• Superfluid Helium pixel-art physics T-shirt, black tee, front view

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Superfluid

Quantum Fluid at Near Absolute Zero Physics

\[v_c = \min_{p}\frac{\varepsilon(p)}{p}\]
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Cool helium enough and part of it stops feeling friction. Below 2.17 K, liquid helium-4 becomes helium II, which pours through gaps no normal liquid can pass. In 1938 Kapitsa, and Allen and Misener, reported it; Tisza and Landau explained it as two fluids, one with zero viscosity. Its rotation comes only in quantized vortices. Superfluid on a pixel-art unisex tee; equation, symbols and sources in the panel below.

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Details

Size & fitS–5XL

Unisex heavy cotton (Gildan 5000), classic fit.

SizeWidthLengthSleeve
S182815.1
M202916.5
L223018
XL243119.5
2XL263221
3XL283322.4
4XL303423.7
5XL323525

Per avere un'idea: la taglia M è larga 20 in (50.80 cm), circa 3.143 × 1034 lunghezze di Planck (lP = 1.616255 × 10−35 m). Un metro a nastro fa prima.

Measurements in inches, ±1.5 in tolerance.

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The physics

Landau showed that a fluid can flow with no friction at all as long as it moves too slowly to create any excitations; that sets a critical speed. In superfluid helium, circulation also comes in whole-number multiples of h/m, so it can only swirl in quantized vortices.

Superfluid flows without friction, but only part of them. The rest is ordinary, and the temperature decides the split.

Cool liquid helium-4 below 2.17 kelvin and it changes into a phase called helium II. In 1938 Pyotr Kapitsa in Moscow, and John Allen and Don Misener in Cambridge, published side by side in Nature: helium II pours through gaps far too narrow for any normal liquid, as if it had no viscosity at all. Yet other experiments, measuring the drag on a body moving through the liquid, found a viscosity that was clearly not zero.

The resolution is the two-fluid picture, proposed by László Tisza in 1938 and developed by Lev Landau. Helium II behaves as if it were two interpenetrating liquids. The superfluid component has exactly zero viscosity and carries no entropy. The normal component is viscous and carries the heat. Narrow channels let only the superfluid through; a moving object is dragged by the normal part. As the temperature falls toward absolute zero, the superfluid fraction grows toward all of it.

Even the superfluid has limits. Landau showed it flows without dissipation only below a critical speed, the first formula on this page: below it, there is no way for the moving fluid to create an excitation and lose energy. Kapitsa received the 1978 Nobel Prize, Landau the 1962 prize.

The second formula is the strangest. Try to set the superfluid spinning and it refuses to rotate smoothly. Its circulation can only take whole-number multiples of Planck's constant divided by the mass of a helium atom, about 10⁻⁷ square metres per second. It rotates by filling with tiny quantized whirlpools, vortices, and in 1961 W. F. Vinen detected single quanta of circulation.

Superfluid never slows down. The super part, anyway, and only below the speed limit.

Equations

\[v_c = \min_{p}\frac{\varepsilon(p)}{p}\]
\[\oint \vec v_s\cdot d\vec\ell = n\,\frac{h}{m_4}\]
SymbolMeaningUnit
\(v_c\) Landau critical velocity \(\mathrm{m/s}\)
\(\varepsilon(p)\) excitation energy at momentum p \(\mathrm{J}\)
\(p\) excitation momentum \(\mathrm{kg·m/s}\)
\(\vec v_s\) superfluid velocity \(\mathrm{m/s}\)
\(n\) integer winding number \(\mathrm{1}\)
\(h\) Planck constant \(\mathrm{J·s}\)
\(m_4\) mass of a helium-4 atom \(\mathrm{kg}\)

Sources

  1. Kapitza (1938) Viscosity of Liquid Helium below the λ-Point, Nature 141, 74
  2. Allen & Misener (1938) Flow of Liquid Helium II, Nature 141, 75
  3. NobelPrize.org — Pyotr Kapitsa, Physics 1978, facts
  4. NobelPrize.org — Lev Landau, Physics 1962, facts
  5. Vinen (1961) Detection of single quanta of circulation in liquid helium II, Proc. R. Soc. A 260, 218
  6. Encyclopaedia Britannica — Superfluidity
  7. Tisza (1938) Transport Phenomena in Helium II, Nature 141, 913

Source Landau, L. (1941). Theory of the Superfluidity of Helium II. Physical Review 60, 356–358.doi.org/10.1103/physrev.60.356

Field notes

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