• Hideki Yukawa Meson Theory pixel-art physics sticker, 3" × 3" / White

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Hideki Yukawa Sticker

Heavy Ball. Short Game. | Physics

\[\begin{gathered} V(r) = -\frac{g^{2}}{4\pi}\,\frac{e^{-r/R}}{r} \\ R = \frac{\hbar}{m c} \end{gathered}\]
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Why does the nuclear force reach only across a nucleus? In 1935 Hideki Yukawa proposed that nucleons trade a heavy particle, so the nuclear force is short-ranged. The pion was found in 1947. A pixel-art portrait on a 3" × 3" sticker; the physics panel below has the equation, every symbol and the sources.

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Size3″ × 3″

3″ × 3″ kiss-cut sticker.

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The physics

Yukawa proposed that nuclear forces come from exchanging a massive particle, so they fade out exponentially beyond a range of about ħ/mc. Turning the observed short range around predicted the mass of the carrier, later identified as the pion.

The sticker says Heavy Ball. Short Game. That is the whole idea behind Hideki Yukawa's meson.

A nucleus packs positively charged protons together, and they repel. Something stronger must hold them, and it has to switch off fast, because nuclear forces barely reach beyond a few femtometres. In November 1934 Yukawa, then 27, presented a theory, published in 1935: protons and neutrons interact by exchanging a new, electrically charged particle, much as charges interact through light.

The catch is the mass. A heavy particle can only be borrowed briefly, so it cannot carry the force far, and the range comes out as ħ/mc. Taking a range parameter of 5×10¹² per centimetre, a range of about 2 femtometres, he got a mass about 200 times the electron's. The ball on this sticker is stamped with that estimate.

In 1937 a particle of roughly that mass turned up in cosmic rays, and for a decade it was taken to be his. It was the muon, which turned out to feel no strong force at all. The real thing, the charged pion, was found in 1947 by Cecil Powell's group at Bristol in photographic emulsions exposed on mountains. It weighs about 273 electron masses.

Yukawa received the 1949 Nobel Prize in Physics for predicting the existence of mesons, the first Japanese Nobel laureate. Today the strong force is described by quantum chromodynamics, but pion exchange is still the right picture of the nuclear force at long range.

Heavy ball. Short game.

Equations

\[\begin{gathered} V(r) = -\frac{g^{2}}{4\pi}\,\frac{e^{-r/R}}{r} \\ R = \frac{\hbar}{m c} \end{gathered}\]
\[\begin{aligned} R &= \frac{1}{\lambda} = 2\ \text{fm} \\ \Rightarrow\; mc^{2} &= \frac{\hbar c}{R} \approx 99\ \text{MeV} \\ &\approx 2\times10^{2}\,m_e c^{2} \end{aligned}\]
SymbolMeaningUnit
\(V(r)\) potential energy between two nucleons \(\mathrm{J}\)
\(g\) coupling strength (g²/4π has units of energy times length here) \(\mathrm{J^{1/2}·m^{1/2}}\)
\(r\) separation \(\mathrm{m}\)
\(R\) range of the force \(\mathrm{m}\)
\(\lambda\) Yukawa's range parameter, the inverse of R (he took 5×10¹² per centimetre) \(\mathrm{m⁻¹}\)
\(\hbar\) reduced Planck constant \(\mathrm{J·s}\)
\(m\) mass of the exchanged particle \(\mathrm{kg}\)
\(c\) speed of light \(\mathrm{m/s}\)
\(m_e\) electron mass \(\mathrm{kg}\)

Sources

  1. Yukawa (1935) On the Interaction of Elementary Particles. I, Proc. Phys.-Math. Soc. Japan 17, 48 (mass estimate on p. 53)
  2. NobelPrize.org — Hideki Yukawa, Physics 1949 (facts)
  3. NobelPrize.org — Physics 1949, award ceremony speech
  4. Neddermeyer & Anderson (1937) Note on the Nature of Cosmic-Ray Particles, Phys. Rev. 51, 884
  5. Lattes, Muirhead, Occhialini & Powell (1947) Processes involving charged mesons, Nature 159, 694
  6. NobelPrize.org — Physics 1950, C. F. Powell (pion discovery)
  7. Particle Data Group (2024) — charged pion listing, mass 139.57 MeV

Source Yukawa, H. (1935). On the Interaction of Elementary Particles. I. Proceedings of the Physico-Mathematical Society of Japan, 3rd Series, 17, 48–57.

Field notes

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