• Corekeeper Nuclear Force pixel-art physics T-shirt, white tee, front view

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Corekeeper

Strong Nuclear Force / Nuclear Binding Physics

\[V(r) = -\frac{g^{2}}{4\pi}\frac{e^{-r/\lambda}}{r}, \qquad \lambda = \frac{\hbar}{m_\pi c}\]
$29.99 USD
$29.99 USD
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Size guide S–5XL · inches

Every nucleus heavier than hydrogen packs protons that repel each other, yet it holds together. The strong force binds protons and neutrons in every nucleus, overcoming proton repulsion at femtometre range. Corekeeper wears it in pixel art on a unisex tee. The physics panel below has the equation, every symbol and the sources.

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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

スケール感の目安：サイズ M の幅は 20 in（50.80 cm）で、プランク長のおよそ 3.143 × 1034 倍です（lP = 1.616255 × 10−35 m）。メジャーで測るほうが早いです。

Measurements in inches, ±1.5 in tolerance.

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Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

Full refund & replacement policy

The physics

Yukawa proposed that nucleons attract by exchanging a massive particle. The force then dies off exponentially, with a range set by that particle's mass, which is why nuclear forces only act at about the size of a nucleus. The mass that disappears when a nucleus forms is its binding energy.

Corekeeper holds on hard, but only up close. Step back a couple of femtometres and he lets go.

Inside a nucleus, protons repel one another electrically. In 1935 Hideki Yukawa proposed that nucleons hold each other by trading a particle, and saw that the mass of that particle would set the reach of the force. A briefly borrowed particle of mass m cannot travel far before it must be paid back, about ħ/mc. Make the reach the size of a nucleus and the carrier should weigh a couple of hundred electron masses. That is the first formula on this page: an attraction that dies away exponentially beyond the range λ.

The particle turned up in 1947 in photographic emulsions exposed to cosmic rays, and was named the pion. Cecil Powell received the 1950 Nobel Prize for the photographic method and its meson discoveries; Yukawa had received the 1949 prize for the prediction. Put the measured charged-pion mass, 139.57 MeV, into Yukawa's range and you get 1.41 femtometres, about the spacing of nucleons in a nucleus. Today this pull is understood as a residue of the strong force between quarks, much as the forces between neutral molecules are residues of the electric forces inside them.

The second formula measures how tightly he holds. Weigh a helium-4 nucleus and it comes up short. Two protons and two neutrons, weighed separately, total 28.3 MeV more than the nucleus they form: about 0.75 percent of the mass, missing. That missing mass is the binding energy, released when the nucleus was made, and the energy needed to pull it apart. Building helium from hydrogen releases energy on this scale, which is why it powers the Sun.

Corekeeper does not chase anyone. His grip is fierce at one femtometre and nearly gone at five.

Equations

\[V(r) = -\frac{g^{2}}{4\pi}\frac{e^{-r/\lambda}}{r}, \qquad \lambda = \frac{\hbar}{m_\pi c}\]
\[B = \left(Z m_p + N m_n - M_{\text{nuc}}\right)c^{2}\]
SymbolMeaningUnit
\(V(r)\) Yukawa potential between nucleons \(\mathrm{J}\)
\(g\) coupling constant (Gaussian-like units: g²/r has units of energy) \(\mathrm{J^{1/2}·m^{1/2}}\)
\(\lambda\) range of the force \(\mathrm{m}\)
\(m_\pi\) pion mass \(\mathrm{kg}\)
\(B\) nuclear binding energy (quoted in MeV) \(\mathrm{J}\)
\(Z, N\) proton and neutron numbers \(\mathrm{1}\)
\(m_p, m_n\) proton, neutron masses \(\mathrm{kg}\)
\(M_{\text{nuc}}\) nuclear mass \(\mathrm{kg}\)

Sources

  1. NobelPrize.org — Hideki Yukawa, Physics 1949, facts
  2. Encyclopaedia Britannica — Strong force
  3. NIST CODATA 2022 — proton mass energy equivalent in MeV
  4. NIST CODATA 2022 — neutron mass energy equivalent in MeV
  5. NobelPrize.org — Cecil Powell, Physics 1950, facts
  6. PDG 2024 listings — charged pion (mass 139.57039 MeV)
  7. NIST CODATA 2022 — reduced Planck constant times c in MeV fm
  8. NIST CODATA 2022 — alpha particle mass energy equivalent in 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.

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