What Holds the Nucleus Together? The Strong Force from Yukawa’s Mesons to Quarks and Gluons

An atomic nucleus is a strange object. It packs positively charged protons into a tiny space at the centre of the atom, and like charges repel. By the logic of electromagnetism alone, every nucleus heavier than hydrogen should fly apart. It doesn’t, because a second, stronger force acts between protons and neutrons at very short range. Our Corekeeper tee is the guardian of that bond. Here is how physicists came to understand it, and what is still hard about it.

Why don’t protons in the nucleus repel each other apart?

They do repel; something just pulls harder. NobelPrize.org’s summary of Hideki Yukawa’s work begins with the key fact: “Atomic nuclei consist of protons and neutrons held together by a strong force” (NobelPrize.org^[1]). That force is attractive between any two nucleons (protons or neutrons), it is much stronger than the electric repulsion at nuclear distances, and it fades away very quickly beyond them (Britannica^[2]). Neutrons help: they add strong-force attraction without adding electric repulsion, which is why heavier stable nuclei need more neutrons than protons.

What is the strong force?

It is one of the four fundamental interactions, alongside gravity, electromagnetism and the weak force. CERN’s overview of the Standard Model places it with the other forces and their carrier particles (Britannica^[2]). We have a lighter-hearted tour of all four in The Four Fundamental Forces Explained by Overworked Pixel Warriors. This post goes deeper into one.

Yukawa’s idea: a force carried by a particle

In 1934 Hideki Yukawa predicted (Nobel presentation speech^[3]) that the nuclear force is carried by a new particle, as NobelPrize.org describes: he “assumed that this force is borne by particles” (NobelPrize.org^[1]). The short range of the force was the clue. A force carried by a particle with mass m has a range of roughly

λ = ħ / (mc)

and from the known range of nuclear forces Yukawa estimated that his particle should be about 200 times heavier than the electron (Nobel presentation speech^[3]). Yukawa won the 1949 Nobel Prize in Physics “for his prediction of the existence of mesons on the basis of theoretical work on nuclear forces” (NobelPrize.org^[1]). The particle, the pion, was found in cosmic-ray tracks on photographic plates by Cecil Powell’s group; Powell won the 1950 prize for “his discoveries regarding mesons made with this method” (NobelPrize.org^[4]). Using today’s charged-pion mass of about 139.6 MeV (PDG^[5]), the formula gives a range of about 1.4 femtometres. That is the formula on our Yukawa sticker: heavy ball, short game.

What are quarks and gluons?

From the 1950s onward, accelerators turned up dozens of new strongly interacting particles. Murray Gell-Mann brought order to the zoo and received the 1969 Nobel Prize “for his contributions and discoveries concerning the classification of elementary particles and their interactions” (NobelPrize.org^[6]). The modern picture is that protons and neutrons are made of quarks, held together by gluons, and the theory of their interaction is called quantum chromodynamics, or QCD (NobelPrize.org^[7]).

In QCD the “charge” of the strong force comes in three kinds, whimsically called colours. A proton or neutron combines three quarks so that their colours cancel. That is the idea behind our Quarkbinders tee.

Why can’t you pull a quark out on its own?

The strong force behaves oddly with distance. At very short distances quarks interact only weakly, a property called asymptotic freedom; David Gross, David Politzer and Frank Wilczek shared the 2004 Nobel Prize “for the discovery of asymptotic freedom in the theory of the strong interaction” (NobelPrize.org^[7]). At larger distances the interaction grows strong. Try to pull quarks apart and the energy you put in creates new quark–antiquark pairs, so you get new particles instead of a free quark. This is called confinement. A full mathematical proof of confinement from QCD is still an open problem, though it is supported by experiment and by computer simulations. That stubborn tug-of-war is Hadron Brawler.

So what holds protons and neutrons to each other?

If the strong force acts between colour charges, and protons and neutrons are colour-neutral, why do they attract? The answer is a residual strong force, a leftover effect of the interactions of the quarks inside them, somewhat like the way neutral atoms can still attract each other through leftover electrical effects. The Particle Adventure, an educational site from Lawrence Berkeley National Laboratory, explains the residual strong interaction this way (Particle Adventure^[8]). Yukawa’s pion exchange is now understood as a good description of this leftover force at nuclear distances.

