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Size
3" × 3"
Surface
White
Made
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Surface

Details

Size3″ × 3″

3″ × 3″ kiss-cut sticker.

Mix & match pricing
  • Pick any 3: $5.25 each (3 for $15.75)
  • Pick any 5: $4.20 each (5 for $21.00)

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Free US shipping. Free international shipping with 4+ stickers, a sheet, or any tee.

Outside the US, orders of only 1–3 single stickers ship for $10.99.

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The physics

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

\[\begin{gathered} V(r) = -\frac{g^{2}}{4\pi}\,\frac{e^{-r/R}}{r} \\ R = \frac{\hbar}{m c} \end{gathered}\]
\[\begin{aligned} R &= \frac{1}{\lambda} = 2\ \text{fm} \\ \Rightarrow\; mc^{2} &= \frac{\hbar c}{R} \approx 99\ \text{MeV} \\ &\approx 2\times10^{2}\,m_e c^{2} \end{aligned}\]
Symbols
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 \(\mathrm{kg}\)
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