy of a quantum harmonic oscillator, obtained here with nothing but the uncertainty relation and a minimisation. What the figure shows and the formula does not is why there is a floor at all: it is not that the particle happens to keep moving, but that every way of stopping it costs more than it saves.

The motion that cannot be stopped

A particle in a well cannot sit at the bottom of it. Squeezing it into a smaller region costs kinetic energy faster than it saves potential energy, so there is a width that minimises the total — and the minimum is not zero. Helium never freezes because of it.

quantum · Uncertainty Circulation against how fast the bucket turns. The circulation round the rim of a bucket of radius 1 mm, against the angular velocity it is spun at. An ordinary liquid ends up rotating with the bucket, and its circulation is 2Ω times the area — the straight dashed line, continuous in Ω and with no special value anywhere on it. A superfluid's velocity is the gradient of a phase, so it can carry circulation only in whole units of h/m = 9.969e-8 m²/s. Below 0.256 radians per second it carries none at all: the bucket turns and the liquid does not, which is what Hess and Fairbank measured. Above it the circulation is a staircase of 13 steps, each exactly one quantum high and each 1.59e-2 radians per second wide. The staircase runs below the classical line by the ln(R/a) quanta the threshold costs, a fixed lag: at 102 radians per second the two agree to 0.24 per cent, which is why a rotating superfluid looks like a rotating liquid at any speed a bucket is normally spun at.

The whirlpool that comes in one size

Spin a bucket of ordinary liquid and it ends up turning with the bucket. Spin a bucket of superfluid helium slowly and it does not turn at all. Spin it faster and it does not turn either — until a threshold, at which a single line of circulation appears, carrying not some amount but exactly h/m. There is nothing in between, because the velocity is the gradient of a phase and a phase has to come back to itself.

fluids · Superfluidity 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.

The fountain a lamp can drive

Below two degrees above absolute zero, liquid helium behaves as though it were two fluids occupying the same space — one carrying all the entropy and all the viscosity, the other carrying neither. It is not a metaphor and not a mixture. Shine a light on one side of a fine plug and the liquid jets out of the tube, because a temperature difference of a thousandth of a degree is a pressure of a hundred pascals.

fluids · Superfluidity Helium's excitations, and the line that decides how fast it can flow. The energy of an elementary excitation in superfluid helium-4 against its momentum, as a smooth fit through published neutron-scattering values: a phonon branch rising at the speed of sound, 238 m/s, a maximum near 13.9 K, and the roton minimum of 8.62 K at 1.92 Å⁻¹. A straight line from the origin has a slope that is a velocity. The steepest line that just touches the spectrum is Landau's critical velocity, 57 m/s, and it touches at 1.98 Å⁻¹, just past the roton minimum. A body moving slower than that cannot make any excitation at all and so cannot lose energy to the liquid; the dashed line is the speed of sound, far above it.

The speed below which nothing can be made

An object moving through an ordinary liquid always feels drag, because it can always pass some of its energy to the liquid. In a superfluid there may be nothing it can pass it to. Landau's argument turns the question into geometry: an excitation of momentum p and energy ε can be made only by a body moving faster than ε/p, so a liquid flows without friction below the smallest value of that ratio. For helium the smallest value is not the speed of sound but 57 m/s, set by the dip in its spectrum called the roton — and an ideal gas, with no dip and no sound, has no such speed at all.

quantum · Superfluidity The melting curve that turns back, then lies flat. The pressure at which liquid and solid helium-3 coexist, against temperature from 0.9 millikelvin to 1 kelvin on a logarithmic axis, from the international provisional low-temperature scale PLTS-2000, which is defined by this curve. Above 315.2 mK the curve behaves like any other: a higher temperature needs a higher pressure to freeze the liquid. Below it the curve turns back — at its minimum, 2.93113 MPa, it takes a higher pressure to freeze colder helium — and it rises to 3.4391 MPa at 1 mK, where it lies almost flat, as the third law requires of every coexistence line. The kinks that mark the liquid becoming superfluid and the solid's spins ordering sit on that flat end, at 2.444, 1.896 and 0.902 mK.

The melting curve that has to arrive flat

The slope of every line on which two phases coexist is their difference in entropy divided by their difference in volume. The third law says that entropy differences vanish at absolute zero, so every such line — melting, boiling, a superconductor's critical field — must reach the bottom of the temperature scale lying flat. Helium-3 reaches it in the strangest way available: below a third of a kelvin its solid is more disordered than its liquid, the melting curve runs backwards, and squeezing the liquid into solid absorbs heat. That backwards curve was the refrigerator in which superfluid helium-3 was found, and it is now the definition of temperature below one kelvin.

thermodynamics · Third law A beaker that empties at a steady rate. The height of liquid helium inside a beaker 1 cm across, held 1 cm above the level of the bath it stands in, against time, as the film carries liquid up the inner wall, over the rim and down into the bath. The film carries a fixed volume per centimetre of rim — 7.5·10⁻⁵ cm³/s at 1.4 K — whatever the difference of levels, so the level falls in a straight line, 180 μm a minute, and the beaker is level with the bath in 56 minutes. At 1.9 K, with less superfluid, it takes 95 minutes; at 2.1 K, 279. Dashed, a flow driven by the difference of levels, like a siphon, starting at the same speed: it slows exponentially as the difference shrinks and never quite finishes. The film's straight line is the signature of a flow set by a critical speed, not by the push.

The film whose thinnest point sets the flow

Leave a beaker of superfluid helium standing in a bath and it empties itself, the liquid creeping up the inside wall as a film eighty atoms thick, over the rim and down the outside. The film itself is nothing special: ordinary liquid helium coats the wall just as thickly. What is special is that it flows without friction, and so its flow is set not by how hard it is pushed but by how fast it can go. The level falls in a straight line rather than slowing as it nears the bath's, and the whole flow is decided at one place, the rim, where the film is thinnest and has to run fastest.

quantum · Superfluidity

Named alongside it

The objects these essays reach for when they reach for this one.

Phase transitionViscosityCirculationCondensateEntropyQuantisationThird lawTwo-fluid modelWavefunctionZero-point energyAngular momentumBose einstein condensation

All concepts