What is nuclear binding energy?

A nucleus weighs less than the separate protons and neutrons that make it. The missing mass, times c², is the binding energy:

B = (Z m_p + N m_n − M_nuc) c²

The simplest example is the deuteron, one proton plus one neutron. Using CODATA values, a proton is 938.272 MeV, a neutron 939.565 MeV and a deuteron 1875.613 MeV (NIST^[9]; NIST^[10]; NIST^[11]). The difference is about 2.22 MeV, matching Feynman’s figure for the deuteron’s binding (Feynman Lectures III-4^[12]). Feynman also explains why two protons do not form a bound “helium-2”: the exclusion principle combined with the spin dependence of the nuclear force, the subject of our pillar Why Can’t Two Electrons Share a State?

Binding energy is the fuel of stars. Fusing light nuclei into heavier, more tightly bound ones releases energy; our Stellar Core tee and the older post How Does the Sun Actually Work? cover that side.

Why do nuclei have “magic numbers”?

Nuclei with certain “magic” numbers of protons or neutrons are unusually stable and more abundant in nature; when both kinds of shell are full, as in oxygen-16 and calcium-40, the nucleus is “doubly magic” (Scientific American^[13]). Maria Goeppert Mayer and J. Hans D. Jensen explained this with the nuclear shell model, sharing half of the 1963 Nobel Prize “for their discoveries concerning nuclear shell structure” (NobelPrize.org^[14]). Her key insight, at Enrico Fermi’s prompting, was that each nucleon’s orbit and spin are coupled (Scientific American^[13]). See her on our Goeppert Mayer sticker.

Where does a proton’s mass come from?

Here is a surprise: the quarks themselves account for only a small part of a proton’s mass. The rest comes from the energy of the strong interaction inside it. A 2008 lattice-QCD study led by Stephan Dürr opened its abstract with the point that more than 99% of the mass of the visible universe is made up of protons and neutrons, which are much heavier than their quark and gluon constituents, and showed that QCD computed from first principles reproduces the masses of light hadrons (Dürr et al.^[15]). The Higgs field gives the quarks their own small masses, which is a different story (Vevette’s).

Common misconceptions

• “Gluons hold protons and neutrons together directly.” Gluons bind quarks inside each nucleon; nucleons are held to each other by the residual effect, well described at nuclear distances by meson exchange (Particle Adventure^[8]).
• “The strong force is just a stronger version of electromagnetism.” It has three colour charges, carriers that themselves carry charge, and gets stronger with distance at the scale of a proton.
• “Mass is mostly quarks.” Mostly not; it is mostly strong-interaction energy (Dürr et al.^[15]).

Key terms in plain English

• Nucleon: a proton or a neutron, the building blocks of nuclei.
• Meson: a particle made of a quark and an antiquark. Yukawa’s predicted particle, the pion, is one (NobelPrize.org^[1]).
• Quark: a fundamental particle that feels the strong force. Protons and neutrons each contain three.
• Gluon: the carrier of the strong force between quarks.
• Colour charge: the strong-force version of electric charge, with three kinds instead of one.
• Asymptotic freedom: quarks interact weakly at very short distances (NobelPrize.org^[7]).
• Confinement: quarks are never seen alone; pulling them apart creates new particles.
• Binding energy: the energy released when nucleons bind, equal to the mass difference times c².

Why is iron so stable, and what does that mean for stars?

Binding energy per nucleon rises from the lightest nuclei, peaks around iron and nickel, and slowly falls for heavier ones. Fusing light nuclei therefore releases energy, and splitting very heavy nuclei does too. This single curve explains why stars shine by fusion and why nuclear reactors run on fission. It is also why stars cannot make energy by fusing past iron: the reactions cost energy instead of releasing it. The heavier elements come from other processes, as we explain in How Did the First Elements Form?

Why is the strong force so hard to calculate?

For electromagnetism, physicists can calculate by starting with a simple answer and adding small corrections, because the interaction is weak. The strong force is weak only at very short distances; at the scale of a proton, it is strong, and that simple approach fails. The alternative is lattice QCD: putting space and time on a grid and simulating the theory on supercomputers. That is how Dürr and colleagues computed the light hadron masses from first principles (Dürr et al.^[15]).

