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Superfluid

Quantised Vortices Physics

\[\oint \mathbf{v}_s\cdot d\boldsymbol{\ell} = n\,\frac{h}{m_4}\]
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Meet Superfluid. Below 2.17 K, part of liquid helium flows with no viscosity at all. Spin it and it forms a lattice of tiny whirlpools, each carrying exactly one quantum of circulation. On the mug: ∮ v_s · dℓ = nh/m₄. Pixel-art full-wrap black ceramic mug in 11 or 15 oz; equation, symbols and sources in the physics panel below.

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Details

Size & material11oz · 15oz

Black glossy ceramic mug with a C-handle. Lead- and BPA-free.

  • 11oz: 0.33 l
  • 15oz: 0.44 l
CareDishwasher & microwave safe

Dishwasher safe (top rack) or hand wash. Microwave safe.

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The physics

In superfluid helium-4 circulation is quantised in units of h/m₄ ≈ 9.97×10⁻⁸ m²/s, so a rotating sample fills with a lattice of quantised vortex lines.

Superfluid flows without friction, and when you spin her she answers in whirlpools.

Cool liquid helium-4 below 2.17 K and it becomes helium II. Part of it, the superfluid, flows with no viscosity at all. Pyotr Kapitsa, and separately John Allen and Don Misener, reported this in 1938. Kapitsa shared the 1978 Nobel Prize in Physics, and Lev Landau, who explained it, received the 1962 prize.

The superfluid is one quantum wave, so it cannot rotate like an ordinary liquid. Its circulation around any loop must come in whole quanta: ∮v_s·dℓ = n h/m₄, the formula on this mug. Spin the container and the fluid forms a lattice of thin vortex lines, each carrying one quantum of circulation, h/m₄ ≈ 9.97×10⁻⁸ m²/s. William Vinen measured single quanta in 1961, and in 2006 Bewley, Lathrop and Sreenivasan imaged the vortex lines directly.

A worked number: a bucket of helium II turning at 1 radian per second holds about 2Ω/(h/m₄) ≈ 2×10⁷ vortex lines per square metre, about 2000 per square centimetre.

“Spin me if you like,” says Superfluid. “I'll turn only in whole quanta.”

Equations

\[\oint \mathbf{v}_s\cdot d\boldsymbol{\ell} = n\,\frac{h}{m_4}\]
\[\frac{h}{m_4} \approx 9.97\times10^{-8}\ \text{m}^{2}/\text{s}\]
\[v_c = \min_p \frac{\varepsilon(p)}{p}\quad\text{(Landau)}\]
SymbolMeaningUnit
\(\mathbf{v}_s\) superfluid velocity \(\mathrm{m/s}\)
\(d\boldsymbol{\ell}\) element of a closed loop in the fluid \(\mathrm{m}\)
\(n\) winding number (integer) \(\mathrm{dimensionless}\)
\(h\) Planck constant \(\mathrm{J·s}\)
\(m_4\) mass of a helium-4 atom \(\mathrm{kg}\)
\(v_c\) Landau critical velocity \(\mathrm{m/s}\)
\(\varepsilon(p)\) energy of an excitation with momentum p \(\mathrm{J}\)
\(p\) momentum of the excitation \(\mathrm{kg·m/s}\)

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. Vinen (1961) The detection of single quanta of circulation in liquid helium II, Proc. R. Soc. A 260, 218
  4. Bewley, Lathrop & Sreenivasan (2006) Visualization of quantized vortices, Nature 441, 588
  5. Landau (1941) Theory of the Superfluidity of Helium II, Phys. Rev. 60, 356
  6. NobelPrize.org — The Nobel Prize in Physics 1978 (Kapitsa)
  7. NIST CODATA 2022 — Planck constant

Source Vinen, W. F. (1961). The detection of single quanta of circulation in liquid helium II. Proceedings of the Royal Society A 260, 218–236.doi.org/10.1098/rspa.1961.0029

Field notes

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