What is the Strong Nuclear Force?

When thinking about atoms, one of the obvious puzzles is how a nucleus stays together when it is packed with positively charged particles (protons). Why don’t the positive protons simply repel each other, causing the nucleus to disintegrate?

The fact that there are also neutral particles (neutrons) in the nucleus may “dilute” the positive charge but an overall repulsive force must remain and the nucleus ought to self-destruct. The only exception is hydrogen, which contains just a single proton that can’t repel itself. But all other nuclei contain multiple protons and should therefore be inherently unstable.

The first successful solution to this problem was proposed by Hideki Yukawa, who developed a theory in which smaller particles are swapped between neutrons and protons. This exchange creates a dynamic equilibrium that gives the nucleus its stability. (For more about Hideki Yukawa’s theory, including an animation, see https://physbang.com/2024/01/19/hideki-yukawa-and-meson-theory/.)

In 1963, Murray Gell-Mann gave order to the idea that protons and neutrons contain even smaller particles, which he called quarks. Protons and neutrons each contain three quarks;

• protons comprise two Up quarks and one Down quark
• neutrons comprise two Down quarks and one Up quark

The quark model provided a mechanism for Yukawa’s theory by identifying the exchange particles as pairs of quarks: one quark is matter and the other is antimatter, such as a Down quark combined with an anti-Down quark. These pairs, known as pions, ought to self-annihilate but they can survive long enough to travel between densely-packed particles inside the nucleus, and their interactions create a binding force that acts independently of electrical charge.

Given that electrical repulsion between protons should cause a nucleus to fall apart, it is clear that the attractive force holding the nucleus together must be much stronger than the electrical force. Unsurprisingly, it is known as the strong nuclear force – or, in recognition of the exchange mechanism, the strong nuclear interaction.

The strong nuclear interaction is so short-range that it exists only between the nearest neighbours within a nucleus. It must therefore act in exactly the same way between neutral neutrons, positive protons and neutron-proton pairs. A graph displaying the scale of the interaction out to 2.5 femtometres (2.5 fm, or 2.5 x10^-15 m) is shown below;

By way of extra detail, it should be noted that the zero-force position (about 0.7 fm in the graph above) is slightly greater for proton-proton interactions than it is for proton-neutron or neutron-neutron interactions. This is because the electrical repulsion that exists between two positively charged particles requires the strong nuclear force to be attractive (have a negative non-zero value) in order for the opposing forces to cancel-out.

As an aside it is worth noting that particles containing trios of quarks form a family known as baryons, where the “bary” prefix means heavy and indicates high mass; particles that contain pairs of quarks are in the family of mesons, a name that indicates medium mass; the lightest particles don’t contain any quarks at all and are categorised as leptons, the most common example being an electron.

Despite its appeal and the Nobel prize it won for Hideki Yukawa, the pion model is no longer taken as being correct and the binding mechanism is now better explained using gluons, which carry a property known as colour. Importantly, gluons are also required to explain the mass values for neutrons and protons since the masses of their component quarks is much less than the masses of the baryons themselves. Unfortunately, gluons have no mass of their own and the answer to the mystery of nuclear mass is bound up (sorry) in energy and momentum calculations.

Things get even more complicated when we include the expectation, inherent in quantum mechanics, that the range of a force is inversely proportional to the mass of its force carriers. Photons, which carry the electromagnetic forces, have zero rest mass and this allows electrical forces act continuously (but increasingly weakly) “out to infinity”. But the strong nuclear interaction is extremely short-range, so we would expect it to be mediated by force carriers that have a lot of mass. This is at odds with the idea that gluons are massless – and this contradiction remains something of a puzzle today.

Finally, there is the matter of how the packing separation of particles within the nucleus compares with the distances shown in the strong nuclear interaction graph.

Typically, atoms are said to have a diameter of about 10^-10 m whereas nuclei are said to have a diameter of about 10^-14 m. This means an atom is about ten thousand times bigger than its nucleus. The diameter of a proton is about one-tenth that of a typical nucleus, at 10^-15 m.

To be a bit more exact, the diameter of a lone proton is about 0.84 fm whereas the diameter of a lone neutron is about 0.86 fm (https://pdg.lbl.gov/2024/tables/contents_tables.html). When inside a nucleus, however, nucleons occupy a slightly greater volume, with an effective diameter that is roughly double these figures (https://profmattstrassler.com/articles-and-posts/particle-physics-basics/the-structure-of-matter/the-nuclei-of-atoms-at-the-heart-of-matter/what-holds-nuclei-together/).

Referring back to the strong nuclear force graph, it is clear that the strongest attraction exists at a distance of about 0.9 fm, which is very similar to the diameters mentioned above. Nucleons are therefore very likely to be packed together so tightly that they are very nearly “touching” each other.

This in turn forms the basis of the “liquid drop” model of the nucleus, in which nucleons are densely packed into a spherical volume and the radius of the nucleus varies with the cube-root of the number of nucleons it contains. The constant of proportionality is 1.2 fm so even a high-mass nucleus has a diameter of no more than about 15 fm, which is broadly consistent with the general rule that atomic nuclei have diameters in the region of 10^-14 m.

Further Reading (and viewing)

• There are two nice video explanations of the strong nuclear force on the Energy Education website, at https://energyeducation.ca/encyclopedia/Strong_nuclear_force.
• Readers who want to extend their knowledge of fundamental particles further may wish to study the excellent summary published by Scientific American magazine in 2015: it is available at https://www.scientificamerican.com/article/the-mysteries-of-the-world-s-tiniest-bits-of-matter/. There is also a more recent (and more detailed) article about the strong nuclear interaction, available at https://www.scientificamerican.com/article/physicists-finally-know-how-the-strong-force-gets-its-strength/. Both articles can be accessed free of charge.
• Quark masses and their contribution to nucleon masses are given in a very readable article (https://physics.aps.org/articles/v11/118) summarising a detailed mathematical paper by Yang et al ( https://link.aps.org/doi/10.1103/PhysRevLett.121.212001). Thanks to CBP for highlighting these sources!
• Professor Matt Strassler has published a good series of articles about particle physics basics. As well as the article mentioned in the text above, there is a nice explanation of nuclei as a whole, which is available at https://profmattstrassler.com/articles-and-posts/particle-physics-basics/the-structure-of-matter/the-nuclei-of-atoms-at-the-heart-of-matter/.
• Mathematically-inclined readers may want to digest Alex Howe’s short and clear explanation of the Cornell Potential, which gives rise to the equation for the strong nuclear interaction: the article is available at https://sciencemeetsfiction.com/2020/12/29/what-is-the-equation-for-the-strong-nuclear-force/