A short timeline of the strong force

• 1934: Hideki Yukawa predicts a meson as the carrier of nuclear forces (Nobel presentation speech^[3]).
• Late 1940s: Cecil Powell’s group finds the pion in cosmic-ray tracks; Powell wins the 1950 Nobel Prize (NobelPrize.org^[4]).
• 1949: Goeppert Mayer and Jensen explain nuclear magic numbers with the shell model; they share the 1963 prize (NobelPrize.org^[14]).
• 1960s: Murray Gell-Mann classifies the particle zoo, leading to the quark model (NobelPrize.org^[6]).
• 1973: Gross, Wilczek and Politzer discover asymptotic freedom (NobelPrize.org^[7]).
• 2008: lattice QCD reproduces light hadron masses from first principles (Dürr et al.^[15]).

The bottom line

Nuclei hold together because a short-range force between nucleons beats electric repulsion at close quarters. Yukawa pictured that force as meson exchange; we now see it as the leftover of the quark-and-gluon interaction inside each nucleon. The same force supplies most of the mass of everything you can touch.

FAQ

What force holds the nucleus together?

The strong nuclear force, acting between nucleons at very short range (NobelPrize.org^[1]).

Who predicted the pion?

Hideki Yukawa, in 1934, winning the 1949 Nobel Prize; Cecil Powell’s group found it, and Powell won in 1950 (NobelPrize.org^[4]).

What is the range of the strong force?

Between nucleons, roughly a femtometre, set by the pion mass through λ = ħ/(m_πc).

Is confinement proven?

It is supported by experiment and simulation, but a complete mathematical proof from QCD is still open.

See every design in The Strong Force collection, the Fine Print sticker sheet, and next: How Did the First Elements Form?

References

1. NobelPrize.org, Hideki Yukawa, Physics 1949, facts. https://www.nobelprize.org/prizes/physics/1949/yukawa/facts/
2. Encyclopaedia Britannica, “Strong force”. https://www.britannica.com/science/strong-force
3. NobelPrize.org, Award ceremony speech, Physics 1949. https://www.nobelprize.org/prizes/physics/1949/ceremony-speech/
4. NobelPrize.org, The Nobel Prize in Physics 1950 (Powell). https://www.nobelprize.org/prizes/physics/1950/summary/
5. Particle Data Group (2024), π± listing. https://pdg.lbl.gov/2024/listings/rpp2024-list-pi-plus-minus.pdf
6. NobelPrize.org, Murray Gell-Mann, Physics 1969, facts. https://www.nobelprize.org/prizes/physics/1969/gell-mann/facts/
7. NobelPrize.org, The Nobel Prize in Physics 2004 (asymptotic freedom). https://www.nobelprize.org/prizes/physics/2004/summary/
8. The Particle Adventure (LBNL), “Residual strong force”. https://particleadventure.org/residualstrong.html
9. NIST CODATA, proton mass energy equivalent in MeV. https://physics.nist.gov/cgi-bin/cuu/Value?mpc2mev
10. NIST CODATA, neutron mass energy equivalent in MeV. https://physics.nist.gov/cgi-bin/cuu/Value?mnc2mev
11. NIST CODATA, deuteron mass energy equivalent in MeV. https://physics.nist.gov/cgi-bin/cuu/Value?mdc2mev
12. Feynman Lectures on Physics, Vol. III, Ch. 4: Identical Particles. https://www.feynmanlectures.caltech.edu/III_04.html
13. Scientific American, “The Last Woman to Win a Physics Nobel”. https://www.scientificamerican.com/article/the-last-woman-to-win-a-physics-nobel1/
14. NobelPrize.org, The Nobel Prize in Physics 1963 (Wigner, Goeppert Mayer, Jensen). https://www.nobelprize.org/prizes/physics/1963/summary/
15. Dürr et al. (2008), “Ab Initio Determination of Light Hadron Masses,” Science 322, 1224, arXiv:0906.3599. https://arxiv.org/abs/0906.3599

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.